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Title: Crystals

Author: A. E. H. Tutton

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Language: English

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                    INTERNATIONAL SCIENTIFIC SERIES

                             VOLUME XCVIII.

  DIRECT REPRODUCTIONS OF AUTOCHROME PHOTOGRAPHS OF SCREEN PICTURES IN
                            POLARISED LIGHT.

[Illustration:

  FIG. 90.—Screen Picture in Polarised Light, with Nicols crossed, of a
    thick Plate perpendicular to the Axis of a naturally twinned Crystal
    of Quartz, the left half being of right-handed Quartz and the right
    half of alternately left and right-handed Quartz, the Planes of
    Demarcation being oblique to the Plate.
]

[Illustration:

  FIG. 97.—Crystals of Benzoic Acid in the Act of Growth, as seen on the
    Screen in Polarised Light with crossed Nicols.
]

                  THE INTERNATIONAL SCIENTIFIC SERIES




                                CRYSTALS


                                    BY
                             A. E. H. TUTTON

                 D.Sc., M.A. (NEW COLLEGE, OXON.), F.R.S.

  VICE-PRESIDENT OF THE MINERALOGICAL SOCIETY MEMBER OF THE COUNCILS OF
 THE CHEMICAL SOCIETY AND THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF
                                 SCIENCE


                          WITH 120 ILLUSTRATIONS


                                  LONDON
                 KEGAN PAUL, TRENCH, TRÜBNER & CO. L^{TD}
                     DRYDEN HOUSE, GERRARD STREET, W.
                                   1911




                                PREFACE


The idea underlying this book has been to present the phenomena of
crystallography to the general reading public in a manner which can be
comprehended by all. In the main the sequence is that of the author’s
evening discourse to the British Association at their meeting at
Winnipeg in the summer of 1909. It is hoped, however, that the book
combines the advantages of sufficient amplification of the story there
told to make it an adequately detailed account of the development of the
subject, and of the immense progress which has been made in it during
recent years, with a full description of the numerous experimental
illustrations given in the lecture, involving some of the most beautiful
phenomena displayed by crystals in polarised light. Such an account has
not been otherwise published, the brief abstract appearing in the Report
of the British Association for 1909 giving no account of the
experiments, which were a feature of the lecture, owing to the
employment of a fine projection polariscope of more or less novel
construction, and including two magnificent large Nicol prisms, a pair
of the original ones made by Ahrens. The author has been frequently
requested to publish a fuller account of this discourse, and as the
general plan of it so fully embodies the present aspect of this
fascinating science, it was determined, when invited by the publishers
to write a generally readable book on “Crystals,” to comply with these
requests.

There is also included an account of the remarkable work of Lehmann and
his fellow workers on “Liquid Crystals,” and the bearing of these
discoveries on the nature of crystal structure is discussed in so far as
the experimental evidence has gone. Similarly, the theory of Pope and
Barlow, connecting crystalline structure with the chemical property of
valency, is referred to and explained, as this theory has called forth
deep and widespread interest. In both cases, however, the author has
been careful to avoid any expression of opinion on purely theoretical
questions for which there is as yet no definite experimental evidence,
and has confined himself strictly to indicating how far such interesting
theories are supported by actual experimental facts.

No forbidding mathematical formulæ and no unessential technical terms
will be found in the book, the aim of the author being to make any
ordinarily cultured reader feel at the conclusion that the story has
been readily comprehensible, and that crystallography is not the
abstruse and excessively difficult subject which it has so generally
been supposed to be, but that, on the contrary, it is both simple and
straightforward, and full of the most enthralling interest, as well for
the exquisite phenomena with which it deals, as for the exceedingly
important bearing which it has on the nature, both chemical and
physical, of solid matter.

If any of its readers should be so impressed with the value of work in
this domain of science as to be desirous of joining the very thin ranks
of the few who are engaged in it, they will find a guide to practical
goniometry and to the experimental investigation of crystals in all its
branches and details, as well as the necessary theoretical help, in the
author’s book on “Crystallography and Practical Crystal Measurement”
(Macmillan & Co., 1911), and also an account of the author’s own
contributions to the subject in a monograph entitled “Crystalline
Structure and Chemical Constitution” (Macmillan & Co., 1910).

                                                        A. E. H. TUTTON.

  _January 1911._




                                CONTENTS


                                                                    PAGE

 Preface                                                               v

 CHAPTER

      I. Introduction                                                  1

     II. The Masking of Similarity of Symmetry and Constancy of
           Angle by Difference of Habit, and its Influence on Early
           Studies of Crystals                                        10

    III. The prescient Work of the Abbé Haüy                          22

     IV. The Seven Styles of Crystal Architecture                     33

      V. How Crystals are Described. The Simple Law limiting the
           Number of possible Forms                                   50

     VI. The Distribution of Crystal Faces in Zones, and the Mode
           of Constructing a Plan of the Faces                        60

    VII. The Work of Eilhardt Mitscherlich and his Discovery of
           Isomorphism                                                70

   VIII. Morphotropy as distinct from Isomorphism                     98

     IX. The Crystal Space-Lattice and its Molecular Unit Cell. The
           230 Point-Systems of Homogeneous Crystal Structure        111

      X. Law of Variation of Angles in Isomorphous Series. Relative
           Dimensions of Unit Cells. Fixity of Atoms in Crystal      121

     XI. The Explanation of Polymorphism and the Relation between
           Enantiomorphism and Optical Activity                      133

    XII. Effect of the Symmetry of Crystals on the Passage of Light
           through them. Quartz, Calcite, and Gypsum as Examples     162

   XIII. Experiments in Convergent Polarised Light with Quartz, as
           an Example of Mirror-Image Symmetry and its accompanying
           Optical Activity                                          183

    XIV. Experiments with Quartz and Gypsum in Parallel Polarised
           Light. General Conclusions from the Experiments with
           Quartz                                                    201

     XV. How a Crystal Grows from a Solution                         236

    XVI. Liquid Crystals                                             255

   XVII. The Chemical Significance of Crystallography. The Theory
           of Pope and Barlow—Conclusion                             283

 Index                                                               295




                                CRYSTALS

                      (INCLUDING LIQUID CRYSTALS)




                               CHAPTER I
                             INTRODUCTION.


It is a remarkable fact that no definition of life has yet been advanced
which will not apply to a crystal with as much veracity as to those
obviously animate objects of the animal and vegetable world which we are
accustomed to regard in the ordinary sense as “living.” A crystal
_grows_ when surrounded by a suitable environment, capable of supporting
it with its natural food, namely, its own chemical substance in the
liquid or vaporous state or dissolved in a solvent. Moreover, when a
crystal is broken, and then surrounded with this proper environment, it
grows much more rapidly at the broken part than elsewhere, repairing the
damage done in a very short space of time and soon presenting the
appearance of a perfect crystal once more. In this respect it is quite
comparable with animal tissue, the wonderful recuperative power of which
after injury, exhibited by special growth at the injured spot, is often
a source of such marvel to us. Indeed, a crystal may be broken in half,
and yet each half in a relatively very brief interval will grow into a
crystal as large as the original one again. The longevity and virility
of the spores and seeds of the vegetable kingdom have been the themes of
frequent amazement, although many of the stories told of them have been
unable to stand the test of strict investigation. The virility of a
crystal, however, is unchanged and permanent.

A crystal of quartz, rock-crystal, for instance—detached, during the
course of the disintegration of the granitic rock of which it had
originally formed an individual crystal, by the denuding influences at
work in nature thousands of years ago, subsequently knocked about the
world as a rounded sand grain, blown over deserts by the wind, its
corners rounded off by rude contact with its fellows, and subjected to
every variety of rough treatment—may eventually in our own day find
itself in water containing in solution a small amount of the material of
which quartz is composed, silicon dioxide SiO_{2}. No sooner is this
favourable environment for continuing its crystallisation presented to
it, than, however old it may be, it begins to sprout and grow again. It
becomes surrounded in all probability by a beautiful coating of
transparent quartz, with exterior faces inclined at the exact angles of
quartz, although no sign of exterior faces had hitherto persisted
through all the stages of its varied adventures. Or it may grow chiefly
at two or three especially favourable places, and in the course of a few
weeks, under suitable conditions, at each place a perfect little quartz
crystal will radiate out from the sand grain, composed of a miniature
hexagonal prism terminated by the well-known pyramid, really consisting
of a pair of trigonal (rhombohedral) pyramids more or less equally
developed, and together producing an apparently hexagonal one. Four such
grains of sand, from which quartz crystals are growing, are shown in
Fig. 1, as they appear under a microscope magnifying about fifty
diameters. One of them shows a perfectly developed doubly terminated
crystal of quartz growing from the tip of a singly terminated one,
attached to and growing directly out of the grain.

[Illustration:

  FIG. 1.—Sand Grains with Quartz Crystals growing from them.
]

This marvellously everlasting power possessed by a crystal, of silent
imperceptible growth, that is, of adding to its own regular structure
further accretions of infinitesimal particles, the chemical molecules,
of its own substance, is one of the strangest functions of solid matter,
and one of the fundamental facts of science which is rarely realised,
compared with many of the more obvious phenomena of nature.

A crystal in the ordinary sense of the word is solid matter in its most
perfectly developed and organised form. It is composed of the chemical
molecules of some definitely constituted substance, which have been laid
down in orderly sequence, in accordance with a specific architectural
plan peculiar to that particular chemical substance. The physical
properties of the latter are such that it assumes the solid form at the
ordinary temperature and pressure, leaving out of consideration for the
present the remarkable viscous and liquid substances which will be
specially dealt with in Chapter XVI. of this book, and which are
currently known as “liquid crystals.” This term is not perhaps a very
appropriate one. For the word “crystal” had much better be left to
convey the idea of rigidity of polyhedral form and internal structure,
which is the very basis of crystal measurement.

The solid crystal may have been produced during the simple act of
congealment from the liquid state, on the cooling of the heated
liquefied substance to the ordinary temperature. Sulphur, for instance,
is well-known to crystallise in acicular crystals belonging to the
monoclinic system under such conditions, a characteristic crop being
shown in Fig. 2 (Plate I.); they were formed within an earthenware
crucible in which the fusion had occurred, and became revealed on
pouring out the remainder of the liquid sulphur when the crystallisation
had proceeded through about one-half of the original amount of the
“melt.”

[Illustration:

  _PLATE I._

  FIG. 2.—Monoclinic Acicular Crystals of Sulphur produced by
    Solidification of Liquid.
]

[Illustration:

  FIG. 3.—Octahedral Crystals of Arsenious Oxide produced by
    Condensation of Vapour.

  CRYSTALS FORMED BY DIFFERENT PROCESSES.
]

[Illustration:

  _PLATE II._

  FIG. 4.—Cubic Octahedral Crystals of Potash Alum growing from
    Solution.
]

[Illustration:

  FIG. 10.—Micro-Chemical Crystals of Gypsum (Calcium Sulphate) produced
    by Slow Precipitation (see p. 14).

  CRYSTALS FORMED BY DIFFERENT PROCESSES.
]

Or the substance may be one which passes directly from the gaseous to
the solid condition, on the cooling of the vapour from a temperature
higher than the ordinary down to the latter, under atmospheric pressure.
Oxide of arsenic, As_{2}O_{3}, is a substance exhibiting this property
characteristically, and Fig. 3 (Plate I.) is a reproduction of a
photograph of crystals of this substance thus produced. The white solid
oxide was heated in a short test tube over a Bunsen flame, and the
vapour produced was allowed to condense on a microscope glass slip, and
the result examined under the microscope, using a 1½ inch objective.
Fig. 3 represents a characteristic field of the transparent octahedral
crystals.

Or again, the crystal may have been deposited from the state of solution
in a solvent, in which case it is a question of the passage of the
substance from the liquid to the solid condition, complicated by the
presence of the molecules of the solvent, from which the molecules of
the crystallising solid have to effect their escape. Fig. 4 (Plate II.)
represents crystals of potash alum, for instance, growing from a drop of
saturated solution on a glass slip placed on the stage of the
microscope, the drop being spread within a hard ring of gold size and
under a cover-glass, in order to prevent rapid evaporation and avoid
apparent distortion by the curvature of an uncovered drop. The crystals
are of octahedral habit like those of oxide of arsenic, but many of them
also exhibit the faces of the cube.

In any case, however it may be erected, the crystal edifice is produced
by the regular accretion of molecule on molecule, like the bricks or
stone blocks of the builder, and in accordance with an architectural
plan more elaborate and exact than that of any human architect. This
plan is that of one of the thirty-two classes into which crystals can be
naturally divided with respect to their symmetry. Which specific one is
developed, and its angular dimensions, are traits characteristic of the
substance. The thirty-two classes of crystals may be grouped in seven
distinctive systems, the seven styles of crystal architecture, each
distinguished by its own elements of symmetry.

A crystal possesses two further fundamental properties besides its style
of architecture. The first is that it is bounded externally by plane
faces, arranged on the definite geometrical plan just alluded to and
mutually inclined at angles which are peculiar to the substance, and
which are, therefore, absolutely constant for the same temperature and
pressure. The second is that a crystal is essentially a homogeneous
solid, its internal structure being similar throughout, in such wise
that the arrangement about any one molecule is the same as about every
other. This structure is, in fact, that of one of the 230 homogeneous
structures ascertained by geometricians to be possible to crystals with
plane faces. The first property, that of the planeness of the crystal
faces, and their arrangement with geometrical symmetry, is actually
determined by the second, that of specific homogeneity. For, as with
human nature developed to its highest type, the external appearance is
but the expression of the internal character.

When nature has been permitted to have fair play, and the crystal has
been deposited under ideal conditions, the planeness of its faces is
astonishingly absolute. It is fully equal to that attained by the most
skilled opticians after weeks of patient labour, in the production of
surfaces on glass or other materials suitable for such delicate optical
experiments as interference-band production, in which a distortion equal
to one wave-length of light would be fatal. In all such cases of ideal
deposition, those interfacial angles on the crystal which the particular
symmetry developed requires to be equal actually are so, to this same
high degree of refinement. This fact renders possible exceedingly
accurate crystal measurement, that is, the determination of the angles
of inclination of the faces to each other, provided refined measuring
instruments (goniometers), pure chemical substances, and the means of
avoiding disturbance, either material or thermal, during the deposition
of the crystal, are available.

The study of crystals naturally divides itself into two more or less
distinct but mutually very helpful branches, and equally intimately
connected with the internal structure of crystals, namely, one which
concerns their exterior configuration and the structural morphology of
which it is the eloquent visible expression, and another which relates
to their optical characters. For the latter are so definitely different
for the different systems of crystal symmetry that they afford the
greatest possible help in determining the former, and give the casting
vote in all cases of doubt left after the morphological investigation
with the goniometer. It is, of course, their brilliant reflection and
refraction of light, with production of numerous scintillations of
reflected white light and of refracted coloured spectra, which endows
the hard and transparent mineral crystals, known from time immemorial as
gem-stones, with their attractive beauty. Indeed, their outer natural
faces are frequently, and unfortunately usually, cut away most
sacrilegiously by the lapidary, in order that by grinding and polishing
on them still more numerous and evenly distributed facets he may
increase to the maximum the magnificent play of coloured light with
which they sparkle.

An interesting and very beautiful lecture experiment was performed by
the author in a lecture a few years ago at the Royal Institution, which
illustrated in a striking manner this fact that the light reaching the
eye from a crystal is of two kinds, namely, white light reflected from
the exterior faces and coloured light which has penetrated the crystal
substance and emerges refracted and dispersed as spectra. Two powerful
beams of light from a pair of widely separated electric lanterns were
concentrated on a cluster of magnificent large diamonds, kindly lent for
the purpose by Mr Edwin Streeter, and arranged in the shape of a crown,
it being about the time of the Coronation of His late Majesty King
Edward VII. The effect was not only to produce a blaze of colour about
the diamonds themselves, but also to project upon the ceiling of the
lecture theatre numerous images in white light of the poles of the
electric arc, derived by reflection from the facets, interspersed with
equally numerous coloured spectra derived from rays which had penetrated
the substance of the diamonds, and had suffered both refraction and
internal reflection.




                               CHAPTER II
    THE MASKING OF SIMILARITY OF SYMMETRY AND CONSTANCY OF ANGLE BY
  DIFFERENCE OF HABIT, AND ITS INFLUENCE ON EARLY STUDIES OF CRYSTALS.


[Illustration:

  FIG. 5.—Natural Rhombohedron of Iceland Spar with Subsidiary Faces.
]

Nothing is more remarkable than the great variety of geometrical shapes
which the crystals of the same substance, derived from different
localities or produced under different conditions, are observed to
display. One of the commonest of minerals, calcite, carbonate of lime,
shows this feature admirably; the beautiful large rhombohedra from
Iceland, illustrated in Fig. 5, or the hexagonal prisms capped by low
rhombohedra from the Bigrigg mine at Egremont in Cumberland, shown in
Fig. 6, appear totally different from the “dog-tooth spar” so
plentifully found all over the world, a specimen of which from the same
mine is illustrated in Fig. 7. No mineral specimens could well appear
more dissimilar than these represented on Plate III. in Figs. 6 and 7,
when seen side by side in the mineral gallery of the British Museum
(Natural History) at South Kensington. But all are composed of similar
chemical molecules of calcium carbonate, CaCO_{3}; and when the three
kinds of crystals are investigated they are found to be identical in
their crystalline system, the trigonal, and indeed further as to the
subdivision or class of that system, which has come to be called the
calcite class from the importance of this mineral.

[Illustration:

  _PLATE III._

  FIG. 6.—Hexagonal Prisms of Calcite terminated by Rhombohedra.
]

[Illustration:

  FIG. 7.—Scalenohedral Crystals of Calcite, “Dog-tooth Spar.”

  CRYSTALS OF CALCITE FROM THE SAME MINE, ILLUSTRATING DIVERSITY OF
    HABIT.

  (Photographed from Specimens in the Natural History Department of the
    British Museum, by kind permission.)
]

Moreover, many of the same faces, that is, faces having the same
relation to the symmetry, are present on all three varieties, the
“forms” to which they equally belong being the common heritage of
calcite wherever found. A “form” is the technical term for a set of
faces having an equal value with respect to the symmetry. Thus the
prismatic form in Fig. 6 is the hexagonal prism, a form which is common
to the hexagonal and trigonal systems of symmetry, and the form
“indices” (numbers[1] inversely proportional to the intercepts cut off
from the crystal axes by the face typifying the form) of which are
{2̄1̄1}; the large development of this form confers the elongated
prismatic habit on the crystal. The terminations are faces of the flat
rhombohedron {110}. The pyramidal form of the dog-tooth spar shown in
Fig. 7 is the scalenohedron {20̄1}, and it is this form which confers
the tooth-like habit, so different from the hexagonal prism, upon this
variety of calcite. But many specimens of dog-tooth spar, notably those
from Derbyshire, consist of scalenohedra the middle portion of which is
replaced by faces of the hexagonal prism {2̄1̄1}, and the terminations
of which are replaced by the characteristic rhombohedron {100} of
Iceland spar; indeed, it is quite common to find crystals of calcite
exhibiting on the same individual all the forms which have been
mentioned, that is, those dominating the three very differently
appearing types. The author has quite recently measured such a crystal,
which, besides showing all these four forms well developed, also
exhibited the faces of two others of the well-known forms of calcite,
{3̄1̄1} and {310}, and a reproduction of a drawing of it to scale is
given in Fig. 8. Instead of indices the faces of each form bear a
distinctive letter; _m_ = {2̄1̄1}, _r_ = {100}, _e_ = {110}, _v_ =
{20̄1} (the faces of the scalenohedron are of somewhat small dimensions
on this crystal), _n_ = {3̄1̄1}, and _t_ = {310}.

[Illustration:

  FIG. 8.—Measured Crystal of Calcite.
]

It is obviously then the “habit” which is different in the three types
of calcite—Iceland spar, prismatic calc-spar, and dog-tooth
spar—doubtless owing to the different local circumstances of growth of
the mineral. Habit is simply the expression of the fact that a specific
“form,” or possibly two particular forms, is or are much more
prominently developed in one variety than in another. Thus the principal
rhombohedron _r_ = {100}, parallel to the faces of which calcite cleaves
so readily, is the predominating form in Iceland spar, while the
scalenohedron _v_ = {20̄1} is the habit-conferring form in dog-tooth
spar. Yet on the latter the rhombohedral faces are frequently developed,
blunting the sharp terminations of the scalenohedra, especially in
dog-tooth spar from Derbyshire or the Hartz mountains; and on the former
minute faces of the scalenohedron are often found, provided the
rhombohedron consists of the natural exterior faces of the crystal and
not of cleavage faces. In the same manner the prismatic crystals from
Egremont are characterised by two forms, the hexagonal prism _m_ =
{2̄1̄1} and the secondary rhombohedron _e_ = {110}, but both of these
forms, as we have seen on the actual crystal represented in Fig. 8, are
also found developed on other crystals of mixed habit.

This illustration from the naturally occurring minerals might readily be
supplemented by almost any common artificial chemical preparation,
sulphate of potash for instance, K_{2}SO_{4}, the orthorhombic crystals
of which take the form of elongated prisms, even needles, on the one
hand, or of tabular plate-like crystals on the other hand, according as
the salt crystallises by the cooling of a supersaturated solution, or by
the slow evaporation of a solution which at first is not quite
saturated. In both cases, and in all such cases, whether of minerals or
chemical preparations, the same planes are present on the crystals of
the same substance, although all may not be developed on the same
individual except in a few cases of crystals particularly rich in faces;
and these same planes are inclined at the same angles. But their
relative development may be so very unlike on different crystals as to
confer habits so very dissimilar that the fact of the identity of the
substance is entirely concealed.

[Illustration:

  FIG. 9.—Crystal of Gypsum.
]

A further example may perhaps be given, that of a substance, hydrated
sulphate of lime, CaSO_{4}.2H_{2}O, which occurs in nature as the
beautiful transparent mineral gypsum or selenite—illustrated in Fig. 9,
and which is found in monoclinic crystals often of very large size—and
which may also be chemically prepared by adding a dilute solution of
sulphuric acid to a very dilute solution of calcium chloride. The
radiating groups of needles shown in Fig. 10 (Plate II.) slowly
crystallise out when a drop of the mixed solution is placed on a
microscope slip and examined under the microscope, using the one-inch
objective. These needles, so absolutely different in appearance from a
crystal of selenite, are yet similar monoclinic prisms, but in which the
prismatic form is enormously elongated compared with the other
(terminating) form.

This difference of facial development, rendering the crystals of one and
the same substance from different sources so very unlike each other, was
apparently responsible for the very tardy discovery of the fundamental
law of crystallography, the constancy of the crystal angles of the same
substance. Gessner, sometime between the years 1560 and 1568, went so
far as to assert that not only are different crystals of the same
substance of different sizes, but that also the mutual inclinations of
their faces and their whole external form are dissimilar.

What was much more obvious to the early students of crystals, and which
is, in fact, the most striking thing about a crystal after its regular
geometric exterior shape, was the obviously homogeneous character of its
internal structure. So many crystals are transparent, and so clear and
limpid, that it was evident to the earliest observers that they were at
least as homogeneous throughout as glass, and yet that at the same time
they must be endowed with an internal structure the nature of which is
the cause of both the exterior geometric regularity of form, so
different from the irregular shape of a lump of glass, and of the
peculiar effect on the rays of light which are transmitted through them.
From the earliest ages of former civilisations the behaviour of crystals
with regard to light has been known to be different for the different
varieties of gem-stones.

About the year 1600 Cæsalpinus observed that sugar, saltpetre, and alum,
and also the sulphates of copper, zinc and iron, known then as blue,
white and green vitriol respectively, separate from their solutions in
characteristic forms. Had he not attributed this to the operation of an
organic force, in conformity with the curious opinion of the times
concerning crystals, he might have had the credit of being the pioneer
of crystallographers. The first two real steps in crystallography,
however, with which in our own historic times we are acquainted, were
taken in the seventeenth century within four years of each other, one
from the interior structural and the other from the exterior geometrical
point of view. For in 1665 Robert Hooke in this country made a study of
alum, which he appears to have obtained in good crystals, although he
was unacquainted with its true chemical composition. He describes in his
“Micrographia” how he was able to imitate the varying habits of the
octahedral forms of alum crystals by building piles of spherical musket
bullets, and states that all the various figures which he observed in
the many crystals which he examined could be produced from two or three
arrangements of globular particles. It is clear that the homogeneous
partitioning of space in a crystal structure by similar particles
building up the crystal substance was in Hooke’s mind, affording another
testimony to the remarkably prescient insight of our great countryman.

Four years later, in 1669, Nicolaus Steno carried out in Florence some
remarkable measurements, considering the absence of proper instruments,
of the angles between the corresponding faces of different specimens of
rock-crystal (quartz, the naturally occurring dioxide of silicon,
concerning which there will be much to say later in this book), obtained
from different localities, and published a dissertation announcing that
he found these analogous angles all precisely the same.

In the year 1688 the subject was taken up systematically by Guglielmini,
and in two memoirs of this date and 1705 he extended Steno’s conclusions
as to the constancy of crystal angles in the case of rock-crystal into a
general law of nature. Moreover, he began to speculate about the
interior structure of crystals, and, like Hooke, he took alum as his
text, and suggested that the ultimate particles possessed plane faces,
and were, in short, miniature crystals. He further announced the
constancy of the cleavage directions, so that to Guglielmini must be
awarded the credit for having, at a time when experimental methods of
crystallographic investigation were practically _nil_, discovered the
fundamental principles of crystallography.

The fact that a perfect cleavage is exhibited by calcite had already
been observed by Erasmus Bartolinus in 1670, and in his “Experimenta
Crystalli Islandici” he gives a most interesting account of the great
discovery of immense clear crystals of calcite which had just been made
at Eskifjördhr in Iceland, minutely describing both their cleavage and
their strong double refraction. Huyghens in 1690 followed this up by
investigating some of these crystals of calcite still more closely, and
elaborated his laws of double refraction as the result of his studies.

There now followed a century which was scarcely productive of any
further advance at all in our real knowledge of crystals. It is true
that Boyle in 1691 showed that the rapidity with which a solution cools
influences the habit of the crystals which are deposited from it. But
neither Boyle, with all his well-known ability, so strikingly displayed
in his work on the connection between the volume of a gas and the
pressure to which it is subjected, nor his lesser contemporaries Lemery
and Homberg, who produced and studied the crystals of several series of
salts of the same base with different acids, appreciated the truth of
the great fact discovered by Guglielmini, that the same substance always
possesses the same crystalline form the angles of which are constant.
Even with the growth of chemistry in the eighteenth century, the opinion
remained quite general that the crystals of the same substance differ in
the magnitude of their angles as well as in the size of their faces.

We begin to perceive signs of progress again in the year 1767, when
Westfeld made the interesting suggestion that calcite is built up of
rhombohedral particles, the miniature faces of which correspond to the
cleavage directions. This was followed in 1780 by a treatise “De formis
crystallorum” by Bergmann and Gahn of Upsala, in which Guglielmini’s law
of the constancy of the cleavage directions was reasserted as a general
one, and intimately connected with the crystal structure. It was in this
year 1780 that the contact goniometer was invented by Carangeot,
assistant to Romé de l’Isle in Paris, and it at once placed at the
disposal of his master a weapon of research far superior to any
possessed by previous observers.

[Illustration:

  FIG. 11.—Contact Goniometer as used by Romé de l’Isle.
]

In his “Crystallographie,” published in Paris in 1783, Romé de l’Isle
described a very large number of naturally occurring mineral crystals,
and after measuring their angles with Carangeot’s goniometer he
constructed models of no less than 500 different forms. Here we have
work based upon sound measurement, and consequently of an altogether
different and higher value than that which had gone before. It was the
knowledge that his master desired to faithfully reproduce the small
natural crystals which he was investigating, on the larger scale of a
model, that led Carangeot to invent the contact goniometer, and thus to
make the first start in the great subject of goniometry. The principle
of the contact goniometer remains to-day practically as Carangeot left
it, and although replaced for refined work by the reflecting goniometer,
it is still useful when large mineral crystals have to be dealt with. An
illustration of a duplicate of the original instrument is shown in Fig.
11, by the kindness of Dr H. A. Miers. This duplicate was presented to
Prof. Buckland by the Duke of Buckingham in the year 1824, and is now in
the Oxford Museum.

From the time that measurement of an accurate description was possible
by means of the contact goniometer, progress in crystallography became
rapid. Romé de l’Isle laid down the sound principle, as the result of
the angular measurements and the comparison of his accurate models with
one another, that the various crystal shapes developed by the same
substance, artificial or natural, were all intimately related, and
derivable from a primitive form, characteristic of the substance. He
considered that the great variety of form was due to the development of
secondary faces, other than those of the primitive form. He thus
connected together the work of previous observers, consolidated the
principles laid down by Guglielmini by measurements of real value, and
threw out the additional suggestion of a fundamental or primitive form.

About the same time Werner was studying the principal forms of different
crystals of the same substance. The idea of a fundamental form appears
to have struck him also, and he showed how such a fundamental form may
be modified by truncating, bevelling, and replacing its faces by other
derived forms. His work, however, cannot possess the value of that of
Romé de l’Isle, as it was not based on exact measurement, and most of
all because Werner appears to have again admitted the fallacy that the
same substance could, in the ordinary way, and not in the sense now
termed polymorphism, exhibit several different fundamental forms.

But a master mind was at hand destined definitely to remove these doubts
and to place the new science on a firm basis. An account of how this was
achieved is well worthy of a separate chapter.




                              CHAPTER III
                  THE PRESCIENT WORK OF THE ABBÉ HAÜY.


The important work of Romé de l’Isle had paved the way for a further and
still greater advance which we owe to the University of Paris, for its
Professor of the Humanities, the Abbé Réné Just Haüy, a name ever to be
regarded with veneration by crystallographers, took up the subject
shortly after Romé de l’Isle, and in 1782 laid most important results
before the French Academy, which were subsequently, in 1784, published
in a book, under the auspices of the Academy, entitled “Essai d’une
Théorie sur la Structure des Crystaux.” The author happens to possess,
as the gift of a kind friend, a copy of the original issue of this
highly interesting and now very rare work. It contains a brief preface,
dated the 26th November 1783, signed by the Marquis de Condorcet,
perpetual secretary to the Academy (who, in 1794, fell a victim to the
French revolution), to the effect that the Academy had expressed its
approval and authorised the publication “under its privilege.”

The volume contains six excellent plates of a large number of most
careful drawings of crystals, illustrating the derivation from the
simple forms, such as the cube, octahedron, dodecahedron, rhombohedron,
and hexagonal prism, of the more complicated forms by the symmetrical
replacement of edges and corners, together with the drawings of many
structural lattices. In the text, Haüy shows clearly how all the
varieties of crystal forms are constructed according to a few simple
types of symmetry; for instance, that the cube, octahedron, and
dodecahedron all have the same high degree of symmetry, and that the
apparently very diverse forms shown by one and the same substance are
all referable to one of these simple fundamental or systematic forms.
Moreover, Haüy clearly states the laws which govern crystal symmetry,
and practically gives us the main lines of symmetry of five of the seven
systems as we now classify them, the finishing touch having been
supplied in our own time by Victor von Lang.

Haüy further showed that difference of chemical composition was
accompanied by real difference of crystalline form, and he entered
deeply into chemistry, so far as it was then understood, in order to
extend the scope of his observations. It must be remembered that it was
only nine years before, in 1774, that Priestley had discovered oxygen,
and that Lavoisier had only just (in the same year as Haüy’s paper was
read to the Academy, 1782) published his celebrated “Elements de
Chimie”; and further, that Lavoisier’s memoir “Reflexions sur le
Phlogistique” was actually published by the Academy in the same year,
1783, as that in which this book was written by Haüy. Moreover, it was
also in this same year, 1783, that Cavendish discovered the compound
nature of water.

Considering, therefore, all these facts, it is truly surprising that
Haüy should have been able to have laid so accurately the foundations of
the science of crystallography. That he undoubtedly did so, thus
securing to himself for all time the term which is currently applied to
him of “father of crystallography,” is clearly apparent from a perusal
of his book and of his subsequent memoirs.

The above only represents a small portion of Haüy’s achievements. For he
discovered, besides, the law of rational indices, the generalisation
which is at the root of crystallographic science, limiting, as it does,
the otherwise infinite number of possible crystal forms to comparatively
few, which alone are found to be capable of existence as actual
crystals. The essence of this law, which will be fully explained in
Chapter V., is that the relative lengths intercepted along the three
principal axes of the crystal, by the various faces other than those of
the fundamental form, the faces of which are parallel to the axes, are
expressed by the simplest unit integers, 1, 2, 3, or 4, the latter being
rarely exceeded and then only corresponding to very small and altogether
secondary faces.

This discovery impressed Haüy with the immense influence which the
structure of the crystal substance exerts on the external form, and how,
in fact, it determines that form. For the observations were only to be
explained on the supposition that the crystal was built up of structural
units, which he imagined to be miniature crystals shaped like the
fundamental form, and that the faces were dependent on the step-like
arrangement possible to the exterior of such an assemblage. This brought
him inevitably to the intimate relation which cleavage must bear to such
a structure, that it really determined the shape of, and was the
expression of the nature of, the structural units. Thus, before the
conception of the atomic theory by Dalton, whose first paper (read 23rd
October 1803), was published in the year 1803 in the Proceedings of the
Manchester Literary and Philosophical Society, two years after the
publication of Haüy’s last work (his “Traité de Minéralogie,” Paris,
1801), Haüy came to the conclusion that crystals were composed of units
which he termed “_Molécules Intégrantes_,” each of which comprised the
whole chemical compound, a sort of gross chemical molecule. Moreover, he
went still further in his truly original insight, for he actually
suggested that the _molécules intégrantes_ were in turn composed of
“_Molécules Elémentaires_,” representing the simple matter of the
elementary substances composing the compound, and hinted further that
these elementary portions had properly orientated positions within the
_molécules intégrantes_.

He thus not only nearly forestalled Dalton’s atomic theory, but also our
recent work on the stereometric orientation of the atoms in the molecule
in a crystal structure. Dalton’s full theory was not published until the
year 1811, in his epoch-making book entitled “A New System of Chemical
Philosophy,” although his first table of atomic weights was given as an
appendix to the memoir of 1803. Thus in the days when chemistry was in
the making at the hands of Priestley, Lavoisier, Cavendish, and Dalton
do we find that crystallography was so intimately connected with it that
a crystallographer well-nigh forestalled a chemist in the first real
epoch-making advance, a lesson that the two subjects should never be
separated in their study, for if either the chemist or the
crystallographer knows but little of what the other is doing, his work
cannot possibly have the full value with which it would otherwise be
endowed.

The basis of Haüy’s conceptions was undoubtedly cleavage. He describes
most graphically on page 10 of his “Essai” of 1784 how he was led to
make the striking observation that a hexagonal prism of calcite,
terminated by a pair of hexagons normal to the prism axis, similar to
the prisms shown in Fig. 6 (Plate III.) except that the ends were flat,
showed oblique internal cleavage cracks, by enhancing which with the aid
of a few judicious blows he was able to separate from the middle of the
prism a kernel in the shape of a rhombohedron, the now well-known
cleavage rhombohedron of calcite. He then tried what kinds of kernels he
could get from dog-tooth spar (illustrated in Fig. 7) and other
different forms of calcite, and he was surprised to find that they all
yielded the same rhombohedral kernel. He subsequently investigated the
cleavage kernels of other minerals, particularly of gypsum, fluorspar,
topaz, and garnet, and found that each mineral yielded its own
particular kernel. He next imagined the kernels to become smaller and
smaller, until the particles thus obtained by cleaving the mineral along
its cleavage directions _ad infinitum_ were the smallest possible. These
miniature kernels having the full composition of the mineral he terms
“_Molécules Constituantes_” in the 1784 “Essai,” but in the 1801
“Traité” he calls them “_Molécules Intégrantes_” as above mentioned. He
soon found that there were three distinct types of _molécules
intégrantes_, tetrahedra, triangular prisms, and parallelepipeda, and
these he considered to be the crystallographic structural units.

[Illustration:

  FIG. 12.
]

Having thus settled what were the units of the crystal structure, Haüy
adopted Romé de l’Isle’s idea of a primitive form, not necessarily
identical with the _molécule intégrante_, but in general a
parallelepipedon formed by an association of a few _molécules
intégrantes_, the parallelepipedal group being termed a “_Molécule
Soustractive_.” The primary faces of the crystal he then supposed to be
produced by the simple regular growth or piling on of _molécules
intégrantes_ or _soustractives_ on the primitive form. The secondary
faces not parallel to the cleavage planes next attracted his attention,
and these, after prolonged study, he explained by supposing that the
growth upon the primitive form eventually ceased to be complete at the
edges of the primary faces, and that such cessation occurred in a
regular step by step manner, by the suppression of either one, two, or
sometimes three _molécules intégrantes_ or _soustractives_ along the
edge of each layer, like a stepped pyramid, the inclination of which
depends on how many bricks or stone blocks are intermitted in each layer
of brickwork or masonry. Fig. 12 will render this quite clear, the face
AB being formed by single block-steps, and the face CD by two blocks
being intermitted to form each step. The plane AB or CD containing the
outcropping edges of the steps would thus be the secondary plane face of
the crystal, and the _molécules_ _intégrantes_ or _soustractives_ (the
steps can only be formed by parallelepipedal units) being
infinitesimally small, the re-entrant angles of the steps would be
invisible and the really furrowed surface appear as a plane one. Haüy is
careful to point out, however, that the crystallising force which causes
this stepped development (or lack of development) is operative from the
first, for the minutest crystals show secondary faces, and often better
than the larger crystals.

[Illustration:

  FIG. 13.
]

An instance of a mineral with tetrahedral _molécules intégrantes_ Haüy
gives in tourmaline, and the primitive form of tourmaline he considered
to be a rhombohedron, conformably to the well-known rhombohedral
cleavage of the mineral, made up of six tetrahedra. Again, hexagonal
structures formed by three prismatic cleavage planes inclined at 60° are
considered by him as being composed of _molécules intégrantes_ of the
form of 60° triangular prisms, or _molécules soustractives_ of the shape
of 120° rhombic prisms, each of the latter being formed by two
_molécules intégrantes_ situated base to base. This will be clear from
Figs. 13 and 14, the former representing the structure as made up of
equilateral prismatic structural units, and the latter portraying the
same structure but composed of 120°-parallelepipeda by elimination of
one cleavage direction; each unit in the latter case possesses double
the volume of the triangular one, and being of parallelepipedal section
is capable of producing secondary faces when arranged step-wise, whereas
the triangular structure is not. The points at the intersections in
these diagrams should for the present be disregarded; they will shortly
be referred to for another purpose.

[Illustration:

  FIG. 14.
]

Probably, the most permanent and important of Haüy’s achievements was
the discovery of the law of rational indices. At first this only took
the form of the observation of the very limited number of rows of
_molécules intégrantes_ or _soustractives_ suppressed. In introducing it
on page 74 of his 1784 “Essai” he says: “_Quoique je n’aie observé
jusqu’ici que des décroissemens qui se sont par des soutractions d’une
ou de deux rangées de molécules, et quelquefois de trois rangées, mais
très rarement, il est possible qu’il se trouve des crystaux dans
lesquels il y ait quatre ou cinq rangées de molécules supprimées à
chaque décroissement, et même un plus grand nombre encore. Mais ces cas
me semblent devoir être plus rares, à proportion que le nombre des
rangées soutraites sera plus considérable. On conçoit donc comment le
nombre des formes secondaires est néçessairement limité._”

The essential difference between Haüy’s views and our present ones,
which will be explained in Chapter IX., is that Haüy takes cleavage
absolutely as his guide, and considers the particles, into which the
ultimate operation of cleavage divides a crystal, as the solid
structural units of the crystal, the unit thus having the shape of at
least the _molécule intégrante_. Now every crystalline substance does
not develop cleavage, and others only develop it along a single plane,
or along a couple of planes parallel to the same direction, that of
their intersection and of the axis of the prism which two such cleavages
would produce, and which prism would be of unlimited length, being
unclosed.

Again, in other cases cleavage, such as the octahedral cleavage of
fluorspar, yields octahedral or tetrahedral _molécules intégrantes_
which are not congruent, that is to say, do not fit closely together to
fill space, as is the essence of Haüy’s theory. Hence, speaking
generally, partitioning by means of cleavage directions does not
essentially and invariably yield identical plane-faced molecules which
fit together in contact to completely fill space, although in the
particular instances chosen from familiar substances by Haüy it often
happens to do so. Haüy’s theory is thus not adequately general, and the
advance of our knowledge of crystal forms has rendered it more and more
apparent that Haüy’s theory was quite insufficient, and his _molécules
intégrantes_ and _soustractives_ mere geometrical abstractions, having
no actual basis in material fact; but that at the same time it gave us a
most valuable indication of where to look for the true conception.

This will be developed further into our present theory of the
homogeneous partitioning of space, in Chapter IX. But it may be stated
here, in concluding our review of the pioneer work of Haüy, that in the
modern theory all consideration of the shape of the ultimate structural
units is abandoned as unnecessary and misleading, and that each chemical
molecule is considered to be represented by a point, which may be either
its centre of gravity, a particular atom in the molecule (for we are now
able in certain cases to locate the orientation of the spheres of
influence of the elementary atoms in the chemical molecules), or a
purely representative point standing for the molecule. The only
condition is that the points chosen within the molecules shall be
strictly analogous, and similarly orientated. The dots at the
intersections of the lines in Figs. 13 and 14 are the representative
points in question. We then deal with the distances between the points,
the latter being regarded as molecular centres, rather than with the
dimensions of the cells themselves regarded as solid entities. We thus
avoid the as yet unsolved question of how much is matter and how much is
interspace in the room between the molecular centres. In this form the
theory is in conformity with all the advances of modern physics, as well
as of chemistry. And with this reservation, and after modifying his
theory to this extent, one cannot but be struck with the wonderful
perspicacity of Haüy, for he appears to have observed and considered
almost every problem with which the crystallographer is confronted, and
his laws of symmetry and of rational indices are perfectly applicable to
the theory as thus modernised.




                               CHAPTER IV
               THE SEVEN STYLES OF CRYSTAL ARCHITECTURE.


It is truly curious how frequently the perfect number, seven, is endowed
with exceptional importance with regard to natural phenomena. The seven
orders of spectra, the seven notes of the musical octave, and the seven
chemical elements, together with the seven vertical groups to which by
their periodic repetition they give rise, of the “period” of
Mendeléeff’s classification of the elements, will at once come to mind
as cases in point. This proverbial importance of the number seven is
once again illustrated in regard to the systems of symmetry or styles of
architecture displayed by crystals. For there are seven such systems of
crystal symmetry, each distinguished by its own specific elements of
symmetry.

It is only within recent years that we have come to appreciate what are
the real elements of symmetry. For although there are but seven systems,
there are no less than thirty-two classes of crystals, and these were
formerly grouped under six systems, on lines which have since proved to
be purely arbitrary and not founded on any truly scientific basis. It
was supposed that those classes in any system which did not exhibit all
the faces possible to the system owed this lack of development to the
suppression of one-half or three-quarters of the possible number, and
such classes were consequently called “hemihedral” and “tetartohedral”
respectively. As in the higher systems of symmetry there were usually
two or more ways in which a particular proportionate suppression of
faces could occur, it happened that several classes, and not merely
three—holohedral (possessing the full number of faces), hemihedral, and
tetartohedral—constituted each of these systems.

Thanks largely to the genius of Victor von Lang, who was formerly with
us in England at the Mineral Department of the British Museum, and to
his successor there, Nevil Story Maskelyne, we have at last a much more
scientific basis for our classification of crystals, and one which is in
complete harmony with the now perfected theory of possible homogeneous
structures. Victor von Lang showed that the true elements of symmetry
are planes of symmetry and axes of symmetry. A crystal possessing a
plane of symmetry is symmetrical on both sides of that plane, both as
regards the number of the faces and their precise angular disposition
with respect to one another.

It is quite possible, and even the usual case, that the relative
development of the faces, that is their actual sizes, may prevent the
symmetry from being at first apparent; but when we come to measure the
angles between the faces, by use of the reflecting goniometer, and to
plot their positions out on the surface of a sphere, or on a plane
representation of the latter on paper, the exceedingly useful
“stereographic projection,” we at once perceive the symmetry perfectly
plainly.

[Illustration:

  FIG. 15.—Crystal of Potassium Nickel Sulphate.
]

[Illustration:

  FIG. 16.—Projection of Potassium Nickel Sulphate and its Isomorphous
    Analogues.
]

Thus in Fig. 15 is represented a crystal of the salt potassium nickel
sulphate, K_{2}Ni(SO_{4})_{2}.6H_{2}O, belonging to the monoclinic
system of symmetry, and which, therefore, possesses only one plane of
symmetry. In Fig. 16 its stereographic projection is shown, in which
each face in one of the symmetrical halves is represented by a dot, the
plane of symmetry, parallel to the face _b_, being the plane of the
paper, so that each dot not on the circumference really represents two
symmetrical faces, one above and one below the paper, while the
circumferential dots represent faces perpendicular to the symmetry plane
and paper. The mode of arriving at such a useful projection, or plan of
the faces, will be discussed more fully later in Chapter VI. But for the
present purpose it will be sufficient to note that the right and left
halves of the crystal shown in Fig. 15 are obviously symmetrical to each
other, and that the plan of either half, projected on the dividing plane
of symmetry itself, may be taken as given in Fig. 16; that is, we may
imagine the crystal shown in Fig. 15 to be equally divided by a section
plane which is vertical and perpendicular to the paper when the latter
is held up behind the crystal and in front of the eye, this section
plane being the plane of symmetry and parallel to the face _b_ = (010).
It may thus be imagined as the plane of projection of Fig. 16.

An axis of symmetry is a direction in the crystal such that when the
latter is rotated for an angle of 60°, 90°, 120°, or 180° around it, the
crystal is brought to look exactly as it did before such rotation. When
a rotation for 180° is necessary in order to reproduce the original
appearance, the axis is called a “digonal” axis of symmetry, for two
such rotations then complete the circle and bring the crystal back to
identity, not merely to similarity. When the rotation into a position of
similarity is for 120°, three such rotations are required to restore
identity, and the axis is then termed a “trigonal” one. Similarly, four
rotations to positions of similarity 90° apart are essential to complete
the restoration to identity, and the axis is then a “tetragonal” one,
each rotation of a right angle causing the crystal to appear as at
first, assuming, as in all cases, the ideal equality of development of
faces. Lastly, if 60° of rotation bring about similarity, six such
rotations are required in order to effect identity of position, and the
axis is known as a “hexagonal” one.

Now, there is one system of symmetry which is characterised by the
presence of a single hexagonal axis of symmetry, and this is the
_hexagonal system_. A crystal of this system, one of the naturally
occurring mineral apatite, which has been actually measured by the
author, is shown in Fig. 17. There is another system, the chief property
of which is to possess a tetragonal axis of symmetry, and which is
therefore termed the _tetragonal system_. A tetragonal crystal of
anatase, titanium dioxide, TiO_{2}, which has likewise been measured on
the goniometer by the author, is shown in Fig. 18. And there is yet
another system, the trigonal, the chief attribute of which is the
possession of a single trigonal axis of symmetry, and which is
consequently named the trigonal system. In Fig. 19 is shown a crystal of
calcite, within which the directions of the three rhombohedral
crystallographic axes of the trigonal system, and that of the vertical
trigonal axis of symmetry, are indicated in broken-and-dotted lines.

[Illustration:

  FIG. 17.—Measured Crystal of Apatite.
]

[Illustration:

  FIG. 18.—Measured Crystal of Anatase.
]

But there is one system of symmetry, the highest possible, and which has
already been referred to as the _cubic system_, which combines in itself
all but one (the hexagonal axis) of the elements of symmetry. Indeed,
not only does it possess a tetragonal, a trigonal, and a digonal axis of
symmetry, but also ten other symmetry axes; for these three
automatically involve altogether the presence of no less than three
tetragonal, four trigonal, and six digonal axes of symmetry, together
with nine planes of symmetry, twenty-two elements of symmetry being thus
present in all.

The perfections of the cube, the simple lines of which are illustrated
in Fig. 20, as the expression of the highest kind of symmetry, with
angles all right angles and sides and edges all equal, were so fully
appreciated by the geometrical minds of the ancient Greek philosophers,
imbued with the innate love of symmetry characteristic of their nation,
that to them the cube became the emblem of perfection. We are reminded
of this interesting fact in the Book of Revelation, which, in describing
in its inimitable language the wonders of the Holy City, speaks of it as
“lying foursquare,” and attributes to it the properties of the cube,
that “The length and the breadth and the height of it are equal.”

[Illustration:

  FIG. 19.—Crystal of Calcite.
]

[Illustration:

  FIG. 20.—The Cube.
]

[Illustration:

  FIG. 21.—The Hexakis Octahedron.
]

The full symmetry of the cubic system is not realised, however, by a
study of the cube alone; we only appreciate it when we come to examine
the general form of the cubic system, that which is produced by starting
with a face oblique to all three axes, and with different amounts of
obliquity to each, and seeing how many repetitions of the face the
symmetry demands. The presence of such a face involves as a matter of
fact, when all the elements of symmetry are satisfied, the presence also
of no less than forty-seven others, symmetrically situated, the
forty-eight-sided figure produced being the hexakis octahedron shown in
Fig. 21, and which is occasionally actually found developed in nature as
the diamond. All diamonds do not by any means exhibit this form so
wonderfully rich in faces, but diamonds are from time to time found
which do show all the forty-eight faces well developed.

[Illustration:

  FIG. 22.—Measured Crystal of Topaz.
]

Besides these four more highly symmetrical systems or styles of crystal
architecture, a fifth, the _monoclinic system_, characterised by a
single plane of symmetry and one axis of digonal symmetry perpendicular
thereto, has already been alluded to, and a typical crystal illustrated
in Fig. 15. A sixth, the _rhombic system_, perhaps in some ways the most
interesting of all, and certainly so optically, possesses three
rectangular axes of symmetry, identical in direction with the
crystallographic axes, and three mutually rectangular planes of
symmetry, coincident with the axial planes and intersecting each other
in the axes. The lengths of the three crystal axes are unequal, however,
and herein lies the essential difference from the cube. A very typical
rhombic substance is topaz, a crystal of which, about three millimetres
in diameter, is shown very much enlarged in Fig. 22. Every face on this
crystal has been actually investigated on the goniometer, and the
interfacial angles measured.

[Illustration:

  FIG. 23.—Measured Crystal of Copper Sulphate.
]

Lastly, there is the seventh, the _triclinic system_, in which there are
neither planes nor axes of symmetry, but, even in its holohedral class,
only symmetry about the centre, each face having a parallel fellow.
Sulphate of copper, blue vitriol, CuSO_{4}.5H_{2}O, shows this type of
symmetry, or rather lack of it, very characteristically, and a crystal
of this beautiful deep blue salt, measured by the author, is represented
in Fig. 23.

Hence, we have arrived logically at seven systems of symmetry or styles
of crystal architecture, distinguished by the nature of their essential
axes of symmetry, and the planes of symmetry which may accompany them.
Now the full degree of symmetry of each system may be reduced to a
certain minimum without lowering the system, and in all the systems but
the triclinic there are several definite stages of reduction before the
minimum is reached, each stage corresponding to one of the thirty-two
classes of crystals. Thus in the cubic system there are four classes
besides the holohedral, in the tetragonal six, in the hexagonal four, in
the trigonal six, in the rhombic and monoclinic two each, and in the
triclinic one.

[Illustration:

  _PLATE IV._

  FIG. 24.—Octahedra of Potassium Cadmium Cyanide.
]

[Illustration:

  FIG. 25.—Octahedra of Cæsium Alum.

  CUBIC CRYSTALS GROWING FROM SOLUTION.
]

We have thus attained at length to a truly scientific classification of
crystal forms, by using axes and planes of symmetry as _criteria_. There
is no occasion whatever to imagine suppression of faces in the classes
of lower than the holohedral or highest symmetry of any system. In these
classes it is simply the fact that less than the full number of elements
of symmetry possible to the system are present and characterise the
class, which still conforms, however, to the minimum symmetry absolutely
essential to the system.

The drawings of crystals of the seven systems in the foregoing
illustrations will have given a correct idea of the nature of the
symmetry in each case. But now it may be much more interesting to
present a series of reproductions of photographs of some actual crystals
of the different systems. Such a series is given in Figs. 24 to 33,
Plates IV. to VIII. They were taken with the aid of the microscope, the
substances being crystallised from a slightly supersaturated solution in
each case, on a microscope slip. A ring of gold size was first laid on
the slip, and allowed to dry for several days. The drop of solution, in
the metastable supersaturated condition (corresponding to the region of
solubility which lies between the solubility and supersolubility curves,
Fig. 98, page 240), was placed in the middle of the ring, and
crystallisation just allowed to start, either owing to evaporation and
consequent production of the labile condition for spontaneous
crystallisation, or by access of a germ crystal from the air. It was
then covered with a cover-glass, which had the desired effect of
enclosing the solution in a parallelsided cell, a film of the thickness
of thick paper, suitable for undistorted microscopic observation and
photomicrography, and also the effect of arresting evaporation and
therefore the rapidity of the growth of the crystals, so that a
photomicrograph taken with the minimum necessary exposure was quite
sharp.

The crystals shown in the accompanying photographic reproductions, Figs.
24 to 33 (Plates IV. to VIII.), as well as Fig. 4 (Plate II.), already
described, were thus photographed in the very act of slow growth,
employing a one-inch objective very much stopped down. Such photographs
are infinitely sharper and more beautifully and delicately shaded than
those taken of dry crystals.

Fig. 24, Plate IV., represents cubic octahedra of the double cyanide of
potassium and cadmium, 2KCN.Cd(CN)_{2}, a salt which crystallises out in
relatively large and wonderfully transparent and well-formed single
octahedra on a micro-slip, and is particularly suitable for
demonstrating the character of this highest system, the cubic, of
crystal symmetry. Special development of the pair of faces of the
octahedron parallel to the glass surfaces has occurred, owing to greater
freedom of growth at the boundaries of these faces, as is usual in such
circumstances of deposition, but the other pairs of faces are quite
large enough to show their nature clearly.

Fig. 25, on the same Plate IV., shows a slide of cæsium alum,
Cs_{2}SO_{4}.Al_{2}(SO_{4})_{3}.24H_{2}O, in which the octahedra are
smaller, and some of them, notably one in the centre of the field, are
perfectly proportioned.

[Illustration:

  _PLATE V._

  FIG. 26.—Octahedra of Ammonium Iron Alum crystallising on a Hair.
]

[Illustration:

  FIG. 27.—Tetragonal Crystals of Potassium Ferrocyanide.

  CRYSTALS GROWING FROM SOLUTION.
]

[Illustration:

  _PLATE VI._

  FIG. 28.--Rhombic Crystals of Potassium Hydrogen Tartrate.
]

[Illustration:

  FIG. 29.--Rhombic Crystals of Ammonium Magnesium Phosphate, showing
    Special Growth along Line of Scratch.

  RHOMBIC CRYSTALS GROWING BY SLOW PRECIPITATION.
]

Fig. 26, Plate V., represents octahedra of ammonium iron alum (formula
like that of cæsium alum, but with NH_{4} replacing Cs and Fe replacing
Al) crystallising on a hair. It illustrates the interesting manner in
which crystallisation will sometimes occur, under conditions of
quietude, when some object or other on which the crystals can readily
deposit themselves is present or introduced, such as a silk or cotton
thread, or a hair as in this case.

Fig. 27, on the same Plate V., represents tetragonal crystals of
potassium ferrocyanide, K_{4}Fe(CN)_{6}, composed of tabular crystals
parallel to the basal pinakoid, bounded by faces of one order, first or
second, of tetragonal prism, the corners being modified at 45° by
smaller faces of the other order of tetragonal prism.

Fig. 28, Plate VI., is a photograph of large rhombic crystals of
hydrogen potassium tartrate, HKC_{4}H_{4}O_{6}, obtained by addition of
tartaric acid to a dilute solution of potassium chloride. They are
rectangular rhombic prisms capped by pyramidal forms, and also modified
by other prismatic and domal forms.

Fig. 29, also on Plate VI., represents another rhombic substance,
ammonium magnesium phosphate, NH_{4}MgPO_{4}.6H_{2}O, obtained by very
slow precipitation of a dilute solution of magnesium sulphate containing
ammonium chloride and ammonia with hydrogen disodium phosphate. It
illustrates in an interesting manner how, when a saturated solution is
kept quiet, and then the surface of the vessel containing it is
scratched by a needle point, a line of small crystals at once starts
forming along the line of scratch, even although the latter has made no
actual impression on the glass itself. Such a line of crystals will be
observed running across the middle of the slide.

Fig. 30, Plate VII., shows a monoclinic substance, ammonium magnesium
sulphate (NH_{4})_{2}Mg(SO_{4})_{2}.6H_{2}O, which crystallises out
splendidly on a micro-slip. The field includes several very well-formed
typical crystals of the salt, which is one of the same exceedingly
important isomorphous series to which potassium nickel sulphate, Fig.
15, belongs; it is obtained by mixing solutions containing molecularly
equivalent quantities of ammonium and magnesium sulphates. The primary
monoclinic prism is the chief form, terminated by clinodome faces and
smaller strip-faces of the basal plane, the latter, however, being
occasionally the chief end form. Small pyramid faces are also seen here
and there modifying the solid angles.

Another beautifully crystallising monoclinic substance is shown in the
next slide, Fig. 31, on the same Plate VII., namely, potassium sodium
carbonate, KNaCO_{3}.6H_{2}O, obtained from a solution of molecular
proportions of potassium and sodium carbonates. Numerous forms of the
monoclinic system are developed, on relatively large and perfectly
transparent and delicately shaded individuals.

A triclinic substance is represented in Fig. 32, Plate VIII., potassium
ferricyanide, K_{6}F_{2}(CN)_{12}. The triply oblique nature of the
symmetry is clearly exhibited by this salt, the absence of any right
angles being very marked.

[Illustration:

  _PLATE VII._

  FIG. 30.—Monoclinic Crystals of Ammonium Magnesium Sulphate.
]

[Illustration:

  FIG. 31.—Monoclinic Crystals of Sodium Potassium Carbonate.

  MONOCLINIC CRYSTALS GROWING FROM SOLUTION.
]

[Illustration:

  _PLATE VIII._

  FIG. 32.—Triclinic Crystals of Potassium Ferricyanide.
]

[Illustration:

  FIG. 33.—Tetrahedral Crystals of Sodium Sulphantimoniate, Cubic Class
    28.

  CRYSTALS GROWING FROM SOLUTION.
]

Fig. 33, also on Plate VIII., illustrates more particularly a class of
one of the systems, the cubic, which is of lower than holohedral (full)
systematic symmetry. This is the case also with hydrogen potassium
tartrate and ammonium magnesium phosphate, but the forms shown of those
salts on the slides represented in Figs. 28 and 29 are chiefly those
which are also common to the holohedral classes of their respective
systems, and the lower class symmetry is not emphasised. But here in
Fig. 33, representing Schlippe’s salt, sodium sulphantimoniate,
Na_{3}SbS_{4}.9H_{2}O, we have very clear development of the
tetrahedron, belonging to the lowest of the five classes (class 28) of
the cubic system. The crystals are almost all combinations of two
complementary tetrahedra, one of which is developed so very much more
than the other that the faces of the latter only appear as minute
replacements at the corners of the predominating tetrahedron.

This is the last for the present of these fascinating growths of
crystals under the microscope, but three more will be given
subsequently, in Figs. 99 and 100, on Plate XXI., and Fig. 101, Plate
XI., to illustrate crystallisation from metastable and labile solutions.

Fig. 34, Plate IX., represents another kind of phenomenon, equally
instructive. It shows a field in a crystal of quartz, as seen under the
same power of the microscope, a one-inch objective with small stop and
an ordinary low power eyepiece. Just above and to the left of the centre
of the field is a cavity, the shape of which is remarkable, for it is
that of a quartz crystal, a hexagonal prism terminated by rhombohedral
faces. The cavity is filled with a saturated solution of salt, except
for a bubble of water vapour, and a beautiful little cube of sodium
chloride which has crystallised out from the solution. This slide,
therefore, gives us an example of a natural cubic crystal, and also an
indication of the shape of quartz crystals, the cavity itself being a
kind of negative quartz crystal. The crystal in which it occurs must
have been formed very deep down in a reservoir of molten material
beneath a volcano, under the great pressure of superincumbent rock
masses. It was probably one of the quartz crystals of a granite rock
which had crystallised under these conditions. Almost every crystal of
quartz found in such granite rocks displays thousands of small cavities
filled with liquid and a bubble, although it is very rare to find one
with so good a cube of salt and having the configuration of a quartz
crystal for the shape of the cavity. Many such cavities, however,
contain as the liquid compressed carbonic acid, the very fact of the
carbonic acid being in the liquefied state affording ample evidence of
the pressure under which the crystal was formed. The proof that the
liquid is carbonic acid in these cases is afforded by the fact that when
the crystal is warmed to 32°C., the critical temperature of carbon
dioxide, under which it can no longer remain liquid, but must become a
gas, the bubble disappears and the cavity becomes filled with gas.
Carbonic acid cavities are readily recognised, inasmuch as the bubble is
extremely mobile, and is normally in a state of movement on the very
slightest provocation.

[Illustration:

  _PLATE IX._

  FIG. 34.—Liquid Cavities in quartz Crystal (Trigonal) containing
    Saturated Solution and Cubic Crystals of Sodium Chloride.
]

[Illustration:

  FIG. 36.—Two characteristic Forms of Snow Crystals (Trigonal).
]

[Illustration:

  FIG. 35.—Negative Ice Crystals, or “Water Flowers,” in Ice.
]

The liquid cavity in the remarkable quartz crystal illustrated in Fig.
34, and the bubble of vapour formed on cooling, and consequent
contraction of the liquid more than the solid quartz (the thermal
dilatation of liquids being usually greater than that of solids) when it
was no longer able to fill the cavity, remind one of the beautiful water
flowers formed for the contrary reason in ice on passing a beam of light
through a slab, owing to the warming effect of the accompanying heat
rays. Water crystallises like quartz, in the trigonal system, its normal
forms being the hexagonal prism and the rhombohedron. A slab of lake ice
is generally a huge crystal plate perpendicular to the trigonal axis, or
in the case of disturbed growth an interlacing mass of such crystals,
all perpendicular to the optic axis, the axis of the hexagonal prism and
of trigonal symmetry. When the heat rays from the lantern pass through
such a slab of ice, the surface of which is focussed on the screen by a
projecting lens, they cause the ice to begin to melt in numerous spots
in the interior of the slab simultaneously; and the structure of the
crystal is revealed by the operation occurring with production of
cavities taking the shape of hexagonal stars, which when focussed appear
on the screen as shown in Fig. 35. They are filled with water except for
a bubble (vacuole), which contains only water vapour. For the liquid
water occupies less room than did the ice from which it was produced,
owing to the well-known fact that water expands on freezing. This
abnormal expansion with cooling begins at the temperature of the maximum
density of water, 4° C., and proceeds steadily until the freezing point
0° is reached, when, at the moment of crystallisation, the mass suddenly
increases in volume by as much as 10 per cent. This expansive leap when
the molecules of water marshal themselves into the organised order of
the homogeneous structure, that of the space-lattice of the trigonal
(rhombohedral) system, is one of the most remarkable phenomena in
nature, and its exceptional character, so contrary to the usual
contraction on solidification of a liquid, is of vital moment to aquatic
life. For the layer of ice formed, being lighter than water, floats on
the surface of the latter, and thus forms a protective layer and
prevents to a large extent further freezing, except as a slow thickening
of the layer, the total freezing of the water of a lake or river being
rendered practically impossible, an obvious provision for the security
of life of the piscatorial and other inhabitants of the waters.

Hence, as the molecules of the substance H_{2}O are one by one detached
from their solid assemblage as ice, and become more loosely associated
as the less voluminous liquid water, they cannot occupy the whole of the
cavity formed in the solid ice, and a small vacuous space, occupied only
by water vapour at its ordinary low tension corresponding to the low
temperature, is formed and appears as the bubble. Moreover, the cavity
itself takes the shape of a hexagonal star-shaped flower, the bubble
showing at its centre, the cavity being thus a kind of negative ice
crystal, like the negative quartz crystal shown in Fig. 34. Apparently
in the production of these cavities, just as in the production of the
well-known etched figures on crystal faces by the application of a
minute quantity of a solvent for the crystal substance, the crystal
edifice is taken down, molecule by molecule, in a regular manner,
resulting in the formation of a cavity showing the symmetry of the
space-lattice which is present in the crystal structure.

[Illustration:

  _PLATE X._

  FIG. 37.—Piz Palü and Snow-field of the Pers Glacier, from the
    Diavolezza Pass, Upper Engadine.

  (From a Photograph by the author.)
]

The water flowers of Fig. 35 remind one very much of snow crystals, two
of which, re-engraved from the wonderfully careful drawings of the late
Mr Glaisher, are represented in Fig. 36, Plate IX. They all exhibit the
symmetry of the hexagonal prism, which is equally a form of the trigonal
system as it is of the hexagonal system. The snow crystals, being formed
from water vapour condensed in the cold upper layers of the atmosphere,
appear more or less as skeleton crystals, owing to the rarity of the
semi-gaseous material condensed, compared with the extent of the space
in which the crystallisation occurs. Indeed the exquisite tracery of
these snow crystals appears to afford a visual proof of the existence of
the trigonal-hexagonal space-lattice as the framework of the crystal
structure of ice. When one considers the countless numbers of such
beautiful gems of nature’s handiwork massed together on an extensive
snow-field of the higher Alps—such as that of the Piz Palü in the Upper
Engadine, shown in Fig. 37, Plate X., as seen from the Diavolezza
Pass—produced in the pure air of the higher regions of the atmosphere,
and frequently seen by the early morning climber lying uninjured in all
their beauty on the surface of the snow-field, one is lost in amazement
at the prodigality displayed in the broadcast distribution of such
peerless gems.




                               CHAPTER V
   HOW CRYSTALS ARE DESCRIBED. THE SIMPLE LAW LIMITING THE NUMBER OF
                            POSSIBLE FORMS.


The most wonderful of all the laws relating to crystals is the one
already briefly referred to which limits and regulates the possible
positions of faces, within the lines of symmetry which have been
indicated in the last chapter. Having laid down the rules of symmetry,
it might be thought that any planes which obey these laws, as regards
their mode of repetition about the planes and axes of symmetry, would be
possible. But as a matter of fact this is not so, only a very few planes
inclined at certain definite angles, repeated in accordance with the
symmetry, being ever found actually developed. The reason for this is of
far-reaching importance, for it reveals to us the certainty that a
crystal is a homogeneous structure composed of definite structural units
of tangible size, probably the chemical molecules, built up on the plan
of one of the fourteen space-lattices made known to us by Bravais, and
to be referred to more fully in Chapter VIII. In order to render this
fundamental law comprehensible, it will be essential to explain in a few
simple words how the crystallographer identifies and labels the numerous
faces on a crystal, in short, how he describes a crystal, in a manner
which shall be understood immediately by everybody who has studied the
very simple rules of the convention.

It is a matter of common knowledge that the mathematical geometrician
defines the position of any point in space with reference to three
planes, which in the simplest case are all mutually at right angles to
each other like the faces of a cube, and which intersect in three
rectangular axes _a_, _b_, _c_, the third _c_ being the vertical axis,
_b_ the lateral one, and _a_ the front-and-back axis. The distances of
the point from the three reference planes, as measured by the lengths of
the three lines drawn from the point to the planes parallel to the three
axes of intersection, at once gives him what he calls the “co-ordinates”
of the point, which absolutely define its position. In the same way we
can imagine three axes drawn within the crystal, by which not only the
position of any point on any face of the crystal may be located, but
which may be used more simply still to fix the position of the face
itself. The directions chosen as those of the three axes are the edges
of intersection of three of the best developed faces.

If there are three such faces inclined at right angles they would be
chosen in preference to all others, as they would certainly prove to be
faces of prime significance as regards the symmetry of the crystal. If
there are no such rectangularly inclined faces developed on the crystal,
then the three best developed faces nearest to 90° to each other are
chosen, the two factors of nearness to rectangularity and excellence of
development being simultaneously borne in mind in making the choice of
axial planes, and discretion used.

[Illustration:

  FIG. 38.—The Cube and its Three Equal Rectangular Axes.
]

[Illustration:

  FIG. 39.—Tetragonal Prism and its three Rectangular Axes.
]

If the crystal belong to the cubic, tetragonal, or rhombic systems, for
instance, three faces meeting each other rectangularly are possible
planes on the crystal, and will very frequently be found actually
developed; such would obviously be chosen as the axial planes. The edges
of the cube, or of the tetragonal or rectangular rhombic prism, will be
the directions of the crystallographic axes in this case, and we can
imagine them moved parallel to themselves until the common centre of
intersection, the “origin” of the analytical geometrician, will occupy
the centre of the crystal, and the faces of the latter be built up
symmetrically about it. When the crystal is cubic, the three axes will
be of equal length as shown in Fig. 38; if tetragonal, the two
horizontal axes will be equal, but will differ in length from the
vertical axis, as represented in Fig. 39. If the crystal be rhombic, all
three axes will be of different lengths, as indicated in Fig. 40, which
represents the axes and axial planes of an actual rhombic substance,
topaz, for which the lateral axis _b_ and vertical axis _c_ are nearly
but not quite equal, while the front-and-back axis _a_ is very
different.

When the crystal is of monoclinic symmetry, as in Fig. 41, three axes
will similarly be found as the intersection of three principal parallel
pairs of faces, but two of them will be inclined at an angle other than
90° to each other, while the third, the lateral one in Fig. 41, will be
at right angles to those first two and to the plane containing them;
moreover, all three are unequal in length. In the case of a triclinic
crystal, shown in Fig. 42, however, there can be no right angles, and
the intersections of three important faces meeting each other at angles
as near 90° as possible are chosen as the axes, regard being had to both
factors of approximation to rectangularity and importance of
development. These triclinic axes are the most general type of crystal
axes, for not only are the angles not right angles, but the lengths of
the axes are also unequal.

[Illustration:

  FIG. 40.—Axial Planes of a Rhombic Crystal.
]

[Illustration:

  FIG. 41.—Axial Planes of a Monoclinic Crystal.
]

[Illustration:

  FIG. 42.—Axial Planes of a Triclinic Crystal.
]

[Illustration:

  FIG. 43.—Hexagonal Prism of the First Order and its Four Axes.
]

[Illustration:

  FIG. 44.—Hexagonal Prism of the Second Order.
]

[Illustration:

  FIG. 45.—The Rhombohedron and its Three Equal Axes.
]

The cases of the hexagonal and trigonal systems are somewhat special.
The hexagonal has four such axes, as represented in Fig. 43, the lines
of intersection of the faces of the hexagonal prism closed by a pair of
perpendicular terminal planes, namely, one vertical axis parallel to the
vertical edges, and three horizontal axes inclined at 120° to each
other, and parallel to the pair of basal plane faces, equal to each
other in length, but different from the length of the vertical axis. The
hexagonal axial-plane prism shown in Fig. 43 is known as one of the
first order. The hexagonal prism corresponding to the tetragonal one of
Fig. 39, in which the axes emerge in the centres of the faces, is said
to be of the second order, and is shown in Fig. 44. The trigonal system
of crystals is best described with reference to three equal but not
rectangular axes, parallel to the faces of the rhombohedron, one of the
principal forms of the system, so well seen in Iceland spar, and
illustrated in Fig. 45. The rhombohedron may be regarded as a cube
resting on one of its corners (solid angles), with the diagonal line
joining this to the opposite corner vertical, and the cube then deformed
by flattening or elongating it along the direction of this diagonal. The
edges meeting at the ends of this vertical diagonal are then the
directions of the three trigonal crystallographic axes.

In this last illustration the vertical direction of the altered diagonal
is that of the trigonal axis of symmetry, for the rhombohedron is
brought into apparent coincidence with itself again if rotated for 120°
round this direction. But although a symmetry axis, this is not a
crystallographic axis of reference. It is not shown in Fig. 45,
therefore, but is given in Fig. 19. On the other hand, the singular
vertical axis of reference of the tetragonal and hexagonal systems is
identical with the tetragonal or hexagonal axis of symmetry of these
systems, and the three crystallographic axes of reference of the cube
are identical with the three tetragonal axes of symmetry of the cubic
system. In the rhombic system also, the three rectangular axes of
reference are identical with the three digonal axes of symmetry, and in
the monoclinic system the one axis of reference which is normal to the
plane of the two inclined axes is the unique digonal axis of symmetry of
that system.

Having thus evolved a scientific scheme of reference axes for the faces
of a crystal, it is necessary in all the systems other than the cubic
and trigonal, in which the axes are of equal lengths, to devise a mode
of arriving at the relative lengths of the axes; for on this depends the
mode of determining the positions of the various faces, other than the
three parallel pairs (or four in the case of the hexagonal system)
chosen as the axial planes. This is very simply done by choosing a
fourth important face inclined to all three axes, when one of this
character is developed, as very frequently happens, as the determinative
face or plane fixing the unit lengths of the axes. When no such face is
present on the crystal, two others can usually be found, each of which
is inclined to two different axes, so that between them all three axial
lengths are determined. The faces of the octahedron, of the primary
tetragonal pyramid and the primary rhombic pyramid, and of the
corresponding forms of the other systems, are such determinative planes,
fixing the lengths of the axes. This fact will be clear from the typical
illustration of the most general of these primary or “parametral” forms,
the triclinic equivalent of the octahedron, given in Fig. 46, the faces
being obviously obtained by joining the points marking unit lengths of
the three axes.

[Illustration:

  FIG. 46.—Triclinic Equivalent of the Octahedron.
]

Having thus settled the directions of the crystallographic axes and
their lengths, it is the next step which reveals the remarkable law to
which reference was made at the opening of this chapter. For we find
that all other faces on the crystal, however complicated and rich in
faces it may be, cut off lengths from the axes which are represented by
low whole numbers, that is, either 2, 3, 4, or possibly 5, and very
rarely more than 6 unit lengths. By far the greater number of faces do
not cut off more than three unit lengths from any axis. Prof. Miller of
Cambridge, in the year 1839, gave us a most valuable mode of labelling
and distinguishing the various faces by a symbol involving these three
values, employed, however, not directly but in an indirect yet very
simple manner. If _m_, _n_, _r_ be the three numbers expressing the
intercepts cut off by a face on the three axes, _a_, _b_, _c_
respectively, and if the Millerian index numbers be represented by _h_,
_k_, _l_, then—

                _m_ = _a_/_h_, _n_ = _b_/_k_, _r_ = _c_/_l_,
            or, _h_ = _a_/_m_, _k_ = _b_/_n_, _l_ = _c_/_r_.

Each figure or “index” of the Millerian symbol is thus inversely
proportional to the length of the intercept on the axis concerned. The
intercepts themselves are used as symbols in another mode of labelling
crystal faces, suggested by Weiss, but this method proves too cumbersome
in practice.

The Millerian symbol of a face is always placed within ordinary curved
brackets (  ), but if the symbol is to stand for the whole set of faces
composing the form, the brackets are of the type {  }. Thus the
Millerian symbol of the fourth face (that in the top-right front
octant), determinative of the unit axial lengths, is (111), as shown in
Fig. 46, the face in question being marked with this symbol; while the
symbol {111} indicates the set of faces of the whole or such part of the
double pyramid as composes the unit form. In the triclinic system this
form only consists of the face (111) and the parallel one (̄1̄1̄1), but
in the case of the regular octahedron of the cubic system it embraces
all the eight faces. The triclinic octahedron, Fig. 46, is thus made up
of four forms of two faces each. A negative sign over an index indicates
interception on the axis _a_ behind the centre, on the axis _b_ to the
left of the centre, or on the vertical axis _c_ below the centre.

To take an actual example, suppose a face other than the primary one to
make the intercepts on the axes 4, 2, 1; in this case _h_ = _a_/4, _k_ =
_b_/2, and _l_ = _c_/1, that is, when referred to the fundamental
primary form for which _a_, _b_, _c_ are each unity, _h_ = ¼, _k_ = ½,
_l_ = 1, or, bringing them to whole numbers by multiplying by 4, _h_ =
1, _k_ = 2, _c_ = 4, and the symbol in Millerian notation is (124).
Again, suppose we wish to find the intercepts on the three cubic axes
made by the face (321) of the hexakis octahedron shown in Fig. 21. To
get each intercept we multiply together the two other Millerian indices,
and if necessary afterwards reduce the three figures obtained to their
simplest relative values. For the face (321) we obtain 2, 3, 6. This
means that the face (321) in the top-right-front octant of the hexakis
octahedron cuts off two unit lengths of axis _a_, three unit lengths of
axis _b_, and six unit lengths of axis _c_. No fractional parts thus
ever enter into the relations of the axial lengths intercepted by any
face on a crystal, and the whole numbers representing these relations
are always small, the number 6 being the usual limit.

This important law is known as the “Law of Rational Indices,” and is the
corner-stone of crystallography. A forecast of it was given in Chapter
III., in describing how it was first discovered by Haüy, and it was
shown how impressed Haüy was with its obvious significance as an
indication of the brick-like nature of the crystal structure. What the
“bricks” were, Haüy was not in a position to ascertain with certainty,
as chemistry was in its infancy, and Dalton’s atomic theory had not then
been proposed.

That Haüy had a shrewd idea, however, that the structural units were the
chemical molecules, and that while the main lines of symmetry were
determined by the arrangement of the molecules its details were settled
by the arrangement of the atoms in the molecules, is clear to any one
who reads his 1784 “Essai” and 1801 “Traité,” and interprets his
_molécules intégrantes_ and _élémentaires_ in the light of our knowledge
of to-day.

Before we pass on, however, to consider the modern development of the
real meaning of the law of rational indices, as revealed by recent work
on the internal structure of crystals, it will be well to consider
first, in the next chapter, a few more essential facts as to crystal
symmetry, and the current mode of constructing a comprehensive, yet
simple, plan of the faces present on a crystal.




                               CHAPTER VI
THE DISTRIBUTION OF CRYSTAL FACES IN ZONES, AND THE MODE OF CONSTRUCTING
                          A PLAN OF THE FACES.


It will have been clear from the facts related in the previous chapters
that the salient property possessed by all crystals, when ideal
development is permitted by the circumstances of their growth, and the
substance is not one of unusual softness or liable to ready distortion,
is that the exterior form consists of and is defined by truly plane
faces inclined to each other at angles which are specific and
characteristic for each definite chemical substance; and that these
angles are in accordance with the symmetry of some particular one of the
thirty-two classes of crystals, and are such as cause the indices of the
faces concerned to be rational small numbers.

It will also be clear that, given the presence of any face other than
the three axial planes, the symmetry of the class—supposing the crystal
to exhibit some development of symmetry and not to belong to class 32,
the general case possessing no symmetry—will require the repetition of
this face a definite number of times on other parts of the crystal. Such
a set of faces possessing the same symmetry value we have already learnt
to call a “Form,” and the faces composing it will have the same
Millerian index numbers in their symbols, but differently arranged and
with negative signs over those which relate to the interception of the
back part of the _a_ axis, the left part of the _b_ axis, or the lower
part of the vertical _c_ axis; that is, parts to the front and right,
and above, the centre of intersection of the three crystal axes are
considered as the positive parts of those axes.

A form, if of general character, that is, if composed of faces each of
which is inclined to all three axes, will comprise more faces the higher
the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the
hexakis octahedron, comprises as many as forty-eight faces, all covered
by the form symbol {321}; while in the rhombic system the highest number
of faces in a form is eight, in the monoclinic only four, and in the
triclinic system two. It will also have become clear that the law of
rational indices limits the number of forms possible of any one type.
For instance, very few hexakis octahedra are known, the most frequently
occurring ones besides {321} being {421}, {531}, and {543}. Forms, of
any class, possessing higher indices than these are very rare,
especially in the systems of lower symmetry.

[Illustration:

  FIG. 47.—The Spherical Projection.
]

We next come to a further very interesting fact about crystals. Let us
imagine a crystal, on which the faces are fairly evenly developed, to be
placed in the middle of a sphere of jelly, as indicated in Fig. 47
(reproduced from a Memoir by the late Prof. Penfield), so that the
centre or origin of the axial system of the crystal and the centre of
the sphere coincide. Let us now further imagine that long needles are
stuck through the jelly and the crystal, one perpendicular to each
crystal face, and so as to reach the centre. The crystal represented in
Fig. 47 is a combination of the cube _a_, octahedron _o_, and rhombic
dodecahedron _d_. If such a thing as we have imagined were possible, we
should find that the needles would emerge at the surface of the sphere
in points which would lie on great circles, that is, on circles which
represent the intersection of the sphere by planes passing through the
centre. Moreover, the points would be distributed along these circles at
regularly recurring angular positions, corresponding to the symmetry of
the crystal. If the crystal belonged to one of the higher systems of
symmetry, it would happen that four of the points on at least one of
these great circles, and possibly on three of them, would be 90° apart,
that is, would be at the ends of rectangular diameters, which would most
likely be the axes of reference. The other points would be distributed
symmetrically on each side of these four points.

The great circles on which the points are thus symmetrically
distributed—and they may legitimately be taken to represent the faces,
for tangent planes to the sphere at these points would be parallel to
the faces—are known as “zone circles,” and the faces represented by the
points on any one of them form a “zone.” Now a zone of faces has this
practical property, that when the crystal is supported so as to be
rotatable about the zone axis—which is parallel to the edges of
intersection of all the faces composing the zone, and is the normal to
the plane of the great circle representing the zone—and a telescope is
directed towards the crystal perpendicularly to the zone axis, while a
bright object such as an illuminated slit is arranged conveniently so as
to be reflected from any face of the crystal into the telescope, an
image of it being thus visible in the latter, then it will be found that
on rotating the crystal a similar image will be seen reflected in the
telescope from every face of the zone in turn. Moreover, when the
crystal is mounted on a graduated circle, the angle of rotation between
the positions of adjustment to the cross-wires of the telescope of any
two successive images, reflected from adjacent faces of the crystal, is
actually the angle between the two points representing the faces
concerned on the zone circle, and is the supplement of the internal
dihedral angle between the two crystal faces themselves. It is, in fact,
the angle between the normals (perpendiculars) to the two faces, the
angle which is measured on the goniometer.

This is, indeed, the very simple principle of the reflecting goniometer,
invented by Wollaston in the year 1809, and which in its modern improved
form is the all-important principal instrument of the crystallographer’s
laboratory. The work with it consists largely in the measurement of the
angles between the faces in all the principal zones developed on the
crystal. The very fact, however, that crystal faces occur so absolutely
accurately in zones immeasurably lightens the labours of the
crystallographer, and is one of prime importance.

[Illustration:

  FIG. 48.—The Reflecting Goniometer.
]

The most accurate and convenient modern form of reflecting goniometer,
reading to half-minutes of arc, and provided with a delicate adjusting
apparatus for the crystal, is shown in Fig. 48. It is constructed by
Fuess of Berlin.

The graduated circle _a_ is horizontal and is divided directly to 15′,
the verniers enabling the readings to be carried further either to
single minutes, which is all that is usually necessary, or to
half-minutes in the cases of very perfect crystals. The divided circle
is rotated by means of the ring _b_ situated below, and the reading of
the verniers is accomplished with the aid of the microscopes _c_. The
circle which carries the verniers is not fixed, except when this is done
deliberately by means of the clamping screw _d_, but rotates with the
telescope _e_ to which it is rigidly attached by means of an arm and a
column _f_. A fine adjustment is provided with the clamping arrangement,
so that the telescope can be adjusted delicately with respect to the
divided circle. Both telescope and collimator are rigidly fixed at about
120° from each other during the actual measurements. The collimator _g_
is carried on a column _h_ definitely fixed to one of the legs (the back
one in Fig. 48) of the main basal tripod of the instrument. The signal
slit of the collimator is carried at the focus of the objective about
the middle of the tube _g_, the outer half of the latter being an
illumination tube carrying a condensing lens to concentrate the rays of
light from the goniometer lamp on the slit. The latter is not of the
usual rectilinear character, but composed of two circular-arc jaws, so
that, while narrow in the middle part like an ordinary spectroscope
slit, it is much broader at the two ends in order to be much more
readily visible; the central part is narrow in order to enable fine
adjustment to the vertical cross-wire of the telescope to be readily and
accurately carried out. The shape of this signal-slit will be gathered
from the images of the slit shown in Fig. 61 (page 126) in Chapter X.
The telescope carries an additional lens _k_ at its inner, objective,
end, in order that when this lens is rotated into position the telescope
may be converted into a low power microscope for viewing the crystal and
thus enabling its adjustment to be readily carried out.

The crystal _l_ is mounted on a little cone of goniometer wax (a mixture
of pitch and beeswax) carried by the crystal holder. The latter fits in
the top of the adjusting movements, which consist of a pair of
rectangularly arranged centring motions, and a pair of cylindrical
adjusting movements; the milled-headed manipulating centring screws of
the former are indicated by the letters _m_ and _n_ in Fig. 48, and
those which move the adjusting segments are marked _o_ and _p_. The top
screw fixes the crystal holder.

The crystal on its adjusting apparatus can be raised or lowered to the
proper height, level with the axes of the telescope and collimator, by
means of a milled head at the base of the instrument, there being an
inner crystal axis moving (vertically only) independently of the circle.
Moreover, a second axis outside this enables the crystal to be rotated
independently of the circle, the conical axis of which is outside this
again. The two can be locked together when desired, however, by a
clamping screw provided with a fine adjustment _q_. Freedom of movement
of the crystal axis, unimpeded by the weight of the circle, is thus
permitted for all adjusting purposes, the circle being only brought into
play when measurement is actually to occur. With this instrument the
most accurate work can be readily carried out, and for ease of
manipulation and general convenience it is the best goniometer yet
constructed.

The idea of regarding the centre of the crystal as the centre of a
sphere, within which the crystal is placed (Fig. 47, page 62), gives
crystallographers a very convenient method of graphically representing a
crystal on paper, by projecting the sphere on to the flat surface of the
paper, the eye being supposed to be placed at either the north or south
pole of the sphere, and the plane of projection to be that of the
equatorial great circle. The faces in the upper hemisphere are
represented by dots which are technically known as the “poles” of the
faces, corresponding to the points where the needles normal to the faces
emerge from the imaginary globe, and all these points or poles lie on a
few arcs of great circles, which appear in the projection either also as
circular arcs terminating at diametrically opposite points on the
circumference of the equatorial circle, which forms the outer boundary
of the figure and is termed the “primitive circle,” or else, when the
planes of the great circles are at right angles to the equatorial
primitive circle, they appear as diametral straight lines passing
through the centre of the primitive circle.

Such a stereographic projection offers a comprehensive plan of the whole
of the crystal faces, which at once informs us of the symmetry in all
cases other than very complicated ones. A typical one, that of the
rhombic crystal of topaz shown in Fig. 22 (page 40), is given in Fig.
49.

It will happen in all cases of higher symmetry, as in that of topaz, for
instance, that the poles in the lower hemisphere will project into the
same points as those representing the faces in the upper hemisphere; but
in cases of lower symmetry, where they are differently situated, they
are usually represented by miniature rings instead of dots. From the
interfacial angles measured on the goniometer the relative lengths and
angular inclinations (if other than 90°) of the crystal axes can readily
be calculated, by means of the simple formulæ of spherical trigonometry;
and the stereographic projection constructed from the measurements as
just described proves an inestimable aid to these calculations, by
affording a comprehensive diagram of all the spherical triangles
required in making the calculations.

[Illustration:

  FIG. 49.—Stereographic Projection of Topaz.
]

The relative axial lengths _a_ : _b_ : _c_ (in which _b_ is always
arranged to be = 1), and the axial angles α (between _b_ and _c_), β
(between _a_ and _c_), and γ (between _a_ and _b_), form the “elements”
of a crystal. These, together with a list of the “forms” observed, and a
table of the interfacial angles, define the morphology of the crystal,
and are included in every satisfactory description of a crystallographic
investigation. They are preceded by a statement of the name and chemical
composition and formula of the substance, the system and the class of
symmetry, and the habit or various habits developed by crystals from a
considerable number of crops. An example of the mode of setting out such
a description will be found on pages 157 to 160.

Having thus made ourselves acquainted with the real nature of the
distribution of faces on a crystal, and learnt how the crystallographer
measures the angles between the faces by means of the reflecting
goniometer, plots them out graphically on a stereographic projection,
and calculates therefrom the “elements” of the crystal, it will be
convenient again to take up the historical development of the subject so
far as it relates to crystal forms and angles, and their bearing on the
chemical composition of the substance composing the crystal, by
introducing the reader to the great work of Mitscherlich, whose
influence in the domain of chemical crystallography was as profound as
that of Haüy proved to be as regards structural crystallography.




                              CHAPTER VII
  THE WORK OF EILHARDT MITSCHERLICH AND HIS DISCOVERY OF ISOMORPHISM.


During the height of the French Revolution, which caused the work of the
Abbe Haüy to be suspended for a time, although he was fortunately not
one of the many scientific victims of that terrible period, there was
born, on the 7th of January 1794, in the village of Neuende, near Jever,
in Oldenburg, the man who was destined to continue that work on its
chemical side. Eilhardt Mitscherlich was the son of the village pastor,
and nephew of the celebrated philologer, Prof. Mitscherlich of
Göttingen. His uncle’s influence appears to have given young
Mitscherlich a leaning towards philological studies, for during his
later terms at the Gymnasium at Jever, where he received his early
education, he devoted himself with great energy to the study of history
and languages, for which he had a marked talent, under the able
direction and kind solicitude of the head of the Gymnasium at that time,
the historian Schlosser. He eventually specialised on the Persian
language, and when Schlosser was promoted to Frankfort young
Mitscherlich accompanied him, and there prosecuted these favourite
studies until the year 1811, when he went to the university of
Heidelberg.

For some time now he had cherished the hope of proceeding to Persia and
conducting philological investigations on the spot, and in 1813, an
opportunity presenting itself in the prospect of an embassy being
despatched to Persia by Napoleon, he transferred himself to the
university of Paris, with the object of obtaining permission from
Napoleon to accompany the embassy. This visit to Paris must have been
one of Mitscherlich’s most exciting and interesting experiences. For
Napoleon had just returned from the disastrous Russian campaign of 1812,
and was feverishly engaged in raising a new army wherewith to stem the
great rise of the people which was now re-awakening patriotic spirit
throughout the whole of Germany, and which threatened to sweep away, as
it eventually did, the huge fabric of his central European Empire.

Indeed Mitscherlich appears to have been detained in Paris during the
exciting years 1813 and 1814, and with the abdication of Napoleon on
April 4th of that year, he was obliged to give up all idea of proceeding
to Persia. He decided that the only way of accomplishing his purpose was
to attempt to travel thither as a doctor of medicine. He therefore
returned to his native Germany during the summer of 1814, and proceeded
to Göttingen, which was then famous for its medical school. Here he
worked hard at the preliminary science subjects necessary for the
medical degree, while still continuing his philology to such serious
purpose as to enable him to publish, in 1815, the first volume of a
history of the Ghurides and Kara-Chitayens, entitled “Mirchondi historia
Thaheridarum.” It is obvious from the sequel, however, that he very soon
began to take much more than a merely passing interest in his scientific
studies, and he eventually became so fascinated by them, and
particularly chemistry, as to abandon altogether his cherished idea of a
visit to Persia. Europe was now settling down after the stormy period of
the hundred days which succeeded Napoleon’s escape from Elba,
terminating in his final overthrow on June 18th, 1815, at Waterloo, and
Mitscherlich was able to devote himself to the uninterrupted prosecution
of the scientific work now opening before him. He had the inestimable
advantage of bringing to it a culture and a literary mind of quite an
unusually broad and original character; and if the fall of Napoleon
brought with it the loss to the world of an accomplished philologist, it
brought also an ample compensation in conferring upon it one of the most
erudite and broad-minded of scientists.

In 1818 Mitscherlich went to Berlin, and worked hard at chemistry in the
university laboratory under Link. It was about the close of this year or
the beginning of 1819 that he commenced his first research, and it
proved to be one which will ever be memorable in the annals both of
chemistry and of crystallography. He had undertaken the investigation of
the phosphates and arsenates, and his results confirmed the conclusions
which had just been published by Berzelius, then the greatest chemist of
the day, namely, that the anhydrides of phosphoric and arsenic acids
each contain five equivalents of oxygen, while those of the lower
phosphorous and arsenious acids contain only three. But while making
preparations of the salts of these acids, which they form when combined
with potash and ammonia, he observed a fact which had escaped Berzelius,
namely, that the phosphates and arsenates of potassium and ammonium
_crystallise in similar forms_, the crystals being so like each other,
in fact, as to be indistinguishable on a merely cursory inspection.

Being unacquainted with crystallography, and perceiving the importance
of the subject to the chemist, he acted in a very practical and sensible
manner, which it is more than singular has not been universally imitated
by all chemists since his time. He at once commenced the study of
crystallography, seeing the impossibility of further real progress
without a working knowledge of that subject. He was fortunate in finding
in Gustav Rose, the Professor of Geology and Mineralogy at Berlin, not
merely a teacher close at hand, but also eventually a life-long intimate
friend. Mitscherlich worked so hard under Rose that he was very soon
able to carry out the necessary crystal measurements with his newly
prepared phosphates and arsenates. He first established the complete
morphological similarity of the acid phosphates and arsenates of
ammonium, those which have the composition NH_{4}H_{2}PO_{4} and
NH_{4}H_{2}AsO_{4} and crystallise in primary tetragonal prisms
terminated by the primary pyramid faces; and then he endeavoured to
produce other salts of ammonia with other acids which should likewise
give crystals of similar form. But he found this to be impossible, and
that only the phosphates and arsenates of ammonia exhibited the same
crystalline forms, composed of faces inclined at similar angles, which
to Mitscherlich at this time appeared to be identical. He next tried the
effect of combining phosphoric and arsenic acids with other bases, and
he found that potassium gave salts which crystallised apparently exactly
like the ammonium salts.

He then discovered that not only do the acid phosphates and arsenates of
potassium and ammonium, H_{2}KPO_{4}, H_{2}(NH_{4})PO_{4},
H_{2}KAsO_{4}, and H_{2}(NH_{4})AsO_{4} crystallise in similar
tetragonal forms, but also that the four neutral di-metallic salts of
the type HK_{2}PO_{4} crystallise similarly to each other.

He came, therefore, to the conclusion that there do exist bodies of
dissimilar chemical composition having the same crystalline form, but
that these substances are of similar constitution, in which one element,
or group of elements, may be exchanged for another which appears to act
analogously, such as arsenic for phosphorus and the ammonium group
(although its true nature was not then determined) for potassium. He
observed that certain minerals also appeared to conform to this rule,
such as the rhombohedral carbonates of the alkaline earths, calcite
CaCO_{3}, dolomite CaMg(CO_{3})_{2}, chalybite FeCO_{3}, and dialogite
MnCO_{3}; and the orthorhombic sulphates of barium (barytes, BaSO_{4}),
strontium (celestite, SrSO_{4}), and lead (anglesite, PbSO_{4}).
Wollaston, who, in the year 1809, had invented the reflecting
goniometer, and thereby placed a much more powerful weapon of research
in the hands of crystallographers, had already, in 1812, shown this to
be a fact as regards the orthorhombic carbonates (witherite,
strontianite, and cerussite) and sulphates (barytes, celestite, and
anglesite) of barium, strontium, and lead, as the result of the first
exact angular measurements made with his new instrument; but his
observations had been almost ignored until Mitscherlich reinstated them
by his confirmatory results.

While working under the direction of Rose, Mitscherlich had become
acquainted with the work of Haüy, whose ideas were being very much
discussed about this time, Haüy himself taking a very strong part in the
discussion, being particularly firm on the principle that every
substance of definite chemical composition is characterised by its own
specific crystalline form. Such a principle appeared to be flatly
contradicted by these first surprising results of Mitscherlich, and it
naturally appeared desirable to the latter largely to extend his
observations to other salts of different groups. It was for this reason
that he had examined the orthorhombic sulphates of barium, strontium,
and lead, and the rhombohedral carbonates of calcium, magnesium, iron,
and manganese, with the result already stated that the members of each
of these groups of salts were found to exhibit the same crystalline
form, a fact as regards the former group of sulphates which had already
been pointed out not only by Wollaston but by von Fuchs (who appears to
have ignored the work of Wollaston) in 1815, but had been explained by
him in a totally unsatisfactory manner. Moreover, about the same time
the vitriols, the sulphates of zinc, iron, and copper, had been
investigated by Beudant, who had shown that under certain conditions
mixed crystals of these salts could be obtained; but Beudant omitted to
analyse his salts, and thus missed discovering the all-important fact
that the vitriols contain water of crystallisation, and in different
amounts under normal conditions. Green vitriol, the sulphate of ferrous
iron, crystallises usually with seven molecules of water of
crystallisation, as does also white vitriol, zinc sulphate; but blue
vitriol, copper sulphate, crystallises with only five molecules of water
under ordinary atmospheric conditions of temperature and pressure.
Moreover, copper sulphate forms crystals which belong to the triclinic
system, while the sulphates of zinc and iron are dimorphous, the common
form of zinc sulphate, ZnSO_{4}.7H_{2}O, being rhombic, like Epsom
salts, the sulphate of magnesia which also crystallises with seven
molecules of water, MgSO_{4}.7H_{2}O, while that of ferrous sulphate,
FeSO_{4}.7H_{2}O, is monoclinic, facts which still further complicate
the crystallography of this group and which were quite unknown to
Beudant and were unobserved by him. But Beudant showed that the addition
of fifteen per cent. of ferrous sulphate to zinc sulphate, or nine per
cent. to copper sulphate, caused either zinc or copper sulphate to
crystallise in the same monoclinic form as ferrous sulphate. He also
showed that all three vitriols will crystallise in mixed crystals with
magnesium or nickel sulphates, the ordinary form of the latter salt,
NiSO_{4}.7H_{2}O, being rhombic like that of Epsom salts.

The idea that two chemically distinct substances not crystallising in
the cubic system, where the high symmetry determines identity of form,
can occur in crystals of the same form, was most determinedly combated
by Haüy, and the lack of chemical analyses in Beudant’s work, and the
altogether incorrect “vicarious” explanation given by von Fuchs, gave
Haüy very grave cause for suspicion of the new ideas. The previous
observations of Rome de l’Isle in 1772, Le Blanc in 1784, Vauquelin in
1797, and of Gay-Lussac in 1816, that the various alums, potash alum,
ammonia alum, and iron alum, will grow together in mixed crystals or in
overgrowths of one crystal on another, when a crystal of any one of them
is hung up in the solution of any other, does not affect the question,
as the alums crystallise in the cubic system, the angles of the highly
symmetric forms of which are absolutely identical by virtue of the
symmetry itself.

It was while this interesting discussion was proceeding that
Mitscherlich was at work in Berlin, extending his first researches on
the phosphates and arsenates to the mineral sulphates and carbonates.
But he recognised, even thus early, what has since become very clear,
namely, that owing to the possibility of the enclosure of impurities and
of admixture with isomorphous analogues, minerals are not so suitable
for investigation in this regard as the crystals of artificially
prepared chemical salts. For the latter can be prepared in the
laboratory in a state of definitely ascertained purity, and there is no
chance of that happening which Haüy was inclined to think was the
explanation of Mitscherlich’s results, namely, that certain salts have
such an immense power of crystallisation that a small proportion of them
in a solution of another salt may coerce the latter into crystallisation
in the form of that more powerfully crystallising salt. Mitscherlich
made a special study, therefore, of the work of Beudant, and repeated
the latter observer’s experiments, bringing to the research both his
crystallographic experience and that of a skilful analyst. He prepared
the pure sulphates of ferrous iron, copper, zinc, magnesium, nickel and
cobalt, all of which form excellent crystals. He soon cleared up the
mystery in which Beudant’s work had left the subject, by showing that
the crystals contained water of crystallisation, and in different
amounts. He found what has since been abundantly verified, that the
sulphates of copper and manganese crystallise in the triclinic system
with five molecules of water, CuSO_{4}.5H_{2}O and MnSO_{4}.5H_{2}O; in
the case of manganese sulphate, however, this is only true when the
temperature is between 7° and 20°, for if lower than 7° rhombic crystals
of MnSO_{4}.7H_{2}O similar to those of the magnesium sulphate group are
deposited, and if higher than 20° the crystals are tetragonal and
possess the composition MnSO_{4}.4H_{2}O. The Epsom salts group
crystallising in the rhombic system with seven molecules of water
consists of magnesium sulphate itself, MgSO_{4}.7H_{2}O, zinc sulphate
ZnSO_{4}.7H_{2}O, and nickel sulphate NiSO_{4}.7H_{2}O. The third group
of Mitscherlich consists of sulphate of ferrous iron FeSO_{4}.7H_{2}O
and cobalt sulphate CoSO_{4}.7H_{2}O, and both crystallise at ordinary
temperatures with seven molecules of water as indicated by the formulæ,
but in the monoclinic system. Thus two of the groups contain the same
number of molecules of water, yet crystallise differently. But
Mitscherlich next noticed a very singular fact, namely, that if a
crystal of a member of either of these two groups be dropped into a
saturated solution of a salt of the other group, this latter salt will
crystallise out in the form of the group to which the stranger crystal
belongs. Hence he concluded that both groups are capable of
crystallising in two different systems, rhombic and monoclinic, and that
under the ordinary circumstances of temperature and pressure three of
the salts form most readily the rhombic crystals, while the other two
take up most easily the monoclinic form. Mitscherlich then mixed the
solutions of the different salts, and found that the mixed crystals
obtained presented the form of some one of the salts employed. Thus even
so early in his work Mitscherlich indicated the possibility of
dimorphism. Moreover, before the close of the year 1819 he had satisfied
himself that aragonite is a second distinct form of carbonate of lime,
crystallising in the rhombic system and quite different from the
ordinary rhombohedral form calcite. Hence this was another undoubted
case of dimorphism.

During this same investigation in 1819, Mitscherlich studied the effect
produced by mixing the solution of each one of the above-mentioned seven
sulphates of dyad-acting metals with the solution of sulphate of potash,
and made the very important discovery that a double salt of definite
composition was produced, containing one equivalent of potassium
sulphate, one equivalent of the dyad sulphate (that of magnesium, zinc,
iron, manganese, nickel, cobalt, or copper), and six equivalents of
water of crystallisation, and that they all crystallised well in similar
forms belonging to the monoclinic system. Some typical crystals of one
of these salts, ammonium magnesium sulphate, are illustrated in Fig. 30
(Plate VII., facing page 44). This is probably the most important series
of double salts known to us, and is the series which has formed the
subject of prolonged investigation on the part of the author, no less
than thirty-four different members of the series having been studied
crystallographically and physically since the year 1893, and many other
members still remain to be studied. An account of this work is given in
a Monograph published in the year 1910 by Messrs Macmillan & Co., and
entitled, “Crystalline Structure and Chemical Constitution.”

This remarkable record for a first research was presented by
Mitscherlich to the Berlin Academy on the 9th December 1819. During the
summer of the same year Berzelius visited Berlin, and was so struck with
the abilities of Mitscherlich, then twenty-five years old, that he
persuaded him to accompany him on his return to Stockholm, and
Mitscherlich continued his investigations there under the eye of the
great chemist. His first work at Stockholm consisted of a more complete
study of the acid and neutral phosphates and arsenates of potash, soda,
ammonia, and lead. He showed that in every case an arsenate crystallises
in the same form as the corresponding phosphate. Moreover, in 1821 he
demonstrated that sodium dihydrogen phosphate, NaH_{2}PO_{4},
crystallises with a molecule of water of crystallisation in two
different forms, both belonging to the rhombic system but with quite
different axial ratios; this was consequently a similar occurrence to
that which he had observed with the sulphates of the iron and zinc
groups.

It was while Mitscherlich was in Stockholm that Berzelius suggested to
him that a name should be given to the new discovery that analogous
elements can replace each other in their crystallised compounds without
any apparent change of crystalline form. Mitscherlich, therefore, termed
the phenomenon “isomorphism,” from ἰσός, equal to, and μορφή, shape. The
term “isomorphous” thus strictly means “equal shaped,” implying not only
similarity in the faces displayed, but also absolute equality of the
crystal angles. The fact that the crystals of isomorphous substances are
not absolutely identical in form, but only very similar, was not likely
to be appreciated by Mitscherlich at this time. For the reflecting
goniometer had only been invented by Wollaston in 1809, and accurate
instruments reading to minutes of arc were mechanical rarities. It will
be shown in the sequel, as the result of the author’s investigations,
that there _are_ angular differences, none the less real because
relatively very small, between the members of such series. But
Mitscherlich was not in the position to observe them. It must be
remembered, moreover, that he was primarily a chemist, and that he had
only acquired sufficient crystallographic knowledge to enable him to
detect the system of symmetry, and the principal forms (groups of faces
having equal value as regards the symmetry) developed on the crystals
which he prepared. His doctrine of isomorphism, accepted in this broad
sense, proved of immediate and important use in chemistry. For there
were uncertainties as to the equivalents of some of the chemical
elements, as tabulated by Berzelius, then the greatest authority on the
subject, and these were at once cleared up by the application of the
principle of isomorphism.

The essence of Mitscherlich’s discovery was, that the chemical nature of
the elements present in a compound influences the crystalline form by
determining the number and the arrangement of the atoms in the molecule
of the compound; so that elements having similar properties, such for
instance as barium, strontium, and calcium, or phosphorus and arsenic,
combine with other elements to form similarly constituted compounds,
both as regards number of atoms and their arrangement in the molecule.
Number of atoms alone, however, is no criterion, for the five atoms of
the ammonium group NH_{4} replace the one atom of potassium without
change of form.

This case of the base ammonia had been one of Mitscherlich’s greatest
difficulties during the earlier part of his work, and remained a
complete puzzle until about this time, when its true chemical character
was revealed. For until the year 1820 Berzelius believed that it
contained oxygen. Seebeck and Berzelius had independently discovered
ammonium amalgam in 1808, and Davy found, on repeating the experiment,
that a piece of sal-ammoniac moistened with water produced the amalgam
with mercury just as well as strong aqueous ammonia. Both Berzelius and
Davy came to the conclusion that ammonia contains oxygen, like potash
and soda, and that a metallic kind of substance resembling the alkali
metals, potassium and sodium, was isolated from this oxide or hydrate by
the action of the electric current, which Seebeck had shown facilitated
the formation of the so-called ammonium amalgam. Davy, however, accepted
in part the views of Gay-Lussac and Thénard, who, in 1809, concluded
from their experiments that ammonium consisted of ammonia gas NH_{3}
with an additional atom of hydrogen, the group NH_{4} then acting like
an alkali metal, views which time has substantiated. But their further
erroneous conclusion that sodium and potassium also contained hydrogen
was rejected by him. Berzelius, however, set his face both against this
latter fallacy and the really correct NH_{4} theory, and it was not
until four years after Ampère, in 1816, had shown that sal-ammoniac was,
in fact, the compound of the group NH_{4} with chlorine, that Berzelius,
about the year 1820, after thoroughly sifting the work of Ampère,
accepted the view of the latter that in the ammonium salts it is the
group NH_{4}, acting as a radicle capable of replacing the alkali
metals, which is present.

The fact that this occurred at this precise moment, four years after the
publication of Ampère’s results, leads to the conclusion that the
observation of Mitscherlich, that the ammonium compounds are isomorphous
with the potassium compounds, was the compelling argument which caused
Berzelius finally to admit what has since proved to be the truth.

While still at Stockholm Mitscherlich showed that the chromates and
manganates are isomorphous with the sulphates, and also that the
perchlorates and permanganates are isomorphous with each other. Although
these facts could not be properly explained at the time, owing to the
inadequate progress of the chemistry of manganese, it was seen that
potassium chromate, K_{2}CrO_{4}, contained the same number of atoms as
potassium sulphate, K_{2}SO_{4}, and that potassium permanganate
KMnO_{4} and perchlorate KClO_{4} likewise resembled each other in
regard to the number of atoms contained in the molecule.

As a good instance of the use of the principle of isomorphism, we may
recall that when Marignac, in 1864, found himself in great difficulty
about the atomic weights of the little known metals tantalum and niobium
which he was investigating, he discovered that their compounds are
isomorphous; the pentoxides of the two metals occur together in
isomorphous mixture in several minerals, and the double fluorides with
potassium fluoride, K_{2}TaF_{7} and K_{2}NbF_{7} are readily obtained
in crystals of the same form. The specific heat of tantalum was then
unknown, so that the law of Dulong and Petit connecting specific heat
with atomic weight could not be applied, and the vapour density of
tantalum chloride, as first determined by Deville and Troost with impure
material, did not indicate an atomic weight for tantalum which would
give it the position among the elements that the chemical reactions of
the metal indicated. Yet Marignac was able definitely to decide, some
time before the final vapour density determinations of Deville and
Troost with pure salts, from the fact of the isomorphism of their
compounds, that the only possible positions for tantalum and niobium
were such as corresponded with the atomic weights 180 and 93
respectively. Time has only confirmed this decision, and we now know
that niobium and tantalum belong to the same family group of elements as
that to which vanadium belongs, and the only difference which modern
research has introduced has been to correct the decimal places of the
atomic weights, that of niobium (now also called columbium, the name
given to it by its discoverer, Hatchett, in 1801) being now accepted as
92.8 and that of tantalum 179.6, when that of hydrogen = 1.

Applying the law of isomorphism in a similar manner, Berzelius was
enabled to fix the atomic weights of copper, cadmium, zinc, nickel,
cobalt, iron, manganese, chromium, sulphur, selenium, and chlorine, the
numbers accepted to-day differing only in the decimal places, in
accordance with the more accurate results acquired by the advance of
experimental and quantitative analytical methods. But with regard to
several other elements, owing to inadequate data, Berzelius made serious
mistakes, showing how very great is the necessity for care and for ample
experimental data and accurate measurements, before the principle of
isomorphism can be applied with safety. Given these, and we have one of
the most valuable of all the aids known to us in choosing the correct
atomic weight of an element from among two or three possible
alternatives. We are only on absolutely sure ground when we are dealing
not only with a series of compounds consisting of the same number of
atoms, but when also the interchangeable elements are the intimately
related members of a family group, such as we have since become familiar
with in the vertical groups of elements in the periodic table of
Mendeléeff.

Before leaving Stockholm Mitscherlich showed, from experiments on the
crystallisation of mixtures of the different sulphates with which he had
been working, that isomorphous substances intermix in crystals in all
proportions, and that they also replace one another in minerals in
indefinite proportions, a fact which has of recent years been
wonderfully exemplified in the cases of the hornblende (amphibole) and
augite (pyroxene) groups.

In November 1821 Mitscherlich closed these memorable labours at
Stockholm and returned to Berlin, where he acted as extraordinary
professor of the university until 1825, when he was elected professor in
ordinary. His investigations for a time were largely connected with
minerals, but on July 6th, 1826, he presented a further most important
crystallographic paper to the Berlin Academy, in which he announced his
discovery of the fact that one of the best known chemical elements,
sulphur, is capable of crystallising in two distinct forms. The ordinary
crystals found about Etna and Vesuvius and in other volcanic regions
agree with those deposited from solution in carbon bisulphide in
exhibiting rhombic symmetry. But Mitscherlich found that when sulphur is
fused and allowed to cool until partially solidified, and the still
liquid portion is then poured out of the crucible, the walls of the
latter are found to be lined with long monoclinic prisms. These have
already been illustrated in Fig. 2, Plate I., in Chapter I.

Here was a perfectly clear case of an element—not liable to any charge
of difference of chemical composition such as might have applied to the
cases of sodium dihydrogen phosphate, carbonate of lime, and iron
vitriol and its analogues, which he had previously described as cases of
the same substance crystallising in two different forms—which could be
made to crystallise in two different systems of symmetry at will, by
merely changing the circumstances under which the crystallisation
occurred. His explanation being thus proved absolutely, he no longer
hesitated, but at once applied the term “dimorphous” to these substances
exhibiting two different forms, and referred to the phenomenon itself as
“dimorphism.” The case of carbonate of lime had given rise to prolonged
discussion, for the second variety, the rhombic aragonite, had been
erroneously explained by Stromeyer, after Mitscherlich’s first
announcement in 1819, as being due to its containing strontia as well as
lime, and the controversy raged until Buchholz discovered a specimen of
aragonite which was absolutely pure calcium carbonate, so that
Mitscherlich’s dimorphous explanation was fully substantiated.

Dimorphism is very beautifully illustrated by the case of the trioxide
of antimony, Sb_{2}O_{3}, a slide of which, obtained by sublimation of
the oxide from a heated tube on to the cool surface of a glass
microscope slip, is seen reproduced in Fig. 50, Plate XI. The two forms
are respectively rhombic and cubic. The rhombic variety usually takes
the form of long needle-shaped crystals, which are shown in Fig. 50
radiating across the field and interlacing with one another; the cubic
variety crystallises in octahedra, of which several are shown in the
illustration, perched on the needles, one interesting individual being
poised on the end of one of the needles. The two forms occur also in
nature as the rhombic mineral valentinite and the cubic mineral
senarmontite, which latter crystallises in excellent regular octahedra.
Antimonious oxide, moreover, is not only isomorphous, but isodimorphous
with arsenious oxide, a slide of octahedra of which has already been
reproduced in Fig. 3, Plate I., in Chapter I. For besides this common
octahedral form of As_{2}O_{3} artificial crystals of arsenious oxide
have been prepared of rhombic symmetry, resembling valentinite. Hence
the two lower oxides of arsenic and antimony afford us a striking case
of the simultaneous display of Mitscherlich’s two principles of
isomorphism and dimorphism.

Thus the position in 1826 was that Mitscherlich had discovered the
principle of isomorphism, and had also shown the occurrence of
dimorphism in several well-proved specific cases, and that he regarded
at this time isomorphism as being a literal reality, absolute identity
of form.

[Illustration:

  _PLATE XI._

  FIG. 50.—Rhombic Needles and Cubic Octahedra of Antimony Trioxide
    obtained by Sublimation. An interesting Example of Dimorphism.
]

[Illustration:

  FIG. 101.—Ammonium Chloride crystallising· from a Labile
    Supersaturated Solution (see p. 248).

  REPRODUCTIONS OF PHOTOMICROGRAPHS.
]

These striking results appeared at once to demolish the theory that any
one substance of definite chemical composition is characterised by a
specific crystalline form, which was Haüy’s most important
generalisation. Mitscherlich, however, soon expressed doubts as to the
absolute identity of form of his isomorphous crystals, and saw that it
was quite possible that in the systems other than the cubic (in which
latter system the highly perfect symmetry itself determines the form,
and that the angles shall be identically constant), there might be
slight distinctive differences in the crystal angles. For he caused to
be constructed, by the celebrated optician and mechanician, Pistor, the
most accurate goniometer which had up till then been seen, provided with
four verniers, each reading to ten seconds of arc, and with a telescope
magnifying twenty times, for viewing the reflections of a signal,
carried by a collimator, from the crystal faces. Unfortunately in one
respect, he was almost at once diverted, by the very excess of
refinement of this instrument, to the question of the alteration of the
crystal angles by change of temperature, and lost the opportunity, never
to recur, of doing that which would at once have reconciled his views
with those of Haüy in regard to this important matter, namely, the
determination of these small but real differences in the crystal angles
of the different members of isomorphous series, and the discovery of the
interesting law which governs them, a task which in these later days has
fallen to the lot of the author.

Another remarkable piece of crystallographic work, this time in the
optical domain, which has rendered the name of Mitscherlich familiar,
was his discovery of the phenomenon of crossed-axial-plane dispersion of
the optic axes in gypsum. (The nature and meaning of “optic axes” will
be explained in Chapter XIII., page 185.) During the course of a lecture
to the Berlin Academy in the year 1826 Mitscherlich, always a brilliant
lecturer and experimenter at the lecture table, exhibited an experiment
with a crystal of gypsum (selenite) which has ever since been referred
to as the “Mitscherlich experiment.” He had been investigating the
double refraction of a number of crystalline substances at different
temperatures, and had observed that gypsum, hydrated calcium sulphate,
CaSO_{4}.2H_{2}O, was highly sensitive in this respect, especially as
regards the position of its optic axes. At the ordinary temperature it
is biaxial, with an optic axial angle of about 60°, but on heating the
crystal the angle diminishes, until just above the temperature of
boiling water the axes become identical, as if the crystal were
uniaxial, and then they again separate as the temperature rises further,
but in the plane at right angles to that which formerly contained them;
hence the phenomenon is spoken of as “crossed-axial-plane dispersion.”
Mitscherlich employed a plate of the crystal cut perpendicularly to the
bisectrix of the optic axial angle, and showed to the Academy the
interference figures (see Plate XII.) which it afforded in convergent
polarised light with rising temperature. At first, for the ordinary
temperature, the usual rings and lemniscates surrounding the two optic
axes were apparent at the right and left margins of the field; as the
crystal was gently heated (its supporting metallic frame being heated
with a spirit lamp) the axes approached each other, with ever changing
play of colour and alteration of shape of the rings and lemniscates,
until eventually the dark hyperbolic brushes, marking by their well
defined vertices the positions of the two optic axes within the
innermost rings, united in the centre of the field to produce the
uniaxial dark rectangular cross; the rings around the centre had now
become circles, the lemniscates having first become ellipses which more
and more approximated, as the temperature rose, to circles. Then the
dark cross opened out again, and the axial brushes separated once more,
but in the vertical direction, and the circles became again first
ellipses and then lemniscates, eventually developing inner rings around
the optic axes; if the source of heat were not removed at this stage the
crystal would suddenly decompose, becoming dehydrated, and the field on
the screen would become dark. If, however, the spirit lamp were removed
before this occurred, the phenomena were repeated in the reverse order
as the crystal cooled.

This beautiful experiment is now frequently performed, as gypsum is
perhaps the best example yet known which exhibits the phenomenon of
crossed-axial-plane dispersion by change of temperature alone. A
considerable number of other cases are known, such as brookite, the
rhombic form of titanium dioxide TiO_{2}, and the triple tartrate of
potassium, sodium, and ammonium, but these are more sensitive to change
of wave-length in the illuminating light than to change of temperature.

[Illustration:

  FIG. 51.—The Mitscherlich Experiment with Gypsum.
]

[Illustration:

  _PLATE XII._

  FIG. 52.—Appearance of the Interference Figure half a Minute after
    commencing the Experiment. Temperature of Crystal about 40° C.
]

[Illustration:

  FIG. 53.—Appearance a Minute or so later, the Axes approaching the
    Centre. Temperature of Crystal about 85° C.
]

[Illustration:

  FIG. 54.—The Two Optic Axes coincident in the Centre of the Figure,
    two or three Minutes from the commencement. Temperature of Crystal
    106° C.
]

[Illustration:

  FIG. 55.—The Axes re-separated in the Vertical Plane a Minute or two
    later. Temperature of Crystal about 125° C.

  THE MITSCHERLICH EXPERIMENT WITH GYPSUM.

  FOUR STAGES IN THE TRANSFORMATION OF THE INTERFERENCE FIGURE IN
    CONVERGENT POLARISED LIGHT, FROM HORIZONTALLY BIAXIAL THROUGH
    UNIAXIAL TO VERTICALLY BIAXIAL, ON RAISING THE TEMPERATURE TO 125°
    C.

  (From Photographs by the author.)
]

The author has recently exhibited the “Mitscherlich experiment” to the
Royal Society,[2] and also in his Evening Discourse to the British
Association at their 1909 meeting in Winnipeg, in a new and more elegant
manner, employing the large Nicolprism projection polariscope shown in
Fig. 51, and a special arrangement of lenses for the convergence of the
light, which is so effective that no extraneous heating of the crystal
is required. The convergence of the rays is so true on a single spot in
the centre of the crystal plate about two millimetres diameter, that a
crystal plate not exceeding 6 mm. is of adequate size, mounted in a
miniature holder-frame of platinum or brass with an aperture not more
than 3 mm; the thickness of the crystal should remain about 2 mm., in
order that the rings round the axes may not be too large and diffuse,
the crystal being endowed with very feeble double refraction, which is
one of the causes of the phenomenon. Such a small crystal heats up so
rapidly in the heat rays accompanying the converging light rays—even
with the essential cold water cell two inches thick between the lantern
condenser and the polarising Nicol, for the protection of the balsam of
the latter—that any extraneous heating by a spirit or other lamp is
entirely unnecessary. The moment the electric arc of the lantern is
switched on, the optic axial rings appear at the right and left margins
of the screen, when the crystal is properly adjusted and the arc
correctly centred, and they march rapidly to the crossing point in the
centre, where the dark hyperbolæ unite to produce the rectangular St
Andrew’s cross, the rings, figure-eight curves, and other lemniscates
passing through the most exquisite evolutions and colour changes all the
time until they form the circular Newton’s rings, around the centre of
the cross; after this the cross and circles again open out, but along
the vertical diameter of the screen, into hyperbolæ and rings and
loop-like lemniscates surrounding two axes once more. It is wise as soon
as the separation in this plane is complete and the first or second
separate rings have appeared round the axes, to arrest the heating by
merely interposing intermittently a hand screen between the lantern and
polariser, or by blowing a current of cool air past the crystal, which
will cause the axes to recede again, and the phenomena to be reversed,
the crossing point being repassed, and the axes brought into the
original horizontal plane again. By manipulation of the screen, or
air-current, the axes can thus be caused to approach or to recede from
the centre at will, along either the horizontal or vertical diameter.
Four characteristic stages of the experiment are shown in Figs. 52 to
55, Plate XII. Fig. 52 exhibits the appearance just after commencing the
experiment, the optic axes being well in the field of view. Fig. 53
shows the axes horizontally approaching the centre. Fig. 54 shows the
actual crossing, which occurs for different crystals at temperatures
varying from 105°.5 to 111°.5 C.; and Fig. 55 represents the axes again
separated, but vertically.

The experiment as thus performed is one of the most beautiful
imaginable, and it can readily be understood how delighted were
Mitscherlich’s audience on the occasion of its first performance by him.
The author has since discovered no less than six other cases of
substances which exhibit crossed-axial-plane dispersion of the optic
axes, in the course of his investigations, one of which is illustrated
in Plate XIII., facing page 108; and, moreover, has arrived at a general
explanation of the whole phenomenon, the main points of which are that
such substances, besides showing very feeble double refraction (the two
extreme of the three refractive indices being very close together), also
exhibit very close approximation of the intermediate refractive index β
to either the minimum index α or the maximum index γ. Also, change of
temperature, or of wave-length, or most usually both, must so operate as
to bring the two indices closest together into actual identity and then
to pass beyond each other, these two indices thus exchanging positions,
the extreme one becoming the intermediate index. In other words, the
uniaxial cross and circular rings are produced owing to two of the three
refractive indices (corresponding to the directions of the three
rectangular axes of the ellipsoid which, in general, expresses the
optical properties of a crystal) becoming equal at the particular
temperature at which the phenomenon is observed to occur, and for light
of the specific wave-length in question. The ellipsoid of general form
which represents the optical properties of a biaxial crystal thus
becomes converted into a rotation ellipsoid corresponding to a uniaxial
crystal. Brookite and the triple tartrate are excellent examples of the
predominance of the effect of change of wave-length, for the optic axes
are separated in both cases widely in one plane for red light and almost
equally widely in the perpendicular plane for blue light. The new cases
observed by the author are sensitive both to change of wave-length and
to change of temperature, and so fall midway between the cases just
quoted and the case of gypsum. The cause of it, in four of these new
instances, is a very interesting one, connected with the regular change
of the refractive indices in accordance with the law of progression in
an isomorphous series according to the atomic weight of the alkali metal
present, which will be discussed in Chapter X.

A further most important discovery was made by Mitscherlich in the year
1827, which also profoundly concerns the work of the author, namely,
that of selenic acid, H_{2}SeO_{4}, analogous to sulphuric acid, and of
the large group of salts derived from it, the selenates, analogous to
the sulphates. He showed first that potassium selenate, K_{2}SeO_{4}, is
isomorphous with potassium sulphate, K_{2}SO_{4}, and subsequently that
the selenates in general are isomorphous with the corresponding
sulphates; consequently it followed that selenium is a member of the
sulphur family of elements. This element selenium had only been
discovered ten years previously by his friend Berzelius, and doubtless
Mitscherlich had seen a great deal of the work in connection with it
during the two years which he spent in the laboratory of Berzelius at
Stockholm, and was deeply interested in it.

The discovery has proved a most fruitful one, for the selenates are
beautifully crystalline salts, particularly suitable for
crystallographic researches, and their detailed investigation has
afforded a most valuable independent confirmation of the important
results obtained for the sulphates.

Again in 1830 Mitscherlich, following up the preliminary work already
referred to, definitely established another fact bearing on the same
series, namely, the isomorphism of potassium manganate K_{2}MnO_{4} with
the sulphate and selenate of potash; moreover, on continuing his study
of the manganese salts he further substantiated the isomorphism of the
permanganates with the perchlorates, and isolated permanganic acid. This
also proved a most important step forward, as these salts likewise
afford admirable material for crystallographic investigation, and such
an examination, carried out by Muthmann and Barker, has yielded most
valuable results.

Much later in his career Mitscherlich also described the dimorphous
iodide of mercury, HgI_{2}, one of the most remarkable and interesting
salts known to us, the unstable yellow rhombic modification being
converted into the more stable red tetragonal form by merely touching
with a hard substance. Also we are indebted to him at the same later
period for our knowledge of the crystalline forms of the elements
phosphorus, iodine, and selenium, when crystallised from solution in
bisulphide of carbon.

From the record of achievements which has now been given in this chapter
it will be obvious how much chemical crystallography owes to
Mitscherlich. The description of his work has taken us into almost every
branch of the subject, morphological, optical, and thermal, and although
it has consequently been necessary to refer to phenomena which have not
yet been explained in this book, it has doubtless proved on the whole
most advantageous thus to present the life work of this great master as
a complete connected story.




                              CHAPTER VIII
               MORPHOTROPY AS DISTINCT FROM ISOMORPHISM.


It has been shown in the last chapter how Mitscherlich discovered the
principle of isomorphism, as applying to the cases of substances so
closely related that their interchangeable chemical elements are members
of the same family group; and also how the principle enabled him to
determine the chemical constitution of two hitherto unknown acids which
he isolated, selenic H_{2}SeO_{4} and permanganic HMnO_{4}. For he
observed that the selenates were isomorphous with the sulphates, and the
permanganates with the perchlorates. It was further made clear that the
principle as bequeathed to us by Mitscherlich was only defined in very
general terms, and its details have only recently been precisely
decided.

Before proceeding further (in Chapter X.) with the elucidation of the
true nature of isomorphism, however, some important crystallographic
relationships between substances less closely related than family
analogues must be referred to, as the outcome of a series of
investigations by von Groth, chiefly between the derivatives of the
hydrocarbon benzene. Also, some suggestive results obtained by the
author from an investigation of an organic homologous series, that is,
one the members of which differ by the regular addition of a CH_{3}
group, may be briefly referred to.

The interval between the work of Mitscherlich and that of von Groth was
one of doubt, discouragement, and somewhat of discredit for chemical
crystallography. The chemists Laurent[3] and Nicklès[4] carried out
during the years from 1842 to 1849 measurements of numerous organic
substances and of some inorganic compounds, the former chiefly halogen
or other derivatives of particular hydrocarbons or salts of homologous
fatty acids. Laurent, for instance, found that naphthalene
tetrachloride, C_{10}H_{8}.Cl_{4}, and chloronaphthalene tetrachloride,
C_{10}H_{7}Cl.Cl_{4}, crystallise in different systems, the former in
the monoclinic and the latter in the rhombic system. Yet the primary
prism angles of the two are less than a degree different, namely, 109°
0′ and 109° 45′. Laurent named this kind of similarity “hemimorphism,” a
most unfortunate term as it was already employed in crystallography in
its other well-known geometrical significance, that is, to denote a
crystal differently terminated at the two ends of an axis. Many other
like similarities were discovered by Laurent, and he again coined an
objectionable term, now discarded, to represent the cases of similarity
extending over more than the same system, namely, “isomeromorphism.”

Nicklès observed similar facts in connection with the barium salts of
the fatty acids, which crystallise in different systems with different
amounts of water of crystallisation. But their prism angles are all
within a couple of degrees of each other, varying from 98° to 100°. Thus
the phenomenon of “isogonism,” a term much less objectionable than those
invented by Laurent, appears to be a common observance not only for
different kinds of derivatives of the same original hydrocarbon or other
organic nucleus, but also for the case of homologous series. But Nicklès
missed the real point by including salts with different amounts of
water, which, it will be shown later, entirely upset the crystalline
structure. When this is eliminated the resemblance between true
similarly constituted homologues, differing by regular increments of
CH_{3}, is very much closer than would appear from Nicklès’ results.

Unfortunately, some of the work of Laurent and Nicklès was not carried
out with the care and accuracy which is indispensable for researches
which are to retain permanent value, and critics were not slow to arise.
Kopp,[5] in 1849, unmercifully exposed these failings, so that the real
kernel of the work, which was of considerable value, came into
discredit.

Pasteur,[6] however, in 1848, besides the important observations
regarding enantiomorphism, to be described in Chapter XI., had noticed
similar zonal likenesses between related tartrates, amounting only
therefore to isogonism and not to isomorphism; for here again the system
often differed, particularly when the members of a series compared
differed in their water of crystallisation. Thus there was ample
evidence of a really significant series of facts in the work of these
authors, but they were not properly arranged and explained.

So high was the feeling against the whole subject carried, however,
after Kopp’s memoir, that had it not been for the steadying influence of
Rammelsberg and Marignac, who themselves carried out many
crystallographic measurements as new substances continued to be
discovered with great rapidity, the science would have suffered a
serious set-back. Moreover, even Rammelsberg was led astray in the
direction of the views of the chemists of the time, that isomorphism
could be extended over the crystal system. Frankenheim, whose discovery
of the space-lattice, to be referred to in the next chapter, will ever
render his name famous, strongly opposed this view. Delafosse, on the
other hand, recognised some truth in both views, and assumed that there
were two kinds of isomorphism, that of Mitscherlich on the one hand, and
the broader one of Laurent on the other hand, and that in the case of
the latter kind the overstepping of the system is no bar.

Hjortdahl,[7] in the year 1865, supported the views of Delafosse more or
less, at any rate so far as to assume the possibility of the existence
of partial isomorphism, that is, of isogonism. He was very definite,
however, against accepting the proposition that any general law could be
applied. He himself discovered a partial similarity of angles in several
homologous series of organic compounds.

About this time Sella[8] uttered a warning which is one worthy of being
prominently posted in every research laboratory, namely, that _It is
unwise to make hasty generalisations from the results of a small number
of observations_. Were this principle more generally followed, much
greater progress would in the end be achieved, and without the
discouragement and discredit which inevitably follows the detection of
errors due to lack of broad experimental foundation. It is certainly an
incontrovertible fact that only such generalisations as find themselves
in accordance with all new but well-verified experimental facts as they
are revealed can stand the test of time and become accepted universally
as true laws of nature. And it is unreasonable to expect any
generalisation to be of such a character unless it is already based on
so large a number of facts that there is little fear of other new ones
upsetting them.

Some order was, however, introduced into this chaotic state of chemical
crystallography in the year 1870 by P. von Groth.[9] He investigated
systematically the derivatives of the hydrocarbon benzene, C_{6}H_{6},
many of which are excellently crystallising solids suitable for
goniometrical measurement. He showed that although the crystal system
may be and often is altered, yet there is a striking similarity in the
angles between the faces of certain zones, which for the purposes of
comparison he arranged to be parallel to each other in his descriptions
of the crystals, so that the relationship would then consist in an
elongation or a shortening of this particular zone axis, which was
usually a crystallographic axis. He recognised that this was a totally
different phenomenon from isomorphism, and called it “morphotropy.”
Although it may possibly be permissible from one point of view to regard
isomorphism as a particular case of complete morphotropy along all
zones, such a course is not advisable, as morphotropic similarities are
frequently of a comparatively loose and often indeed of a somewhat vague
character, while isomorphous relationships are governed by very precise
laws.

Thus von Groth showed first that benzene, C_{6}H_{6}, crystallises in
the rhombic system with axial ratios _a_ : _b_ : _c_ = 0.891 : 1 :
0.977. Next, that when one or two of the hydrogen atoms are replaced by
hydroxyl OH groups the substances produced, phenol C_{6}H_{5}.OH and
resorcinol C_{6}H_{4}(OH)_{2}, are found also to crystallise in the
rhombic system, and in the second case, for which alone the axial ratios
could be determined, the ratio _a_ : _b_ proved to be very similar, but
the ratio _c_ : _b_ was different, the actual values being _a_ : _b_ :
_c_ = 0.910 : 1 : 0.540. Pyrocatechol, the isomer of resorcinol, also
crystallises in the rhombic system, but the crystals have not been
obtained sufficiently well formed to enable any deductions to be made
from any measurements carried out with them.

Similarly, the nitro-derivatives of phenol, orthonitrophenol
C_{6}H_{4}.OH.NO_{2}, dinitrophenol C_{6}H_{3}.OH.(NO_{2})_{2}, and
trinitrophenol C_{6}H_{2}.OH.(NO_{2})_{3}, also crystallise in the
rhombic system, and with the following respective axial ratios: 0.873 :
1 : 0.60; 10.933 : 1 : 0.753; 0.937 : 1 : 0.974. Again, the value for
the ratio _a_ : _b_ is not very different from that of benzene itself,
while the ratio _c_ : _b_ differs considerably in the first two cases.
Similar relations were also found to hold good in the cases of
meta-dinitrobenzene, C_{6}H_{4}(NO_{2})_{2}, axial ratios 0.943 : 1 :
0.538, and trinitrobenzene, C_{6}H_{3}(NO_{2})_{3}, which possesses the
axial ratios 0.954 : 1 : 0.733.

The introduction of a chlorine or bromine atom or a CH_{3} group in
place of hydrogen was found by von Groth to produce more than the above
effect, the symmetry being often lowered to monoclinic, a fact which had
also been observed to occur in the cases of certain isomers of the
substances quoted above, ortho-dinitrobenzene for instance. But it was
nevertheless observed that the angles between the faces in the prism
zone remained very similar, the angles between the faces of the primary
prism (110) and (1̄10), for instance, only varying in eight such
derivatives of all three types, whether rhombic or monoclinic, from 93°
45′ to 98° 51′.

The crystallographic relationships of organic substances, however, are
very much complicated by the possibilities of isomerism, the ortho,
meta, and para compounds—corresponding to the replacement of the two
hydrogen atoms attached to two adjacent, alternate, or opposite carbon
atoms respectively, of the six forming the benzene ring—generally
differing extensively and sometimes completely in crystalline form.
Consequently, the phenomenon of morphotropy is best considered quite
independently of isomorphism.

An interesting intermediate case between morphotropy and true
isomorphism was investigated by the author in the year 1890, namely, a
series of homologous organic compounds differing by regular increments
of the organic radicle CH_{3}. They were prepared by Prof. Japp and Dr
Klingemann, and consisted of the methyl, CH_{3}, ethyl, C_{2}H_{5}, and
propyl, C_{3}H_{7}, derivatives of the substance triphenyl pyrrholone,
all of them being solids crystallising well. The problem was somewhat
complicated by the development of polymorphism, the methyl, ethyl, and
propyl compounds having each been found to be dimorphous, and not
improbably trimorphous, but only two varieties of each salt were
obtained in crystals adequately perfect for measurement. That the
production of these different forms was due to polymorphism and not to
chemical isomerism (different arrangement of the chemical atoms in the
molecule) was shown by the fact that one variety could be obtained from
the other by simply altering the conditions of crystallisation from the
same solvent. Their identical chemical composition was established by
direct analysis.

The methyl (CH_{3}) compound crystallised in rhombohedra and in
triclinic prisms. The ethyl (C_{2}H_{5}) derivative was deposited in
triclinic prisms exactly resembling those of the methyl compound in
habit and disposition of faces. A crystal of the triclinic methyl
derivative which would represent equally well the ethyl compound is
shown in Fig. 56. The angles also of the crystals of the two substances
are so similar that one might infer the existence of true and complete
isomorphism. The actual angular differences rarely exceeded three
degrees.

[Illustration:

  FIG. 56.—Crystal of Methyl Triphenyl Pyrrholone.
]

Besides the triclinic form the ethyl derivative was also obtained in
monoclinic crystals, one of which is represented in Fig. 57. This
illustration might serve equally well, however, for a corresponding
monoclinic form of the propyl (C_{3}H_{7}) derivative, and the angles of
these two monoclinic ethyl and propyl compounds are even closer than
those of the triclinic methyl and ethyl derivatives, the closeness
increasing with the advent of symmetry.

[Illustration:

  FIG. 57.—Crystal of Ethyl Triphenyl Pyrrholone.
]

This similarity of angles in the cases of the two pairs of triclinic and
monoclinic compounds is not only true about particular zones, but about
all the zones, so that it is a case isomorphism rather than of isogonism
(morphotropy). The similarity of optical properties is also very close,
and so much so in the cases of the monoclinic crystals of ethyl and
propyl triphenyl pyrrholone that both exhibit very high dispersion of
the optic axes. In the case of the propyl derivative the difference
between the apparent angle of the optic axes for red lithium light and
for green thallium light amounts to 11°. In the case of the ethyl
compound this difference is enhanced so considerably that the crystals
afford a remarkable instance of dispersion of the optic axes in crossed
axial planes, resembling the case of gypsum discovered by Mitscherlich
and described in the last chapter, except that the sensitiveness is to
change of wave-length in the illuminating light rather than to change of
temperature. The optic axial plane is perpendicular to the symmetry
plane for lithium and sodium light, as it is also in the case of the
propyl compound; but in the ethyl derivative it crosses over for
thallium light and rays beyond that towards the violet, into a plane at
right angles to the former plane, namely, the symmetry plane itself. The
total dispersion between the two axes as separated in the one plane for
red light, and as separated in the other perpendicular plane for blue
light, is more than 70°. Fig. 58, Plate XIII., shows the nature of the
interference figures afforded in convergent polarised light of different
wave-lengths by a section-plate perpendicular to the first median line.
The figure at _f_ represents what is observed in white light, as far as
is possible by a drawing in black and white. It consists of a series of
concave coloured curves, falling in between the arms of the cross, and
looping round the axes, a figure very much like that afforded by
brookite and triple tartrate of ammonium, potassium, and sodium, the
substances already mentioned in Chapter VII. as being similarly very
sensitive to change of wave-length. The figure in red monochromatic
lithium light is shown at _a_ in Fig. 58, and that for yellow sodium
light at _b_, the axes being now much closer together. On changing to
green thallium light the line joining the optic axes becomes vertical
instead of horizontal, as shown at _d_.

When, instead of employing monochromatic flames, the spectroscopic
monochromatic illuminator (Fig. 75, page 193), described by the author
some years ago to the Royal Society, is employed to illuminate the
polariscope, the source of light being the electric arc, the change of
the figure from that given by the extreme red of the spectrum to that
afforded by the violet may be beautifully followed, and the exact
wave-length in the greenish yellow determined for which the crossing
occurs and an apparently uniaxial figure of circular rings and
rectangular cross is produced. For it is possible with the aid of this
illuminator directly to observe the production of the uniaxial figure.
The wave-length is either directly afforded by the graduation of the
fine-adjustment micrometric drum or is obtained from a curve of
wave-lengths, constructed to correspond to the circle readings of the
illuminator. The appearance of the interference figure for this critical
wave-length is shown at _c_ in Fig. 58. The remaining figure at _e_
represents the appearance when a mixture of sodium and thallium light is
employed, which clearly indicates the four extreme axial positions, and
assists in elucidating the nature of the figure _f_ exhibited in white
light.

The second form of the propyl derivative belongs to the rhombic system,
and a similar rhombic form of the ethyl compound was once obtained, but
lost again on attempting to recrystallise.

These interesting relationships of the homologous methyl, ethyl, and
propyl derivatives of triphenyl pyrrholone thus appear to form a
connecting link between cases of isogonism or morphotropy and of true
isomorphism.

[Illustration:

  _PLATE XIII._

  FIG. 58.—Interference Figures in Convergent Polarised Light of
    different Wave-lengths afforded by the Monoclinic Variety of Ethyl
    Triphenyl Pyrrholone; _a_, in Red Lithium Light; _b_, in Yellow
    Sodium Light; _c_, in Greenish-Yellow Light of the Critical
    Wave-length for Production of the Uniaxial Figure; _d_, in Green
    Thallium Light; _e_, in mixed Sodium and Thallium Light; and _f_, in
    White Light.

  (Reproductions of Drawings by the author.)
]

We are now, therefore, in a position to approach the question of true
isomorphism, and as leading up to the fuller treatment of the subject in
Chapter X. we may conclude this chapter by referring first to one
important investigation in which the necessity for extreme accuracy of
measurement and perfection of material was fully appreciated. This was
an admirable research carried out in the years 1887 and 1888 by H. A.
Miers[10] on the red silver minerals, proustite, sulpharsenite of
silver, Ag_{3}AsS_{3}, and pyrargyrite, the analogous sulphantimonite of
silver, Ag_{3}SbS_{3}, which afforded a further indication of the
existence of real small differences of angle between the members of
truly isomorphous series. These two minerals form exceptionally
beautiful crystals belonging to the trigonal system, the hexagonal prism
being always a prominent form, terminated by the primary and other
rhombohedra, scalenohedra and various pyramidal forms, many of the
crystals being exceedingly rich in faces. When the crystals are freshly
obtained from the dark recesses of the silver mine they are very
lustrous and transparent, but they are gradually affected by light, like
many silver compounds, and require to be stored in the dark in order to
preserve their transparency. A magnificent crystal of proustite from
Chili is one of the finest objects in the British Museum at South
Kensington, but is rarely seen on account of the necessity for
preservation from light. Pyrargyrite is generally dark grey in
appearance, and affords a reddish-purple “streak” (colour of the powder
on scratching or pulverising). Proustite, however, possesses a beautiful
scarlet-vermilion colour, and affords a very bright red streak.

Now these two beautiful minerals are obviously analogous compounds of
the same metal, silver, with the sulpho-acid of two elements, arsenic
and antimony, belonging strictly to the same family group, the
nitrogen-phosphorus group, of the periodic classification of the
elements according to Mendeleéff. Consequently, they should be perfectly
isomorphous. Miers has shown in a most complete manner that they are so,
that they occur in very perfect crystals of similar habit belonging to
the same class of the trigonal system, the ditrigonal polar class, both
minerals being hemimorphic, that is, showing different forms at the two
terminations, in accordance with the symmetry of the polar class of the
trigonal system. But the angles of the two substances were not found to
be identical, although constant for each compound within one minute of
arc, there being slight but very real differences, which are very well
typified by the principal angle in each case, that of the primary
rhombohedron. In the case of proustite it is 72° 12′, while the
rhombohedron angle of pyrargyrite is 71° 22′.

This interesting and beautiful investigation of Miers thus gave us an
inkling of the truth, that small angular differences do exist between
the members of isomorphous compounds. It paved the way for, and indeed
partly suggested, the author’s systematic investigation of the
sulphates, selenates, and double salts of the alkali series of metals, a
brief account of the main results of which will be given in Chapter X.




                               CHAPTER IX
     THE CRYSTAL SPACE-LATTICE AND ITS MOLECULAR UNIT CELL. THE 230
            POINT-SYSTEMS OF HOMOGENEOUS CRYSTAL STRUCTURE.


The interval between the morphotropic work described in the last chapter
and the present time has been remarkable for the completion of the
geometrical and mathematical investigation, and the successful
identification, of all the possible types of homogeneous structures
possessing the essential attributes of crystals. It has now been
definitely established that there are 230 such types of homogeneous
structures possible, and the whole of them conform to the conditions of
symmetry of one or other of the thirty-two classes of crystals. This
fact is now thoroughly agreed upon by all the authorities who have made
the subject their special study, and may truly be considered as
fundamental.

There has long been a consensus of opinion that the crystal edifice is
built up of structural units which can be likened to the bricks or stone
blocks of the builder, but which in the case of the crystal are so small
as to be invisible even under the highest power of the microscope. The
conceptions of their nature, however, have been almost as numerous as
the investigators themselves, everyone who has thought over the subject
forming his own particular ideas concerning them. We have had the
“Molécules intégrantes” of Haüy, the “Polyhédres” of Bravais, the
“Fundamentalbereich” of Schönflies, the “Parallelohedra” of von Fedorow,
and the fourteen-walled cell, the “Tetrakaidecahedron” of Lord Kelvin,
and again the “Polyhedra” of Pope and Barlow. Ideas have thus been
extremely fertile, and indeed almost every variety of speculation has
been indulged in as to the shape and nature of the unit of the structure
which can exhibit such remarkable evidences of organisation and such
extraordinary optical and other physical properties as those of a
crystal.

There is one inherent difficulty, however, which renders all such
speculations more or less chimerical, until we know very much more as to
the structure of the chemical atom, and the organisation of the
corpuscles composing it. Such speculations, however, are deeply
interesting, and the difficulty alluded to accounts largely for the
great variety of conception possible. It is this, that the matter of the
molecules, and again that of the atoms composing them, is not
necessarily, nor even probably, continuous and in contact throughout,
but that on the contrary the space which may legitimately be assigned to
the unit of the structure is partly void. How much of this unit space is
matter and how much is unoccupied, and how the one is related to the
other as regards its position or distribution in space, we have yet no
means of knowing, although there are signs that the day is not far
distant when we shall know at least something concerning it. The recent
brilliant work of Sir J. J. Thomson and his school of physicists has
rendered it clear that the chemical atom is composed of cycles of
electronic corpuscles, the orbital motions of which determine its
boundaries.

In this condition of our knowledge obviously the only safe course is
to consider each atom of the chemical molecule as occupying a “sphere
of influence,” within the limits of which the material parts of the
atom, the corpuscles in organised motion, are confined. The
“Fundamentalbereich” of Schönflies and the “Sphere of Influence” of
Barlow, are the conceptions which in all probability have the greatest
value in the present state of our knowledge, and if we adopt the
latter we shall not be committing ourselves to anything more than the
experimental facts fully warrant.

It may be quite definitely stated, however, that there is a considerable
amount of experimental evidence that the unit of the space-lattice of
the crystal structure is certainly not more complex than the chemical
molecule, the idea of an aggregation of chemical molecules to form a
“physical molecule” acting as a structural unit having proved to be a
misleading myth.

Fortunately, however, there is no necessity whatever to introduce the
subject of the actual shape of the unit, and the greatest progress has
been effected by disregarding it altogether, and agreeing to the
representation of the unit by a point. This leads us at once to perceive
the importance of the brilliant work of the geometricians, who have now
completed their theory of the homogeneous partitioning of space into
point-systems possible to crystals, the structural units of the latter
being regarded as points. The investigations extend from those of
Frankenheim in the year 1830 to the finishing touches given by Barlow in
1894, and prominently standing forth as those of the greatest
contributors to the subject, besides the two investigators just
mentioned, are the names of Bravais, Sohncke, Schönflies and von
Fedorow.

Bravais, perfecting the work of his predecessor Frankenheim, made us
acquainted with the fourteen fundamentally important space-lattices, or
same-ways orientated arrangements of points. If we regard each chemical
molecule as represented by a point, disregarding the separate atoms of
which it is composed, then these fourteen space-lattices represent the
possible arrangements of the molecules in the crystal in all the simpler
cases; three of these lattices have cubic symmetry; the tetragonal,
hexagonal, trigonal, rhombic and monoclinic systems claim two
space-lattices each; while one space-lattice conforms to the lack of
symmetry of the triclinic system.

The fourteen space-lattices of Bravais thus represent the arrangement of
the chemical molecules in the crystal, and determine the systematic
symmetry. The points being taken absolutely analogously in all the
molecules, and the whole assemblage being homogeneous, that is, such
that the environment about any one spot is the same as about every
other, the arrangement is obviously a same-ways orientated one, the
molecules being all arranged parallel-wise to each other.

[Illustration:

  FIG. 59.—Triclinic Space-Lattice.
]

But the fact that the structure is that of a space-lattice also causes
the crystal to obey the law of rational indices. To enable us to see how
this comes about it is only necessary to regard a space-lattice. In Fig.
59 is represented the general form of space-lattice, that which
corresponds to triclinic symmetry. It is obviously built up of
parallelepipeda, the edges of which are proportional to the lengths of
the three triclinic axes, and their mutual inclinations are those of the
latter. As we may take our representative point anywhere in the
molecule, so long as the position chosen is the same for all the
molecules of the assemblage, we may imagine the points occupying the
centres of the parallelepipeda instead of the corners if we choose, for
that would only be equivalent to moving the whole space-lattice slightly
parallel to itself. Hence, each cell may be regarded as the habitat of
the chemical molecule.

Now the faces of the crystal parallel to each two of the three sets of
parallel lines forming the space-lattice will be the three pairs of
axial-plane faces, and any fourth face inclined to them must be got by
removing parallelepipedal blocks in step-wise fashion, precisely like
bricks, as already shown in Fig. 12 (page 28) in Chapter III., in order
to illustrate the step by step removal of Haüy’s unit blocks. It will
readily be seen that if one more cell be removed from each row than from
the row below it, the line of contact touching the projecting corner of
the last block of each row will be inclined more steeply than if two
more cells were removed from each row. Moreover, the angle varies
considerably between the two cases, and if three blocks are removed at a
time the angle gets very small indeed. Hence, there cannot be many such
planes possible, and we see at once why the indices of the faces
developed on a crystal are composed of low whole numbers and why the
forms are so relatively few in number. Owing to the minuteness of a
chemical molecule, all the irregularities of such a surface are
submicroscopic, and the general effect to the eye is that of a smooth
plane surface.

The space-lattice arrangement of the molecules in the crystal structure
thus causes the crystal to follow the law of rational indices, by
limiting and restricting the number of possible facial forms which can
be developed. It also determines which one of the seven systems of
symmetry or styles of crystal architecture the crystal shall adopt. It
does not determine the details of the architecture, however, that is, to
which of the thirty-two classes it shall conform, this not being the
function of the molecular arrangement but of the atomic arrangement that
is, of the arrangement of the cluster of atoms which form the molecule,
and this leads us to the next step in the unravelling of the internal
structure of crystals.

The credit of this next stage of further progress is due to Sohncke,
whose long labours resulted in the discrimination and description of
sixty-five “Regular Point-Systems,” homogeneous assemblages of points
symmetrically and identically arranged about axes of symmetry, which are
sometimes screw axes, that is, axes about which the points are spirally
distributed. Sohncke’s point-systems express the number of ways in which
symmetrical repetition can occur. Moreover, the points may always be
grouped in sets or clusters, the centres of gravity of which form a
Bravais space-lattice.

This latter fact is of great interest, for it means that Sohncke’s
points may represent the chemical atoms, and that the stereometric
arrangement of the atoms in the molecule is that which produces the
point-system and determines the crystal class, while the whole cluster
of atoms forming the molecule furnishes, as above stated, the
representative point of the space-lattice.

This, however, is not the whole story, for the sixty-five Sohnckian
regular point-systems only account for twenty-one of the thirty-two
crystal classes, the remaining eleven being those of lower than full
holohedral systematic symmetry, and which are characterised by showing
complementary right and left-handed forms. In other words, they exhibit
two varieties, on one of which faces of low symmetry are developed on
the right, while on the other symmetrically complementary faces are
developed on the left; that is, these little faces modify on the right
and left respectively the solid angles formed by those faces of the
crystal which are common to both the holohedral class of the system and
to the lower symmetry class in question. In some cases, moreover, these
two complementary forms are known to exist alone, without the presence
of faces common to both the holohedral class and the class of lower
symmetry. The two varieties of the crystals are the mirror images of
each other, being related as a right-hand glove is to a left-hand one.

Further, the crystals of these eleven classes very frequently exhibit
the power of rotating the plane of polarised light to the right or to
the left, and complementarily in the cases of the two varieties of any
one substance, corresponding to the complementariness of the two crystal
forms. The converse is even more absolute, for no optically active
crystal has yet been discovered which does not belong to one or other of
these eleven classes of lower than holohedral symmetry.

The final step of accounting for the structure of these highly
interesting eleven classes of crystals was taken simultaneously by a
German, Schönflies, a Russian, von Fedorow, and an Englishman, Barlow,
who quite independently and by totally different lines of reasoning and
of geometrical illustration showed that they were entirely accounted for
by the introduction of a new element of symmetry, that of mirror-image
repetition, or “enantiomorphous similarity” as distinguished from
“identical similarity.” These three investigators all united in finally
concluding that when the definition of symmetrical repetition is thus
broadened to include enantiomorphous similarity, 165 further
point-systems are admitted, and the whole 230 point-systems then account
for the whole of the thirty-two classes of crystals.

Schönflies’ simple definition of the nature of the structure is that
every molecule is surrounded by the rest collectively in like manner,
when likeness may be either identity or mirror-image resemblance. Von
Fedorow finds the extra 165 types to be comprised in “double systems”
consisting of two “analogous systems” which are the mirror images of
each other. Barlow proceeds to find in how many ways the two
mirror-image forms can be combined together, there being in general
three distinct modes of duplication, including the insertion of one
inside the other. He also shows that all homologous points in a
structure of the type of one of these additional 165 point-systems
together form one of the 65 Sohnckian point-systems, the structure being
capable of the same rotations or translations, technically known as
“coincidence movements” (movements which bring the structure to exhibit
the same appearance as at first), as those which are characteristic of
that point-system.

This fascinating subject of mirror-image symmetry, and the optical
activity connected with it, will be reverted to and the latter explained
in Chapter XI.

We have thus seen how satisfactorily the geometrical theory of the
homogeneous partitioning of space has been worked out, and how admirably
it agrees with our preliminary supposition that a crystal is a
homogeneous structure. The fact that the 230 homogeneous point-systems
all fall into and distribute themselves among the thirty-two classes of
crystals, the symmetry of which has also now been fully established,
affords undeniable proof that as regards this branch of the subject
something like finality and clearness of vision has now been arrived at.




                               CHAPTER X
LAW OF VARIATION OF ANGLES IN ISOMORPHOUS SERIES. RELATIVE DIMENSIONS OF
                UNIT CELLS. FIXITY OF ATOMS IN CRYSTAL.


We are now in a position to approach the conclusion of the long
controversy as to the constancy or otherwise of crystal angles in
the cases of greatest similarity, those of isomorphous substances,
and to appreciate how the conflicting views of Haüy and Mitscherlich
and their schools of thought have at length been reconciled. As the
result of a comprehensive study, on the part of the author, of the
sulphates and selenates of the rhombic series R_{2}S/SeO_{4}, and of
the double sulphates and selenates of the monoclinic series
R_{2}M(S/SeO_{4})_{2}.6H_{2}O, in which R represents the alkali
metals, potassium, rubidium and cæsium, and in which M may be
magnesium, zinc, iron, nickel, cobalt, manganese, copper or cadmium,
four facts of prime significance have been definitely established.

(1) The crystals of the different members of an isomorphous series
exhibit slight but real differences in their interfacial angles, the
magnitude of the angle changing regularly with the alteration of the
atomic weight of the interchangeable metals or negative elements of the
same family group which give rise to the series, as one metal or
acid-forming element is replaced by another. The amount of the
difference increases as the symmetry of the system diminishes. Thus the
maximum difference for the more symmetrical rhombic series of sulphates
and selenates is 56′, which occurs in the case of one angle between
potassium and cæsium selenates, and it is usually much less than this;
in the case of the less symmetrical monoclinic series of double salts
the maximum angular difference observed was 2° 21′, between potassium
and cæsium magnesium sulphates.

(2) The physical properties of the crystals, such as their optical and
thermal constants, are also functions of the atomic weights of the
elements of the same family group which by their interchange produce the
series.

(3) The dimensions of the elementary parallelepipedon of the
space-lattice, or in other words, the separation of the molecular
centres of gravity, the points or nodes of the space-lattice, along the
three directions of the crystal axes, also vary with the atomic weight
of the interchangeable elements.

(4) Specific chemical replacements are accompanied by clearly defined
changes in the crystal structure along equally specific directions.
Thus, when the metal, say potassium, in an alkali sulphate or selenate
is replaced by another of the same alkali-family group, rubidium or
cæsium, there is a marked alteration in the crystal angles and in the
dimensions of the space-lattice, corresponding to elongation of the
vertical axis; and when the acid-forming element sulphur is replaced by
selenium, its family analogue, a similar very definite change occurs,
but the expansion in this case takes place in the horizontal plane of
the crystals.

Confirmatory results have also been obtained as regards the
morphological constants, the investigations not extending to the optical
or thermal properties, by Muthmann for the permanganates, and by Barker
for the perchlorates, of the alkali metals. Hence, there can be no doubt
whatever that, as regards the various series investigated, which are
such as would be expected to afford the most definite results owing to
the electro-positive nature of metals being at its maximum strength in
the alkali group, the above rules are definite laws of nature.

Thus it is clear that in the cases of isomorphous substances, which were
the only possible exceptions to the generalisation that _to every
chemically distinct solid substance of other than perfect cubic symmetry
there appertains a specific crystalline form, endowed with its own
particular angles and morphological crystal elements, which are
absolutely constant for the same temperature_, the law does really hold,
and isomorphous substances are no exceptions. The law of progression of
the crystal properties according to the atomic weight of the
interchangeable elements affords indeed at the same time both an
amplification of the generalisation and a precise explanation of its
mode of operation in these cases.

The discovery of the local effect produced by the two kinds, positive
and negative, of chemical replacement, has a profound bearing on crystal
structure. For it is thereby rendered certain that the atoms are fixed
in the crystal edifice, and therefore in the molecule in the solid
state. It becomes obvious that the atoms—in their stereometric positions
in the molecule, being thus fixed in the solid crystal when the
molecules set themselves rigidly in the regular organisation of the
space-lattice—form the points of the regular point-system of the crystal
structure, which determines to which of the thirty-two classes of
symmetry the crystal shall belong. Any movement of the atoms in the
crystal, other than that which accompanies change of temperature, and
possibly change of pressure, is thus improbable; and this experimental
proof of their fixity, afforded by the fact that definitely orientated
changes accompany the replacement of particular atoms, also doubtless
indicates that the latter are located in the particular directions along
which the changes of exterior angle and of internal structural
dimensions are observed to occur. Stereo-chemistry, which has made such
enormous advances during the last few years, thus becomes of even
greater importance than Wislicenus and its other originators ever dreamt
of.

Within the atoms in the crystal the constituent electronic corpuscles
may be and probably are in rapid movement, and such physical effects as
have hitherto been ascribed to movement of the atoms within the crystal
are doubtless due to movement of the electronic corpuscles within them,
the sphere of influence of the atom itself being fixed in space in the
solid crystal, and being doubtless defined by the area within which the
corpuscular movements occur.

[Illustration:

  FIG. 60.—Diagram illustrating Progressive Change of Crystal Angles in
    Isomorphous Series.
]

Three illustrations of the law of change of the crystal properties with
variation of the atomic weight of the determinative elements of an
isomorphous series may be given, and will serve to render the practical
meaning of the generalisation clearer. The first is a diagrammatic
representation, in Fig. 60 (in a very exaggerated manner as the real
change would be inappreciable on the scale drawn), of the change of
angle on replacing the potassium in potassium sulphate, K_{2}SO_{4}, or
selenate, K_{2}SeO_{4}, by rubidium or cæsium. The inner crystal
outline, a vertical section, is that of the potassium salt. The vertical
lines represent the intersections of the two faces of the brachypinakoid
_b_ = {010} with a vertical plane parallel to the macropinakoid _a_ =
{100}; the horizontal lines represent the intersection of the two faces
of the basal plane _c_ = {001} with the same vertical plane; and the
oblique lines represent the intersection of the vertical plane with the
four faces of the dome form _q_ = {011}, which are inclined to both _b_
and _c_ planes. The diagram is thus designed to show the variation of
the inclination of these latter dome faces to the two rectangular axial
plane faces _b_ and _c_. The outer crystal outline represents a similar
section of a crystal of the corresponding cæsium salt, and the middle
outline that of a crystal of the rubidium salt.

The progressive alteration of the angle of the q-face will be obvious,
the direction of the change being correct, but the amount of change, as
already stated, being much exaggerated; in reality it never reaches a
degree between the two extreme (potassium and cæsium) salts. It will be
remembered that the respective atomic weights of potassium, rubidium,
and cæsium are 38·85, 84·9 and 131·9, when hydrogen equals 1, that of
rubidium being almost exactly the mean.

[Illustration:

  FIG. 61.—Diagram illustrating Progressive Change of Double Retraction
    in Isomorphous Series.
]

The second illustration is taken from the optical properties. Fig. 61
represents graphically the regular diminution of double refraction (the
difference between the two extreme indices of refraction α and γ) which
accompanies increase of the atomic weight of the metal present. The
diagram exhibits the closing up of the two spectra afforded by three
analogously orientated 60°-prisms, one of each of the three salts, such
as was used in determining two of the refractive indices of the salt.
Each prism produces two refracted rays from the single ray furnished by
the collimator of the spectrometer, and consequently two images of the
signal-slit of the collimator when monochromatic light is used, or two
spectra if white light be employed. The Websky signal-slit is narrow at
the centre to enable an accurate allocation to the vertical cross-wire
of the telescope to be made, but wide at its top and bottom ends, in
order to transmit ample light, and Fig. 61 shows four images of this
signal produced by each prism, namely, one R in red C-hydrogen light and
another B in greenish-blue F-hydrogen light belonging to each of the two
spectra, in order to locate the two ends of each of the latter, coloured
monochromatic light of each of the two colours in turn and of the exact
C and F wave-lengths having been fed to the spectrometer from the
spectroscopic illuminator. It will be observed in the case of the top
row that the two spectra, each indicated by the adjacent red and
greenish-blue images, are well apart, the relative distance being about
that actually observed in the case of potassium sulphate. They are
nearer together, however, in the second row, which indicates what is
observed in the case of the analogous rubidium salt, and in the lowest
row representing the relative distances of the two spectra apart in the
case of the cæsium salt, they are so close together as to overlap; for
in this latter case the greenish-blue image of the left-hand spectrum,
corresponding to the a index of refraction, occupies the same position
as the image for yellow sodium light of the right-hand spectrum
corresponding to γ would occupy in the case of cæsium sulphate, the a
refractive index for F-light being 1·5660 and the γ index for Na-light
being 1·5662. The progression of the alteration of the amount of the
double refraction is thus very striking, as the atomic weight of the
metal is varied.

The third illustration of the law of progression with atomic weight is
also an optical one, and is taken from the monoclinic series of double
sulphates and selenates. It indicates the rotation, with increase of the
atomic weight of the metal, of the ellipsoid which graphically
represents the optical properties, about the unique axis of symmetry,
which is likewise an axis of optical symmetry, of the crystal. In the
potassium salt the ellipsoid occupies the position indicated by the
ellipse drawn in continuous line in Fig. 62, the section of the
ellipsoid by the symmetry plane; the outline of a tabular crystal
parallel to the symmetry plane is also given, as well as the axes of the
crystal and of the ellipsoid lying in that plane.

[Illustration:

  FIG. 62.—Diagram illustrating Progressive Rotation of Optical
    Ellipsoid in Monoclinic Isomorphous Series.
]

In the rubidium salt the ellipsoid has rotated over to the left, as
indicated by the dotted ellipse, for a few degrees, the number of which
varies slightly for the different groups of double salts; while in the
cæsium salt it has swung over much more still, to the place marked by
the ellipse drawn in broken line. In both this and the last illustration
it will be remarked that the optical change is greater between the
rubidium and cæsium salts than it is between the potassium and rubidium
salts, the reason being that the optical properties are usually
functions (of the atomic weight of the interchangeable elements) which
are of an order higher than the first corresponding to simple
proportionality.

These three ocular illustrations may serve to render this interesting
law of progression, according to the atomic weight of the
interchangeable elements which give rise to the isomorphous series,
clearer to the mind, by placing before it concrete instances of the
operation of the law.

The generalisation itself may be very concisely expressed in the
statement that:

_The whole of the properties, morphological and physical, of the
crystals of an isomorphous series of salts are functions of the atomic
weights of the interchangeable chemical elements of the same family
group which give rise to the series._

The fact that this law extends to the structural dimensions, equally
with all other morphological properties, as stated under (3) at the
beginning of this chapter, is of especial interest. For it has actually
been found possible to determine the relations of the dimensions of the
unit parallelepipeda of the space-lattices of the various salts, that
is, the separation of the molecular points of the space-lattice in the
directions of the three crystal axes, for the various salts of the
isomorphous series. This is achieved by combining in suitable formulæ
the volume of the unit cell of the space-lattice with the relative
lengths of the three crystal axes, _a_, _b_, _c_.

The axial ratios _a_ : _b_ : _c_ are calculated from the measurements of
the crystal angles, as explained in Chapter VI., page 68, and the volume
is the physical constant long known as “molecular volume,” but now for
the first time understood as regards its meaning in the case of solid
substances. It is the quotient of the chemical constant molecular weight
(the sum of the atomic weights, taking into account the number of atoms
of each element present) by the specific gravity of the substance, here
the solid crystal. Very great care has been taken to obtain absolutely
accurate determinations of the specific gravities of the salts, as much
depends on this now very valuable physical constant, and all the values
obtained were reduced to the constant reference temperature of 20°, as
the density notoriously alters rapidly with change of temperature.

We have thus arrived at morphological constants of very considerable
importance, which are best termed “_Molecular Distance Ratios_,” as they
express the relative distances apart in the three directions of space of
the centres of gravity or other representative points of contiguous
chemical molecules. They are dependent on three experimental
determinations, atomic weight, specific gravity, and crystal angles, all
of which have now been brought to the highest pitch of refinement and
accuracy; hence the molecular distance ratios are particularly
trustworthy constants. If it were only known how much is matter and how
much is space in the molecular parallelepipedal cell, we should actually
have in these constants a relative measure of the sizes of the
molecules. They do give us, however, the relative directional dimensions
of the molecular unit parallelepipedal cells of the space-lattices of
the various members of the isomorphous series, just as the molecular
volumes give us the relative volumes of these cells. For in an
isomorphous series we are absolutely sure that the plan on which the
space-lattice is constructed, its style of architecture, is identical
for all the members of the isomorphous series. Hence, the molecular
distance ratios are in these cases absolutely valid and strictly
comparable. The ratios are generally expressed by the Greek letters χ :
ψ : ω.

On comparing the molecular distance ratios for a potassium, a rubidium,
and a cæsium salt of any of the series of sulphates, selenates,
permanganates, perchlorates, double sulphates or double selenates
investigated, we invariably find that the values of χ, ψ, and ω for the
rubidium salt (rubidium having the intermediate atomic weight) lie
between the analogous sets of three values for the potassium and cæsium
salts respectively, in complete accordance with the law.

For the generalisation to apply absolutely it is essential that the
interchangeable elements shall belong strictly to the same family group
of the periodic classification of Mendeleéff. Potassium, rubidium, and
cæsium fulfil this condition absolutely, and so the law of progression
of the crystal properties with the atomic weight of the interchangeable
elements applies rigidly to their salts. Now there are two bases, the
metal thallium and the complex radicle group ammonium NH_{4}, which are
not thus related to the group of three alkali metals just mentioned, but
which are yet capable of replacing those metals isomorphously in their
crystals without more change of angle or of structural constants than is
provoked by the replacement of potassium by cæsium; and often indeed the
amount of change has been singularly like the lesser amount observed
when rubidium has been interchanged for potassium. But although this is
so, the directions of the changes are irregular, being sometimes the
same as when rubidium or cæsium is introduced, and sometimes
contrariwise, and in the case of thallium there are also striking
optical differences, the thallium salts being exceptionally highly
refractive. Still, morphologically the ammonium and thallium salts may
legitimately be included in the same isomorphous series with the salts
of potassium, rubidium, and cæsium, and a somewhat wider interpretation
has to be given to the term “isomorphism” in order to admit these cases.
To distinguish the inner group formed by family analogues, that is, the
more exclusive group obeying the law of progression according to the
atomic weight, the term “eutropic” is employed.

Thus the “isomorphous series” of rhombic sulphates, selenates,
permanganates, and perchlorates, and the monoclinic series of double
sulphates and double selenates, comprise the potassium, rubidium,
cæsium, thallium and ammonium salts and double salts of sulphuric,
selenic, permanganic, and perchloric acids, while the inner more
exclusive “eutropic series,” following the law absolutely, comprises in
each case only the salts containing the family analogues, potassium,
rubidium, and cæsium.

In this beautiful manner has the controversy between the schools of Haüy
and Mitscherlich now been settled, the interesting law described in this
chapter having definitely laid down the true nature and limitations of
isomorphism, while at the same time absolutely proving as a law of
nature the constancy and specific character of the crystal angles of
every definitely chemically constituted substance.




                               CHAPTER XI
       THE EXPLANATION OF POLYMORPHISM, AND THE RELATION BETWEEN
                 ENANTIOMORPHISM AND OPTICAL ACTIVITY.


_Polymorphism._ It has been shown in Chapter VII. that Mitscherlich had
in several instances proved the possibility of the occurrence of the
same substance in two different forms, notably sodium dihydrogen
phosphate NaH_{2}PO_{4}.H_{2}O, calcium carbonate CaCO_{3} (as calcite
and aragonite), the metallic sulphates known as vitriols, and the
chemical element sulphur, and that he gave to the phenomenon the name
“dimorphism.” Since that time large numbers of dimorphous substances
have been discovered, and several which occur in three forms and even a
few in no less than four totally distinct forms. Until the establishment
of the geometrical theory of crystal structure, as expounded in Chapter
IX., this phenomenon of polymorphism gave rise to endless fruitless
discussion. It was most generally attributed to the different nature of
the so-called “physical molecule,” which was supposed to be an aggregate
of chemical molecules and the unit of the space-lattice determining the
crystal system; the different polymorphous varieties were supposed to be
built up of structural units or physical molecules consisting of an
aggregation of a different number of chemical molecules. Several
attempts were made by various investigators, notably by Muthmann and by
Fock, to determine the number of chemical molecules constituting the
physical molecule.

All these efforts, however, ended unsatisfactorily, and in the year 1896
the author showed, in a memoir[11] on “The Nature of the Structural
Unit,” that in general the physical molecule is a myth, and that the
chemical molecule is the only structural unit possessing the full
chemical composition of the substance in question; and that its centre
of gravity, or better, any representative point within it, such as a
particular atom, is the unit point of the Bravais space-lattice of the
crystal structure, while the atoms of which the chemical molecule are
composed, arranged stereometrically identically similarly in all the
molecules, are the points of the individual point-systems which make up
the combined point-system. This does not imply a necessarily parallel
and identically orientated arrangement of all the molecules, as at first
postulated by Sohncke and which is a fact for his sixty-five
point-systems; for in accordance with the conclusions of Schönflies, von
Fedorow, and Barlow discussed in Chapter IX., cases are possible in
which alternate molecules may be arranged as each other’s mirror images.
Such are the cases of external molecular compensation or molecular
combination, two oppositely enantiomorphous sets of molecules balancing
each other within the structure, but by exterior compensation as regards
the molecule itself. Moreover, the principle of mirror-image symmetry
enters, as stated in Chapter IX, altogether into the constitution of no
less than 165 of the 230 types of homogeneous structure possible to
crystals.

Hence the conception of a physical molecule is totally unnecessary and,
moreover, erroneous. The alkali sulphates and selenates exhibit
dimorphism, one member of the series, ammonium selenate, having only
hitherto been observed in the pure state in the second, monoclinic,
form, and never in the ordinary rhombic form; and the author has
conclusively proved for these salts, and also for the double salts which
they form with the sulphates and selenates of magnesium, zinc, iron,
nickel, cobalt, manganese, copper, and cadmium, that the chemical
molecule is the only kind of molecule present, and that its
representative points are, as just stated, the nodes of the Bravais
space-lattice of the crystal structure, determining both the system of
the crystal and its obedience to the law of rational indices.

The explanation of polymorphism thus proves, in the light of the results
which have now been laid before the reader, to be a remarkably simple
one. Special pains were taken in explaining those results to show that
the temperature had a great deal to do with the conditions of
equilibrium of the crystal structure, for it determines the
intermolecular distances, that is, the amount of separation of the
molecules, and thus controls their possibility of movement with respect
to one another. Now the behaviour of the chemical molecules on the
advent of crystallisation is undoubtedly largely influenced by the
stereometric arrangement of the atoms composing them, and it is possible
for the latter to be such that the molecules may take up several
different parallel or enantiomorphously related positions; or as we have
just seen, a regular alternation within the crystal structure of such
mirror-image positions may be taken up. These different arrangements,
whether parallel or enantiomorphously opposite, may be, and probably
will be, of different degrees of stability, each of these different
forms finding its maximum stability of equilibrium at some particular
temperature, which is different for the different varieties. Hence, at a
series of ascending or descending temperatures, assuming the pressure to
remain the ordinary atmospheric, these different types of homogeneous
crystal structures will be most liable to be produced, each at its own
particular temperature, for which stable equilibrium of that crystal
structure occurs.

These different assemblages are as a rule quite dissimilar, certainly in
the crystal elements, often in class and not infrequently in system.
Generally two such different crystalline forms are all that are possible
within the life-range of temperature of the substance. But occasionally
three or even as many as four such different forms are found to be
capable of existence within the temperature life-limits of the
substance.

Polymorphism is thus completely and simply explained as a direct result
of the establishment of the geometrical theory of crystal structure as
laid down in Chapter IX. The equilibrium of the homogeneous structure is
a function of the temperature, and the stereometric arrangement of the
atoms in the chemical molecule of a substance may be such as permits of
two or more homogeneous arrangements of the molecules in assemblages of
varying degrees of stability, but each of which has a maximum stability
at a particular temperature. Hence, within any given range of
temperature such a substance will assume that type of homogeneous
arrangement of its molecules in a crystal which corresponds to the
stablest equilibrium within these temperature limits, assuming the
pressure constant within the bounds of the usual atmospheric variations.
Employing the language of physical chemistry, such a substance will thus
present two or more different solid “phases,” each characterised by its
specific crystalline form, the elementary parallelepipedon of which is
quite a distinct one. Each phase possesses also its own specific optical
and other physical properties, such as melting point, solubility,
thermal expansion, and elasticity.

It would appear as if the element sulphur is also polymorphous in this
sense, for the monoclinic prismatic form (Fig. 2, Plate I.)—the best
known and most easily prepared, from the state of fusion, of all the
forms other than the common rhombic form, in which sulphur is found in
the neighbourhood of volcanoes and in which it is also deposited from
solution in carbon bisulphide—is of distinctly lower stability, the
crystals passing in a few days into powder composed of minute crystals
of the stable rhombic variety. But in the case of carbon, with its
totally different and apparently at ordinary temperatures equally stable
varieties of octahedral-cubic diamond (Fig. 82, Plate XVI.) and
hexagonal graphite, there is some doubt; for although the diamond is
converted into graphite at a red heat in the electric arc, it is
doubtful whether we are not in the presence of a case of chemical
polymerism or allotropy, like the case of ozone, where three atoms of
oxygen compose the molecule, instead of the two atoms in the molecule of
ordinary oxygen. The fact that the negatively electrified electronic
corpuscles of the Crookes tube cause the same conversion of diamond into
graphite, producing according to Parsons and Swinton a temperature of
4,890° C. in the act, is evidence in favour of allotropy, as the charged
corpuscles are a very likely agent for breaking down such atomic
combinations. Moreover, diamond is volatilised out of contact with air
at 3,600° C. without liquefaction, and the vapour when cold condenses as
graphite. But there is reason to believe, from experiments by Sir Andrew
Noble and Sir William Crookes, that under great pressure carbon does
liquefy at 3,600° C., and that the liquid drops on cooling crystallise
as diamond.

The yellow and red varieties of phosphorus may also be due to a similar
cause, the yellow variety, which forms excellent crystals, corresponding
to P_{4}, while the red variety may correspond to a molecule composed of
a different number of atoms than four.

Another view of the nature of polymorphism has lately been brought
forward by Lehmann, as the result of his remarkable experimental
discovery of “liquid crystals,” to which fuller reference will be made
in Chapter XVI. This new view is, however, but an amplification of the
foregoing explanation of polymorphism, indicating the possible mode in
which the stereometric position of the atoms in the molecule does
actually influence and even determine the particular homogeneous
structure which shall be erected, and explains why the temperature plays
such an important rôle. Lehmann’s theory is that any one definitely
stereometrically constituted chemical molecule can only display one
particular homogeneous structure and form of crystal, and that when at a
particular temperature the system or class of symmetry is altered, this
occurs because the stereometric arrangement of the atoms within the
molecule is altered, that is, a new form of molecule is produced, which
naturally gives rise to a new form of crystal. As far as the author
understands it, this does not mean an isomeric change from the chemical
point of view, the chemical compound remaining the same, but that the
stereometric positions of the atoms have been changed, without altering
their chemical attachments, but sufficiently to change the nature of the
point-system which they produce. A significant fact in support of this
view is that the molecules of the substances forming liquid crystals are
usually very complicated and extended ones, comprising a large number of
atoms, the molecules, in fact, corresponding in length with the long
names of the organic substances of which they are generally composed.

Lehmann’s work has certainly proved that the molecule is endowed
with more individuality than has hitherto been ascribed to it, and
he even shows that there is some ground for believing that his
liquid crystals are such because this directive orientative force
resident in the molecules themselves maintains them in their
mutually crystallographically orientated positions even in the
liquid state, which may be and sometimes is as mobile as water. It
thus appears that any general acceptance of Lehmann’s ideas will
only tend to amplify and further explain the nature of polymorphism
on the lines here laid down, the temperature of conversion of one
form into another being merely that at which either a different
homogeneous packing is possible, or that at which the stereometric
relations of the atoms in the molecule are so altered as to produce
a new form of point-system without forming a new chemical compound.

_Enantiomorphism of Crystalline Form and Optical Activity._ It has
already been stated that two supplementary forms which are similar but
not identical, the one being the inverse or mirror-image reflection of
the other, as a right-hand glove is to a left-hand one, are termed
“enantiomorphous.” Also it has been shown that all those crystal forms
which have no plane of symmetry, either of simple symmetry or
alternating symmetry (which is equivalent to saying that no centre of
symmetry is present in addition to no plane of symmetry), are
enantiomorphous, and that such forms belong to eleven specific classes.
It has further been shown that the introduction of this principle of
mirror-image symmetry or enantiomorphism into the conditions already
laid down by Bravais and Sohncke for a homogeneous structure, by von
Fedorow, Schönflies, and Barlow, enabled those investigators to derive
the remaining 165 of the 230 possible types of homogeneous structures
compatible with crystal structure, over and above the 65 already
established by Bravais and Sohncke, and thus to complete the geometry of
crystal structure, when the units of such structure are represented by
points. Sohncke subsequently accepted the new principle, and modified
his own theory so as to bring it into line with it. He exhibited some
disinclination, however, at first, to accept the idea—which is a part of
the assumption of the other three authors just referred to, and which
appears to be absolutely necessary to explain one or two of the most
complicated of the crystal classes—of the possibility of two
enantiomorphous kinds of molecule being present in the crystal of the
same single substance, the balancing of the two sets having the effect
of producing mirror-image symmetry of the whole crystal, that is, the
development of a plane of symmetry.

Now the whole subject is of deep interest, both physical and chemical as
well as crystallographical, inasmuch as it is precisely such substances
as show enantiomorphism,—and can thus exist in two forms, one of which
is the mirror-image of the other and not its identical counterpart, the
two being like a pair of gloves,—which are found to possess the property
of rotating the plane of polarised light and which are therefore said to
be “optically active.” Moreover, the property may be displayed by both
the crystals and their respective solutions, or by the crystals only.
If, therefore, two optical antipodes of the same substance are known,
one rotating the plane of polarisation to the right and the other
rotating it to the same extent to the left, their crystals invariably
exhibit mirror-image symmetry with respect to each other. The converse
does not necessarily hold good, however, that a crystal possessing the
symmetry of one of these eleven classes will always exhibit optical
activity.

Pasteur[12] was the first to recognise this important relation between
enantiomorphous crystalline form and optical activity, in the case of
tartaric acid, which has the empirical formula C_{4}H_{6}O_{6} and the
constitution:

                                  COOH
                                  |
                                  CHOH
                                  |
                                  CHOH
                                  |
                                  COOH

Tartaric acid was isolated by Scheele in 1769, and its discovery was
described in the very first memoir of that distinguished chemist.
Another very similar acid, as regards some of its more apparent
properties, was afterwards, in 1819, described by John of Berlin, and
investigated by Gay-Lussac in 1826; the latter obtained it from the
grape juice deposits of the wine manufactory of Kestner at Thann in the
Vosges. It was still more fully investigated by Gmelin in 1829, who
called it racemic acid (Traubensäure). But it needed the genius of
Berzelius to prove that it really had the same composition as tartaric
acid, although so different to that acid in some of its properties.

We have here as a matter of fact, the first instance brought to light
involving the principle of isomerism, the existence of two or more
distinct compounds having the same chemical composition as regards the
numbers of atoms of the same elements present, but differing in chemical
or physical properties, or both, owing to the different arrangement of
those atoms within the molecule. The “isomers” may be chemical or purely
physical; the latter involves no alteration of the linking of the atoms,
but merely of their disposition in space, and is the kind met with in
the case of the tartaric acids.

Biot, so noted for his optical researches, showed afterwards that
tartaric and racemic acids behave optically differently in solution, an
aqueous solution of the former rotating the plane of polarisation to the
right whilst that of racemic acid is optically inactive, not rotating
the plane of polarisation at all. That is, if the dark field be produced
in the polariscope, by crossing the polarising and analysing Nicol
prisms at right angles, tartaric acid solution will restore the light
again, and the analyser will have to be rotated to the right in order to
reproduce darkness. In the case of tartaric acid, the crystals
themselves also rotate the plane of polarisation, the amount being as
much as 11°.4 in sodium fight for a plate of the crystal one millimetre
thick. On the other hand, neither the solution nor the crystals of
racemic acid rotate the plane of polarisation at all.

Pasteur’s discovery, made in the year 1848, consisted in finding that
racemic acid is really a molecular compound of two physical “isomers,”
namely, of ordinary tartaric acid, which, as we have seen, rotates the
plane of polarisation to the right, and of another variety of tartaric
acid which rotates the beam of polarised fight to the same extent to the
left. The latter and ordinary tartaric acid he therefore distinguished
as lævo tartaric acid and dextro-tartaric acid respectively. Pasteur
went even further than this, in discovering yet a fourth variety of
tartaric acid, which is optically inactive like racemic acid, but which
cannot be split up into two optically active antipodes.

Indeed, it has since been shown that there are three varieties of this
truly inactive tartaric acid; they are cases of isomerism of the
chemical molecule itself, that is, the stereometric arrangement of the
atoms in the molecule is different in the three cases. For the molecule
of tartaric acid—in common with the molecules of all carbon compounds
the solutions of which, or which themselves in the liquid state, rotate
the plane of polarisation—possesses an asymmetric carbon atom, an atom
of carbon which is linked by its four valency attachments to four
different kinds of atoms or radicle groups; indeed, the molecule of
tartaric acid contains two such asymmetric carbon atoms, namely, the two
in the pair of CHOH groups. For each of these carbon atoms is linked by
one attachment to the carbon atom of the outer COOH group, by another to
an atom of hydrogen, by a third to the oxygen of the group OH, and by
its fourth attachment to the carbon atom of the other group CHOH, which
carries the rest of the molecule, that is, this attachment is to the
other half-molecule CHOH.COOH. Hence, it is quite obvious that there can
be two different dispositions of the atoms in space, one of which would
be the mirror-image of the other, while leaving the arrangement of the
atoms about the two asymmetric carbon atoms dissimilar and not
symmetrical in mirror-image fashion. That is, the two dispositions would
render the molecules in the two cases enantiomorphous with respect to
each other, and these two would be the arrangements respectively in the
two optically active varieties. That this is the correct explanation of
the ordinary dextro variety and the lævo variety of tartaric acid can
now admit of no doubt.

But if the groups round the two asymmetric carbon atoms are symmetrical
in mirror-image fashion, there will be compensation within the molecule
itself, and the substance will be optically inactive from internal
reasons. This is the explanation of the optically inactive variety which
is unresolvable into any components. The different varieties of this
inactive form are doubtless due to the different possibilities of
arrangement of the atoms in each half, while leaving the two halves
round each asymmetric carbon atom symmetrical to each other.

We now know that the decomposable inactive variety, racemic acid, may
be readily obtained by· dissolving equal weights of the ordinary
dextro and lævo varieties in water and crystallising the solution by
slow evaporation at the ordinary temperature. For further
investigation has fully borne out the conclusion of Pasteur, that
racemic acid simply consists of a molecular compound of the two active
varieties. It is thus itself inactive because it is externally
compensated, the two kinds of enantiomorphous molecules being
alternately regularly distributed throughout the whole crystal
structure, the very case which von Fedorow, Schönflies, and Barlow
assumed to be possible, and which Sohncke only tardily admitted. The
crystalline form of racemic acid is, as was to be expected, quite
different from the monoclinic form of the active tartaric acids, being
triclinic; and indeed it is not crystallographically comparable with
the active form, inasmuch as the crystals of racemic acid contain a
molecule of water of crystallisation, whereas the active varieties
crystallise anhydrous.

Ordinary dextro and lævo tartaric acids crystallise in identical forms
of the sphenoidal or monoclinic-hemimorphic class of the monoclinic
system, the class which is only symmetrical about a digonal axis, the
unique symmetry plane of the monoclinic system, which also operates when
full monoclinic symmetry is developed, being absent in this class. Hence
the interfacial crystal angles, the monoclinic axial angle, and the
axial ratios are identical for the two varieties. But the crystals are
hemimorphic, owing to the absence of the symmetry plane, and
complementarily so, the dextro variety being distinguished by the
presence of only the right clino-prism {011}, while the lævo variety is
characterised by the presence only of the left-clino-prism {0̄11}, these
two complementary forms, each composed of only two faces and which on a
holohedral crystal exhibiting the full symmetry of the monoclinic system
would both be present as a single form of four faces, being never both
developed on the same optically active crystal.

This hemimorphism of the two kinds of crystals will be rendered clear by
Figs. 63 and 64, representing typical crystals of dextro and lævo
tartaric acids which are obviously the mirror images of each other.

[Illustration:

  FIG. 63.
]

[Illustration:

  FIG. 64.

  Crystals of Dextro and Lævo Tartaric Acids.
]

A remarkable discovery was made by Pasteur in connection with one
of the salts of racemic acid, sodium ammonium racemate,
Na(NH_{4})C_{4}H_{4}O_{6}, or

                               COONH_{4}
                               |
                               CHOH
                               |
                               CHOH
                               |
                               COONa

which is obtained by adding ammonia to the readily procurable salt
hydrogen sodium racemate. Sodium ammonium racemate was found by Pasteur
to be decomposable into the salts of dextro and lævo tartaric acids, on
crystallisation of a solution saturated at 28° C. by inoculation with a
crystal of either of those active salts. The solution on cooling being
in the state of slight supersaturation, which we now know from the work
of Ostwald and of Miers as the metastable condition, corresponding to
the interval between the solubility and supersolubility curves (see Fig.
98), if a crystal say of sodium ammonium lævo-tartrate be introduced,
this variety crystallises out first and can be separated from the
residual dextro-salt, which can then be subsequently crystallised.
Moreover, in certain direct crystallisations of sodium ammonium racemate
without such specialised inoculation, Pasteur found all the crystals
hemimorphic, some right-handed and some left-handed, and he was actually
able to isolate from each other crystals of the two varieties. On
separate recrystallisation of these two sets of crystals, he found them
to retain permanently their right or left-handed character, indicating
that the molecules themselves composing these crystals were
enantiomorphous. Their solutions correspondingly rotated the plane of
polarisation of light in opposite directions. Pasteur afterwards
obtained from the dextro-salt pure ordinary (dextro) tartaric acid, and
from the lævo-salt the lævo-acid, by converting them first into the lead
salts and then precipitating the lead as sulphide by sulphuretted
hydrogen.

In the case of lævo tartaric acid, this was its first isolation, as it
had hitherto been unknown. Gernez afterwards independently found that a
saturated solution of sodium ammonium racemate affords crystals of the
lævo-salt just as readily as of the dextro-salt; if a crystal of either
salt be introduced, crystals corresponding to that variety are produced.

Another most fruitful observation of Pasteur, the principle of which has
since been the means of isolating one of the two constituents of many
racemic compounds, was that when the spores of _Penicillium glaucum_ are
added to a solution of racemic acid containing traces of phosphates the
ordinary dextro component is destroyed by the organism, while the lævo
component is unattacked so long as any dextro remains; hence, if the
fermentation operation be stopped in time the lævo-acid may be isolated
and crystallised. Why a living organism thus eats up by preference one
variety only, possessing a particular right or left-handed screw
structure, of a compound containing the same elementary constituents
chemically united in the same manner, remains a most interesting
biological mystery.

The crystals of both dextro and lævo tartaric acids prove to be
pyro-electric, that is, develop electric excitation when slightly
heated. The end which exhibits the development of the clinodome develops
positive electricity in each case, when the crystal is allowed to cool
after warming, so that the two varieties are oppositely pyro-electric,
just as they are oppositely optically active. The most convenient method
of demonstrating the fact is to dust a little of Kundt’s powder, a
mixture of finely powdered red lead and sulphur, through a fine muslin
sieve on to the crystal as it cools. The sulphur becomes negatively
electrified and the red lead positively by mutual friction of the
particles in the sifting, and the sulphur thus attaches itself to the
positively electrified part of the crystal and the red lead to the
negatively electrified end. This phenomenon of the development by the
two varieties of an optically active substance of opposite electrical
polarity has since been shown to be a general one.

Finally, on mixing concentrated solutions containing equivalent weights
of dextro and lævo tartaric acid Pasteur observed that heat was evolved,
a sign of chemical combination, and the solution afterwards deposited on
cooling crystals of racemic acid. Hence, the only conclusion possible is
that racemic acid must be a molecular compound of the two oppositely
optically active tartaric acids. It thus partakes of the character of a
double salt, analogous to potassium magnesium sulphate for instance.
Consequently the crystal structure is one in which alternating molecules
of the two acids are uniformly distributed, and the case is actually
presented of two oppositely enantiomorphous sets of molecules producing
a homogeneous structure.

This interesting pioneer case of tartaric acid has been the cause of
the term “racemic” being applied to the inactive form of a substance
when it is decomposable into two oppositely optically active
enantiomorphous varieties of the substance. No well authenticated
exception has been found, in all the many instances which have been
observed of the phenomenon since Pasteur’s time, to the fact that
optically active substances exhibit what was formerly termed
hemihedrism; that is, expressing the case in accordance with our later
more accurate ideas of crystal structure as elucidated in previous
chapters, such substances invariably belong to classes of symmetry
possessing less than the full number of elements of symmetry possible
to the system to which the class belongs. These classes are eleven in
number, those possessing no plane of symmetry; they are, namely, the
asymmetric class of the triclinic system, the sphenoidal class of the
monoclinic system (to which the two tartaric acids, dextro and lævo,
belong), the bisphenoidal class of the rhombic system, the pyramidal
and trapezohedral classes of the trigonal, tetragonal, and hexagonal
systems, and the tetrahedral-pentagonal-dodecahedral and
pentagonal-icositetrahedral classes of the cubic system.

The optical activity has been proved by Le Bel and Van t’Hoff to be due
in most cases to enantiomorphism of the chemical molecules, that is, to
the enantiomorphous stereometric arrangement of the atoms in the
molecules, and therefore also,—as we have just seen, in accordance with
the geometrical theory of crystal structure,—of the combined
point-system in the case of each of the two varieties.

The point-systems are probably of a spiral screw-like character, either
right-handed or left-handed, as has been shown by Sohncke to be the case
for the two varieties of quartz, which crystallises in the trapezohedral
class of the trigonal system, one of the eleven classes just enumerated.
The example afforded by quartz will be developed fully in the next two
chapters, as this beautifully crystallised mineral enables us to study
and to demonstrate the phenomena of optical activity in a unique manner
and on the large scale.

The solutions as well as the crystals are usually optically active in
the cases where, as in the instance of the tartaric acids, the
substances are soluble in water or other solvent. Occasionally, however,
the optical activity is lost by dissolving in a solvent, and in such
cases it is the point-system only, and not the molecules themselves,
which is enantiomorphous. Sodium chlorate, NaClO_{3}, is an instance of
this kind. Moreover, a crystal can belong, as already mentioned, to one
of the eleven above enumerated classes of symmetry without displaying
optical activity, as all the point-systems possessing the symmetry of
these eleven classes do not exhibit screw-coincidence movements. Barium
nitrate, Ba(NO_{3})_{2}, is such a case.

The two “optical antipodes,” as the dextro and lævo varieties are
conveniently termed, of an optically active substance thus possess an
enantiomorphous crystal structure; but they are alike in their physical
properties such as density, melting point, optical refraction and optic
axial angle, cleavage, and elasticity. The crystal angles are identical
for the forms which are developed in common by them, and which are
usually those which the particular low class of symmetry possesses in
common with the holohedral class of the system. The crystallographic
difference between the two varieties comes in with respect to the
specific forms characteristic of the particular class of lower than full
systematic symmetry, and these forms are never displayed in common by
the two varieties, this being the essence of the enantiomorphism. When
the crystals are not rich in faces, however, it frequently happens that
only the common forms of higher symmetry just referred to are developed
on the crystals, and the two varieties are then indistinguishable in
exterior configuration; it is only on testing their rotatory power,
either by means of a section-plate of the crystal or by means of a
solution, or their pyro-electric properties, or, lastly, their
etch-figures afforded by a trace of a solvent (which etchings on the
crystal faces are enantiomorphous and an excellent indication of the
true symmetry), that their real character can be ascertained. Many
mistakes have been made in the past, and crystals assigned to a higher
than their true class of symmetry, owing to the investigation of only a
single crop of crystals fortuitously poor in the number of forms
displayed.

In the racemic form, if one should be deposited from the mixed solutions
of the two optical antipodes as a molecular compound of the latter, we
have an occurrence akin to polymerism, that is, the combination into a
single whole entity of a number of molecules, essentially two in the
case of racemism. Just as polymeric varieties of organic substances are
always found to have quite different crystalline forms, so an optically
inactive racemic form of a substance is generally quite different
crystallographically to the dextro and lævo varieties. But there is
usually some similarity along specific zones of the crystals, a kind of
isogonism or morphotropy being developed, such as has been shown to
occur, for instance, by Armstrong and Pope in the case of the substance
sobrerol.[13]

Besides the true racemic form it is often observed that under certain
conditions crystals are obtained which appear to combine the characters
of both the dextro and lævo varieties, exhibiting both series of
distinguishing hemimorphic or hemihedral forms on the same crystal; that
is, they show the full, holohedral, symmetry of the system. This has
been shown by Kipping and Pope[14] to be due to repeated twinning, thin
layers of the right and left-handed varieties being alternated, just, in
fact, as in the interesting form of quartz known as amethyst, to which
reference with experimental demonstration will be made in Chapter XIV.;
the whole structure assumes in consequence the simulated higher symmetry
which usually accompanies laminated twinning. Such forms have been
termed “pseudo-racemic.” In their memoir (_loc. cit._, p. 993) Kipping
and Pope summarise a large amount of highly interesting work on this
chemico-crystallographic subject which has been carried out by them, and
it may be useful to quote their precise definition of the relationship
between racemic and pseudo-racemic substances. They say:

“We define a pseudo-racemic substance as an intercalation of an equal,
or approximately equal, proportion of two enantiomorphously related
components, each of which preserves its characteristic type of
crystalline structure, but is so intercalated with the other as to form
a crystalline individual of non-homogeneous structure. A solid racemic
compound, on the other hand, may be defined as a crystalline substance
of homogeneous structure which contains an equal proportion of two
enantiomorphously related isomerides.

“The relations holding between a mere mixture of optical antipodes, a
pseudo-racemic substance, and a racemic compound, are closely parallel
to those existing between a crystalline mixture, an isomorphous mixture,
and a double salt. The crystallographic methods, by which a double salt
can be distinguished from an isomorphous mixture, may be directly
applied to distinguish between racemic and pseudo-racemic substances.
Thus, according as the crystalline substance obtained from a mixture of
two salts resembles or differs from either of its components
crystallographically, it is regarded either as an isomorphous mixture or
a double salt; similarly, an inactive externally compensated substance,
which closely resembles its active isomerides crystallographically, is
to be considered as pseudo-racemic, whereas when the contrary is true,
it is to be regarded as racemic.”

The work of Kipping and Pope may be regarded as having finally
vindicated and substantiated the law of Pasteur, that substances of
enantiomorphous molecular configuration develop enantiomorphous
crystalline structures, and that the crystal structures assumed by
enantiomorphously related molecular configurations are themselves
enantiomorphously related.

This subject, the main results and principles of which have now been
elucidated, may well be closed with a reference to an interesting case
of enantiomorphism and optical activity which the author has himself
investigated,[15] and which is very similar to the case of the tartaric
acids. It had been previously shown[16] by P. F. Frankland and W. Frew,
that when calcium glycerate was submitted to the fermenting action of
the _Bacillus ethaceticus_ one-half only of the glyceric acid was
destroyed, and that the remaining half was optically active, rotating
the plane of polarisation to the right.

Now glyceric acid,

                               CH_{2}.OH
                               |
                               CH.OH,
                               |
                               COOH

has manifestly one so-called asymmetric carbon atom (that is, a carbon
atom the four valencies of which are satisfied by attachment to four
different monad elements or groups), that belonging to the CHOH group.
There are consequently two possible arrangements of the molecule in
space, probably corresponding to the two optically active varieties,
namely, those represented, as far as is possible in one plane, as below,
the asymmetric carbon atom (not shown in the graphic representation)
being supposed to be at the centre of the tetrahedron, which is usually
taken to represent a carbon atom with its four valencies.

[Illustration]

Dextro-glyceric acid itself proved to be an uncrystallisable syrup, but
the calcium salt, Ca(C_{3}H_{5}O_{4})_{2}.2H_{2}O, was obtained in
crystals sufficiently well-formed to permit of a complete
crystallographic investigation, which the author undertook by friendly
arrangement with Prof. Frankland. Although the acid itself is
dextro-rotatory, aqueous solutions of the calcium salt are lævo-rotatory
to the extent of –12.09 units of “specific rotation” for sodium light.

The crystals were colourless well-formed prisms which proved to be of
monoclinic symmetry, the best individuals being formed by very slow
evaporation of the aqueous solution. They were terminated at both ends
by pyramid and dome faces, and sometimes grew to the length of a
centimetre. The actual crystal elements found after a full series of
measurements were as under:—


                       CALCIUM DEXTRO-GLYCERATE.

_Crystal system_: monoclinic.

_Class of Monoclinic System_: sphenoidal or monoclinic-hemimorphic.

_Habit_: prismatic.

_Monoclinic axial angle_: β=69° 6′.

_Ratio of axes_: _a_ : _b_ : _c_ = 1.4469 : 1 : 0.6694.

_Forms observed_:

          _a_ = {100}, _c_ = {001}, _r′_ = {̄201}, _p_ = {110},
          _m_ = {011}, _o_ = {111}  _s_ = {̄1̄11},  _n_ = {̄2̄11}.

It will thus be seen that the system and the class are precisely those
of the two active tartaric acids, which renders the case the more
interesting. The usual appearance of the crystals is shown in Fig. 65,
and the stereographic projection is given in Fig. 66, which will
elucidate the symmetry more clearly, the plane of projection being the
plane of symmetry The latter, however, in this class is inoperative, the
two ends of the digonal symmetry axis, which runs perpendicularly to the
plane of the paper, being differently terminated, as in the tartaric
acids. The faces of the forms _o_ = {111} and _m_ = {011} were never
found developed on the left side of the symmetry plane, that is, on the
left side of the crystal as drawn in Fig. 65, the symmetry plane running
perpendicularly to the paper vertically from front to back; they were
only present on the right. Conversely, the faces of _s_ = {̄1̄11} and
_n_ = {̄2̄11} were never found developed on the right, but only on the
left of the plane of possible symmetry.

Moreover, it was frequently observed that the right-hand faces (110) and
(̄110) of the primary prismform _p_ were much more brilliant and truly
plane than those on the left hand, (1̄10) and (̄1̄10), which were
usually dull and often curved, as were also frequently the faces of the
left-hand forms _s_ and _n_. The right-hand distinguishing forms _m_ and
_o_, on the contrary, were generally most brilliant and gave admirable
reflections of the goniometer signal-slit.

[Illustration:

  FIG. 65.—Crystal of Calcium Dextro-Glycerate.
]

[Illustration:

  FIG. 66.—Stereographic Projection of Calcium Dextro-Glycerate.
]

The following table represents the results of the angular measurements,
twelve different well-formed individual crystals having been employed.
The angles marked with an asterisk were the important angles the mean
observed values of which were accepted as correct, being the best
measured angles, and which were therefore used as the basis of the
calculations.

            _Table of Interfacial Angles of Calcium Glycerate._

  Angle measured.      No. of           Limits.         Mean    Calculated.
                    measurements.                     observed.
 {_ap_  = 100 : 110            42  52° 32′ −  54° 16′   53° 29′      *
 {_pp_  = 110 : ̄110            20   72   7 −   73  33    73   4     73°  2′

 {_ac_  = 100 : 001            13   68  22 −   69  42    69   3      69   6
 {_cr′_ = 001 : ̄201            13   52   4 −   52  31    52  13      *
 {_r′a_ = ̄201 : ̄100            13   58  35 −   58  46    58  41      *

  _cm_  = 001 : 011            10   31  47 −   32  19    32   3      32   2

  _r′n_ = ̄201 : ̄2̄11             2   29  43 −   29  48    29  45      29  47

 {_ao_  = 100 : 111             7   53  59 −   54  10    54   3      53  54
 {_om_  = 111 : 011             7   18  20 −   18  35    18  26      18  29
 {_ma_  = 011 : ̄100            13  107  22 −  108  24   107  41     107  37
 {_an_  = ̄100 : ̄2̄11            11   62  32 −   63  44    63   6      63  10
 {_ns_  = ̄2̄11 : ̄1̄11             1          −             21  35      21  27
 {_sa_  = ̄1̄11 : 100             3   94  49 −   95  34    95  18      95  23

 {_po_  = 110 : 111             9   43  51 −   44  44    44  35      44  38
 {_oc_  = 111 : 001             9   32  56 −   33  15    33   7      33   7
 {_cs_  = 001 : ̄1̄11             7   41  32 −   43   2    42  10      42  17
 {_sp_  = ̄1̄11 : ̄1̄10             7   59  15 −   60  54    59  59      59  58
 {_pc_  = ̄1̄10 : 00̄1            14   77   2 −   78  21    77  42      77  45
 {_cp_  = 00̄1 : 110            16  101  39 −  103  36   102  16     102  15

 {_pm_  = 110 : 011             9   52  15 −   53  25    52  42      52  41
 {_mn_  = 011 : ̄2̄11             5   79   5 −   79  26    79  15      79  12
 {_np_  = ̄2̄11 : ̄1̄10             5   47  41 −   48  23    48   4      48   7

 {_pr′_ = 110 : ̄201            14  106  35 −  108  42   108   5     108   1
 {_r′p_ = ̄201 : ̄1̄10            26   70  54 −   73  32    71  55      71  59
 {_ps_  = ̄1̄10 : 1̄1̄1             5   66  42 −   67  17    67   4      67   5
 {_sr′_ = 1̄1̄1 : 20̄1             6   40  36 −   41  29    41   5      40  56

  _pm_  = ̄110 : 011             3   75   5 −   76  21    75  37      75  45

There is a moderately good cleavage parallel to the basal plane _c_ =
{001}.

The optical properties afford conclusive proof of the monoclinic nature
of the symmetry. The plane of the optic axes is perpendicular to the
possible symmetry plane, _b_ = {010}, and the first median line makes an
angle of 23° with the vertical axis c, emerging consequently nearly
normal to the basal plane _c_ = {001}, so that a section-plate parallel
to the _c_-faces, or a tabular crystal or cleavage plate parallel to
_c_, shows the optic axial rings and brushes well. The values of the
apparent optic axial angle in air, 2E, and of the true optic axial angle
within the crystal, 2V_{a}, the latter measured with the aid of a pair
of accurately ground section-plates perpendicular to the first and
second median lines and immersed in oil, are given in the next table.

                                        2E    2V_{a}
                   For lithium  light 51° 35′ 34° 56′
                    „   sodium    „   52° 30′ 35° 28′
                    „  thallium   „   53° 50′ 36° 16′

The intermediate refractive index β was found to be as under—

                      For red lithium light 1.4496
                       „  yellow sodium  „  1.4521
                       „  green thallium „  1.4545

The double refraction was also determined and found to be of positive
sign.

The optical properties of calcium dextro-glycerate thus confirm
absolutely the monoclinic nature of the symmetry, as regards the crystal
system. And it was conclusively demonstrated by the goniometrical part
of the investigation that the exterior symmetry was not such as agreed
with holohedral monoclinic symmetry, but with that of the sphenoidal
class, in which the only one of the two elements of monoclinic symmetry
(the plane of symmetry and the digonal axis of symmetry) in operation is
the digonal axis, thus leaving the two terminations of that axis, at
opposite sides, right and left, of the possible symmetry plane,
unsymmetrical. And this is precisely the symmetry which is
characteristic of an enantiomorphous optically active substance.

Unfortunately, the corresponding lævo-salt has not yet been obtained in
measurable crystals, but there can be no doubt that whenever such are
forthcoming they will display enantiomorphism in the precisely opposite
and complementary sense, the facial forms characteristic in this
dextro-salt of the right termination of the digonal axis being absent on
that side of the systematic symmetry plane but developed on the left
side instead, and _vice versa_, and that the two enantiomorphous forms
will together make up the whole of the faces required by the full
symmetry of the monoclinic system.

A concrete instance like this, worked out practically in the laboratory,
brings home the precise nature of this interesting relationship, between
crystallographic and molecular enantiomorphism on the one hand and
optical activity on the other hand, in a particularly clear and forcible
manner. It is hoped that this brief account of it will also consequently
have been of assistance to the reader, in more clearly appreciating the
main points of this chapter.




                              CHAPTER XII
EFFECT OF THE SYMMETRY OF CRYSTALS ON THE PASSAGE OF LIGHT THROUGH THEM.
                QUARTZ, CALCITE, AND GYPSUM AS EXAMPLES.


The action of transparent crystals on the rays of light which they
transmit is a subject not only of the deepest interest, but also of the
utmost importance. For it is immediately possible to detect a cubic
crystal, and to discriminate between two groups, optically uniaxial and
biaxial respectively, of the other six systems of symmetry, three
systems going to each group, by this means alone. For a cubic crystal is
singly refractive in all directions. A 60°-prism, for instance, cut from
a cube of rock-salt, for the purpose of obtaining the refractive index
of the mineral by the ordinary method of producing a spectrum and
arranging it for minimum deviation of the refracted rays, affords but a
single spectrum, or a single sharp image of the spectrometer slit when
the latter is fed by pure monochromatic light instead of ordinary white
light. This is true however the prism may have been cut, as regards its
orientation with respect to the natural crystal faces.

But a 60°-prism cut from a crystal belonging to the optically biaxial
group, composed of the rhombic, monoclinic, and triclinic systems of
symmetry, will always afford two images of the slit or two spectra,
corresponding to two indices of refraction; and, when the orientation of
the prism is arranged so that the refracting angle is bisected by a
principal plane of the ellipsoid which represents the optical
properties, and the refracting edge is parallel to one of the principal
axes of the optical ellipsoid, the prism, when arranged for minimum
deviation of the light rays, will at once afford two of the three
refractive indices, α, β, γ, corresponding to light vibrations along two
of the three principal axial directions of the ellipsoid. The two
indices which the prism affords will be (1) the one which corresponds to
vibrations parallel to the refracting edge, and (2) that which
corresponds to undulations perpendicular to the edge and to the
direction of transmission of the light through the prism (the third axis
of the ellipsoid). For the vibrations of the light in the two rays into
which the beam is divided on entering the crystal are both perpendicular
to the direction of transmission and to each other; the two images or
spectra produced owing to the double refraction, that is, owing to the
different velocities of the two mutually rectangularly vibrating rays,
thus correctly afford the means of determining two of the three
principal (axial) refractive indices.

Gypsum, the monoclinic hydrated sulphate of lime, CaSO_{4}.2H_{2}O,
already referred to in connection with the Mitscherlich experiment in
Chapter VII., is an excellent substance to employ for the demonstration
of this fact, by cutting and polishing a 60°-prism out of a clear
transparent crystal of the mineral as above described; and if a Nicol
prism be introduced in the path of the rays, one spectrum or
monochromatic image will be extinguished when the Nicol is arranged at
its 0° position, and the other when the Nicol is rotated 90° from this
position. This proves that the two rays affording the two refractive
indices are polarised in planes at right angles to each other, and,
moreover, enables us to verify that the planes in which the vibrations
of the two rays occur are actually parallel and perpendicular
respectively to the refracting edge of the prism. For the two
extinctions occur when the vibration plane of the Nicol is either
vertical, parallel to the prism edge, or horizontal, perpendicular
thereto.

If a second prism be cut complementarily to the first, that is, so that
the refracting edge is parallel to the third axis of the ellipsoid (the
direction of transmission through the first prism) and the bisecting
plane again parallel to one of the three axial planes of the ellipsoid,
such a prism will also yield two refracted images corresponding to two
indices; one of them, that particular image the vibrations of which are
parallel to the refracting edge, will correspond to that one of the
three principal indices which was not given by the first prism, while
the other one will afford a duplicate determination of one of the two
indices afforded by the first prism. Hence, a couple of such axially
orientated prisms of a rhombic, monoclinic, or triclinic crystal will
enable us to determine all three refractive indices, and one of them in
duplicate, which latter fact will enable us to check the accuracy of our
work.

If the 60°-prism be cut from a crystal of the uniaxial group, that is,
from a hexagonal, tetragonal, or trigonal crystal—quartz or calcite
being admirable examples of the latter and particularly suitable for
demonstration purposes—it will generally afford two spectra in the same
manner as a crystal of the three birefringent systems of lower symmetry.
But there is one special mode of cutting which results in the prism
exhibiting only a single spectrum, namely, when the hexagonal,
tetragonal, or trigonal axis of symmetry, which is also the unique
“optic axis” of the crystal along which there is no double refraction,
is arranged to be perpendicular to the bisecting plane of the 60°-prism.
For then the light is transmitted along this unique axial direction when
the prism is arranged for the minimum deviation of the refracted rays
out of their original path, and as it may vibrate in any direction
perpendicular thereto with equal velocity there is no separation into
two rays, that is, no double refraction, and thus only a single spectrum
is afforded by such a prism in white light, or a single image of the
slit in monochromatic light, and this latter will at once yield the
refractive index which is generally indicated conventionally by the
letter ω, corresponding to light vibrations perpendicular to the axis.

Spectroscopists take advantage of this interesting fact, when they
employ a train of quartz prisms so cut in order to explore the violet
and ultra-violet region of the spectrum; for quartz transmits many of
the ultra-violet rays which glass absorbs. Each prism gives only a
single image like glass, whereas if it were otherwise cut it would give
two spectra, which would so complicate matters as to render quartz
useless for the purpose.

When the prism of quartz or calcite, or of any hexagonal, tetragonal, or
trigonal substance, is cut so that the rays of light are transmitted
through it perpendicularly to the axis, and so that the refracting edge
is parallel to the axis, the light is broken up into two rays, one of
which is composed of light vibrating parallel to the edge and therefore
to the axis, and the other of light vibrating perpendicularly to the
axis. Such a prism consequently affords the two principal extreme
refractive indices of the crystal, ω and ε, the latter letter being
always assigned to the refractive index of a uniaxial crystal
corresponding to vibrations parallel to the axis.

A uniaxial crystal, one belonging to the hexagonal, tetragonal, or
trigonal systems, has thus two principal refractive indices, ω and ε,
while a biaxial crystal, one belonging to the rhombic, monoclinic, or
triclinic systems of symmetry, has three, α, β, γ, corresponding to
vibrations respectively parallel to the three rectangular axial
directions of the optical ellipsoid, which are also the crystallographic
axial directions in the case of a rhombic crystal. The index α is the
minimum, and γ the maximum refractive index,the β index being
intermediate; when the latter lies nearer to α in value, the crystal is
said to be a positive one, but when nearer to γ the crystal is
conventionally supposed to be negative. Similarly, when in a uniaxial
crystal ε is the greater, as it is in the case of quartz, the crystal is
termed positive, but if ω be the greater index, as happens in the case
of calcite, then the crystal is by convention considered negative.

Just as in the case of gypsum, which is a positive biaxial crystal (the
reason for the term biaxial will presently be more fully explained),
when the two spectra afforded by a prism of calcite or quartz cut to
afford both ε and ω are examined in plane polarised light, by
introducing a Nicol prism somewhere in the path of the light, the two
images corresponding respectively to ε and ω will be found to be
produced by light polarised in two planes at right angles to each other.
For when the Nicol is at its 0° position one will be extinguished, and
when it is at 90° the other will be quenched. At the 45° position of the
Nicol both images will be visible with their partial intensities, as
happens also in the cases of biaxial prisms.

This behaviour of 60°-prisms of crystals belonging to the seven
different styles of crystal architecture, as compared with a prism of
glass or other transparent non-crystalline substance, is extremely
instructive. For not only is the optical constant refractive index—the
measure of the power exhibited by the crystal of bending light,
corresponding to its effect in retarding by the nature of its internal
structure the velocity of the light vibrations—the most important of all
the optical constants, but also in the course of its determination we
learn more of the behaviour of crystals towards light than from any
other type of optical experiment.

[Illustration:

  FIG. 67.—Experiment to show Rectangular Polarisation of the two
    Spectra afforded by a 60°-Prism of a Doubly Refracting Crystal cut
    to afford two Indices of Refraction.
]

In Fig. 67 is shown a convenient mode of demonstrating the experiment
with the aid of the electric lantern and one of the large Nicol prisms
of the projection polariscope, already briefly described in Chapter VII.
in connection with the Mitscherlich experiment. The 60°-prism is
arranged on a small adjustable stand nearest the screen; then comes the
Nicol polarising prism of 2½ to 3 inches clear aperture, behind which is
the projecting lens, at the focus of which is placed the adjustable slit
on a separate stand. The slit is filled with light from the condenser of
the electric lantern, and in the lantern front a thick water cell is
arranged, in order to remove sufficient of the heat rays which accompany
the light beam to avoid damage to the balsam joint of the calcite Nicol.
When all the parts are properly arranged a sharp image of the slit
should first be thrown on the screen directly, in the temporary absence
of the 60°-prism, and then on replacing the latter at the proper angle
for minimum deviation, when the light traverses the prism parallel to
its third unused side, a spectrum or pair of spectra—according to the
position of the Nicol and to the nature of the 60°-prism as explained in
the foregoing discussion of the possibilities—will be projected on a
second screen (or the same one if movable) arranged at the proper angle
to receive the refracted rays.

If a single spectrum be afforded, which remains single on rotation of
the Nicol, the prism is of glass or of a uniaxial crystal cut so that
the light passes along the optic axis. If two spectra be shown when the
Nicol is arranged in the neighbourhood of its 45° position, the crystal
is a doubly refracting one, and if orientated so that the single optic
axis, if the crystal be uniaxial, is parallel to the refracting edge,
or, if the crystal be biaxial, so that the refracting edge is parallel
to one of the three principal axes of the optical ellipsoid and its
bisecting plane is parallel not only to this but also to a second
principal axis, then one spectrum, corresponding to one principal
refractive index, will extinguish when the Nicol is rotated to its 0°
position, and the other spectrum, corresponding to a second principal
refractive index, will be quenched on rotation of the Nicol to its 90°
position.

The separation of the two spectra on the screen depends on the amount of
the double refraction, and in the case of calcite this is exceptionally
large, so that the two spectra are widely separated on the screen. They
differ also considerably in dispersion. In the case of quartz the double
refraction is very small, and the spectral images of the slit are
consequently so close together as almost to touch one another. The pair
of spectra afforded by gypsum are similarly very close together, owing
also to weak double refraction. The amount of the double refraction is
measured by the difference between the uniaxial indices ε and ω, or that
between the minimum and maximum biaxial indices α and γ. The two spectra
given by quartz and calcite will correspond to ε and ω, and the greatest
separation of spectra occurs in the case of gypsum when the spectra are
those corresponding to α and γ, and not to α and β or β and γ.

It will now be useful and very helpful to examine more closely into the
nature of the beautiful mineral quartz, in order that a series of
interesting experiments may be described with it, which will assist
largely in rendering the optical characters of crystals clear to us.

_Quartz_, rock-crystal, although perhaps the commonest and best known of
all crystallised substances, the naturally occurring dioxide of silicon
SiO_{2}, is yet one of the most remarkable and fascinatingly
interesting. To begin with, as explained in the last chapter, quartz
belongs to one of the eleven enantiomorphous classes of lower than full
systematic symmetry, those which exhibit two mirror-image forms related
to one another like a pair of gloves. The particular class of the eleven
to which quartz belongs is the trapezohedral class of the trigonal
system, and two typical left-handed and right-handed crystals are shown
in Fig. 68 and Fig. 69 respectively.

There is one principal form which is common to both the hexagonal and
trigonal systems, namely, the hexagonal prism, and this is the chief
form exhibited by quartz crystals. They are terminated by an apparently
hexagonal pyramid, but which really consists of a pair of complementary
rhombohedra, which are purely trigonal forms; three upper faces of each
rhombohedron are developed at one end of the prism which may be regarded
as the upper, and the three lower faces of each of the two individual
rhombohedra likewise at the lower end of a fully developed doubly
terminated crystal. The rhombohedron is the characteristic form of the
trigonal system of crystal symmetry, the systematic crystallographic
axes being parallel to its edges. It is like a cube deformed by
extension or compression along a diagonal, which latter is arranged
vertically, and becomes the trigonal axis of symmetry (not a
crystallographic axis), as shown in Fig. 70.

[Illustration:

  FIG. 68.
]

[Illustration:

  FIG. 69.

  Left-handed and Right-handed Crystals of Quartz.
]

When two rhombohedra are equally developed, one being rotated with
respect to the other 60° round the vertical trigonal axis of symmetry,
they together resemble a hexagonal pyramid, and crystals of quartz thus
terminated at both ends are not uncommon, so that at first sight a
quartz crystal might be mistaken for a hexagonal prism doubly terminated
by the hexagonal pyramid, and the mineral considered, in error, to
belong to the hexagonal system.

[Illustration:

  FIG. 70.—The Rhombohedron and its Axes.
]

But one alternate set of three faces of the hexagonal pyramid at one
end, and the oppositely alternate set of three similar faces at the
other end, will usually be found to be much less brilliant (indeed often
quite dull) than the other alternate three, and very frequently also the
amount of development is markedly different, both facts indicating that
the terminal faces belong to two different but complementary
rhombohedral forms, and that the system of symmetry is the trigonal and
not hexagonal.

But there is much stronger evidence than this for trigonal symmetry. For
the little faces marked _s_ and _x_ on Figs. 68 and 69 are
characteristic of the trapezohedral class of the trigonal system, and it
will be observed that on one crystal, Fig. 68, these faces occupy and
modify a left-hand corner or solid angle on the crystal, while on the
other crystal, Fig. 69, they occupy and replace a right-hand solid
angle. Now, if a plate be cut out of the former crystal perpendicularly
to the axis of the hexagonal prism, that is, to the optic axis of the
trigonal uniaxial crystal, it will be found to rotate the plane of
polarisation to the left, the direction in which the small faces are
situated; while if a similar plate be cut out of the right-handed
crystal shown in Fig. 69, that is, one which has the small faces on the
right, it will be observed to rotate the plane of polarisation to the
right.

As quartz possesses the symmetry of the trigonal system and is thus
optically uniaxial, its optical properties are expressed, in common with
those of all trigonal, tetragonal, and hexagonal crystals, by an
ellipsoid of revolution, an ellipsoid the section of which perpendicular
to the principal axis—that of revolution, the maximum or minimum
diameter of the ellipsoid—is a circle. The optical properties are
consequently the same in all directions round this axis, which has
already been referred to by its common appellation of the “optic axis.”

The optic axis is identical in direction with the trigonal axis of
symmetry in the case of quartz or other trigonal crystal, and in the
cases of hexagonal and tetragonal crystals with the axes of hexagonal
and tetragonal symmetry, these three axes of specific symmetry being the
distinctive property of these three respective systems, which are thus
known in common as optically “uniaxial.”

Consequently, no double refraction is suffered by a ray transmitted
parallel to the optic axis, and the refractive index is equal in all
directions perpendicular to the optic axis, that is, for all rays
vibrating perpendicularly to the axis; hence the value of the refractive
index obtained along any such direction is one extreme value for the
whole crystal, and as already mentioned is distinguished by the letter
ω. The refractive index along the direction of the axis itself is the
other extreme value, and is labelled ε. It must be clearly appreciated,
however, that it is not the direction of transmission but that of
vibration perpendicular thereto, that is meant when it is said that, for
instance, the direction of the axis corresponds to the index ε. That is
to say, a ray the _vibrations_ of which occur parallel to the optic axis
of a uniaxial crystal is refracted to an amount which corresponds to the
refractive index ε, while a ray the vibrations of which occur
perpendicularly to the axis affords ω. The difference between ε and ω is
the measure of the double refraction of the crystal.

In the case of quartz ε is the greater, being 1.5534 for sodium light,
quartz being thus positive according to the convention already alluded
to; while ω is the smaller, namely, 1.5443. In the case of the other
widely distributed trigonally uniaxial mineral calcite, carbonate of
lime CaCO_{3}, the opposite is the case, ω being the greater, having the
value 1.6583 for sodium light, and ω the less, namely, 1.4864, calcite
being thus a negatively uniaxial substance. The amount of the double
refraction in the cases of the two minerals is very different, ε-ω for
quartz being 0.0091, and ω-ε for calcite being as much as 0.1719.
Calcite is indeed a mineral endowed with an especially large amount of
double refraction, a property which renders it so eminently suitable for
use in demonstrating the phenomenon, and for the construction of the
Nicol polarising prism, in which one of the two mutually perpendicularly
polarised rays, that which affords the index ω, is got rid of by total
reflection at a balsam joint, a large rhomb of calcite being cut in half
along a particular diagonal plane and the two halves cemented together
again with Canada balsam; the other ray, which affords ε (but not at its
minimum value), is transmitted as a beam of perfectly polarised light.

The result of this difference in the amount of the double refraction of
the two minerals quartz and calcite is very interesting as regards their
behaviour with polarised light. A thin plate of quartz, such as is often
found in the slices of rock sections employed for microscopic
investigation, of muscovite granite or quartz porphyry for instance, and
which is usually about one-fiftieth of a millimetre in thickness, shows
brilliant colours in a parallel beam of polarised light, the Nicol
prisms of the polarising microscope being crossed for the production of
the dark field before the introduction of the section-plate on the
stage. This is only true, however, when the plate has not been cut
perpendicularly to the axis, for such a thin plate thus cut does not
perceptibly affect the dark field, there being no double refraction of
rays transmitted along the axis, and the interference colours afforded
by crystal plates in polarised light being due to the interference of
the two rays produced by double refraction, one of which is retarded
behind the other so as to be in a different phase of vibration. Also,
the plate, even when cut obliquely, and best of all parallel, to the
axis, has to be rotated in its own plane (perpendicular to the optical
axis of the microscope), to the favourable position for the production
of the most brilliant colour. This especially favourable position is
halfway between (at 45° to) the positions at which darkness is afforded
by the plate. For on rotating the plate between the crossed Nicols it
becomes four times dark during a complete revolution, and at places
exactly 90° apart, known as the “extinction positions,” whenever, in
fact, that plane perpendicular to the plate which contains the optic
axis is parallel to the plane of polarisation of either the polarising
or analysing Nicol. At the intermediate 45° positions the maximum colour
is produced.

The colour owes its origin, as already mentioned, to the interference of
the two rays, corresponding to the two refractive indices, into which
the light is divided on entering the crystal in any direction except
along the axis. For one of the rays is retarded behind the other owing
to the difference in velocity which is expressed reciprocally
(inversely) by the refractive indices, and thus a difference of phase is
produced between the two light-wave motions, with the inevitable result
of interference when the vibrations have been reduced to the same plane
by the analyser; light of one particular wave-length is then
extinguished, and the plate therefore exhibits a tint in which the
complementary colour to that extinguished predominates. The light which
leaves the polarising Nicol is vibrating in one plane, but on reaching
the crystal this is resolved into two rays vibrating at right angles to
each other, and at 45° on each side of its previous direction of
vibration, supposing the crystal to be arranged for the production of
most brilliant colour. On reaching the analysing Nicol, the function of
which is to bring the two vibrations again into the same plane, these
two rays are each separately resolved back to the planes of vibration of
the two Nicols, and that pair (one from each ray) vibrating parallel to
the analysing Nicol are transmitted, while the other pair are
extinguished. The two former rays thus surviving, one individual ray of
the two having one refractive index and the other individual the other
index, are thus in a position to interfere; for they are composed of
vibrations in the same plane and of practically the same intensity, and
differ only in phase. Extinction occurs when this amounts to half a
wave-length, or an odd multiple of this, to which, however, requires to
be added half a wave difference of phase which is introduced by the
operation of the analyser. This explanation is a general one, applicable
to thin plates of crystals belonging to all the six systems of symmetry
other than the cubic. For plates of the latter, unless they are in an
abnormal condition of strain, do not polarise.

When we take a plate of calcite of the same small thickness as that of
the quartz in a rock section, thinner than a sheet of thin paper, we
find that the calcite does not polarise. So great is the retardation of
one of the two rays behind the other in calcite, that a plate
excessively thin is required in order that colour shall be observed. For
the colours of crystal plates under the polariscope, due to double
refraction, are subject to the same laws as the colours of thin films,
namely, that as the thickness increases—introducing more and more
retardation in the case of a crystal, just as in a thin film greater
length of path is introduced with increase of thickness—the various
tints of all the seven orders of Newton’s spectra are exhibited in turn,
each spectrum differing by one further wave-length of retardation, and
after the seventh the white of the higher orders (white light mixed with
colour, the latter thus appearing only as a faint tint) gives place to
true white light, colour being no longer perceptible. Hence with
calcite, owing to the extremely powerful double refraction, and
therefore very considerable retardation of the slower ray behind the
quicker, a plate a fiftieth of a millimetre only in thickness already
shows the white of the higher orders, that is, appears only very feebly
tinted with colour, and a plate of calcite very much thinner still is
required to show brilliant colours. A plate of calcite, therefore, cut
obliquely or parallel to the optic axis, of the thickness of a rock
section or thicker, simply appears four times dark and four times light
alternately, at positions 45° apart, as the section-plate is rotated in
its own plane perpendicular to the axis of the polariscope.

When a plate of either quartz or calcite one-fiftieth of an inch thick,
cut perpendicularly to the optic axis, is examined under the polariscope
or polarising microscope, the dark field is unaffected by its
introduction on the stage, remaining dark on a complete rotation of the
crystal plate in its own plane. Moreover, the calcite plate continues to
behave similarly however much the thickness is increased, the field
remaining dark. But when quartz is examined as regards the effect of
thickness an extraordinary thing happens. As the plate is thickened,
that is, as a series of plates of gradually increasing thickness are
successively placed on the stage, the dark field begins to brighten, and
eventually colour makes its appearance. Moreover, rotation of the plate
in its own plane—supposing the latter to be strictly perpendicular to
the axis of the polariscope and the plate itself to have been truly cut
perpendicularly to the optic axis of the quartz crystal—produces no
change whatever, the colour remaining the same and evenly distributed
over the plate, thus differing from the previous phenomena of
interference due to double refraction. When monochromatic light is
employed, yellow sodium light for instance, it is found that if the
plate be not too thick, say a millimetre in thickness, the dark field is
restored when the analyser is rotated in a particular direction, either
to the right or to the left, for a specific angle, which is 21° 42′ for
a plate of quartz one millimetre thick. Moreover, if the plate has been
cut from a crystal showing the distinctive trapezohedral-class faces s
and x on the right (Fig. 69) the analysing Nicol requires to be rotated
to the right; whereas if the plate has been cut from a crystal showing
these little determinative faces on the left (Fig. 68) the analyser has
to be rotated to the left in order to quench the light.

It is obvious, therefore, that the colours of these thicker plates of
quartz are due to the phenomenon of “optical activity.” The original
plane of polarisation of the light received from the polarising Nicol is
rotated by the quartz plate, and to an extent which is directly
proportional to the thickness. When white light is used a particular
colour is extinguished for each position of the analyser, and the
complementary colour therefore predominates in the tint actually
exhibited. Now the most intensely luminous part of the spectrum is about
wave-length 0.000550 millimetre in the yellow, and in the case of a
plate of quartz 7.5 millimetres thick this colour is extinguished when
the Nicols are crossed, while a plate of half this thickness, 3.75 mm.,
actually exhibits the colour under crossed Nicols and extinguishes it
under parallel Nicols. For the angle of rotation of the plane of
polarisation for light of this wave-length is 90° for a plate 3.75 mm.
thick, so that the analyser has to be turned through a right angle from
the crossed position, that is, placed parallel to the polariser, in
order to extinguish this colour. A plate of double the thickness, 7.5
mm., will require the analyser to be rotated through 180°, the angle of
rotation for this thickness of plate, in order to extinguish this yellow
ray. But 180° rotation simply brings the Nicol again to the crossed
position, so that no rotation is really necessary at all.

Now the complementary colour to the yellow of wave-length 0.000550 mm.
is the transition violet tint, the well-known “tint of passage” between
the brilliant red end of the first order spectrum of Newton and the deep
blue of the beginning of the second order. Hence, this violet tint is
afforded by a plate of 7.5 mm. thickness when the Nicols are crossed,
and by a plate of 3.75 mm. thickness when they are parallel. When,
therefore, these plates are examined respectively under crossed and
parallel Nicols, and the analysing Nicol is turned ever so little, the
tint changes remarkably rapidly into brilliant red or blue, according to
the direction of the rotation of the Nicol and the nature, whether right
or left-handed, of the quartz. Moreover, when two complementary plates
of each thickness are thus examined, one of each pair being cut from a
right-handed crystal and the other from a left-handed one, the colour
will be red in one case and blue in the other for the same direction of
rotation of the analyser.

A composite plate is frequently found very useful in work in connection
with optical rotation, and is known as a “biquartz,” two plates of
opposite rotations being cemented together by Canada balsam, the plane
of junction being made perpendicular to the plate so as to be almost
invisible when the plate is examined normally. When polarised light is
employed, the least rotation of the analyser from exact crossing with
the polariser, for which the violet transition tint is evenly produced
over the whole composite plate, causes the half on one side of the plane
of junction (appearing as a fine line) to turn red and the other half to
turn blue or green.

This, in essence, is the nature of the optical activity of quartz, and
the secondary effects derived from it influence all the optical
phenomena afforded by this interesting mineral. Owing to the fact that
quartz crystals are practically unendowed with any facility for
cleavage, the natural rhombohedral cleavage being very imperfectly
developed and rarely seen, it is possible to cut, grind, and polish
large plates of this beautiful, colourless, and limpidly transparent
mineral without a trace of flaw. Such quartz plates of large size,
adequate to fill the field of a large projection polariscope, the stage
aperture of which is nearly 2 inches in diameter, form magnificent
polarising objects for the projection on the screen of the effects
observed in polarised light. As many of the optical properties of
crystals may be illustrated with their aid, it is proposed in the next
two chapters to describe a few of the more interesting screen
experiments which can be performed with quartz, first (Chapter XIII.) in
convergent polarised light, and then (Chapter XIV.) in parallel
polarised light, and thus to illustrate the facts relating to the
connection between optical activity and the internal structure of
crystals in a manner which will at the same time be interesting and will
lead to their much clearer comprehension.

The experiments described are largely those with which the author
illustrated his lecture to the British Association for the Advancement
of Science during their 1909 meeting at Winnipeg.




                              CHAPTER XIII
EXPERIMENTS IN CONVERGENT POLARISED LIGHT WITH QUARTZ, AS AN EXAMPLE OF
      MIRROR-IMAGE SYMMETRY AND ITS ACCOMPANYING OPTICAL ACTIVITY.


It has already been shown that crystals are optically divisible into two
classes characterised respectively by single and by double refraction.
Singly refractive crystals belong exclusively to the system of highest
symmetry, the cubic. They afford obviously only one index of refraction,
which is generally symbolised by the Greek letter μ, the value of this
constant being the same for all directions throughout the crystal.
Crystals of the other six systems of symmetry are all doubly refractive.
Those of the trigonal, tetragonal, and hexagonal systems have been shown
in the last chapter to possess two refractive indices, a maximum and a
minimum, one represented by ε corresponding to light vibrating parallel
to the singular axis of the system, the trigonal, tetragonal, or
hexagonal axis of symmetry, and another signified by ω corresponding to
light vibrations perpendicular to that axis. For the properties are
identical in all directions around this axis, which is thus the optic
axis as well as the predominating crystallographic one. Such crystals
are consequently known as “uniaxial.” When ε is the larger refractive
index the crystal is positive, while if ω be the maximum the crystal is
said to be negative. It has been shown in the last chapter that quartz
belongs to the positive category, while calcite is negative. Along the
one direction of the optic axis these uniaxial crystals behave like
singly refractive crystals do in all directions.

Crystals of the rhombic, monoclinic, and triclinic systems of symmetry
have also a minimum refractive index, symbolised by α, and a maximum
index indicated by γ, corresponding to light vibrating parallel to two
directions at right angles to each other; the third direction
perpendicular to both these and normal to their plane does not afford an
index of refraction equal to either of these, however, as in the case of
a uniaxial crystal, but one of an intermediate value, for which the
second letter β of the Greek alphabet is reserved. Whether this value β
is nearer to the minimum α or to the maximum γ determines the
conventional optical sign of the crystal, whether positive or negative.
In the case of the rhombic system the three rectangular directions in
question are identical with the three rectangular crystallographic axes.
In the monoclinic system the single symmetry axis normal to the unique
plane of symmetry is identical in direction with either the α, β, or γ
optical direction, but in the triclinic system there are no coincidences
between the crystal axes and those of the optical ellipsoid. Along none
of these axial directions of the optical ellipsoid which can be imagined
to express graphically the refractive index—an ellipsoid known as the
optical “indicatrix,” and which has been shown by Fletcher to be a more
convenient mode of expressing the optical characters of a crystal than
the vibration-velocity ellipsoid of Fresnel—do the optical properties
resemble those of a uniaxial crystal along the optic axis, or of a cubic
singly refractive crystal, the crystal being doubly refractive along all
three axes.

But it is a remarkable fact, nevertheless, that there are two directions
in such a crystal along which the latter is apparently singly
refractive, and these two directions are known as the “optic axes,” and
the crystals of the three systems of lower symmetry are consequently
said to be “biaxial.” These two singular directions are symmetrical to
two of the three rectangular axes of the ellipsoid, those corresponding
to the extreme indices α and γ, in the plane containing which two axes
they lie, and they are perpendicular to the third β. For if we draw the
ellipse of which the minimum and maximum axes are represented in length
by α and γ, there will obviously be four symmetrical positions on the
curve where a line drawn to the centre of the ellipse would be equal to
the intermediate value β. If we join opposite pairs of these four points
by diameters (lines passing through the centre of the ellipse) we have
two directions each of which, together with the perpendicular direction
of the β axis, lies on a circular section of the ellipsoid, for all
radii from the centre lying in each of these sections are alike equal to
β. Consequently, light transmitted along the two directions in the
crystal normal (perpendicular) to these two circular sections will
suffer no apparent double refraction, the refractive index being the
same, namely β, and the velocity of vibration equal in all directions in
the crystal parallel to the two circular sections. Hence, we have two
directions in biaxial crystals in which the optical properties are
similar to those of uniaxial crystals along their singular optic axis.
But the optical properties along the two optic axes of a biaxial crystal
are advisedly stated to be “similar” to, and not “identical” with those
along the optic axis of a uniaxial crystal; for although they are
identical to all ordinary experimental tests, they are not quite so when
we come to ultimate details, which, however, are beyond the purview of
this book, but an account of which will be found in the author’s
“Crystallography and Practical Crystal Measurement” (Macmillan & Co.,
1911).

[Illustration:

  FIG. 71.—Projection Polariscope arranged for Convergent Light.
]

With these prefatory theoretical remarks, which are necessary in order
that the experiments now to be described should be understood, we may
proceed to consider a graduated series of experimental demonstrations
which it is hoped will render clear some of the more important features
of crystal structure which have been dealt with in previous chapters.
Our principal agent will be polarised light, that is, light which has
been reduced to vibration in a single plane by means of the well-known
Nicol’s prism. This latter is a rhomb of calcite which has been cut in
two parts along a specific diagonal direction, and the two parts of
which have been re-cemented together with Canada balsam, in such a
manner that one of the two rays, known as the “ordinary” and which
corresponds to the ω refractive index, into which the doubly refracting
calcite crystal divides the ordinary light which it receives from the
lantern or other source of light, is totally reflected at the layer of
balsam, while the other ray, known as the “extraordinary” and
corresponding to a refractive index of intermediate value between ω and
ε, and composed of vibrations at right angles to those of the totally
reflected ray, is alone transmitted, as a ray of plane polarised light.

We employ a pair of such Nicol prisms (a very large pair being shown in
Fig. 71), together with a convenient system of lenses for focussing
either the object-crystal or the phenomena displayed by it, as a
“polariscope,” which is the most powerful weapon of optical research on
crystals which has ever been invented. When the two prisms are arranged
so that the vibration planes of the polarised light which they would
singly transmit are parallel, we speak of them as “parallel Nicols,” and
light is transmitted unimpeded through the pair thus placed in
succession; but when one of them is rotated the light diminishes, until
when the vibration planes are at right angles no light escapes at all if
the Nicols are properly constructed, there being produced what is known
as the “dark field” of the “crossed Nicols.” For the plane polarised
light reaching the analyser from the polariser cannot get through the
former, its plane of possible light vibration being perpendicular to
that of the already polarised beam.

The phenomena exhibited by crystals in polarised light are of two kinds,
namely, those observed when a parallel (cylindrical) beam of fight is
passed through the crystal, and those exhibited when a converging
(conical) beam of fight is employed and concentrated on the crystal, the
centre of which should occupy the apex of the cone. The disposition of
apparatus in the former case of parallel light will be described in the
next chapter and illustrated in Fig. 79. The arrangement for convergent
light, as employed for projections on the screen, has already been
referred to in connection with the Mitscherlich experiment with gypsum,
and illustrated in Fig. 51 (page 92). The arrangement is shown again
here for convenience, in Fig. 71. The parts of the apparatus are briefly
as follows: (1) the electric lantern with self-adjusting Brockie-Pell or
Oliver arc lamp and a 4½ or 5–inch set of condensers; (2) the water
cell; (3) the polarising Nicol with a parallelising concave lens at its
divided-circle end; (4) a condensing lens; (5) the convergent system of
three lenses closely mounted in succession; (6) the crystal; (7) the
collecting system of three lenses equal and similar to the convergent
system; (8) the field lens; (9) the projection lens; and (10) the
analysing Nicol. The ten parts are separately mounted in the author’s
apparatus, which confers greater freedom in experimenting and more power
of varying the conditions; the converging and collecting lens systems,
however, are mounted in a separately adjustable manner on a common
standard, which carries in the centre complete goniometrical adjustments
for the crystal.

When we place on the stage of the polariscope, the Nicols being crossed,
a plate of a uniaxial crystal cut perpendicularly to the optic axis, and
subsequently a similar plate of a biaxial crystal cut perpendicularly to
that axis of the optical ellipsoid, either α or γ, which is the
bisectrix of the acute angle between the two optic axes, and use the
system of lenses which converges the light rays received from the
polarising Nicol prism on the crystal, as shown in Fig. 71, we observe
in the two cases quite different and very beautiful interference
phenomena, which at once distinguish a uniaxial from a biaxial crystal.
The two appearances are illustrated in Plate XIV., by Figs. 72, 73, and
74, which are reproductions of the author’s direct photographs. Fig. 72
shows the interference figure afforded by uniaxial calcite, which is the
same for all positions of the crystal plate when rotated in its own
plane by the rotation of the stage. Figs. 73 and 74 represent the
interference figures given by biaxial aragonite, the orthorhombic form
of carbonate of lime, calcite and aragonite being the two forms of this
substance, which has been shown in Chapter VII. to be dimorphous. The
effect shown in Fig. 73 is afforded when the line joining the two optic
axes is parallel to the plane of vibration of either of the crossed
Nicols, and the interference figure represented in Fig. 74 is given when
the stage and crystal (or the two Nicols simultaneously) are rotated
45°.

The uniaxial calcite figure (Fig. 72) consists of circular
spectrum-coloured rings resembling the well-known Newton’s rings, but
with a dark cross, fairly sharp near the centre but shading off towards
the margin of the field, marking the directions of the vibration planes
of the Nicols.

The biaxial aragonite figures (Figs. 73 and 74) show two series of rings
surrounding the two optic axes and thus locating the positions of their
emergence, equidistant from the centre of the field, where the bisectrix
emerges. They are not circular, but are curves known as lemniscates,
which are complete rings nearest to the two optic axes, but soon pass
into figure-of-eight loops, and eventually into ellipse-like lemniscates
enveloping both optic axes, and more and more approaching circles in
their curvature as the margin of the field is approached. Moreover, when
the direction of the fine joining the two optic axes is parallel to the
vibration plane of either of the Nicols, as was the case when Fig. 73
was produced and photographed, a black rectangular cross is seen, one
bar, which is much the sharper one, passing through the optic axes and
the other lying between them at right angles to the first bar, the
centre of the cross being in the middle of the field.

[Illustration:

  _PLATE XIV._

  FIG. 72.—Crystal Plate cut perpendicularly to the Axis.

  Uniaxial Interference Figure afforded by Calcite (Trigonal) in
    Convergent Polarized Light, with Crossed Nicols.
]

[Illustration:

  FIG. 73.—Crystal Plate cut perpendicularly to the Bisectrix of the
    Acute Optic Axial Angle.

  Biaxial Interference Figure afforded by Aragonite (Rhombic) in
    Convergent Polarised Light, with Nicols crossed and parallel to the
    Vibration Directions of the Crystal.
]

[Illustration:

  FIG. 74.—The same Plate as for the previous Figure.

  The same when the two Nicols have been rotated in the same direction
    for 45°, still remaining crossed.

  CHARACTERISTIC UNIAXIAL AND BIAXIAL INTERFERENCE FIGURES IN CONVERGENT
    POLARISED LIGHT.

  (Reproductions of direct Photographs by the author.)
]

On rotating the crystal plate in its own plane, while no change occurs
with the calcite, the aragonite figure changes as regards the black
cross, which breaks up into hyperbolic curves currently spoken of as
“brushes,” until when the plate has been rotated 45° the appearance is
that shown in Fig. 74, the eye being supposed to have followed the
rotation. Or, keeping the eye still, the effect shown in Fig. 74 is
equally produced by the simultaneous rotation of both Nicols for 45°.
The vertices of the hyperbolæ now mark the positions of the optic axes,
and the angle between them is the apparent angle of the optic axes as
seen in air, which is considerably different from the true angle between
the optic axes within the crystal, owing to the very different
refraction of light in air and in the crystal substance.

Now some crystals exhibit a very different optic axial angle at
different temperatures, and one of the most beautiful experiments which
have ever been performed is the Mitscherlich experiment with gypsum,
which has already been described in Chapter VII. in connection with the
work of Mitscherlich, and illustrated in Plate XII., Figs. 52 to 55.
Other substances, on the other hand, show a marked change of optic axial
angle as the wave-length of the light is changed, and such a case has
already been described in Chapter VIII. and illustrated in Plate XIII.,
Fig. 58. The figure afforded by such a substance in ordinary white light
is, however, a complicated one, quite different from the normal one of
Fig. 73 afforded by aragonite, as will be clear on reference to the
interference figure shown at _f_ in Fig. 58, which represents the figure
given by ethyl triphenyl pyrrholone in white light.

In order to understand such biaxial interference figures thoroughly,
they should be studied in monochromatic light, when one obtains a clear
and sharp figure consisting of black curves as well as the cross or
brushes, and very sharp vertices to the brushes when the crystal is
arranged as in Fig. 74. The optic axial angle can then be measured for
each important wave-length of light in turn, and the variation for
wave-length followed throughout the whole spectrum. For this purpose it
is very convenient to have a source of monochromatic light of any or
every wave-length always at hand, and the author some years ago devised
a spectroscopic monochromatic illuminator,[17] for use with any
observing instrument, and which is particularly convenient for use with
the polariscopical goniometer which is employed in practice for the
measurement of optic axial angles. It is shown, along with the latter
instrument, in Fig. 75. The spectroscope has a single but very large
prism of heavy but colourless flint glass, and the spectrum produced—the
electric lantern being the source of light, its rays being concentrated
on the slit—is filtered through a second slit at the other end of the
spectroscope, where the detachable eyepiece is situated when the
instrument is used as an ordinary spectroscope, and for the calibration
(with the Fraunhofer solar lines) of the circle on which the prism is
mounted. The escaping narrow slit of monochromatic light includes only
the 250th part of the spectrum, so is monochromatic in a high sense of
the word. It impinges on a little ground glass diffuser carried in a
very short tube in front of this exit slit, and the optic axial angle
polariscope is brought up almost into contact with the ground glass, and
is thus supplied with an even field of pure monochromatic light. With
this apparatus it is easy to observe the exact crossing wave-length in
all cases of crossed-axial-plane dispersion such as that illustrated in
Fig. 58; for the reading of the graduated circle on which the prism is
mounted, and which is rotated in order to cause monochromatic light of
the different wave-lengths in turn to stream through the exit slit,
affords the exact wave-length with the aid of the calibration curve once
for all prepared. This calibration of the graduations is readily carried
out by using sunlight, and determining the readings corresponding to the
adjustment of the principal Fraunhofer lines in the middle of the exit
slit.

[Illustration:

  FIG. 75.—Optic Axial Angle Goniometer and Spectroscopic Monochromatic
    Illuminator.
]

Having thus rendered clear the nature of ordinary interference figures
afforded by crystals of the two types, uniaxial and biaxial, in
convergent polarised light, we may pass on to see what happens when we
take a number of plates of quartz of different thicknesses, cut
perpendicularly to the optic axis in all cases, instead of a plate of
calcite. We will examine first a fine pair of hexagonal quartz plates so
cut, each 1 millimetre thick exactly, and about 2 inches in diameter.
One was cut from a right-handed hexagonal prism, and the other from a
similar left-handed one.

Employing the lantern projection polariscope shown in Fig. 71, arranged
for convergent light just as for the Mitscherlich experiment, and with
the Nicols crossed, we will now see what happens when each of these
plates in turn is placed at the focus of the light rays, between the two
convergent systems of lenses. On the screen we observe in each case a
somewhat similar interference figure to that given by calcite, a black
cross and rainbow coloured circular rings, the smallest ring, however,
being very large relatively to the innermost ring given by calcite, and
the other rings being also further separated from each other. Moreover,
the black cross appears broadened out, this spreading of both rings and
cross being due to the thinness of the plate combined with the low
double refraction of quartz. Further, the right-handed and left-handed
plates both afford apparently identical figures. In order to obtain a
sharp figure like that of calcite we require to add a fourth lens, kept
in reserve for such cases, to each of the two similar convergent lens
systems, one on each side of the crystal plate, in order to increase the
convergence of the light rays. The figure then obtained with one of the
two plates is reproduced in Fig. 76, Plate XV.

[Illustration:

  _PLATE XV._

  FIG. 76.

  Interference Figure afforded by a Quartz Plate, 1 Millimetre thick, in
    strongly Convergent Polarised Light.
]

[Illustration:

  FIG. 77.

  Interference Figure afforded by a Quartz Plate, 3·75 Millimetres
    thick, in moderately Convergent Polarised Light.
]

[Illustration:

  FIG. 78.

  Interference Figure (Airy’s Spirals) afforded by two superposed Quartz
    Plates, 3·75 Millimetres thick, one of Right-handed Quartz and the
    other of Left-handed Quartz, in moderately Convergent Polarised
    Light.

  INTERFERENCE FIGURES IN CONVERGENT POLARISED LIGHT AFFORDED BY QUARTZ
    PLATES CUT PERPENDICULARLY TO THE AXIS.
]

Let us now observe, however, what occurs when a thicker plate of quartz
is used. Taking one of 7.5 mm. thickness, and placing it in the focus of
the converging rays, after removing the two extra lenses, we see on the
screen quite a different effect, an attempt to reproduce which
photographically in black and white is made in Fig. 77 on the same Plate
XV. The rings are closer together (using the same degree of
convergence), and the innermost is smaller; moreover, within it all
signs of the central part of the black cross have disappeared, and
instead a brilliant violet colour is shown, which alters to bright red
of the first order spectrum with the least rotation of the analysing
Nicol in one direction from its crossed position with respect to the
polarising Nicol, while if the rotation be in the opposite direction the
deep blue of Newton’s second order is produced. The arms of the cross,
however, appear towards the margin of the field. The violet colour shown
for the exact position of crossing of the Nicols is the tint of passage
between the first and second orders of Newton’s spectra, and this
illumination of the central part of the interference figure is obviously
the effect of the optical activity of quartz, for the tint is the same
as is produced with the plate in ordinary parallel plane polarised
fight, and is, in fact, due to the central axial rays of the convergent
cone being practically parallel.

On rotating the analysing Nicol for a few more degrees to the right we
observe that the innermost ring widens out and that the red passes into
orange and yellow, the quartz plate being a right-handed one. But when a
similar plate cut from a left-handed quartz crystal is used instead, the
inner ring closes up somewhat for the same rotation of the analyser,
moving inwards instead of outwards, and the blue colour given with the
first slight rotation passes into green and yellow as the rotation is
continued. Moreover, the circular character of the rings is altered, and
so much so that when the rotation has proceeded as far as 45° the shape
of the rings has changed almost to a square. These alterations in the
interference figure are characteristic of the two varieties of quartz
crystals. A useful rule to remember is, that for a right-handed crystal
rotation of the analyser to the right causes the colours to appear in
the order of their refrangibility, namely, the least refrangible red
first, then orange, yellow, green, blue and violet in their order; while
for a left-handed crystal the converse is true when the direction of
rotation of the analyser is the same, that is, to the right, clockwise;
obviously also the colours appear in the opposite order when the
rotation of the analyser is to the left.

It will now prove of interest to examine the effects produced by two
plates of opposite varieties of quartz of half this thickness, namely,
3.75 mm. The phenomena are very similar to those just described, but the
rings are a little wider, and the larger area within the innermost ring
is now filled with yellow light instead of violet, when the analyser is
exactly crossed to the polariser. It passes into a bright green when the
analyser is rotated slightly on one side, and into orange when the Nicol
is rotated in the reverse direction. But the most interesting thing of
all is to observe what occurs when these two plates of 3.75 mm.
thickness, one of right-handed quartz and the other of left-handed, are
superposed and placed in contact together as one plate, of double the
thickness, 7.5 mm., at the convergent focus. A beautiful spiral figure
is produced on the screen, composed of the celebrated “Airy’s spirals”
as if the black cross were being reproduced in the central part, but
with each of its bars distorted into the shape of the letter S, as shown
in Fig. 78 at the foot of Plate XV. The contrary effects of the two
opposing rotations are thus extraordinarily indicated visually in the
interference figure afforded by the composite plate.

Now, it is of great practical interest that certain quartz crystals are
found in nature which show Airy’s spirals directly, on cutting a plate
7.5 mm. thick or thereabouts, perpendicular to the optic axis. For
instance, one in the author’s collection of quartzes, a single plate of
an apparently homogeneous and perfectly limpid crystal, shows the
spirals exceedingly well and clearly defined. As a matter of fact, it is
a twin, a right and a left-handed crystal being twinned together with an
invisible plane of composition, which is only revealed on examining the
crystal in polarised light, as will be demonstrated in the next chapter
by the use of parallel polarised light. The fact of such a plate of
quartz affording Airy’s spirals in convergent polarised light is,
however, of itself an excellent proof of the twinning of two crystal
individuals of the opposite varieties.

Now the very shape of these spiral figures suggests screw action of the
molecular structure of the crystals on the waves of light passing
through them, and moreover, of the action of two screws of opposite
directions of winding, one clockwise and the other anti-clockwise, thus
remarkably confirming the supposition that the point-systems of the
structure of the right and left-handed varieties of quartz are of a
helical nature and respectively of opposite modes of winding.

Another experiment, devised by Reusch, which still further enhances the
probability that this supposition as to the structure of quartz crystals
is correct, may next be introduced. A thin film of biaxial mica has been
cut into twenty-four narrow strips, which have been laid over each other
at angles of 60°, so that a screw-shaped pile has been formed of the
central overlapping parts, consisting of four complete rotations; that
is, there are four repetitions of the “pitch” of the screw, each
composed of six films. On placing this composite plate of mica at the
convergent focus of the lantern polariscope, so that the overhanging
ends of any four identically superposed strips occupy the focus, the
ordinary biaxial interference figure of mica—two sets of rings and
hyperbolic brushes, very much like Fig. 52, Plate XII.—is observed on
the screen. But when the plate is moved so that the central part comes
into the focus, where all the twenty-four films overlap in their six
different orientations 60° apart, and so that all the light rays have to
traverse the whole helical pile of the twenty-four films, a uniaxial
figure exactly like that of quartz is produced, namely, one composed of
circular rings, with a black cross only visible, however, at the
marginal part, and with the inner ring filled with brightly coloured
light. Moreover, on slightly rotating the analysing Nicol the innermost
ring moves outwards or inwards and the colour changes to blue or red,
according to the direction in which the helix had been wound, in exact
accordance with the rule stated above for quartz.

If now a second such helical pile of mica films, but one for which the
opposite manner of winding has been adopted, anti-clockwise if the first
had been clockwise, be examined at the convergent focus, precisely the
same appearance will be observed with crossed Nicols, but the opposite
changes will occur on rotating the analyser. Finally, to complete the
interesting proof of the helical nature of quartz crystals, when these
two oppositely wound composite mica plates are superposed—each being
marked carefully to indicate the direction of the helix and the proper
mode of superposition in order to effect precise oppositeness of
arrangement, mirror-image symmetry, in fact, about the plane of
contact—and placed in the convergent beam near its focus, there is at
once seen on the screen a magnificent display of Airy’s spirals, as
perfect as those afforded by the fine natural twin last experimented
with. Hence, there can be no doubt whatever that the remarkable optical
behaviour of quartz is due to its point-system being of a helical
nature, a right or a left-handed screw structure being apparently
produced in nature with equal facility. The circumstances of environment
during the formation of the crystal probably determine which variety
shall be produced, and when the nature of the environment becomes
changed during the operation of formation either twins are produced of
the two varieties, or separate individual crystals.

This may well conclude our experiments in convergent polarised light,
which—including the beautiful Mitscherlich experiment described in
Chapter VII., of exhibiting the crossing of the optic axial plane in the
case of gypsum, and the production of all the types of interference
figures in succession, as the crystal becomes warmed by the heat rays
accompanying the beam of convergent light—will have introduced the
reader to a typical series of such experiments, and such as were
actually exhibited by the author to the British Association at Winnipeg.
We may pass, therefore, in the next chapter to the consideration of an
equally interesting series in which a parallel beam of polarised light
will be used, which will still further elucidate the internal structure
in the especially instructive case of quartz crystals, and that of
crystals in general.




                              CHAPTER XIV
EXPERIMENTS WITH QUARTZ AND GYPSUM IN PARALLEL POLARISED LIGHT. GENERAL
             CONCLUSIONS FROM THE EXPERIMENTS WITH QUARTZ.


In order to rearrange the projection polariscope for experiments in
parallel light, we simply remove the three lenses on separate stands
(Fig. 71), and the convergent systems of lenses on their special
adjustable stand with goniometrical crystal holder, from between the two
Nicol prisms, and replace them by two other separately mounted lenses,
acting together as an achromatic projecting objective, and a rotatable
object stage. The whole arrangement as thus altered for experiments in
parallel polarised light is shown in position in Fig. 79. The change is
readily made, a gap in the plinth-bed guides near the analysing Nicol
enabling it to be effected without removing either of the prisms, the
analyser being simply drawn along a few inches nearer the end in order
to expose the changing gap. The pair of lenses consists of a
plano-convex lens of 5 inches focus and 2¼ inches diameter, and another
plano-convex lens of 8½ inches focus and 2 inches aperture, with their
convex faces turned towards each other. Together they produce on the
screen an excellent image of the object on the stage, and the size of
the image can be varied at will by regulating the relative positions of
the two lenses with respect to each other and to the object stage. If
found more suitable for the particular screen distance available, the
5–inch lens may be replaced by a 6–inch lens also provided as an
alternative.

[Illustration:

  FIG. 79.—Projection Polariscope arranged for Parallel Light.
]

When the analysing Nicol is arranged with its vibration direction
parallel to that of the polariser, we obtain bright light on the screen
on actuating the electric lantern, and the image of an object on the
stage can thus be projected on the screen on a bright ground. But when
the analyser is crossed to the polariser, that is, rotated to the
position 90° from this parallel position, the two planes of vibration of
the Nicols being then at right angles, the screen is quite dark. Before
continuing in this dark field our experimental study of quartz, which is
obviously a type of the more exceptionally behaving substances owing to
its special structure, it will be wise to examine a more ordinary kind
of crystalline substance. For this purpose gypsum—better known in
optical work as selenite, hydrated sulphate of lime, CaSO_{4}.2H_{2}O,
crystallising in beautifully transparent and often large crystals
belonging to the monoclinic system, a typical one of which has been
illustrated in Fig. 9 (page 14), and which we have already referred to
in connection with the Mitscherlich experiment described in Chapter
VII.—is especially suitable, on account of its clear and colourless
transparency, the large size of crystals available, and the brilliancy
of the polarisation colours which they afford when adequately thin. A
very perfect cleavage being developed parallel to the symmetry plane,
the clinopinakoid {010}, such thin films, of even thickness throughout,
can be readily prepared.

Such a very thin cleavage plate, about 1½ inches in its longest
dimension, is mounted with Canada balsam between a pair of circular
glass plates 1⅞ inches in diameter, the standard size of object plates
for the projection polariscope; the double plate is then supported in a
mahogany frame also of the standard size—4 by 2¼ inches, with clear
aperture of 1⅝ inches diameter and supporting rabbet for the plate 1⅞ to
2 inches diameter—on the rotating stage by a pair of spring clips. The
Nicols being arranged with their vibration directions parallel, in order
to permit light to travel to the screen, and the lenses being arranged
properly for a sharply focussed picture of suitable size, the outline of
the crystal plate will be seen on the screen, and the whole area of the
crystal will either at once appear coloured, or will do so on more or
less rotation of the stage carrying the crystal, which rotates the
latter in its own plane. The crystal outline is of the character shown
in Fig. 80, which also gives the positions of the crystal axes a and c,
and a simple stereographic projection of the faces of the crystal, from
which the nature of the faces bounding the section-plate will be clear.

[Illustration:

  FIG. 80.—Section of Gypsum Crystal showing the Extinction Directions.
]

On rotating the Nicol analyser the colours change, and appear at their
maximum brilliancy when the field is dark and the Nicols crossed.
Leaving the analyser crossed to the polariser, and rotating the stage
and therefore the crystal, the colours again change, and at certain
positions 90° apart during the rotation, marked by the two strong lines
in Fig. 80, they disappear altogether, and the crystal becomes dark like
the rest of the field, while the positions of maximum brilliancy of
colour are found to be situated at the 45°-positions intermediate
between these positions of “extinction.” When the quenching occurs the
vibration planes of the two rays, travelling by virtue of double
refraction through the crystal, are parallel to the planes of vibration
of the rays transmitted through the two Nicols, and the fact is a very
important one, enabling us to determine the directions of light
vibration in the crystal. In the case of our gypsum plate, the cleavage
of gypsum being parallel to the unique plane of symmetry of the
monoclinic crystal, these two positions are the directions of the two
axes of the optical ellipsoid which lie in the symmetry plane, and they
correspond to the vibration directions of rays affording the refractive
indices α and γ. The direction corresponding to γ is that of the “first
median line,” the bisectrix of the acute angle between the optic axes;
while α corresponds to the obtuse bisectrix or “second median line.”
These directions are clearly marked by the strong lines in Fig. 80. The
third axis of the optical ellipsoid is obviously perpendicular to the
plate and to the symmetry plane, and corresponds to the intermediate
refractive index β. Thus this simple observation of the extinction
directions in such a case as gypsum enables us at once to fix completely
the orientation of the optical ellipsoid, a fundamental optical
determination.

A second thin plate of gypsum may next be examined, similarly prepared
and mounted. It is clearly a composite one, being composed of a pair of
twins. For when placed on the stage in the dark field of the crossed
Nicols, and rotated to the position for maximum brilliancy of colour, it
shows different colours in the two halves, as indicated by different
shading in Fig. 81. If, however, the analysing Nicol prism be withdrawn
from the plinth-bed and removed altogether the crystal appears in its
natural colourless condition as a single one, with no indication
whatever of any line of division.

[Illustration:

  FIG. 81.—Twin of Gypsum as seen in Parallel Polarised Light.
]

Some exceedingly brilliant polarisation effects are afforded by a number
of objects exhibited by the author in his lecture at Winnipeg, composed
of selenite (gypsum) twins and triplets, some arranged to cross one
another like the mica films of Reusch described in the last chapter, but
only for a single rotation, three twin strips going to a rotation, at
angular distances of 120°; others are arranged in geometrical patterns,
and in circles overlapping one another, and the whole series afford the
most gorgeous and variegated display of colour imaginable, the colours,
moreover, altering either on rotation of the stage or of the analysing
Nicol, and thus passing through every tint conceivable.

Having thus demonstrated the usual effect afforded by a doubly
refracting crystal plate in parallel polarised light, we may next
illustrate two special cases, which will lead us up to the case of
quartz once more. The first relates to a crystal belonging to the cubic
system, which is theoretically singly refractive or “isotropic”; the
second concerns a plate of a uniaxial crystal cut perpendicularly to the
optic axis, the unique direction of single refraction of such a crystal.
A plate of fluorspar affords a good example of the first case. When
placed on the stage of the polariscope it shows no colour at all in
polarised light, whatever be the position of the two Nicols with respect
to each other, and the field remains dark when they are crossed, the
crystal, in fact, behaving just like so much glass.

A word of caution, however, is here necessary, for natural mineral
crystals are not infrequently formed under conditions of considerable
strain, at high temperatures or under great pressure, as in the case of
the diamond for instance. So that we must be careful to choose a normal
and well-formed crystal of fluorspar for our experiment. This point may
be well illustrated by placing on the stage a thick circular plate of
glass, an inch or more in diameter, which has been purposely heated and
then suddenly cooled in order to evoke such a condition of strain.
Crossing the Nicols so as to obtain the dark field, there is at once
produced on the screen a black cross and circular concentric
spectrum-coloured rings, resembling with wonderful simulation the
interference figure, shown in Fig. 72, Plate XIV., afforded by calcite
or other uniaxial crystal in convergent polarised light. Artificial
double refraction has been produced in the glass by the strained
conditions, in a fashion concentrically symmetrical to the axis of the
cylinder, an interference figure being afforded symmetrical about the
axis of the cylinder as if it were an optic axis.

The diamond crystallises in the cubic system, in octahedra, hexakis
octahedra, or hexakis tetrahedra, and should, therefore, theoretically
be without effect on polarised light. Yet it is rare to find a diamond
which does not show more or less colour in the dark field, owing to the
condition of strain in which it exists. It is notorious that the strain
is occasionally so great that a diamond explodes into powder shortly
after removal from its enveloping matrix of blue clay. The author, by
the great kindness of Sir William Crookes, was enabled to show on the
screen, both in a lecture at the Royal Society and in the Evening
Discourse to the British Association at Winnipeg, the images of ten
magnificent large diamonds, natural, perfectly formed crystals uncut and
unspoilt by the lapidary. They were mounted between two circular glass
plates of the usual 1⅞ inches diameter, the diamonds being attached by
balsam to one of them; each plate was held in a mahogany frame of 1⅝
inches circular aperture, the two frames being then attached face to
face to form a single one, an enclosing cell, which could be placed on
the rotating stage as an object-slide for the projection polariscope.
The appearance of the diamonds on the screen in ordinary light is
reproduced in Fig. 82, Plate XVI., as well as is possible without their
natural colour, for while several of them are brilliantly colourless,
others are tinted, one being a bright green diamond. On producing the
dark field by crossing the analysing Nicol with respect to the
polariser, the darkness was dispelled by brilliant polarisation colours,
at once revealing the diamonds and outlining them clearly against the
dark background. On rotating the analyser the colours changed in the
usual manner of polarising objects, and bright colours were shown by all
the diamonds even when the Nicols were parallel.

It is obvious, then, that both a transparent non-crystalline substance
such as glass, and a cubic crystal, must be free from strain in order
that it shall exhibit no colour in polarised light and, indeed, no
polarisation effects whatever, and behave as an isotropic substance.

[Illustration:

  _PLATE XVI._

  FIG. 82.—Ten Diamonds exhibiting Natural Faces, mounted for the
    Lantern Polariscope, to show Polarisation Colours due to Internal
    Strain.
]

[Illustration:

  FIG. 121.—Doubly Refracting Liquid Crystals of Cholesteryl Acetate,
    projected on the Screen in the Act of Growth (see p. 281).

  TWO FIGURES ILLUSTRATING THE HARDEST (DIAMOND) AND THE SOFTEST (LIQUID
    CRYSTALS) OF CRYSTALS.
]

The second special case to which attention may be called, that of a
plate of an ordinary uniaxial crystal such as calcite, cut
perpendicularly to the optic axis, is also obviously subject to the same
proviso, that the crystal must be free from strain in order to exhibit
the normal phenomena. Such a perfectly normal plate remains quite
obscure in the dark field in parallel light, producing neither colour
nor interference figure, even on rotation of the object stage with the
crystal, in its own plane. For the light traverses the crystal along the
optic axis, the axis of single refraction, and the vibrations occur with
equal velocity in all directions perpendicular to it. Hence there is no
division into two rays, one retarded behind the other on account of less
velocity of vibration, and therefore no interference colour.

And now this leads us back to quartz, for this mineral is also uniaxial,
and we will investigate in the same manner in parallel polarised light
the plates of the mineral cut perpendicularly to the optic axis, which
have already been referred to in connection with the experiments
concerning the interference figures produced in convergent polarised
light. Suppose we take first the large plate of quartz 7.5 mm. thick and
over 2 inches in diameter. Placing it on the stage—instead of finding
the dark field to be unaffected by the introduction of the plate, and to
remain so on rotation of the latter in its own plane, as should
theoretically be the case if quartz were a normal uniaxial crystal, and
as calcite has been actually shown to do—we observe that it polarises in
brilliant colour, the whole hexagonal outline of the plate, clearly
focussed on the screen, being filled with an evenly brilliant violet
tint, the tint of passage, just as the central part of the interference
figure, within the innermost ring, had been coloured in the convergent
light experiment with the same plate. The colour changes with the
slightest rotation of either of the Nicols, passing into red for one
direction of rotation and into blue and green when the Nicol is rotated
in the other direction. The tint also alters when the section-plate is
rotated about its vertical diameter, by rotating the upper adjustable
part of the supporting column of the stage within its outer fixed
tubular column; this latter change is equivalent to a thickening of the
plate, the light beam having to traverse a longer path through the
quartz during such oblique setting of the plate.

This colour is due to the same fact which produced colour in the central
part of the interference figure, namely, the optical activity of quartz,
the fact that the plane of vibration of a beam of plane polarised light
transmitted along the axis of quartz is rotated to the right hand or to
the left. The amount of this rotation is precisely equal, although
opposite in direction, for the two varieties of quartz, but the rotation
varies very considerably for different rays of the spectrum. It also
varies directly proportionally to the thickness of the plate. A plate
one millimetre thick cut perpendicularly to the axis rotates the plane
of polarisation for red hydrogen light (C of the spectrum) to the extent
of 17° 19′, for yellow D sodium light 21° 42′, and for greenish-blue F
hydrogen light 32° 46′. The rotation is a maximum for plates
perpendicular to the axis, and the effect is inappreciable in directions
at right angles thereto. It is clearly due to the oppositely spiral
winding of the regular-point-system of the crystal structure, round the
direction of the optic axis, the trigonal axis of symmetry of the
crystal, a structure which we have proved to be characteristic of quartz
by the beautiful experiments with the helical piles of mica plates,
absolutely reproducing the polarisation effects with quartz, as
described in the last chapter.

The opposite optical rotation of the two varieties of quartz can be well
shown by constructing a “biquartz.” Two plates of equal thickness,
preferably either 7.5 mm. or 3.75 mm., are cut, one from a right-handed
and the other from a left-handed crystal, each exactly perpendicular to
the optic axis. The two edge-surfaces to be subsequently joined together
are also cut, ground and polished as true planes perpendicular to the
plate surfaces, and the two plates are then cemented together with
Canada balsam by these two prepared edge-surfaces, taking care that the
broad plate-surfaces of the two halves are absolutely continuous as if
the whole were a single parallel-surfaced plate of quartz. Such a
composite plate or “biquartz,” is one of the most useful aids to the
study of optical activity, being much used for enhancing the
sensitiveness of the determination of the angle of rotation.

When the image of such a 7.5 mm. biquartz, mounted in the usual mahogany
frame and placed on the object stage of the projection polariscope, is
thrown on the screen—the Nicols being crossed for production of the dark
field, and the stage and crystal plate being strictly perpendicular to
the parallel beam of polarised light—the whole of the screen covered by
the image of the plate appears uniformly coloured with the violet tint
of passage. But the moment the analysing Nicol is rotated for a very few
degrees, one-half turns red and the other blue and then green. If the
Nicol be turned back again to the crossing position with the polariser,
and then rotated further in the opposite direction to the former
rotation, the appearances on the two sides of the sharply focussed fine
line of demarcation between the two halves are inverted, the side which
formerly turned red now becoming green, and _vice versa_. The two
varieties of quartz are thus oppositely affected, and it will be obvious
that the biquartz is a very delicate test for the exact crossing of a
pair of polarising prisms, or for the determination of the mutual
extinction of two rectangularly polarised beams of light in general.

A very striking and beautiful mode of exhibiting this opposite and equal
rotation of the plane of polarisation by the two varieties of quartz may
next be described, an experiment which we owe to Prof. S. P. Thompson. A
composite plate of mica is constructed out of 24 sectors of 15° angle
each, the whole making up a complete circular plate. They are cemented
between two circular glass plates of the usual 1⅞ inch size, with
balsam; the sectors are laid down in succession on one of the plates
first, side by side, with the edge of every one in turn in close contact
with the edge of the next in order, so as to radiate from a common
centre. The second glass plate is only cemented after the arrangement
has been allowed to set for some days, when there is less risk of
disturbing the mounting of the sectors. The latter have all been cut
from the same film of mica, which has a thickness corresponding to a
retardation of one of the two rays produced by the double refraction of
the crystal behind the other equal to one and a half waves. Each sector
is so cut that the line bisecting the 15° angle is parallel to the line
joining the positions of emergence of the two optic axes of the crystal.

[Illustration:

  FIG. 83.—A Disc _b_ of 24 Mica Sectors under Crossed Nicols, showing
    Effects at _a_ and _c_ of Introduction of Left and Right-handed
    Quartz Plates.
]

On placing this wheel of mica on the polariscope stage, the Nicols being
crossed, the effect shown at _b_ in Fig. 83 is observed on the screen.
The four sectors 90° apart, the bisecting lines of which are vertical
and horizontal respectively, parallel to the vibration planes of the
Nicols, appear as a jet black cross; the sectors next to them appear
pale brown, and the next again a still paler delicate shade of sepia,
while the central diagonal ones of each quadrant, at 45° to the black
cross, are brilliantly white.

On now introducing behind or in front of the stage a right-handed quartz
plate one millimetre thick, one of the pair of large ones described in
one of the convergent light experiments of the last chapter, the black
cross is observed to be deflected one sector to the right, as shown at
_c_ in Fig. 83; whereas when the left-handed companion plate is
introduced in like manner the cross moves over one sector to the left,
as indicated at _a_ in Fig. 83. The two quartz plates are mounted on the
same mahogany object frame, a specially long one with two large
apertures carrying the quartzes, so that first one and then the other
can be placed in or out of position, and when this is done rapidly the
movement of the cross from right to left and back again is very marked.

Occasionally a natural biquartz is obtained, on cutting a plate out of a
crystal of quartz perpendicularly to the axis. For it is not uncommon to
find a crystal which, while apparently a single crystal, is really a
twin, the two right and left individuals being joined by an invisible
plane of contact, or “plane of composition” as it is called, so
beautifully have the two grown together. Figs. 84 and 85 show two kinds
of twins of quartz. The former consists of two obviously different
individuals, with the little _s_ and _x_ faces indicating right or
left-handedness clearly developed in an opposite manner. The crystal
shown in Fig. 85, however, appears to be a single individual, yet
differs from either a right-handed or a left-handed crystal in showing
the _s_ and _x_ faces developed on both right and left solid angles. It
is a case of complete interpenetration.

In both cases the plane of twinning is parallel to the optic axis, and
to a pair of faces of the hexagonal prism of the second order,
perpendicular to a pair of the actual first order prism faces shown by
the crystal. They are examples of the well-known “Brazilian twinning” of
quartz, so called because many quartz crystals found in Brazil display
it.

[Illustration:

  FIG. 84.—Pair of Brazilian Twins of Quartz.
]

[Illustration:

  FIG. 85.—Completely Interpenetrated Brazilian Twins of Quartz.
]

A natural biquartz of 3.75 millimetres thickness cut from such a crystal
as is shown in Fig. 85, the plate having a hexagonal outline just as if
the crystal were really a single one, may next be projected on the
screen. The Nicols being crossed, the outline of the crystal is seen
sharply defined, the whole area of the crystal being coloured a uniform
yellow, there being absolutely no trace of any dividing line. But the
moment one commences to turn the analysing Nicol different shades,
orange and green respectively, begin to develop on the two sides of the
line indicating the plane of composition of the twin, the hexagon being
divided by a diametral line joining two corners, which have been
arranged in mounting the plate in its carrier frame to be above one
another, so as to bring the line of composition vertical, as will be
clear from Fig. 86. On rotating the analyser further the difference is
still more marked, and we have blue on one side and orange red on the
other, developing still deeper into red and purple as the analyser
approaches the parallel position with respect to the polariser; when
this latter position is attained the transition violet tint is developed
evenly over the whole plate, and the dividing line has again
disappeared.

[Illustration:

  FIG. 86.—A Natural Biquartz in Parallel Polarised Light.
]

Another natural biquartz, also shown in the author’s lecture at
Winnipeg, introduces us to a new phenomenon. For when the Nicols are
crossed we observe a black band down the centre of the plate, marking
the line of division of the twins. When the analyser is rotated until it
is parallel to the polariser this black band changes to a white one, the
sequence of colours on the different sides of the band, that is, in each
half of the plate, being the same as just described. The effect with
crossed Nicols is more or less simulated in Fig. 87, Plate XVII., which
is a reproduction of a direct photograph of the screen picture. The
reason for this black band in the dark field, and for the white one in
the bright field, is that the two halves of the twin overlap at the
centre, the plane of junction of the two individual crystals being
oblique to the plate, instead of exactly perpendicular thereto as was
the case with the first natural biquartz. We are, in fact, beginning to
get the effect of two superposed wedges of quartz.

[Illustration:

  _PLATE XVII._

  FIG. 87.—Natural Biquartz exhibiting the Black Band (Nicols crossed)
    at the Oblique Junction of the Right-handed and Left-handed Parts.
]

[Illustration:

  FIG. 88.—Artificial Biquartz, the two parts being obliquely joined in
    order to produce the Black Band.

  DIRECT PHOTOGRAPHS OF PICTURES PROJECTED ON THE SCREEN BY THE LANTERN
    POLARISCOPE, USING PARALLEL LIGHT.
]

When the obliquity is greater, or the crystal thicker, a white band
appears on each side of the black central one, the Nicols being crossed,
and when the thickness is as great as 6 to 7.5 mm. a spectrum band
appears on each side of the white one.

That this obliquity of the surface of contact of the two intergrown
individuals (not the plane of twinning, which remains parallel to a pair
of faces of the hexagonal prism of the second order) is the true
explanation can be readily proved by reproducing the effect
artificially. A thick double plate of quartz is constructed, as shown in
Fig. 88, composed of two halves of respectively right-handed and
left-handed quartz, each 6 to 7 millimetres thick, and each of which has
had the edge-face of junction ground and polished obliquely at an angle
of 30° or so, and oppositely so, instead of perpendicularly to the
plates; the two halves are then cemented together in the usual manner
for a biquartz, with Canada balsam, in order to make a continuous plate.
On placing the plate of this construction possessed by the author on the
stage of the projection polariscope, the two halves exhibit on the
screen respectively brilliant red and green colour, with a vertical
central black band, and on each side of it first a white strip and then
a spectrum band, all the bands being parallel to each other, and the
whole effect being precisely what was observed with the thickest natural
biquartz.

Thus, we have imitated the oblique junction of the twin parts of the
second and third biquartzes, and proved that this obliquity is the
reason for the phenomena of bands, the black band occupying the centre
where the two opposite rotations of the right and left quartz are
precisely neutralised. The dark field of the crossed Nicols consequently
prevails along this central strip, for the rotatory effect of the first
individual crystal on the light passing through it is exactly undone by
the subsequent passage of the rays through the other individual. On
either side of this neutral strip there is a little preponderance of
right-handed quartz on one side, and of left-handed quartz on the other,
and the usual effect of a thin plate of quartz is therefore seen,
namely, no colour but a little light, while further accretions of
thickness of the preponderating variety give all the colours of the
spectrum in turn, as with growing thicknesses of ordinary single quartz
plates, thus producing the spectrum band.

The black band is also afforded when the plate is cut somewhat
obliquely, out of a twin crystal with a junction plane truly
perpendicular to the equatorial section, instead of cutting it truly
perpendicularly to the axis, the junction plane being then oblique to
the plate. The polarisation colours are not so strong, however, unless
the plate be made thicker.

[Illustration:

  _PLATE XVIII._

  FIG. 89.—Black Central Band and equidistant Spectrum Bands on each
    side, afforded by Babinet’s Composite Plate of two Quartz Wedges,
    one parallel and the other perpendicular to the Axis. (Direct
    Photograph of Screen Picture as projected by Lantern Polariscope.)
]

[Illustration:

  FIG. 94.—Section-plate of Amethyst, natural size, as seen directly in
    Ordinary Light, showing Alternate Violet Sectors (see p. 223).
]

This effect of a black band with flanking spectra is very similar to
that obtained, due to double refraction and not to optical activity,
when two thin wedges of quartz are cemented together to form a parallel
plate, one wedge being cut so that the optic axis is parallel to the
edge of the wedge, and the other with the optic axis perpendicular to
the edge. When such a composite plate of quartz, often known as a
Babinet plate from the name of its first constructor, is placed on the
stage of the polariscope, and rotated to the 45° position with respect
to the planes of vibration of the crossed Nicols, there is observed on
the screen a deep black band in the centre parallel to the edge of the
wedge, and a number of spectrum bands on each side, separated by white
equal interspaces, the rainbow coloured bands showing the orders of
Newton’s spectra. The effect, as seen on the screen, is reproduced
photographically in black and white in Fig. 89, Plate XVIII.

These experiments lead us naturally to the study of a great variety of
quartz twins, involving some of the most beautiful and gorgeously
chromatic phenomena which it is possible to produce on the screen with
the projection polariscope. They will eventually bring us to the study
of amethyst quartz, in which the twinning is repeated so often that the
laminations of alternate right and left quartz are sometimes countless,
and almost approach molecular dimensions.

The Brazilian twinning of quartz, parallel to a pair of faces of the
second order hexagonal prism {11̄20}, often occurs in a very erratic
manner, as regards the arrangement of the portions of the composite
crystal belonging to the two varieties, the surfaces of contact and
character of the interpenetration being frequently very irregular, and
often remarkably so. Thus Fig. 90, the upper figure of the coloured
frontispiece, gives some faint idea of the appearance presented on the
screen by a very beautiful quartz plate, one-half of which is entirely
composed of left-handed quartz, giving a rich even rose-red colour when
the Nicols are crossed, not very far from the violet transition tint,
the plate being nearly 7.5 mm. thick, while the other half consists of
an alternation of strips of right and left-handed quartz, joined
obliquely to the surface of the plate, the black band and its
accompanying white ones and spectrum bands being repeated two or three
times before the edge is reached. This is a very instructive case, for
it shows in this half of the plate, on a large scale, what occurs in
amethyst in a more minutely structural manner, the broad strips, the
sections of plates upwards of a quarter of an inch thick, of alternating
character becoming in amethyst thin lines, the sections of laminæ or
films of microscopic tenuity, their number being correspondingly
enormously increased.

It may be interesting to state how this Fig. 90, and the lower Fig. 97
of the frontispiece representing the projection on the screen of benzoic
acid in the act of crystallisation, were produced. The pictures on the
screen were directly photographed on the latest Lumière autochrome
plates, a transparency in the actual natural colours being thus obtained
in each case. These transparent colour-photographs were then used as
originals wherewith to reproduce the effects on paper by the most recent
improved three-colour photographic process.

[Illustration:

  _PLATE XIX._

  FIG. 91.—Sectorial 60° or 120° Intrusive Twinning of Right and
    Left-handed Quartz, showing Ribbons with Central Black Band where
    Oblique Overlapping occurs.
]

[Illustration:

  FIG. 92.—Irregular Intrusive Twinning of Right and Left-handed Quartz.

  DIRECT PHOTOGRAPHS OF REMARKABLE SCREEN PICTURES AFFORDED IN PARALLEL
    POLARISED LIGHT BY SECTIONS OF TWINS OF RIGHT AND LEFT QUARTZ.
]

Two other typical cases of irregular quartz twinning may also with
advantage be demonstrated. The first is a plate in which there are
repeated 60° V-shaped or 120° wedge-shaped intrusions of one variety
into a greater mass of the other variety. The border of the V or 120°
wedge is composed of a ribbon, the outer edges of which are
spectrum-coloured and the central line of which is formed by the deep
black band, which is separated on each side from the spectra by a white
strip. Some idea of the beauty of this quartz plate, which was
generously lent to the author by Prof. S. P. Thompson, as projected on
the screen under crossed Nicols, may be gathered from Fig. 91, Plate
XIX., the upper homogeneous part of the plate being coloured a brilliant
green, and the lower part red.

The second is an irregular interpenetration of one variety into the
other, in repeated V-shapes occupying the lower half of the image of the
plate as seen on the screen in the dark field of the projection
polariscope, like a range of sharp mountain peaks, the black bands being
so rapidly repeated as to be nearly continuous. These darker portions
thus appear to form the bulk of the mountains, while the upper untwinned
half of the crystal shows a clear and even sky blue; to make the
resemblance to a range of Alpine mountains even more complete, the wavy
line of demarcation between the twinned and non-twinned portions of the
plate is bordered by a white ribbon, of varying width, giving the
appearance of a snow-cap to each peak, which shows up clearly against
the blue sky. It will be obvious that this quartz plate affords an
altogether very beautiful series of phenomena in parallel polarised
light on the screen, for the colours change with every movement of the
analysing Nicol from the crossed position, the appearance for which has
just been described. Fig. 92, Plate XIX., gives only the faintest idea
of the beauty of the screen picture afforded by this section-plate. The
effect chosen as best for photographic reproduction purposes is one
afforded when the analysing Nicol is rotated somewhat away from the
crossed position with respect to the polariser.

And now we arrive finally at amethyst quartz, three very beautiful
hexagonal plates of which—cut perpendicularly to the optic axis as usual
for quartzes intended to display optical activity, from an apparently
single hexagonal prism in each case—will be taken as typifying the
phenomena exhibited by this especially interesting variety of quartz on
the screen in parallel polarised light. The smaller one affords a screen
picture, with Nicols not quite crossed, such as is portrayed in Fig. 93,
Plate XX. We observe that the area of the hexagon is roughly divisible
into six 60°-sectors, and that alternate ones are uniformly coloured,
indicating that they belong to wholly right-handed or left-handed
quartz; whereas the other alternate sectors are most beautifully marked,
as if by line shading parallel or inclined at 30° to the edges of the
hexagon, by a considerable number of equally spaced dark or slate
coloured bands, close together but separated by white bands, with a
trace of spectrum colours along the middle of the latter. If we rotate
the analysing Nicol somewhat we can readily find a position, which is
not always that of crossed Nicols, for which these parallel bands of
laminar twinning are most clearly defined, as shown in the illustration,
the colours of the other sectors ever changing during the rotation.

[Illustration:

  _PLATE XX._

  FIG. 93.—Section-plate of Amethyst Quartz, showing Sectorial Repeated
    Twinning of the Right and Left Varieties.
]

[Illustration:

  FIG. 95.—Section-plate of large Amethyst Quartz Crystal, showing
    relatively large Area of Sectorial Repeated Twinning of Right and
    Left-handed Quartz (see p. 225).

  DIRECT PHOTOGRAPHS OF SCREEN PICTURES OF AMETHYST PROJECTED BY THE
    LANTERN POLARISCOPE IN PARALLEL LIGHT.
]

It is obvious that we have here to do with the same phenomenon as was
illustrated by the parallel bands shown on the large scale by the
section illustrated in Fig. 90 of the coloured frontispiece, the black,
white, and spectrum-coloured bands being simply repeated very many more
times in the same space, and in alternate sectors of the crystal.

The twinning of amethyst in 60°-sectors is very characteristic of this
variety of quartz, and it is an interesting fact that the sectors which
show the laminar bands in polarised light often appear purple coloured
in ordinary light, the tint from which amethyst derives its name. This
is not necessarily or always so however, and the section just described
and illustrated in Fig. 93 appears quite colourless throughout on casual
inspection in ordinary light, in fact as a clear colourless hexagonal
section of ordinary simple quartz; a trace of the amethyst colour
becomes, however, apparent on closer examination when held obliquely, in
the sectors where the bands become visible in polarised light.

The second plate of amethyst is a magnificent section 9 millimetres
thick and 2½ inches in diameter, of which alternate 60°-sectors are
deeply amethyst coloured, the tint being a pure violet of about the
wave-length of the hydrogen line near G of the spectrum. Moreover, even
to the naked eye when the specimen is held in the hand up to the light,
in certain positions the laminæ become visible as more deeply shaded
violet line markings. On placing it on the stage of the polariscope but
with the analysing Nicol removed, so as to observe the natural
appearance of the section in white light (for, although polarised by the
polarising Nicol, being unanalysed the section exhibits no polarisation
effects), these facts become clear to everyone in the room. The violet
staining of alternate sectors appears very deep, and traces of
lamination in the violet parts are just apparent on close scrutiny, the
other alternate sectors appearing colourless and unmarked except by a
few flaws almost always present in so large a section-plate of amethyst.
The natural appearance of this plate is shown in Fig. 94, Plate XVIII.
(facing page 218), as far as is possible photographically, the violet
sectors being clearly demarcated.

On replacing the analysing Nicol the colourless sectors are seen to
polarise uniformly in brilliant colours, indicating a homogeneous
variety of quartz in each, either right or left-handed. Moreover,
whenever two of these naturally colourless parts touch each other, which
they do as the margin of the plate is approached, an irregular ribbon is
produced, composed of the black band in the centre, with first white and
then spectrum-coloured flanking strips on each side, the spectra forming
the edges of the ribbon. The violet sectors show the laminated twinning,
but, owing to the great thickness of this plate, in too complicated
(overlapping) a manner to be easily followed, a thinner plate being
required to show such fine laminations clearly.

Finally, the third section is such a thinner plate, about 3.5 mm. thick
and nearly 1½ inches in diameter. This section of amethyst is probably
the most beautiful of all, for it not only shows the laminated twinning
to perfection, in three alternate 60°-sectors and in all six in the
middle part of the plate, but also these alternate sectors are
distinctly violet even to the eye when the specimen is held in the hand
against a white background; and the laminations are likewise also
clearly visible on holding the section obliquely up to the light. In
polarised light, either with crossed or parallel or anyway arranged
Nicols, the phenomena on the screen are of the most superb character.
The whole of the middle part of the plate appears made up of six
sectors, _all_ showing the fine laminar bands parallel to the edges of
the second order hexagonal prism {11̄20}, that is, at 30° to the edges
of the section, the crystal being a first order hexagonal prism {10̄10}.
Some idea of the arrangement will be afforded by Fig. 95, Plate XX. The
marginal parts develop into alternately right and left-handed sectors or
half-sectors, polarising in different and very brilliant colours, and
showing the ribbon bands at every junction. On rotating the analysing
Nicol the changes are remarkably beautiful, particularly for the
positions of the analyser when the laminar bands take on their deep
slate colour, with white and marginally spectral interstrips. The whole
phenomena, indeed, afforded by this plate of amethystine quartz, are the
most magnificent which the author has ever seen on the screen, in the
whole of his crystallographic experiences.

The Brazilian twinning law of quartz, according to which the plane of
twinning is parallel to a pair of faces of the second order hexagonal
prism {11̄20}, appears capable of explaining all these varieties of
right and left-handed twins, the interpenetration of the intimate kind
shown in Fig. 85 (page 215) usually resulting in sectorial portions of
space being occupied by each kind, the surfaces of junction of
oppositely optically active parts being, however, very varied in their
distribution and character. Where they happen to be more or less
horizontal, a plate cut perpendicularly to the axis to include both
kinds would show Airy’s spirals in convergent polarised light, as may
readily be demonstrated by such a plate, one of several, in the author’s
collection. Where they are oblique, a plate cut at right angles to the
axis would, as we have seen experimentally, afford the black, white and
spectral ribbon bands in parallel polarised light. Where, however, the
mode of interpenetration is still more intimate, we have the rapidly
alternating laminæ of the two varieties, right and left-handed, building
up the beautiful structure of amethyst in thin layers. A section-plate
of such an intimate blending of the two varieties, cut as usual
perpendicular to the axis in order that any phenomena of optical
activity shall be exhibited at the maximum, affords no indication
whatever of optical rotation, the two varieties simply neutralising each
other’s effects, and the plate behaves as an ordinary uniaxial crystal,
affording in convergent polarised light a black cross like calcite,
complete to the centre. In parallel polarised light it shows of course
the laminated structure, but the tendency to remain dark under crossed
Nicols is shown by the fact that the tints exhibited by the laminations
are slates, greys, and even black, when the Nicols are crossed, the
delightful other colours only making their appearance when the analysing
Nicol is rotated. Thus the simple law of Brazilian twinning is quite
capable of explaining the whole of the phenomena exhibited by composite
crystals of the two varieties of quartz, and such an explanation is the
one accepted by von Groth, in the excellent description of quartz in the
last edition of his _Physikalische Krystallographie_.

[Illustration:

  FIG. 96.—Plan of Amethyst Crystal.
]

An interesting crystal of amethyst very similar to the third of those
just described, the one illustrated in Fig. 95, was described by Prof.
Judd in the year 1892 to the Mineralogical Society.[18] The plan of the
crystal is given in Fig. 96. The wedges marked _x_, _y_, _z_, are of a
pale yellow colour, as are also the three strips, sections of plates,
proceeding from the wedges and meeting at the centre _o_. The wedge _y_
exhibits left-handed polarisation, and the wedge _z_ right-handed. The
large wedge _x_ is composite, the part marked _x__{r} being right-handed
and that marked _x__{l} left-handed. The surface of junction of the two
parts is not perpendicular to the plate, so where the two varieties
overlap, the part marked _x__{rl}, a ribbon band is shown in parallel
light and Airy’s spirals in convergent polarised light. The yellow parts
of the crystal exhibit ordinary rotatory polarisation colours, even
tints; but in the remaining sectors of the crystal, the lines of
division of which are indicated by the radial lines A, B, C, no trace of
circular polarisation is displayed, and the central part, where the
lamellæ are very well developed, gives the ordinary calcite-like
uniaxial interference figure. The more marginal portions, however, show
complicated interference figures, somewhat resembling those of biaxial
crystals, owing to irregular distribution of the two varieties of
quartz, and probable displacement of the optic axis by distortion.

An ingenious theory of the formation of the lamellæ is put forward by
Prof. Judd in the same memoir. He had already shown that quartz is
endowed with planes of gliding, parallel to the rhombohedral faces, and
suggests that the lamellation is the result of the effect of high
pressure and possibly high temperature on the quartz crystal after its
formation. The lamellæ appear to be frequently parallel to the
rhombohedral terminal faces of the crystal, as if they were indeed glide
plane effects. It is quite conceivable that the gliding of layers of
molecules, which when permanent usually involves rotation and inversion
of the molecules, might result in alternately right and left structural
arrangements, and there is considerable evidence that the development of
the purple tint occurred subsequently to the growth of the crystal. It
is probably due to change in the state of oxidation of the trace of
manganese present as a minute impurity in the quartz crystal, and which
is concentrated between the lamellæ, just as the yellow tint is due to a
slight trace of iron (ferric) oxide. The theory is an interesting one,
and throws considerable light on the possible nature of intimate
lamellar twinning.

One last experiment may now be referred to, the concluding experiment of
the Winnipeg lecture, and which is very reminiscent of the beautiful
slate colour of the lamellæ of amethyst. It is the actual
crystallisation, projected on the screen, of a thin film of melted
benzoic acid, which affords radiating closely packed long and narrow
crystals, shooting out on the screen from centres near the margin of the
field, very much like the individual crystals of repeatedly twinned
quartz in the beautiful amethyst crystal illustrated in Fig. 95.
Provided the film of melted benzoic acid be thin enough, the crystals
appear on the screen in parallel polarised light, under crossed Nicols,
tinted with the same beautiful shades of slate colour as amethyst, the
intermediate low-order tint between the black and the grey of Newton’s
first order spectrum. Some idea of the appearance on the screen is
afforded by Fig. 97, the lower of the two coloured figures in the
frontispiece. As in the case of Fig. 90, the screen picture was
photographed directly on a Lumière autochrome plate, and the
transparency in the actual colours thus obtained was employed as an
original wherewith to reproduce the picture on paper by the latest
three-colour photographic process.

In carrying out the experiment a few of the flaky crystals of benzoic
acid are placed on one of the circular glass object plates of the
standard 1⅞-inch size for the projection polariscope; they are covered
by a second similar one, and the two plates are then held in a pair of
tongs and gently warmed over a small spirit lamp, or miniature Bunsen
lamp. As soon as the crystals have fused, and the melted substance is
evenly spread as a thin film between the two glass plates, the latter
are rapidly transferred to a special mahogany object frame, fitted with
a side slide to press the double-plate edge just sufficiently to hold it
in position in the frame, which is then at once placed on the rotating
stage of the polariscope. The screen appears quite dark at first, the
Nicols being crossed, but in a second or two as the slide cools the
benzoic acid begins to crystallise out at the sides, brilliant colours
and the deep greys being both developed, the former chiefly near the
edges of the crystals, rendering the crystallisation wonderfully
distinct and beautiful on the black background. Then long needle
crystals shoot out from various quarters one after another or
simultaneously, in lovely shades of slate or grey tinted with brilliant
colours at the margins and tips, the growing point cutting its way along
like a sharp brilliantly coloured arrowhead. Eventually an arch is
formed of such acicular crystals, radiating simultaneously from many
centres, gorgeously coloured in parts, but showing the yet more æsthetic
slates and greys in the main. Finally, the whole screen picture fills up
with a mass of interlacing yet ever distinct crystals, the last few to
crystallise in the centre usually doing so with a burst of especially
bright colour, as the thickness increases adequately for the double
refraction retardation to reach the more brilliant second order
spectrum, a concluding effect which evokes the emphatic delight of even
the most phlegmatic philosopher, inured to scenes of beauty in natural
phenomena.

The series of experiments with quartz described in this and the previous
chapter, culminating with those revealing the alternate repetition of
extremely fine layers of right and left-handed quartz in amethyst, will,
it is hoped, have illustrated and rendered intelligible the important
structural principle of enantiomorphism or mirror-image symmetry. We
have only to imagine the layers to become thinner and thinner until we
approach ultimately the neighbourhood of the minute dimensions of the
chemical molecule, without as yet penetrating within the range of the
molecular forces; the two such oppositely constructed and intimately
blended structures, built up by atoms arranged oppositely screw-wise,
clockwise and anti-clockwise, will now form an ultramicroscopic mixture
of the two varieties in equal quantities, that is, in equal molecular
proportions.

Such a structure will exhibit the symmetry of the system to which the
two individuals belong, but instead of only displaying that of the
enantiomorphous class of that system, possessing lower than the full
symmetry, as each variety does when crystallised alone, it will now
display the full holohedral symmetry of the system. That is, the
symmetry is enhanced by this intimate blending of the two complementary
enantiomorphous forms, the two together supplying all the possible
elements of symmetry of which the system is capable. Moreover, as we
have seen in the case of the lamellar portion of the amethyst crystals
represented in Figs. 95 and 96, there will now be no sign of optical
activity, for the two opposite rotations are equal and destroy each
other.

Hence, such a compound crystal shows the holohedral symmetry of the
system, and is optically inactive. In such cases we are, in fact,
confronted with the phenomenon of pseudo-racemism, as defined in Chapter
XI. For we know that the two varieties are still present intact,
polarised light revealing them in the case of their grosser development
such as is found in amethyst, and the system of symmetry being clearly
the same, the forms developed being merely the sum of those of the two
individual varieties.

Amethyst thus affords us a gross demonstration of the nature of
pseudo-racemism, and as such has proved an exceedingly illuminating
study.

We can carry the process further, however, in imagination, until the two
differently helical molecules are themselves juxtaposed face to face,
right molecule to left molecule. When, however, this occurs, we have
entered into that most fascinatingly interesting region, the range of
molecular forces, a mysterious sphere of activities of which we are only
just beginning to learn something. Within this region of larger activity
the two oppositely constructed molecules are often known to combine
chemically to produce a molecular compound, just as potassium sulphate
molecules, for instance, will combine with those of magnesium sulphate
to form the well-known double salt. The double molecule now furnishes
the representative point of the space-lattice, in other words, a new
space-lattice is now erected, the units of which may be taken to be the
representative points of the double molecules. Such a space-lattice will
of necessity be of a totally different character to the old one
corresponding to the single molecule of either variety (for each variety
has the same space-lattice, the points, however, representing
differently, enantiomorphously, orientated atomic details). That is to
say, we shall have an entirely new kind of crystal produced, in all
probability belonging to a different crystal system. It is known as a
racemic compound, as described in Chapter XI.

This is exactly what happens in the case of tartaric acid, the two
varieties, dextro or right-handed tartaric acid and lævo or left-handed
tartaric acid, not forming pseudo-racemic crystals of like but enhanced
(holohedral) symmetry, but a truly molecular compound, the well-known
inactive racemic acid, in which the phenomenon of “racemism” was first
discovered and from which it took its name.

Now a molecular compound is notoriously regarded by chemists as a type
of chemical compound of low stability, molecular attraction or affinity
not being nearly so powerful as atomic affinity. Hence, under suitable
conditions it may be possible to induce the two component varieties to
crystallise out separately from the solution of the racemic compound. In
the case of racemic acid itself this does not readily happen, but in the
cases of certain of its metallic salts, sodium ammonium racemate, for
instance, specific conditions are known under which the two varieties of
crystals, right and left-handed respectively, may be separately
crystallised out from the solution, some of which conditions were
referred to in Chapter XI. Racemic acid itself, however, crystallises
quite differently to the two tartaric acids, namely, in triclinic
prismatic crystals. These are, in fact, absolutely different from the
monoclinic crystals of the dextro and lævo varieties of ordinary
tartaric acid, for racemic acid takes up also a molecule of water of
crystallisation on separating from its aqueous solution. There are
certain chemical differences also, due to the chemical union of the two
enantiomorphous molecules into a single double molecule, such, for
instance, as greater facility of reduction by hydriodic acid to succinic
acid.

Thus our experiments with quartz have afforded us the means of acquiring
a clear idea of the nature of this most interesting type of crystal
structure which involves the principle of mirror-image symmetry. Racemic
acid and its similar structures, racemic compounds in general, are known
as “externally compensated” structures, the reflective principle here
acting externally to the single enantiomorphous molecule. It is but
another step, however, to imagine internal compensation of
enantiomorphous parts of a molecule, by mirror-image combination of such
parts, such as in all probability occurs in the case of the truly
inactive fourth variety of tartaric acid, in order to comprehend how the
principle enabled the 165 types of homogeneous structure involving this
kind of repetition to be arrived at, and thus, together with the 65
regular point-systems already known, to afford us the complete set of
230 types of homogeneous structures possible to crystals.




                               CHAPTER XV
                  HOW A CRYSTAL GROWS FROM A SOLUTION.


One of the most deeply interesting aspects of a crystal, especially from
the point of view of the history of crystallographic investigation,
concerns the mysterious process of its growth from a solution (in a
solvent) of the substance composing it. The story of the elucidation, as
far as it has yet been accomplished, of the nature of crystallisation
from solution in water is one of the most romantic which the whole of
scientific progress can furnish. Again we are struck with the
parallelism between crystals and living objects. For just as the
discovery of bacteria, the infinitesimal germs of life, has given an
immense impetus to our knowledge of disease and been blessed with most
beneficent effects in combatting the ravages of the latter, so the
discovery that crystal-germs of most common crystallised substances, of
no larger size than bacteria, are floating about in our atmosphere, and
ready at any time to drop into our solutions and, if the latter are in
the proper receptive condition, to set them crystallising, is little
less marvellous, and has had as profound an effect on our knowledge of
the process of crystallisation. A true story, told to the Royal Society
the other day, may serve to illustrate the point. A new chemical
compound had been discovered, and at the time there could obviously be
no crystal-germs, minute crystallites of the dimensions of possibly only
a comparatively few chemical molecules, of this hitherto unknown
substance floating about in the air. It was found impossible to obtain
the deposition of crystals in the ordinary way, from solutions of the
substance in its ordinary solvent, although they were in the condition
of proper receptivity above referred to, on account of the absence of
such germs in the air. But later on, when the air of the laboratory had
become impregnated with such germs, on account of the daily handling of
the substance in the laboratory, no difficulty was any longer found in
obtaining good crystals quite readily from these solutions.

We are at first inclined to wonder whether such extraordinary statements
can possibly be sober facts. Yet such is, indeed, the case, and it will
be very well worth devoting a chapter to the story of how we have at
length arrived at definite knowledge concerning the process of
crystallisation from the state of solution in water. For water is the
ordinary solvent from which we obtain our crystals, that is, such as are
prepared artificially in the laboratory. The laws which have been
discovered to hold for aqueous solutions are, however, equally
applicable to the cases where other solvents are used, such for instance
as the usual organic solvents like alcohol, ether, chloroform, and
benzene.

The conditions under which crystallisation occurs from the liquid state,
or from solution of the substance in a solvent, have been accurately
determined experimentally by H. A. Miers,[19] and they bear out in the
main the predictions from theoretical considerations which were made by
Ostwald.[20] Taking first the case of crystallisation from solution,
there are two distinct curves representing the degree of solubility of
the solid substance and of supersolubility. The well-known ordinary
solubility curve is obtained by taking the temperature for abscissæ and
concentration for ordinates, so that any point on the curve indicates
the amount of the solid substance which the solvent can hold in solution
at that particular temperature. Now the fact that supersaturation may
occur has long been established, the phenomenon being of frequent
occurrence; and it is common knowledge that a supersaturated solution
may be preserved for a long time without crystals being deposited from
it, provided the liquid be maintained quietly at rest. Obviously,
therefore, this condition of supersaturation ought to be represented by
a second curve a few degrees lower as regards temperature than the
solubility curve, and its conditions were fairly fully predicted by
Ostwald, after collecting together and analysing the results of the
experiments of Gernez, Lecoq de Boisbaudran, J. M. Thomson, de Coppet,
Lefebvre, and Roozeboom. It was reserved for Miers, however, to discover
a means of experimentally tracing this curve, by observations of the
refractive index of the solution. The point at which the deposition of
crystals from the supersaturated solution occurs is immediately
indicated by a sudden change in the refraction of the liquid, the
refractive index attaining its maximum value at the temperature of
spontaneous crystallisation, and then dropping suddenly the moment the
crystals begin to fall. Moreover, the solution at the same time records
its own strength, for the refractive index varies directly as the amount
of salt dissolved. The determination of the strength of the solution at
the critical moment itself had previously proved an impossibility by
ordinary methods.

Fig. 98 gives a general diagrammatic representation of Miers’ results
for a typical crystalline substance soluble in water. S is the ordinary
solubility curve, which may also be termed the “curve of crystallisation
by inoculation.” For as soon as the solution reaches this condition of
normal saturation it is liable to be caused to commence crystallising if
a germ crystal, that is, a miniature crystallite floating in the air as
dust, of the substance itself or of one isomorphous with it or capable
of forming parallel growths with it, fall into the solution from the
air. It has been a revelation to us that such minute crystallites of all
common substances are scattered broadcast in our atmosphere, and that
sooner or later one will introduce itself into any solution set to
crystallise which is not sealed up or placed in a vessel with a
filtering plug of cotton-wool in its neck or other aperture.

[Illustration:

  FIG. 98.—Diagram illustrating the Conditions for Crystallisation from
    Solution or the Liquid State.
]

SS is the supersolubility curve, situated approximately 10° to the left
of the solubility curve as regards temperature, but about as much above
as regards concentration, so that the two curves usually run diagonally
and more or less parallel to each other across the diagram. This
supersolubility curve may be also called the “curve of spontaneous
crystallisation,” for it represents the conditions under which alone
crystals may begin to form without the initiating impulse of inoculation
by a germ-crystallite. On the suggestion of Ostwald it is also termed
the “metastable limit,” and the whole area between the solubility and
supersolubility curves is named the area of metastability, that which
represents the “metastable” condition of the solution. Within this area
the conditions are those for the start of crystallisation by
inoculation. The area beyond the supersolubility curve represents the
“labile” state, in which the conditions are those for spontaneous
crystallisation, inoculation being no longer necessary. These precise
results will, it is hoped, be quite clear with the aid of Fig. 98.

Hence, when a cooling solution not quite saturated at the higher
starting temperature is stirred in an open vessel a slight shower of
crystals, started by inoculation, appears when the saturation point is
reached, which Miers calls a “metastable shower,” corresponding to the
ordinary solubility curve; the liquid then goes on cooling without
depositing the main bulk of the excess which that curve indicates ought
to be deposited, if it represent the whole truth. But when the
temperature of the supersolubility curve about 10° lower is reached, a
much more copious shower falls by spontaneous crystallisation, the
“labile shower.”

In a closed vessel, such for instance as a glass tube sealed with the
aid of the blowpipe after the introduction of the solution, on cooling
after heating to a temperature superior to that of saturation, the first
shower never falls at all, no amount of shaking inducing the deposition
of crystals at the ordinary saturation point, proving that the slight
shower of the experiment in the open vessel is due to crystal-germs
introduced from the atmosphere. The second shower of crystals falls at
the lower temperature just as before, however, at the temperature of the
supersolubility curve, indicating that this shower is due in both cases
to spontaneous crystallisation. Solutions thus enclosed in sealed tubes,
to which inoculating dust crystals cannot have access, can never be made
to crystallise at any temperature higher than that given by the
supersolubility curve, however agitated, although they immediately do
crystallise, if shaken, as soon as that temperature is reached during
the cooling. If allowed to remain absolutely quiet, however, the
temperature may fall considerably lower before any crystallisation
occurs, the labile region being frequently well penetrated before this
happens. When crystallisation does supervene, the temperature usually
rises somewhat. After the labile shower has been deposited, the crystals
continue to grow steadily further, until the metastable region has been
traversed, and the saturation state is eventually reached, when final
equilibrium is produced.

The proof that the crystals deposited in the metastable condition were
started by the advent of atmospheric germ crystals—that is, by
infinitesimal but perfectly structurally developed crystals, carried by
their very lightness like the particles of dust which are only revealed
in the path of a sunbeam as seen against a dark background—was afforded
by a series of experiments with a mixture of two rare organic chemical
preparations, salol (phenyl salicylate) a substance melting at 42.5°,
and betol (β-naphthol salicylate) another melting at 92°, which Miers
assumed were not likely to be present in ordinary air. The assumption
proved well grounded, and the first shower never fell at all in the
earlier experiments in which mixtures of these two substances were
allowed to cool in open vessels, from the state of fusion. But very soon
the air of the laboratory became impregnated with crystallites of both
substances, due to the very operations themselves being carried on in
contact with the air, and in the later experiments the first shower of
crystals did fall. The experiments were really designed to effect the
determination of the solubility curve for salol and betol in each other,
that is, the freezing-point curve of their mixtures, and the discovery
of the so-called “eutectic” point at which a mixture of constant
composition solidifies at a definite temperature. But incidentally the
experiments also served to establish similar laws for the production of
crystals from the liquefied state, by cooling below the melting point,
to those applying to crystallisation from solution. In the case of the
mixtures of the substances the one of lower melting point acted as a
solvent for the one of higher melting point, just as water does for
salt. Two curves corresponding to the ordinary freezing point and to the
limit of superfusion were established, analogous to the solubility and
supersolubility curves. Pure salol alone proved to crystallise
spontaneously at 33°, 9½° below its melting point, and the refractive
index attained a maximum for this temperature. Betol spontaneously
crystallised at 79°, 13° below its melting point.

Two general cases of crystallisation are shown by the dotted curves ABCD
and ABE in Fig. 98. The first, represented by ABCD, is the case of a
supersaturated solution, made by adding the salt to hot water, being
allowed to cool slowly while stirred. The solution cools from A to B
without anything visibly happening, no crystal-germ falling into the
solution until B is reached, somewhere well within the metastable
region. When the germ has fallen in, however, crystals begin to appear
as a slight shower at B, and from B to C they continue to grow slowly.
On reaching the labile condition at C a cloud of crystals, the heavy
shower, is deposited, and the concentration falls rapidly to D on the
solubility curve, generally with slight rise of temperature.

The second case is the important one employed by the author in the
investigations which will be found described in his “Crystalline
Structure and Chemical Constitution” (Macmillan & Co., 1910), for the
purpose of producing crystals of high perfection for goniometrical
investigation. The method can be confidently recommended as the one best
adapted to afford good measurable crystals, and is of quite general
application. The solution is made up so as to be in the metastable
condition, that is, only slightly supersaturated for the ordinary
temperatures. Eventually, while the solution is at rest in a protected
place, free from draughts or vibration, and after it has cooled to the
temperature of the air, a crystal-germ enters, followed probably by
others; each forms a centre of crystal growth, which proceeds very
slowly and deliberately, keeping pace with the evaporation in such a
manner that the labile condition is never reached. The natural result is
the production of very well-formed crystals bounded by excellent faces,
truly plane and free from striation or distortion.

When the operation is arranged to occur during the night, as will
usually be the case, the solution being set out to crystallise in a
quiet and protected place on the previous afternoon or evening, the
slight fall of temperature during the night gently assists the process
and almost ensures a good crop of a few well-formed crystals large
enough for goniometrical purposes next morning. They should be removed
before the temperature begins to rise again with the advent of the sun,
dried with blotting paper and by air exposure for a short time, and
stored in a miniature bottle labelled with the name or formula of the
substance and the date of collection of the crop. In such cases the
labile state is never reached, and the course of the crystallisation is
represented by the curve BE. The whole conditions for the curve ABE,
however, would correspond to much lower temperatures, such as those
given at the foot of the diagram below the word “temperature,” rather
than to the upper row of temperature abscissæ suitable for the other
purposes of the diagram already referred to. Crystallisation might well
begin about 13° or 14°, as shown at B, and the liquid would cool a
couple of degrees or more during the night while crystallisation was
steadily proceeding, until equilibrium was reached at E on the
solubility curve.

The diagram does not represent any substance in particular, but is a
perfectly general one, corresponding to the facts observed with most of
the very varied salts worked with by Miers and those of which the author
has had experience. The exact temperatures and concentrations will, of
course, differ for each substance.

A beautiful experimental demonstration of crystallisation from the
metastable and labile conditions of solution respectively is afforded by
potassium bichromate, K_{2}Cr_{2}O_{7}. When deposited slowly from a
metastable solution under conditions of quietude, this salt is slowly
deposited in bright orange coloured and excellently formed crystals,
often of considerable size, belonging to the triclinic system of
symmetry; they are bounded by good pinakoidal (pairs of parallel) faces
intersecting in sharp edges. But when the crystallisation occurs from a
labile solution, being much more rapid, it takes the form of feathery or
arborescent branching skeletal growths, there being inadequate time for
the formation of well-developed crystals.

Fig. 99, Plate XXI., is a photographic reproduction of well-formed
crystals of potassium bichromate, grown from a solution in the
metastable condition on a microscope slip, just as they are seen through
the microscope in the slow act of formation, employing a 1½ inch
objective. The crystallisation had been started by germ crystals of the
salt falling in from the air, after which the drop, placed within the
ring of hardened gold size on the slide, had been covered with a
cover-glass, under which the crystallisation had proceeded with
sufficient slowness to enable a successful photograph to be taken, when
the camera was subsequently attached above the vertically arranged
microscope. An upright micrographic apparatus had been designed by the
author specially for this photography of growing crystals, many of the
results of which are reproduced in this book.

[Illustration:

  _PLATE XXI._

  FIG. 99.—Potassium Bichromate slowly crystallising from a Metastable
    Solution.
]

[Illustration:

  FIG. 100.—Potassium Bichromate rapidly crystallised from a Labile
    Solution.

  CHARACTERISTIC DIFFERENCE IN THE CRYSTALS DEPOSITED FROM METASTABLE
    AND LABILE SOLUTIONS.
]

Fig. 100 is the reproduction of another photograph taken under similar
conditions, but employing a hot and somewhat more concentrated solution
of potassium bichromate, and making the exposure at the moment when, in
the particular field chosen, a rapid labile crystallisation was just
completing itself, the rapidity of growth of the feathery skeletal
crystals having just become arrested. Indeed, the branches are
frequently terminated by small well-formed crystals, the rapid growth
having been succeeded by a final slow crystallisation where the solution
had discharged its labile excess and attained once more the metastable
condition.

This experiment with potassium bichromate lends itself admirably to
lantern demonstration with the projection microscope. When the drop of
hot concentrated solution is first placed on the warmed microscope slip,
and the latter laid on the stage, nothing visible on the screen happens
for a minute or two, the solution becoming, however, more or less
rapidly cooled. But suddenly, the drop having cooled sufficiently to
bring the solution to the labile condition of supersaturation
corresponding to the conditions for spontaneous crystallisation
indicated by the supersolubility curve, arborescent or feathery growths
begin to shoot out from various points in the field, often near the
margin, and traverse the screen so rapidly that in a moment or two it is
filled with them. The crystallisation then slows down once more, the
labile shower of excess having become exhausted, and the terminations of
the branches and ramifications begin to develop into good little
crystals, which thus hang like fruit on a tree. The experiment is
rendered the more brilliant and beautiful by the bright orange colour of
the crystals.

In Fig. 101, Plate XI., facing page 88, a reproduction of a photograph
of a similar crystallisation from a labile solution of ammonium chloride
is given. This salt is also particularly suitable for screen
demonstrations. The beautiful skeletal ramifications follow the axial
directions of the cubic axes, ammonium chloride crystallising in the
pentagonal-icositetrahedral class of the cubic system. Good crystals
may, however, be very slowly grown from metastable solutions, and they
usually exhibit as the principal forms the icositetrahedron
(predominating), cube, octahedron, rhombic dodecahedron, and the
class-distinguishing pentagonal icositetrahedron. The rapid growths by
spontaneous crystallisation of labile solutions, however, invariably
take the form of the rectangularly branching feathery crystals shown in
Fig. 101.

Further light has been thrown on the act of crystallisation by another
most interesting research of Miers concerning “vicinal faces,”[21] such
as the three very low pyramid faces (forming a very flat triakis
octahedron) which often replace each octahedron face on a crystal of
alum which has been grown somewhat rapidly. The author has frequently
observed this phenomenon in the course of the numerous crystallisations
required for the investigation of the sulphates and selenates. It may be
described in general terms as the replacement of primary faces
possessing the simplest rational indices by faces having such high
indices that it is doubtful whether they ought really to be represented
by indices at all. The number of such vicinal faces which replace the
simple face depends on the symmetry of the crystal, to which, of course,
they conform. Thus, while three such vicinal faces replace an octahedral
face, and two replace the face of a tetragonal prism, the simple primary
prism face of a rhombic or monoclinic crystal would only be replaced by
one, which may have a deformation of as much as even 30′ from the
correct position of the prism face, and on either side of it. Indeed it
is possible for a whole succession of such vicinal faces to be developed
within the degree of arc which may in extreme cases separate the
limiting values on each side of the prism face, and such are often seen
and make up the well-known bundle of images afforded on the goniometer
by a bad face, a face which would cause the author at once to reject the
whole crystal for measurement purposes. One of the faces, even in cases
such as alum or a tetragonal crystal, where three or two might have
equal values as regards the symmetry, generally predominates, and
affords a very much more brilliant image of the goniometer signal than
the others in the bundle, so that an unwary observer might easily come
to the conclusion that this was the really valid image corresponding to
the octahedron face or to the simple primary prism face, or whatever
particular face was expected in the neighbourhood indicated by the
bundle of images. Obviously, however, it might only be one of three or
two equally valid faces of a vicinal form, which had grown
predominatingly during the last period of growth previous to removal
from the mother liquor.

The explanation of this interesting phenomenon of the production of
vicinal faces is one intimately connected with the structure of
crystals, and it forms one of the strongest confirmations of the
correctness of the theory of crystal structure the basis of which is the
molecular space-lattice. Miers is in full agreement with the author in
emphasising the importance of the space-lattice formed by the “points”
representative of the molecules, and analogously chosen in the
molecules. He says: “Whatever structures may be necessary to account for
other features of crystals, there is little doubt that we are justified
in regarding their faces as the planes of a space-lattice.”[22] Now
Wulff,[23] who has contributed very considerably to our knowledge of the
nature of the act of crystallisation, has proved, from his own
investigations and those of Weyberg, carried out at his suggestion in
his laboratory at Warsaw,[24] that faces of greatest reticular density,
that is, those along which the points of the space-lattice are most
thickly strewn, are those which grow the most slowly, and therefore are
the best developed. This latter will be obvious on a little
consideration, for the faces of less reticular density which grow more,
tend in doing so to extend the boundaries of the faces of greatest
reticular density, and thus to enlarge those faces. Hence the usual
planes on a crystal must be those of high reticular density; and these
are such as are represented by the simplest indices, the faces most
dense of all in points being the primary ones.

But it has been shown from the researches of Miers that vicinal faces
are often produced in preference to these simple index planes of high
density, and such vicinal faces, although the nearest (in angular
position) of all possible faces to those simple index planes, are
themselves of excessively low reticular density, so much so that if
represented by indices at all they can only be indicated by very high
numbers, not such as we are accustomed to consider as in keeping with
the simple spirit of the law of rational indices. Taking the example
worked out most fully by Miers, the octahedral crystals of alum, it is a
fact that the cubic faces of highest reticular density are those of the
cube itself, then come in order those of the rhombic dodecahedron and
those of the octahedron. Hence, the density of octahedral faces is very
high. But those of the very low triakis octahedron, which Miers finds to
replace the octahedron faces so frequently as vicinal faces, are of
excessively low reticular density.

Miers explains the appearance of the latter instead of octahedral faces
by assuming that the supersaturated liquid in contact with the growing
crystal consists of the particles (molecules) of salt uniformly mingled
with those of water, the solvent, and that the act of crystallisation
consists of the escape of the water and solidification of the salt.
Consequently, the salt particles just before crystallisation cannot be
so dense as they are along primary planes of the crystal, as they are
separated by the water particles, which are presumably much more
numerous. Hence it is that they are not deposited along the planes of
high reticular density, but along vicinal planes of low density of
points. For instance, he shows that the shower of salt particles upon a
cube face would have to be so dense that there would be insufficient
room for the water particles. The density in a cube face is 114 times as
great as that in one of the vicinal planes observed. Now, 100 cubic
centimetres of solid alum weigh 172 grammes, and 100 c.c. of the
solution depositing crystals contain 9·74 grammes of alum. Thus the
density of the growing crystal of alum is nearly 18 times that of the
alum in the adjacent saturated solution.

Consequently the deposition of the salt particles, in a moderately quick
crystallisation, when insufficient time is afforded for the deliberate
escape of the water particles and for the orderly rearrangement of the
salt particles, occurs along vicinal planes instead of along the primary
planes. For it must not be forgotten that whenever it has the
opportunity of coming into operation there is a directive molecular
force of some kind, which controls the operation of crystallisation, and
which undoubtedly attempts to cause, and given adequate time and scope
succeeds in causing, the production of faces of high reticular density,
the fundamentally important primary faces of lowest indices, and which
are often those along which cleavage occurs. Wulff emphasises this in
saying (_loc. cit._, p. 461): “Bei der Krystallisation orientiren sich
die Molekeln auf den Flächen des Krystalles ganz gleichförmig durch den
Einfluss der Richtkraft der Krystallisation.” The more rapidly the
crystallisation occurs, however, the less chance is there for this force
to attain its ultimate object. More will be said about this directive
force in the next chapter, after we have studied the remarkable “liquid
crystals” discovered by Lehmann.

This highly interesting explanation of Miers, supported as it is by the
work of Wulff, and confirmed also in many respects by the observations
of the author, whose great aim throughout his investigations has been to
avoid the production of such vicinal faces, throws an important light on
the nature of the act of crystallisation. It renders the reason clear
why crystals which are very slowly grown from solutions only feebly
supersaturated and under conditions of absolute rest, protected from
either air currents or preventable earth tremors—conditions which the
author has taken quite exceptional pains to procure for the preparation
of the crystals used in the investigation of the sulphates and selenates
of the rhombic simple salt series and monoclinic double salt series—are
occasionally obtained quite free from any sign of such vicinal faces.
They are small perfect individuals exhibiting primarily the faces of
high reticular density, that is, the faces of the simple forms of low
rational indices; and these faces are absolutely plane, affording one
single brilliant image of the goniometer signal, which can be adjusted
with great precision to the cross-wires of the telescope. For the slower
the growth, the more time is afforded for the escape of the water
molecules, and for the salt molecules to deposit themselves as directed
by the molecular guiding force of crystallisation, along the planes of
high reticular density. In many of his experiments Miers expressly
stirred the solution, to prevent concentration currents, which had been
considered by Wulff of importance in the process, from coming into play
and causing unknown effects. Hence his experiments in which vicinal
faces were produced are not comparable with the author’s slow growths.

The work of Miers assists in the proof that the constancy of angle to
within one or two minutes of arc is a real property of the crystals of a
substance. For previously the frequent presence of vicinal faces rather
than the simple forms of high reticular density, and which had been
mistaken for the latter, had caused Pfaff in 1878[25] and others to
conclude that variations from the true crystal angle amounting to as
much as 30′ were of common occurrence as the result of strain during
deposition.

Brauns[26] in 1887 made a careful series of measurements of very good
octahedral crystals of lead nitrate, and found 13′ 20″ the largest
deviation of a good image from the theoretical position. He imputed it
to the action of gravity as a disturbing cause during deposition. The
researches of Miers have cleared away all this misconception, in proving
that the bright images referred to, taken for those of the simple
primary form, are not such at all, but vicinal faces of very low
reticular density.




                              CHAPTER XVI
                            LIQUID CRYSTALS.


We have seen in the foregoing pages that a crystal is usually a solid,
highly organised in a homogeneous manner, and, unless the symmetry be
developed to its highest extent, the crystal then belonging to the cubic
system, it is also in general anisotropic, that is, it exhibits double
refraction. Section-plates of it, more or less thin according to the
strength of the double refraction, exhibit colours in parallel polarised
light, and show the phenomenon of a single optic axis, or of two optic
axes, in convergent polarised light. Every variety of hardness, however,
is displayed, from that of the diamond down to that of a crystal as soft
as gypsum, and even softer. Moreover, many of the softer crystallised
substances develop the property of permitting one layer to glide over
another by gentle side pressure with a knife blade, when inserted in an
edge or face in an attempt to cut the crystal. Calcite and ice, for
instance, both possess such planes of gliding of the structural units
over one another in layers. There are also the border line cases of
crystals so soft as to be readily bent, and many well-known viscous
substances crystallisable only with great difficulty, some of which form
pliable crystals.

But in the year 1876 Lehmann discovered a new property in an already
remarkable substance, iodide of silver, AgI, namely, that at
temperatures superior to 146° C. it can flow like a viscous liquid,
while exhibiting several of the properties which are characteristic of
crystals. Silver iodide is dimorphous, exhibiting a hexagonal form at
the ordinary temperature, which persists up to 146°. But during the
heating to the latter temperature a regularly accelerating diminution of
volume occurs, the feeble expansion in directions perpendicular to the
axis being overbalanced by a considerable contraction along the axis,
both quantities having been accurately measured so long ago as the year
1867 by Fizeau, by means of his delicate interference dilatometer. This
contraction, so unusual an occurrence with increase of temperature,
culminates at 146°, according to Mallard and Le Chatelier, in a sudden
change of condition into a cubic modification, accompanied by absorption
of heat. Now Lehmann, studying this cubic modification of silver iodide
under a microscope which he had devised—specially adapted for
observations at temperatures higher than the ordinary, by being supplied
with the means of heating the object under observation—found that it was
not only plastic, but actually a liquid.

[Illustration:

  FIG. 102.—Lehmann’s Crystallisation Microscope.
]

The form of Lehmann’s “Crystallisation Microscope,” as now constructed
by Zeiss, is shown in Fig. 102. Its essential features are that the
glass object-plate, which is somewhat wider than the usual microscope 3
by 1 inches slip, is supported by little metallic columns at a height an
inch or more above the ordinary stage, and may be heated from below by a
miniature Bunsen burner, which is provided with a delicate graduated
gas-tap and is adjustable for its position, swinging in or out as
desired. The small Bunsen flame may be converted into a blowpipe flame
if necessary, an air-blast attachment to a mixing reservoir being
provided, to which the arm of the burner is hinged. Two cooling blasts,
connected with a gas-holder of air, are also provided, and are
adjustable to the most suitable symmetrical positions above the slide
for directing the cooling air on the part of the latter where the liquid
is situated. These arrangements enable the substance on the slide to be
rapidly or slowly heated or cooled at will. Electric connections are
also provided, in the event of the observer desiring to study the
behaviour of the liquid crystals under the influence of the electric
current.

Considerably later, in the year 1889, the attention of Lehmann was
called by Reinitzer to another similarly singular substance, cholesteryl
benzoate, which appeared to consist of an aggregate of minute crystals
which flow as readily as oil, while preserving many of the characters of
crystals.

In the next year, 1890, the substance para-azoxyphenetol, then recently
discovered by Gattermann, was observed by Lehmann to form a turbid
“melt” on fusion, which consisted of an aggregate of crystals flowing
with a mobility equal to that of water, and which take the form of
spherical drops showing a dark kernel inside, as shown at _a_ in Fig.
103, quite unlike a drop of ordinary liquid. The kernel disappears on
shaking, but reappears on coming to rest again. In polarised light the
drops show dichroism, that is two different colours in different parts
or directions, being divided into white and yellow parts, the yellow as
a pair of opposite approximately 60°-sectors, as indicated at _c_ in
Fig. 103. Under crossed Nicols they show a black cross, as represented
at _d_ in Fig. 103.

Now obviously these drops are doubly refractive, and their whole optical
behaviour corresponds to the arrangement of the molecules in concentric
circles, such as that suggested at _b_ in Fig. 103.

[Illustration:

  FIG. 103.—Liquid Crystals of Para-azoxy-phenetol arranged in Spherical
    Drops.
]

Another substance of like character, para-azoxy-anisol, was subsequently
found to behave similarly, and forms an excellent substance to use for
demonstration purposes. A reproduction of a photograph, kindly sent to
the author by Prof. Lehmann, of a slide of this substance is given in
Fig. 104, Plate XXII. It shows a characteristic field of such drops,
exhibiting white parts and yellow sectorial parts which photograph dark,
of para-azoxy-anisol mixed with a little para-azoxy-phenetol, oil and
resin (colophony), as seen under the polarising microscope with crossed
Nicols.

The next and possibly most interesting step in this remarkable series of
discoveries was made by Lehmann himself in the year 1894. He alighted on
the fact that ammonium oleate, crystallised from solution in alcohol,
affords a splendid example of flowing crystals, which are sufficiently
large to enable their habits to be studied in detail. The individuals
are almost invisible in ordinary light, owing to the refractive index of
the crystals and of the mother liquor being approximately the same. But
in polarised light, using crossed Nicols, they are clearly revealed as
steep double pyramids with more or less rounded edges. Their section is
nearly circular in consequence, and they exhibit optical properties of a
uniaxial character, the optic axis being that of the double cone or
bipyramid. A characteristic individual is shown at _e_ in Fig. 105. When
two of these flowing crystals approach each other, as at _a_ in Fig.
105, they coalesce to form a larger single individual, as is indicated
in stages at _b_, _c_, and _d_ in the illustration.

[Illustration:

  FIG. 105.—Liquid Crystals of Ammonium Oleate.
]

When the cover-glass, under which they are growing on a microscope 3 by
1 inch slip, is moved to and fro so as to distort these remarkable
bodies, which we may well hesitate to call crystals, the singular effect
is produced of causing them all to become similarly orientated, for the
extinction directions follow the direction of the pressure. They at once
seek to regain their original form, however, on cessation of the
disturbance. A slide of the bipyramids under pressure is shown in Fig.
106. In the black extinguished portions of the field the flowing
crystals are flattened, according to Lehmann, and arranged so that the
optic axis is in all cases perpendicular to the tabular crystals and the
glass plates and parallel to the axis of the microscope. The black parts
are separated by oily strips, as shown in another slide under
considerable pressure, represented in Fig. 107, which are composed of
the tabular crystals standing on end, with their optic axes parallel to
the plates. These strips polarise the more brightly the more truly the
crystals stand perpendicularly to the plates. The two conditions are
shown diagrammatically at _a_ and _b_ in Fig. 108.

[Illustration:

  FIG. 106.—Liquid Crystals of Ammonium Oleate under slight pressure.
]

[Illustration:

  FIG. 107.—Ammonium Oleate under considerable pressure.
]

[Illustration:

  FIG. 108.—Diagrammatic Representation of Arrangement of Molecules.
]

Lehmann believes the explanation of these singular phenomena to be that
the “liquid crystals” of ammonium oleate are composed of piles or layers
of thin plates perpendicular to the optic axis. Disturbance detaches the
plates from their piled positions over one another, and sets them
parallel to the glass plate, except in places, the oily strips, where
the plates stand upright, perpendicularly to the micro-slip and
cover-glass. Lehmann, indeed, goes further, and asserts that the
molecules themselves are anisotropic, and probably take the form of
plates.

[Illustration:

  FIG. 109.—A Crystal of Ammonium Oleate A, broken at B, each part
    repairing itself at C, perfect again at D.
]

An extremely interesting experimental observation of Lehmann’s with the
bipyramids of ammonium oleate is, that if one of them, for instance A in
Fig. 109, be broken into two parts, as at B, each part grows again and
completely repairs itself, becoming once more a perfect double pyramid,
as indicated in stages at C and D in the figure.

[Illustration:

  FIG. 110.
]

[Illustration:

  FIG. 111.

  Cruciform, Boomerang, and Arrow-head Twins of Ammonium Oleate.
]

[Illustration:

  _PLATE XXII._

  FIG. 104.

  Dichroic Crystal Drops of Para-azoxy-anisol.
]

[Illustration:

  FIG. 112.

  Rectilinear Liquid Prisms of Para-azoxy-benzoic Acid.
]

[Illustration:

  FIG. 115.—Tetragonal Astatic Magnet-system.
]

[Illustration:

  FIG. 116.—Cubic Astatic Magnet-system.
]

Twins of ammonium oleate are also shown in Figs. 110 and 111, the former
figure representing a twin of cruciform character, and the latter
exhibiting twins resembling a boomerang and an arrowhead respectively.

This substance, ammonium oleate, thus appears to be one of the most
remarkable and interesting of all the bodies yet observed to afford
liquid crystals. Many other oleates produce liquid crystals also, but
the ammonium salt is by far the most striking, and very convincing as to
the reality of Lehmann’s discovery.

Another substance of a different nature was discovered by Vorländer in
the year 1904, namely, the ethyl ester of para-azoxy-benzoic acid. A
characteristic microscope slide of it in ordinary light is shown in Fig.
112, Plate XXII., which is a reproduction of an actual photograph most
generously sent to the author by Prof. Lehmann.

The individuals are described by Lehmann, who further studied the nature
of the substance, as almost perfectly rectilinear prisms with nearly
sharply defined basal plane end faces. A singular fact about this
substance is, that when two individuals approach each other they arrange
themselves parallel with a jerk, and then flow into each other,
producing a single larger liquid crystal, and often with such rapidity
that the eye can scarcely follow the movements. These coalescences
appear to be occurring all over the field at once, with the production
of larger and larger crystals. Indeed, Lehmann likens it to a struggle
between the innumerable individuals, in which the smaller ones are being
continually eaten up by the larger.

Vorländer also prepared the ethyl ester of para-azoxy-cinnamic acid, and
Lehmann found it to be similarly interesting. The substance separates
from a solution in monobromonaphthalene in uniaxial prisms or
hemimorphic pyramids, the edges and solid angles of which are more or
less rounded, and which appear colourless in the direction of the axis
and yellow in all other directions. When pressed between the cover-glass
and the micro-slip on which the crystallisation is proceeding,
extinction of the light occurs throughout the whole mass when polarised
light is being employed and the Nicols are crossed. For throughout the
entire substance the particles—whether they are the molecules themselves
as Lehmann asserts or aggregations of them in the form of
ultramicroscopic crystals—arrange themselves with their optic axes (the
crystals being uniaxial) perpendicular to the cover-glass and
micro-slip, as in the case of ammonium oleate. Lehmann’s theory is that
the molecules themselves are tabular perpendicular to the axis, as in
the case just referred to, and that they are thus readily coerced by the
pressure of the flat cover-glass to take up positions parallel to it.

Two further reproductions of photographs, taken in polarised light, of a
somewhat remarkable character, which have been placed at the author’s
disposal by the courtesy of Prof. Lehmann, are given in Figs. 113 and
114, Plate XXIII. Fig. 113 represents numerous doubly refractive and
dichroic strips marking the boundaries of elongated individual crystals
of the substance dibenzal benzidine, and affords a graphic idea of the
real character of the double refraction displayed by liquid crystals.

[Illustration:

  _PLATE XXIII._

  FIG. 113.—Elongated Liquid Crystals of Dibenzal Benzidine, showing
    Double Refraction and Dichroism.
]

[Illustration:

  FIG. 114.—Spherical Liquid Crystals of Para-azoxy-anisol, showing
    Interference Colours under Crossed Nicols as the Effect of
    Compression.
]

Fig. 114 represents the effect of compression on para-azoxy-anisol, and
demonstrates very clearly the distribution of the interference colours
due to double refraction.

We are thus face to face in these remarkable experiments with some new
facts concerning the nature of crystals. For we pass here into the
borderland between ordinary liquids—singly refractive and structureless,
in which the molecules are rolling over each other with every possible
orientation—and solid true crystals possessing homogeneous structure,
and the basis of which is a space-lattice arrangement of the chemical
molecules, determinative of the system of symmetry displayed. In this
wonderful borderland we certainly have had revealed to us, by the genius
and persistency of Lehmann, liquids which possess many of the attributes
of crystals, such as definite orientation of the ultimate particles,
double refraction, and optic axes. These are undoubtedly solid facts
which require to be faced.

Whether Lehmann’s theory is to be accepted in full can only be decided
after much more investigation by several independent investigators. We
are now becoming familiar with the phenomena, as they have naturally
excited immense interest in all scientific circles, and demonstrations
of many of the experiments of Lehmann have been given in this country by
Dr. Miers, Prof. Pope, and others, and particularly by Messrs Zeiss,
with their new high temperature microscope, a description of the use of
which for the projection of liquid crystals on the screen will presently
be given. Prof. Lehmann himself has described the phenomena so clearly
and fully that it is quite easy for others to repeat his experiments,
and doubtless time would often be much better spent in doing so than in
criticising points of theory without observing the phenomena at first
hand. It frequently happens, in the inevitable march of scientific
progress, that striking new facts, such for instance as the discovery of
the composite nature of the chemical atom, are apt to cause either
alarm, even panic, as to cherished theories, or else unreasoning
scepticism. The happy mean between these two modes of receiving such
facts, the open philosophic mind, ever ready to widen the scope of the
horizon when a novel supposition is indubitably proved to be a real
fact, and to assimilate that truth into the theory, widening
correspondingly the scope of the latter if needful, is obviously the
ideal thing to cultivate, and one which eventually finds itself in
harmony with the authenticated final results of the new discoveries. It
usually happens that too sweeping conclusions are at first drawn from
such new facts, but time, with its further wealth of experience,
especially the accumulation of experimental _data_ which it brings in
its train, soon levels these down and relegates the facts to their
proper positions in the great scheme of natural knowledge.

Lehmann’s view is that the ordinary effect of surface tension to cause
truly liquid particles to assume the spherical “drop” form is resisted
by a special force, which he terms “Gestaltungskraft,” and which we may
perhaps translate “Configuration-determining force.” This force he
considers is not identical with that of elasticity, but is that force by
virtue of which a “flowing crystal” continually seeks, while freely
swimming in the mother liquor or fused liquid, to take up its normal
configuration. Even if a spherical drop could be cut out of it, the
sphere would at once become a rod, prism, or pyramid or whatever the
normal configuration of the flowing crystals of the substance in
question might be.

The much debated term “liquid crystal” has been given by Lehmann to the
normal configuration of each of the now considerable number of
substances which have been discovered to exhibit the phenomena of
flowing crystals. The latter appellation “flowing crystal,” which
Lehmann also uses, appears to the author to be in many ways more
suitable, however, and would avoid much of the criticism which has been
levelled at the term “liquid crystal.”

As already indicated, Lehmann attributes the whole phenomena to a
fundamental cause, namely, anisotropy (optical dissimilarity in
different directions) of the molecules themselves, which he considers
must cause self-restoration of the structure after disturbance, a
process which he terms “spontaneous homœotropy.” He considers that it is
the molecular configuration-producing force, connected with the tabular
form of, and directionally differentiated distribution of energy and
force in, the single chemical molecules, which maintains the inner
structure of the flowing crystal in position. The polyhedral outward
form thus appears to be a necessary consequence of the inner structure,
on this basis that it is a force resident in the molecules themselves
which produces the structure.

Now the revelation of new facts, as startling as those which are now
experimentally fully confirmed concerning flowing crystals, must
inevitably cause searching reflection as to whether the magnificent
geometrical work on the 230 homogeneous structures, and their
development in actual fact in the 32 classes of crystals, is to stand or
to be seriously affected. Again the author ventures to express the
opinion, that just what happened in regard to the historic differences
between the schools of Haüy and Mitscherlich, will in all probability
again occur, namely, both extreme views will be shown to depend more or
less on real facts, and other connecting facts will eventually be
revealed which will completely reconcile the two series with each other.
In the author’s opinion, the geometrical work will stand, as the grand
generalisation it really is. But it will be interpreted in the future
without the somewhat arbitrary assumptions which have more or less
accompanied it. From these it will be freed, and then rise purified and
elevated to its real dominating position in regard to crystal
morphology.

Lehmann, with the natural enthusiasm of the discoverer of one of the
most remarkable facts for which the last few decades have been famous,
may have carried his theory too far, and particularly in that part of
his work, to which the author has not hitherto referred, in which he
describes certain phenomena of flowing crystals as akin to the movement
of living organisms such as bacteria, and thus brought even some of the
sound facts under the criticism of the sceptic more than might otherwise
have been the case. He may also have made his theory far more
revolutionary than is essential. But the one incontrovertible thing
stands out plainly, namely, that the “flowing crystals” with which he
has made us acquainted are an indubitable experimental fact. Flowing
crystals are produced, however, by a relatively few substances of very
complex molecular constitution, involving a large number of atoms in the
molecule; they are mostly compounds of carbon, and in number possibly
not one per cent. of the innumerable substances known to produce
ordinary solid crystals. That the theory of crystal structure can
eventually be made to include these few remarkable substances is highly
probable, when many more facts have been accumulated.

Lehmann would appear to have made one point very clear, which at once
removes an objection long felt by the author to the theory of crystal
structure as it stands at present, namely, that the chemical molecule is
endowed with a directive orientative force, which is certainly concerned
in crystallisation. To assume, as has been done, just because it is not
necessary from the point of view of the geometrician in developing his
possible homogeneous structures, that no directive force is operative in
crystallisation, and that all is a mere question of the most convenient
mechanical packing of the molecules, is, in the author’s opinion, going
beyond what the experimental facts justify. If Lehmann’s discovery of
flowing crystals does nothing more than return to the molecule the
property always hitherto attributed to it, of possessing in itself some
directive force by reason of which it arranges itself homogeneously by
mutual accommodation with its similarly endowed fellow molecules, when
its motion in the liquid state has been sufficiently arrested by its
approach to its fellows within the range of molecular action (four or
five molecular diameters), either by cooling or the falling out of
previously separating solvent molecules, it will have achieved a notable
thing.

What does occur at the moment of crystallisation is at the present time
one of the most interesting unsolved questions in crystallography, and
one calling most urgently for solution. Attention was directed to the
problem in the last chapter, in connection with the suggestive work of
Miers on vicinal faces. It was shown that it was only when the directive
force had time to come properly into operation that the primary faces of
fundamental importance were produced, and that when the crystallisation
was rapid vicinal faces formed instead. Lehmann believes that a single
kind of chemical molecule is only capable of producing a single specific
space-lattice, and that polymorphism is due to alteration of the
molecules themselves at the critical temperature of transformation. He
showed so far back as 1872 that this limit could be actually observed
under the microscope, as a definite line of demarcation between the two
varieties as the temperature fell, one side of the field attaining the
critical temperature slightly before the other, and the defining line
between the two kinds thus travelling over the field. Internal friction
did not appear to Lehmann to enter into the question at all, as he
considered it would have done if a rearrangement of the molecules were
the sole cause of the change. The molecules themselves, he states, must
have been undergoing change, and such rearrangement of them as occurred
must have been due to that fact.

[Illustration:

  _PLATE XXIV._

  FIG. 117.—Arrangement of Astatic Magnet-systems in a Plane.
]

[Illustration:

  FIG. 118.—Arrangement of Astatic Magnet-systems in Space.
]

Lehmann suggests a very interesting explanation of the molecular
orientative force of configuration, namely, that it is due to the action
of the electronic corpuscles (forming the elementary atoms) rotating in
the molecule. For the molecules of flowing crystals behave like freely
suspended astatic systems of magnets, which are constantly setting
themselves, even while moving about, in a crystalline space-lattice. He
suggests that the molecules are really magnets the poles of which
mutually attract and repel one another; that two equal magnetic
molecules are arranged alongside with opposite poles against each other,
thus mutually binding each other, or that four horse-shoe magnets may be
arranged with opposite poles together, in a tetragonal astatic system,
as shown in Fig. 115, Plate XXII. The latter may be grouped in space in
a cubic astatic system, as represented in Fig. 116 on the same Plate;
while Figs. 117 and 118, Plate XXIV., are further suggestive of how a
homogeneous structure of such astatically distributed molecules can be
built up, Fig. 117 representing a single plane of them, and Fig. 118 the
complete arrangement in space.

An astatic system of molecules of this nature would have lost all power
of attraction by a magnet, and the fact would thus be accounted for that
no striking crystallographic results have ever attended experiments on
crystallisation in a magnetic field. Astatic systems, however, such as
that shown in Fig. 115, would certainly arrange themselves in
space-lattices. For such parallel arrangements would, in general,
involve differences in different directions, with regard both to
internal friction and to the power of thermal expansion and of such
regular dilatational or other deformational changes as we know are
provoked by different physical conditions of environment. These
differences would naturally, in turn, give rise to external polyhedral
form.

Lehmann then goes on to point out that either electric currents or
mechanically moved quantities of electricity, such as moving negative
electronic corpuscles, can give rise to just such magnetic effects, and
he suggests that these corpuscles are the true cause. He supposes that
the directive forces result in astatic combinations which find their
equilibrium when the latter have taken up their positions at the eight
corners of a cube or other elementary parallelepipedon of one of the
fourteen possible space-lattices, the positive atoms being encircled
spiral-wise by the negative electronic corpuscles in alternately
opposite directions. Such parallelepipeda would seek homogeneous
repetition by virtue of the fact of the corners exhibiting alternating
polarity.

These theoretical ideas of Lehmann have naturally called forth much
discussion, criticism, and scepticism. But, so far, his experimental
facts have been fully substantiated by further investigation. Much more
practical work is urgently required, however, before the subject can be
considered as laid on a secure foundation. So much may be said, however,
that it is clear that we must concede the existence of a directive force
of crystallisation, and not be led by the pure geometry of the subject
of crystal structure to ignore facts of such interest and undoubted
importance as have been brought into prominence by the remarkable work
of Lehmann.

A further interesting contribution has recently been made by
Vorländer[27] to the facts regarding the relationship between chemical
constitution and the formation of liquid crystals. It must have already
struck the reader that most of the substances which exhibit liquid
crystals are composed of a large number of chemical atoms, being either
long-chain compounds of the fatty acids or complex derivatives of the
hydrocarbon benzene, C_{6}H_{6}; also that many of the latter are “para”
compounds, that is, derivatives in which the substitution groups are
inserted in the benzene ring of six carbon atoms in the “para” position,
which is that at the opposite corner of the hexagon to the carbon atom
to which a substitution group has already been attached. This renders
the para compounds the most extended in a straight line of all the
benzene derivatives. Now Vorländer finds that a particularly favourable
condition for the production of liquid crystals is a linear structure of
the molecule. As the para substitution products of benzene derivatives
possess this elongated structure, many of them exhibit the development
of liquid crystals. The more linearly extended the structure becomes,
that is, the longer the straight chain of atoms is, the more favourable
become the conditions. The advent of a third substitution group,
however, which would have the effect of producing a kink in the chain,
or of bending it, appears to destroy the possibility of the production
of liquid crystals. This interesting observation may afford the key to
many of the extraordinary phenomena of liquid crystals which have been
described, and is undoubtedly one of prime importance. Further
favourable conditions for the formation of liquid crystals, according to
Vorländer,[28] are the aromatic character, and the presence of the
doubly-linked carbon and nitrogen groups C:C, C:N, and N:N, which are
usually so rich in energy.

The idea of the formation of a specific crystalline homogeneous
structure, merely because the mechanical fitting-in of the molecules
occurs with the minimum of trouble or maximum of ease for this
particular type of all the 230 possible types, is certainly not
applicable to the case of Lehmann’s liquid crystals. With this,
moreover, is also connected the question of softness or hardness of
crystals, which was referred to at the opening of this chapter. For the
so-called liquid crystals are extreme cases of softness, and yet in
these cases the molecules must still be arranged in accordance with the
internal structure of a crystal, either parallel or enantiomorphously
definitely orientated with respect to each other, for otherwise it is
not possible to account for the optical properties resembling the
orientated ones of a crystal. Yet the condition being that of a liquid,
the molecules must be able readily to pass and roll over each other, and
hence cannot be at the close quarters where mere “fitting-in” comes into
play.

Again, as has been pointed out earlier, many soft crystals, even such as
calcite, which are only relatively soft, attaining the position of as
much as four in the scale of hardness, readily exhibit the property of
being deformable upon glide-planes. The molecules in these cases have
been shown to undergo a movement which has two components, a
transference and a rotation, a fact which has been thoroughly
substantiated by optical investigations of the parts of the crystal
concerned before and after gliding. There cannot, therefore, have been
merely “fitting-in” of the molecules, but their orientated positions
must have been determined and maintained by the organising force, which
is probably purely physical and not chemical, but is nevertheless the
cause of crystallisation; it draws the molecules within a certain range
of each other, corresponding to and dependent upon the temperature,
causes or enables them to arrange themselves in the marshalled order of
the particular type among the 230 possible arrangements, and keeps them
at the same time from approaching nearer to each other than within these
prescribed limits corresponding to the temperature. It is doubtless
within these limits that gliding can occur parallel to such planes as
leave the molecules most room for the purpose, and which are directions
of least resistance.

[Illustration:

  FIG. 119.—Lehmann’s Crystallisation Microscope arranged for
    Projection.
]

Connected with this important question is the principle enunciated by
Bravais, as a result of his discovery of the space-lattice, that
cleavage occurs most readily parallel to those net-planes of the
space-lattice which are most densely strewn with points. The force just
referred to, whether we term it cohesion or otherwise, is obviously at a
maximum within such a plane, and at a minimum in the perpendicular
direction where the points are further off from each other. Moreover, it
has been fairly well proved also, from the experiments of Wulff,
described in the last chapter, that the direction or directions of
maximum cohesion are those of slowest growth of the crystal; so that
faces parallel to those directions become relatively more extended owing
to the more rapid growth of other faces on their boundaries, and thus
become the most largely developed and confer the “habit” on the crystal.
All these are facts so important as evidences of a controlling force at
work in crystallisation, that a _purely_ geometrical theory of the
formation of crystals which would make “facility of fitting-in” of the
molecular particles its chief tenet, obviously does not tell us the
complete story. Hence the author desires to utter a warning against
going too far with the pure geometry of the subject. The geometricians
have done a grand work in providing us with the thoroughly well
established 230 types of homogeneous structures, as a full and final
explanation of the 32 classes of crystals, and so far their results are
wholly and unreservedly acceptable.

The phenomena of “liquid crystals” lend themselves admirably to screen
demonstration, for which purpose an excellent improved form of the
crystallisation microscope of Lehmann, shown in Fig. 119, is constructed
by Zeiss, and its actual use in the projection, with the aid of the
well-known Zeiss electric lantern, but specially fitted for the purpose,
is shown in Fig. 120.

A magnification of 600–700 diameters on the screen is very suitable,
employing a Zeiss 8–millimetre objective without eyepiece. This
objective affords directly a magnification of 30 diameters. For ordinary
eye observation an eyepiece magnifying 6–8 times is added, thus
affording to the eye a magnification of about 200 diameters.

[Illustration:

  FIG. 120.—Zeiss Apparatus for the Projection of Liquid Crystals.
]

The lantern is supplied with a self-feeding electric arc lamp, ensuring
a steady light. A collective lens of extra light-gathering power is
fitted in front as condenser, and from it proceeds a light-tight tube
provided with a water cell to filter out most of the heat rays which
accompany the light. The electric lantern with Brockie-Pell or Oliver
self-feeding arc lamp, shown in Figs. 71 and 79 (pages 186 and 202), is
also equally suitable, and with the water cell, and parallelising
concave lens removed from the large Nicol polariser, affords a parallel
beam of the same character as the Zeiss apparatus. The microscope stands
on a sole plate provided with levelling screws, and is naturally
employed in the vertical position for such work with fused substances. A
mirror inclined at 45° at the foot of the microscope directs the
parallelised rays from the optical lantern through the microscope, and
another above the optical tube reflects them to the screen.

The heating apparatus consists of a form of miniature Bunsen burner
fitted with blowpipe blast, the respective pressures of gas and air
being regulated by means of two taps with graduated arcs for obtaining
greater delicacy of adjustment. The tabular plate seen to the left in
Fig. 119 is the graduated semi-circle of the two taps; below it is seen
the cylindrical mixing chamber for gas and air, in the event of the
necessity for using the Bunsen as a blowpipe. There are two separate
attachments for indiarubber tubes to this cylinder, conveying
respectively gas and air. Above the object stage a double air-blast is
fitted, each tube of which is hinged with a universal joint, so that it
can be readily adjusted to any desired position on either side of and
above the glass plate (supported on little metallic uprights) on which
the experiment is being conducted. A polarising Nicol prism and an
analysing Nicol, both constructed in a manner which protects them from
the effects of heat more effectually than is the case with the usual
form, are provided for obtaining the projections in polarised light. The
objective and analysing Nicol, as well as the substage condenser, are
also specially protected from injury by heat, by being surrounded with a
water jacket, supplied with running water, and a disc-like screen just
above the objective assists in deflecting the heat rays from the optical
tube and its Bertrand lens and other usual fittings. The miniature
Bunsen flame is usually brought about an inch below the object-plate,
and the size of the flame can be regulated with the utmost precision, so
that a fairly constant temperature can be obtained for a considerable
time. With the aid of the blowpipe air-blast temperatures up to 700° C.
can be safely employed.

The microscope shown in position on the projection apparatus in Fig. 120
is a still more recent form introduced by Zeiss, embodying several
further conveniences and improvements.

The following substances lend themselves particularly well to projection
purposes. Para-azoxy-anisol with resin, which exhibits the phenomenon of
rotating drops; cholesteryl acetate, which affords a fine example of
spherical liquid crystals; paraazoxy-phenetol with resin, which gives
beautiful interference colours; and the acetyl ester of
para-azoxy-benzoic acid with resin, which shows the uniting of crystals
to form larger and larger individuals.

Perhaps the most interesting and beautiful of all is cholesteryl
acetate, a characteristic field of which is shown in Fig. 121 on Plate
XVI., facing page 208. It is interesting that on this Plate XVI. there
are represented the very hardest and the softest of crystals, namely,
diamonds and liquid crystals. In order to obtain the finest effect the
heating and cooling should be carried out very slowly. The little Bunsen
burner, with a very minute flame, is first placed under the slide, and
allowed to act until the substance melts and forms a clear liquid. The
gas jet is then removed and the air-blasts, both of which are
simultaneously actuated when the tap controlling them is turned, are
very gently brought into operation, one on each side of the centre of
the slide, there being a good working distance of a quarter of an inch
or more between the slide and the objective. The cooling is thus brought
about very slowly. The Nicols should be crossed, and at this time the
field is quite dark, the liquid substance being at this temperature
(well above 114.5° the ordinary melting point) an ordinary singly
refractive liquid.

As soon as the temperature has become reduced to that at which the
particular modification of cholesteryl acetate is produced which forms
liquid crystals, spots of light make their appearance at various points
in the field, and each expands into a beautiful circular and more or
less coloured disc marked by a rectangular sectorial black cross, which
latter is well shown in the illustration (Fig. 121). These beautiful
apparitions continue to occur, and each to expand to a certain size,
which is rarely exceeded, until the whole field becomes filled with the
wheels or crossed discs, the general effect very much in some respects
resembling that afforded by a slide of the well-known polarising
substance salicine. These discs, however, are liquid, being spherical
drops, of the structure already described and illustrated in Fig. 103,
and that this is so is at once made apparent on touching the cover-glass
with a pen-knife or other hard pointed substance, which immediately
causes them to become distorted. They recover instantly their shape
again when the pressure is removed. When the cooling, moreover, has
proceeded still further, there is a sudden change, and acicular solid
crystals shoot over the screen, tinted with all the colours of the
spectrum, until the field is full of them, the ordinary solid
modification of the substance having then been produced. The experiment
may be repeated with the same specimen of the substance, mounted on the
same slide, covered with the usual thin cover-glass, time after time for
months, at reasonable intervals.

In concluding this chapter it may be mentioned that absolute proof of
the double refraction of the liquid crystals of several different
substances, derivatives of cinnamic acid, has been afforded during the
year 1910. For direct measurements have been carried out by two
independent investigators, Dorn and Stumpf, of the two refractive
indices corresponding to the ordinary and extraordinary rays in each
case, the crystals being uniaxial.




                              CHAPTER XVII
  THE CHEMICAL SIGNIFICANCE OF CRYSTALLOGRAPHY. THE THEORY OF POPE AND
                           BARLOW—CONCLUSION.


Nothing in connection with the subject of crystallography is more
surprising than the neglect and apathy with which it has been for long
treated by the chemical world. That crystalline structure is intimately
related to chemical constitution will have been made abundantly plain
during the course of this book. Yet in spite of the great work of
Mitscherlich, essentially a chemist, and of a large amount of striking
work which has been steadily accumulating during the last thirty years,
with results of vital importance to chemistry, it is only at the
eleventh hour that chemists are really awakening to the vast
significance which crystal structure has for them.

The explanation undoubtedly is, that the long interregnum of conflicting
investigations, doubt, and controversy, which followed the work of Haüy
and Mitscherlich, and preceded the beginning of really accurate and
painstaking investigation of an organised and systematic character, had
caused chemists to regard with more or less of indifference the work of
the crystallographers. Added to this we must remember that the subject
of crystallography has hitherto been taught, when taught at all, merely
as an appanage of mineralogy, although the pure chemical substances
which crystallise well infinitely outnumber the naturally occurring
minerals, and the results afforded by them frequently possess a much
greater value by reason of the purity of the substances and their more
definite chemical constitution. Also the mathematical and geometrical
side has usually been unduly emphasised, and carried on in lectures
without any practical goniometrical work at all. Moreover, the current
text-books have often proved forbiddingly full of calculations and
formulæ, and of the obsolete and unenticing symbols of Naumann.

At last we have come to see that the subject is one of fascinating
interest when its study is commenced in a practical manner from the
beginning, armed from the very first lesson with the goniometer. The
crystal itself is then our main and highly interesting study; its
exterior form unravels itself in a most delightfully simple manner when
we follow the arrangement of its faces in zones on the goniometer
itself; and its symmetry becomes immediately patent to our eyes in all
ordinary simple cases, when we construct for ourselves its plan in a
stereographic projection, drawn at first in freehand while still at the
goniometer. The calculations also become perfectly simple when we have
learnt that only the simplest of the very easy formulæ of spherical
trigonometry are required, and which a knowledge of only elementary
plane trigonometry enables us to apply. Aided by a few very helpful
rules, such as those of Napier for calculating right-angled spherical
triangles, and the rule of the anharmonic ratio of four poles in a
zone—which, when the positions of three crystal faces of the zone are
known, at once enables us to calculate the situation of any fourth face
of the zone—we have at once a stock-in-trade which carries us over all
difficulties in the way of calculation, and relegates this side of the
work to an altogether subordinate position, although accuracy in
carrying it out is, of course, absolutely essential and even vital.

Especial interest has recently been attracted on the part of chemists to
the bearing of crystallography on their science by a remarkable theory
which has been advanced by Pope and Barlow, connecting the internal
structure of crystals with chemical valency, the power which the atom of
a chemical element possesses of combining with the atoms of other
elements, and which is generally expressed by the number of atoms of a
monad element, such as hydrogen or chlorine, with which it is capable of
combining chemically. Thus the electro-positive metal potassium is said
to have monad valency because it is capable of combining with one atom
of the electro-negative element chlorine to form the salt potassium
chloride, KCl; calcium possesses dyad valency because it can unite with
two atoms of chlorine to form calcium chloride, CaCl_{2}; aluminium is
triadic as it can combine with three atoms of chlorine producing
aluminium chloride, AlCl_{3}; carbon is a tetrad because it can take up
four chlorine atoms, forming carbon tetrachloride, CCl_{4}, and
phosphorus a pentad as it can fix five with production of phosphorus
pentachloride, PCl_{5}; while sulphur is a hexad because it can take up
as many as six atoms of chlorine, forming sulphur hexachloride, SCl_{6}.

Occasion has already been taken in Chapter XI. to refer to the able work
of Prof. Pope with regard to optically active carbon compounds, and in
Chapter IX. the important contribution of Mr Barlow to the completion of
the theory of the homogeneous partitioning of space has likewise been
discussed. In collaboration these two investigators have now propounded
a theory which connects the chemical and geometrical sides of
crystallography in a somewhat startling manner, which has naturally
aroused very considerable discussion, and which, whether right or wrong,
cannot fail to have the best results in attracting investigators to the
subject.

Starting from the facts which have now been laid down in this book as
having been firmly established by the most careful measurement and
experimental investigation—notably the constancy of the interfacial
angles of the crystals of the same substance, the fixed positions of the
atoms or their spheres of influence in the molecule and in the crystal,
and the arrangement of the molecules in space-lattices and the atoms in
point-systems—Pope and Barlow assume that valency, the expression of the
relative combining power of the chemical elements, is a question of the
size of the sphere of influence of the atom of an element, and that the
relative sizes of such spheres of influence determine the modes in which
they can be packed, that is, the nature of the homogeneous crystal
structure which they can build up. The theory consequently renders the
chemical phenomenon of valency and the physical phenomenon of
crystalline form mutually interdependent.

It will thus be apparent that the essence of their conception is that
the chemical molecule may be considered as made up of a number of
spheres corresponding to, and representing, the spheres of influence of
the atoms composing it, and that the volume of each sphere is roughly
proportional to the valency of the atom which it represents. They then
assume that the sum of the valencies of the atoms present in the
molecule may be substituted for the molecular volume, and the quantity
thus arrived at is termed by them the “valency volume.” By using the
valency volume instead of the molecular volume in the author’s formulæ
for calculating the molecular distance ratios, which have been shown in
Chapter X. to afford the relative dimensions of the unit cell of the
space-lattice, and the distances of separation of the centres of gravity
of the molecules from each other along the directions of the crystal
axes, Pope and Barlow arrive at new ratios, which they term equivalence
parameters. By the use of these latter they have attempted to account
for the crystalline structure of a number of substances—chiefly organic
compounds, in the investigation of which Prof. Pope has proved so
adept—which are connected morphotropically in the manner described in
Chapter VIII., and of others which are still less intimately connected.

Unfortunately, the equivalence parameters do not make clear the
relationships in an isomorphous series, as do the molecular distance
ratios; for they are, from their very nature and mode of derivation,
almost identical for all the members of an isomorphous series, the
valencies of the interchangeable elements being the same. The molecular
distance ratios have also the great advantage of being derived from the
three measurements which have now been brought to the highest pitch of
experimental accuracy, namely, atomic weight determinations, density
determinations by the Retgers immersion method, and goniometrical and
physical measurements (optical and thermal) with instruments now
available of the utmost refinement. Structural constants thus derived
are obviously of especial value. It would thus appear that the theory
requires modification so as to take account of the experimentally proved
regular increase in volume and in the directional dimensions of the
structural unit cell of the crystal space-lattice, when one element of
the same family group and of the same valency is interchanged for
another. Indeed, as the theory stands at present it entirely ignores and
fails to offer any explanation of the highly important physical property
of density, specific gravity. That this physical constant, and the
equally important constant molecular volume, derived by dividing the
molecular weight by the density, possess a real and significant meaning
in isomorphous series, formed by the interchange of elements of the same
family group, has been clearly proved in Chapter X. This fact is,
indeed, so obvious that any further development of the theory must of
necessity take account of it.

A precise statement of their conception has recently been made by Prof.
Pope in an excellent Report on the Progress of Crystallography, issued
by the Chemical Society early in the year 1909. He states that they
(Pope and Barlow) “regard the whole of the volume occupied by a
crystalline structure as partitioned out into polyhedra, which lie
packed together in such a manner as to fill the whole of that volume
without interstices. The polyhedra can be so selected that each
represents the habitat of one component atom of the material, and are
termed the spheres of atomic influence of the constituent atoms. Up to
this point no assumption is made other than that clearly indicated by
the result of crystallographic measurements, namely, that each atom
present in a crystalline structure exerts a distinct morphological
effect—or, what is the same thing, appropriates a certain definite
volume. The assumption is next made that the crystalline structure,
which is resolvable into individual molecules and ultimately into
individual atoms, exists as such by reason of equilibrium set up between
opposing attractive and repulsive forces operating between the component
atoms, and that this equilibrium results in the polyhedra representing
the spheres of atomic influence assuming shapes which are as nearly as
possible spherical.... The polyhedra thus arrived at may be regarded as
derived by compression of a close-packed assemblage of deformable,
incompressible elastic spheres,[29] the compression sufficing for the
practical extinction of the interstitial space. When such an assemblage
is released from pressure it is evident that in place of polyhedra, the
shapes of which approximate as closely as possible to the spherical,
closely packed spheres are presented; the distances between the sphere
centres can be substantially in the same ratios as the distances between
the centres of the corresponding polyhedra in the unexpanded mass, and
the equilibrium condition of maximum sphericity of the polyhedra will be
presented in the expanded mass of spheres by the existence of the
maximum number of contacts between spheres. The whole method of treating
the primary assumption thus resolves itself into finding close-packed
assemblages of spheres of various sizes representing by their relative
volumes the spheres of influence of the component atoms of any
particular crystalline structure.”

Some very interesting evidence of the validity of their fundamental
assumption of spheres of influence of the component atoms as the
ultimate structural units is brought forward. They show that there are
two modes of closely packing equal spheres, which give rise respectively
to a cubic and a hexagonal crystal structure, the latter having a
specific axial ratio of the vertical to the three equal equatorial
horizontal axes; and that the chemical elements which are solids and
crystallise, and the structural units of which can naturally be assumed
to be equal spheres, being those of the similar atoms of the same
chemical element, do practically all crystallise either in the cubic
system or in the hexagonal system with the specific axial ratio
indicated by them. The theory as it concerns chemical valency is
obviously not affected by these interesting facts, as the spheres of
influence present are those of the identically similar atoms of the same
element. But the theory has received considerable support from the
results of the investigation of a number of carbon compounds, chiefly
derivatives of benzene.

Thus, in spite of the very wise decision of Barlow at the time of
developing his theory of the homogeneous partitioning of space, to keep
quite clear of attributing shape to the structural unit atoms or
molecules, and to consider them as points, he, in common with the other
contributors to that splendid geometrical work, appears driven to
consider the question of shape when, in collaboration with his chemical
colleague, he endeavours to apply his geometrical results to the
practical problems of chemistry. It may be inevitable that we cannot get
away from the idea of shape of the fundamental structural units. Yet the
moment we do admit the idea, and begin to talk of polyhedra, or even of
spheres, in close or any other packing, we enter the debatable land,
concerning which the experimental evidence is as yet but shadowy and
liable to many interpretations. Hence it is that we have the
parallelohedra of Von Fedorow, having volumes proportional to the
molecular volumes, the more general plane-faced cell of fourteen faces,
the tetrakaidecahedron of Lord Kelvin and its deformed derivatives, and
now the polyhedra of Pope and Barlow. The more indefinite
“Fundamentalbereich” of Schönflies appears to be left behind, and we
have embarked on a definite course of attributing shape to the component
atoms or their regions of influence in the crystal structure. Von
Fedorow has developed his particular theory in a very masterly manner,
and with the aid of it professes, and with very considerable success in
many cases, to determine the correct mode of setting up a crystal for
truly comparative descriptive purposes, and has derived therefrom a
remarkable method of crystallochemical analysis.

Moreover, there is yet another view, that of Sollas,[30] that the
packing of the molecules is a more open one altogether, a view to which
he has been guided by consideration of the molecular volume. Sollas has
offered some remarkable explanations of crystal structure, notably in
the case of the dimorphous forms of silver iodide. The abnormal
contraction which occurs on heating this interesting substance, and its
sudden transformation at 146° from the ordinary hexagonal into a cubic
modification as discovered by Lehmann, appear capable of very clear
explanation on the basis of his theory. According to this theory of
Sollas the volumes of the spheres of influence of the atoms of the
different elements of the same family group, such as those of the group
of alkali metals or those of the halogens, chlorine, bromine, and
iodine, vary progressively in a manner which is dependent on the atomic
volumes of the elements, which have a real comparative significance when
the elements belong to the same family group.

It is probable that there is a considerable substratum of truth behind
these various apparently conflicting views, and what is now required is
that the germ of real fact shall be winnowed from the husk of fallacious
speculation, just as occurred in the happy recent settlement of the old
issue between Haüy and Mitscherlich. As in that case, moreover, it will
be experimental work of superlative accuracy which can alone offer the
desirable evidence on which a satisfactory arbitration can be founded.

It is thus obvious that we have now arrived at a stage in the history of
crystallography when more experimental data, and many more measurements
of the most carefully conducted character, on pure materials and
excellently developed crystals, are most urgently needed, in order to
decide these important, indeed fundamental, questions, the present state
of which the author has endeavoured to present with judicial
impartiality. When one looks around, and sees the almost complete lack
of opportunities for the training of investigators in this rapidly
growing branch of science, the importance of which to chemistry and
physics is increasing every day, while the field is ripe for the
harvesters, one is inclined to feel depressed with the thought of the
opportunities which are being lost. Our country has, in this science at
any rate, a fine record, having with few breaks led the van of progress
from the time of Wollaston, the inventor in the year 1809 of the
reflecting goniometer, and of Miller, the originator of our method of
describing crystals and the pioneer of accurate experimental work, down
to the present day. It may be, also, that our country’s reputation is
safe at this moment. But it is in the hands of a band of investigators
so small, and often of the private and not professional nature, carrying
on the work for sheer love of it and deep interest in it, that the
wonder is that so much has been done, and it is the provision for
carrying on our national tradition of leadership in crystallography in
the future that is a matter for the deepest concern.

If the perusal of this book should have awakened sufficient interest in
the minds of some of its readers to prompt them to offer themselves as
recruits to this small band of investigators, and especially if it
should have inspired the zeal and enthusiasm of a few young students
looking around for a promising and fascinating field of work, and,
finally, if it should prove to be of assistance in obtaining the means
of training such recruits with the help of the best and most accurate
experimental apparatus which can be obtained, the author’s main objects
in writing it will have been attained.




                                 INDEX


 Airy’s spirals, 197

 Alpine snow-field, 49

 Alum, ammonium iron, 43;
   cæsium, 42;
   potash, 5, 16, 17, 77;
   vicinal faces of potash, 248, 251

 Amethyst quartz, 222–229;
   as example of pseudo-racemism, 231, 232

 Ammonium chloride labile crystallisation, 248;
   iron alum, 43;
   NH_{4} group, and its isomorphism with alkalies, 82, 83, 131;
   oleate, liquid crystals of, 259–263;
   oleate, twins of, 262

 Ampère’s researches on ammonia, 83

 Anatase, crystal of, 37

 Antimony oxide, dimorphism of, 87, 88

 Apatite, crystal of, 37

 Aragonite, biaxial interference figure of, 189, 190

 Armstrong and Pope on sobrerol, 153

 Arsenic oxide, crystals of, 5, 88

 Astatic systems of molecular magnets of Lehmann, 271, 272

 Asymmetric carbon atom, 144, 145

 Axes and axial planes of crystals, 51, 55, 56

 Axial ratios, 129


 Babinet’s double-wedge quartz plate, 219

 Bacteria, destruction of one enantiomorphous form by, 149

 Barium nitrate, 152

 Barker, researches on perchlorates, 123

 Barlow, discovery of remainder of 230 point-systems, 118, 119, 140;
   and Pope’s theory, 285–292

 Bartolinus, Erasmus, 17

 Benzoic acid, screen experiment on crystallisation of, 229–231

 Bergmann and Gahn’s laws of cleavage, 18

 Berzelius and Mitscherlich, atomic weights and isomorphism, 80, 82, 83,
    85

 Beudant’s researches on the vitriols, 75, 76

 Biaxial crystals, 60°-prisms and refractive indices of, 162, 163, 184;
   optic axes of, 185;
   optic axial angle of, 191

 Biot’s researches on tartaric and racemic acids, 143

 Biquartze, natural and artificial, 181, 211, 214, 216, 217

 Black band of quartz twins, 216–218

 Boisbaudran, Lecoq de, 238

 Boyle, Robert, 17

 Brauns, 254

 Bravais, 114;
   space-lattices, 114

 Brookite, 91


 Cæsalpinus, 15

 Cæsium alum, 42

 Calcite, crystals of, 10–13, 38;
   amount of double refraction of, 174;
   in rock sections, 177, 178;
   uniaxial interference figure of, 189, 190;
   plate perpendicular to axis of, 209;
   refractive indices of, 174;
   60°-prism of, 165, 168

 Calcium carbonate, three habits of crystals of, 11

 Calcium dextro-glycerate, optical activity and crystal form of, 155–160

 Carangeot’s contact goniometer, 18–20

 Carbonate, calcium, 11, 87;
   potassium sodium, 44

 Carbonates of alkaline earths, 74

 Carbon, dimorphism of, 137

 Carbon dioxide, liquid in quartz cavities, 46

 Cavendish, 23

 Chemical significance of crystallography, 73, 283, 284;
   valency and crystalline form, 285–291

 Cholesteryl benzoate, liquid crystals of, 258;
   acetate, spherical liquid crystals of, 281, 282

 Chromate, potassium, 84

 Chromates and manganates, isomorphism with sulphates, 84

 Classes, the 32 crystal, 6, 33

 Cleavage, 17;
   and glide-planes, 275, 276

 Cobalt sulphate, 78

 Conditions for growth of crystals, 240, 244, 245

 Constancy of crystal angles, 6, 13, 14, 17, 23, 132

 Convergent light experiments, 186, 188

 Copper sulphate, 40, 76, 78

 Coppet, de, 238

 Crookes, Sir William, 138, 208

 Crossed-axial-plane dispersion of optic axes, 89, 94, 95;
   of ethyl triphenyl pyrrholone, 106, 107, 108;
   of gypsum, 89

 Crystal, definition of, 4;
   germs and their influence on crystallisation, 236, 237

 Crystals, modes of formation of, 4, 5

 Cube and its perfection of symmetry, 38;
   axes of, 52

 Cubic system, 37

 Cyanide, potassium cadmium, crystals of, 42


 Dalton, 25

 Dark field of polariscope, 188, 202

 Davy, Sir Humphry, researches on ammonia, 82, 83

 Delafosse and morphotropy, 101

 Deville and Troost’s researches on tantalum chloride, 84, 85

 Diamond, 8, 137, 138, 207, 208

 Dibenzal benzidine, liquid crystals of, 264

 Digonal axis of symmetry, 36

 Dimensions of structural parallelepipeda, 129

 Dimorphism, 79, 87;
   of antimony oxide, 87, 88;
   of carbonate of lime, 79, 87;
   of mercuric iodide, 97;
   of sodium dihydrogen phosphate, 80, 81;
   of sulphur, 86;
   of vitriols, 79

 Directive molecular force in crystallisation, 139, 269, 274–277

 Dog-tooth spar, 10–12

 Double refraction, interference colours due to, 176, 177;
   measure of, 169, 170;
   of biaxial crystals, 184;
   of uniaxial crystals, 183

 Double sulphates and selenates, 35, 79, 121, 127, 128, 132

 Dulong and Petit’s law, 84


 Electronic corpuscles, constituents of atoms, 113, 124

 Elements of a crystal, 68, 69

 Enantiomorphism and optical activity, 140;
   11 classes showing, 150

 Epsom salts, 76, 78

 Ethyl triphenyl pyrrholone, 105, 106

 Eutropic series, definition of, 132

 External molecular compensation, 134, 234

 Extinction directions, 204, 205


 Fedorow, von, discovery of remainder of 230 point-systems, 118, 119,
    140;
   theory of, 291, 292

 Ferricyanide, potassium, 44

 Ferrocyanide, potassium, 43

 Ferrous sulphate, 76, 78

 Fletcher, indicatrix of, 184

 Fluorspar, single refraction of, 206

 Form, definition of a, 11, 60, 61

 Frankenheim on morphotropy, 101;
   discovery of space-lattices, 114

 Frankland and Frew, 155

 Fuchs, von, researches on sulphates of barium, strontium, and lead, 75,
    77

 Fuess reflecting goniometer, 64–66

 Fundamentalbereich of Schönflies, 112, 113


 Gattermann, 258

 Gay-Lussac on alums, 77;
   on ammonia, 83;
   on racemic acid, 142

 Gernez, 238

 Gessner, 14

 Gmelin, researches on racemic acid, 142

 Goniometer, contact, 19, 20;
   reflecting, 64–66

 Graphite, 137

 Groth, von, morphotropic researches, 98, 102–104

 Growth of a crystal, 1;
   from solution, 237–254

 Guglielmini, researches on crystal structure, 17

 Gypsum (selenite), cleavage of, 203;
   crystals of, 14;
   extinction directions of, 204;
   60°-prism, experiment with, 163, 168;
   twins of in polarised light, 205, 206


 Habit of crystals, 12, 13

 Hardness of crystals, 255, 274

 Hatchett’s discovery of columbium, 85

 Haüy, 22;
   and Mitscherlich, 70, 75, 77, 88, 132;
   fundamental forms, 22, 23;
   law of constancy of form, 23;
   lattices, 29;
   law of rational indices, 24, 29, 30;
   modernisation of theories of, 30–32;
   molécules intégrantes, soustractives, and élémentaires, 25–28;
   structural units, 24

 Hemihedral classes of crystals, 34

 Hexagonal axes of symmetry, 36;
   prism, 11;
   system, 36, 53

 Hexakis octahedron, 39;
   indices of, 61

 Hjortdahl and morphotropy, 101

 Holohedral classes of crystals, 34

 Homogeneity, 6, 16;
   definition of, 114

 Homogeneous structures, the 230 types of, 6, 111

 Hooke, Robert, 16

 Huyghens, 17;
   discovery of laws of double refraction, 17;
   investigation of calcite, 17


 Ice, crystalline form of, 47

 Iceland spar, discovery of, 17;
   rhombohedron of, 10–12

 Inactive tartaric acid, 144

 Inactivity, true optical, 235

 Indicatrix of Fletcher, 184

 Indices of crystal faces, 11

 Intercepts on crystal axes, 58

 Interference colours due to double refraction, 177;
   figures of biaxial and uniaxial crystals, 189–191

 Internal structure of crystals, 15, 111–120

 Iodide of mercury, dimorphism of, 97

 Isomerism, chemical and physical, 142, 143

 Isomorphism, Mitscherlich’s conferment of the term, 81;
   doctrine of, 81, 82;
   limitations of, 85;
   recent clearer definition of, 121–132

 Isotropic crystals, 206


 John of Berlin’s discovery of racemic acid, 142


 Kipping and Pope, definition of racemism and pseudo-racemism, 153, 154

 Kopp, 100

 Kundt’s powder, 149


 Labile solutions, 241

 Laurent and Nickle’s organic researches, 99

 Lavoisier, 23

 Law of rational indices, 24, 50, 57, 59

 Le Bel and van t’Hoff’s explanation of optical activity, 151

 Le Blanc, researches on alums, 77

 Lehmann, researches on liquid crystals, 256–282

 Liquid crystals, 255–282;
   and polymorphism, 138, 139;
   list of substances forming, 280, 281


 Magnesium sulphate, 76, 78

 Mallard and Le Chatelier on silver iodide, 256

 Manganate, potassium, 96

 Manganates, 84, 96

 Manganese sulphate, 78

 Marignac and isomorphism of tantalum and niobium compounds, 84;
   and morphotropy, 101

 Mercury iodide, dimorphism of, 97

 Metastable solutions, 240

 Methyl triphenyl pyrrholone, 105, 106

 Mica-sectors plate for testing sign of optical rotation, 212, 213

 Microscope, Lehmann’s crystallisation, 256, 257, 276–280

 Miers, H. A., researches on crystallisation, 238–243;
   on red silver ores, 109;
   on vicinal faces, 248–254

 Millerian indices, 57

 Mirror-image symmetry, 118, 119, 134, 135;
   illustrated by quartz, 231

 Mitscherlich, experiment with gypsum, 90–94;
   work of, 70–97

 Mixed crystals, 77, 86

 Molecular compound, racemic acid a, 150

 Molecular volume and distance ratios, 129, 130

 Molecule, individuality and directive force of, 139, 269

 Monochromatic illuminator, 192, 193

 Monoclinic system, 39;
   axes and axial planes of, 53, 54

 Morphotropy, 98–104

 Muthmann, researches on permanganates, 123


 Naphthalene tetrachloride, 99

 Newton’s seven orders of spectra, 177

 Nickel sulphate, 76, 78

 Nicol prism, 174, 175, 187

 Nitrobenzenes, von Groth’s researches on, 104

 Noble, Sir Andrew, experiments on liquefaction of carbon, 138


 Optical activity and mirror-image symmetry, 141;
   antipodes, characters of, 152;
   characters of crystals, 7

 Optically active classes of crystals, 150, 151

 Optic axes of biaxial crystals, 185;
   axis of uniaxial crystals, 165;
   axial angle, 191;
   axial angle of ethyl triphenyl pyrrholone for different wave-lengths,
      107, 191;
   axial angle of gypsum at different temperatures, 90–94, 191

 Ostwald’s predictions of crystallisation phenomena, 238, 240

 Oxides of arsenic and antimony, isodimorphism of, 88


 Para-azoxy-anisol, liquid crystals of, 259, 265;
   -benzoic acid, 263;
   -cinnamic acid, 263;
   -phenetol, 258, 259

 Parametral form, 56

 Pasteur’s law, 155;
   research on morphotropy of tartrates, 100;
   research on tartaric and racemic acids, 142–150

 Penfield’s diagram of spherical projection, 61, 62

 _Penicillium glaucum_, destruction of dextro component of racemic acid,
    148

 Perchlorates and permanganates, isomorphism of, 84, 96

 Pfaff, 254

 Phases, different solid, 137

 Phenol and resorcinol, von Groth’s researches on, 103

 Phosphate, ammonium magnesium, 43;
   sodium dihydrogen, dimorphism of, 80, 81

 Phosphates and arsenates, isomorphism of, 73, 74

 Phosphorus, dimorphism of, 138

 Photomicrographs of growing crystals, mode of obtaining, 41

 Pistor’s goniometer, 89

 Planeness of crystal faces, 6, 7

 Polarisation colours due to optical activity, 179

 Polarisation, rectangular, of spectra from doubly refractive prisms,
    164

 Polariscope, the, 187, 188

 Polymorphism, 133–137

 Pope and Barlow, theory of, 285–291

 Positive and negative uniaxial and biaxial crystals, 166

 Potassium bichromate, metastable and labile crystallisation of, 246,
    247;
   manganate, 96;
   nickel sulphate, 35;
   selenate, 96;
   sulphate, 13

 Priestley, 23

 Primitive form of Romé de l’Isle, 20, 27

 Projection polariscope, for convergent light, 92, 186, 188, 189;
   for parallel light, 201, 202

 Progressive change of crystal angles in isomorphous series, 125;
   of double refraction, 126;
   of molecular distance ratios, 130, 131;
   of position of optical ellipsoid, 128

 Propyl triphenyl pyrrholone, 106, 108

 Proustite, 109, 110

 Pseudo-racemism, 153, 154

 Pyrargyrite, 109, 110

 Pyroelectrical properties of crystals, 149


 Quartz, 170;
   crystalline form of, 171, 172;
   crystals in rock sections, polarisation colours of, 175–178;
   crystals, liquid cavities in, 45;
   crystals on sand grains, 2, 3;
   double refraction of, 174;
   interference figure in convergent polarised light, 194–196;
   optical activity of, 173, 179–181, 210;
   plates, preparation of for polariscope, 181;
   polarisation colours of due to optical activity, 209, 210;
   refractive indices of, 174;
   screw point-systems of, 151;
   60°-prism experiment with, 165, 168;
   Steno’s research on, 16;
   twinning of, 215, 216, 219–221, 225;
   two varieties of as examples of mirror-image symmetry, 171


 Racemate, sodium ammonium, Pasteur’s researches on, 147, 148

 Racemic acid, 142–146, 233, 234

 Racemic forms and racemism, 150, 153, 233

 Rammelsberg and morphotropy, 101

 Rational indices, 24, 50, 59, 116

 Reflection of light by crystal faces, 8

 Refractive index, meaning of, 167

 Reinitzer, 258

 Reusch’s artificial quartzes, 198, 199

 Rhombic system, 39;
   axes and axial planes of, 52, 54

 Rhombohedron and its axes, 10, 11, 54, 55, 172

 Rings and brushes, optic axial, 190, 191

 Rock-salt, cube of in quartz cavity, 45;
   60°-prism experiment with, 162, 168

 Romé de l’Isle, 18–20;
   researches on alums, 77

 Roozeboom, 238

 Rotation of plane of polarisation by quartz of two varieties and
    different thickness, 180, 210, 211

 Royal Institution experiment with diamonds, 8


 Sal-ammoniac, 82, 83

 Salol and betol, Miers’ researches on, 242, 243

 Sand grains with quartz crystals, 3

 Scalenohedron of calcite, 11

 Scheele’s discovery of tartaric acid, 142

 Schlippe’s salt, 45

 Schönflies, discovery of remainder of 230 point-systems, 118, 119, 140

 Seebeck, researches on ammonia, 82, 83

 Selenates, isomorphous with sulphates, 96;
   of alkalies, 121

 Selenic acid, 95

 Selenite, polarisation colours of films of, 203

 Selenium, discovery by Berzelius, 96

 Sella’s warning against hasty generalisation, 102

 Senarmontite, 88

 Silver iodide, 256, 292

 Single refraction of cubic crystals, 162, 183

 Snow crystals, 49

 Sobrerol, 153

 Sodium chlorate, 151;
   sulphantimoniate, 45

 Sohncke, regular point-systems, 117;
   mirror-image molecular arrangement, 141;
   two point-systems of quartz, 151

 Sollas, crystal structure, theory of, 292

 Solubility and supersolubility, 238–241;
   curves of, 240

 Solutions, optical activity of, 151;
   metastable and labile, 240, 241

 Space-lattices, 50, 114–116;
   triclinic illustration of, 115

 Specific gravity, importance of determinations of, 129

 Spheres of influence of atoms, 113

 Steno, 16

 Stereographic projection, 34, 62, 67;
   of topaz, 68;
   of double sulphates, 35

 Stereometric arrangement of atoms in molecule, 124, 136

 Story Maskelyne, 34

 Strain, polarisation colours of glass and diamond due to, 207, 208

 Structural units of crystals, 24, 111–113

 Sulphantimoniate of sodium, 45

 Sulphate, ammonium magnesium, 44

 Sulphates of alkalies, 121;
   of barium, strontium and lead, 74

 Sulphur, dimorphism of, 86, 87, 137;
   monoclinic form of, 4, 86

 Symbol of a face or form, 57

 Symmetry, axes and planes of, 34, 36, 41, 55;
   elements of, 34

 Systems, the crystal, 6, 7, 33


 Tartaric acid, 142–148;
   dextro, 143, 146–148;
   lævo, 144, 146–148;
   pyro-electrical properties of, 149

 Tartrate, hydrogen potassium, 43

 Tetartohedral classes of crystals, 34

 Tetragonal system, 37;
   axes and planes of, 36, 52

 Thallium, relation of to alkali metals, 131

 Thénard and Gay-Lussac, research on ammonia, 83

 Thomson, J. M., 238

 Thomson, Sir J. J., discovery of composition of atoms, 112

 Topaz, crystal of, 40;
   stereographic projection of, 68

 Transition tint, 180

 Triclinic bipyramid, 56;
   system, 40;
   axes and planes of, 53, 54

 Trigonal system, 37;
   axes and planes of, 37, 53, 55

 Triphenyl pyrrholone derivatives, 105–108

 Triple tartrate of sodium, potassium, and ammonium, 91

 Tutton, fixed positions of atoms in crystals, 122–124;
   law of progression of crystal properties in isomorphous series,
      121–123, 129;
   nature of structural units, 134, 135;
   research on calcium dextro-glycerate, 157, 160;
   researches on simple and double sulphates and selenates, 121;
   researches on triphenyl pyrrholones, 105–108


 Uniaxial crystals, optic axis of, 165;
   60°-prisms of, 164;
   two refractive indices of, 166, 183


 Valentinite, 88

 Vanadium family group of elements, 85

 Vauquelin, researches on alums, 77

 Vicinal faces, 248–254

 Vitriols, the, 15, 76, 78

 Von Lang, 23, 34

 Vorländer, 263;
   nature of molecules forming liquid crystals, 273


 Water, exceptional thermal dilatation of, 47

 Water flowers in ice, 47

 Water of crystallisation, 76, 78, 79

 Wave-length of most luminous part of spectrum, 179

 Werner’s fundamental form, 20

 Westfeld, 18

 Weyberg, 250

 White of higher orders, 178

 Wollaston’s reflecting goniometer, 63, 81;
   work on carbonates and sulphates of barium, strontium and lead, 74,
      75

 Wulff, 250, 252, 253


 Zinc sulphate, 76, 78

 Zone of crystal faces, 63;
   circle, 63




                               PRINTED BY
                          TURNBULL AND SPEARS,
                               EDINBURGH

-----

Footnote 1:

  See page 57 for explanation of indices.

Footnote 2:

  _Proc. Roy. Soc._, 1908, A, 81, 40.

Footnote 3:

  _Comptes Rendus_, 1842, 15, 350, and 1845, 20, 357.

Footnote 4:

  _Comptes Rendus_, 1848, 27, 611, and 1849, 29, 339.

Footnote 5:

  _Jahresbericht_, 1849, 19.

Footnote 6:

  _Comptes Rendus_, 1848, 26, 535.

Footnote 7:

  _Journ. für Prakt. Chemie._, 1865, 94, 286.

Footnote 8:

  _Mem. R. Accad. di Torino_, 2A, 17, 337, and 20, 355.

Footnote 9:

  _Pogg. Ann._, 141, 31.

Footnote 10:

  _Mineralogical Magazine_, 1888, 8, 37.

Footnote 11:

  _Journ. Chem. Soc._, 1896, 69, 507.

Footnote 12:

  _Ann. de Chim. et Phys._, 1848, 24, 28 and 38; also 1850, 28, 56;
  _Comptes Rendus_, 1848, 26, 535; also 1849, 29, 297; also 1850, 31,
  480; also 1853, 37, 162, and 1858, 46, 615.

Footnote 13:

  _Journ. Chem. Soc._, 1891, 315.

Footnote 14:

  _Ibid._, 1897, 989.

Footnote 15:

  _Journ. Chem. Soc._, 1891, 59, 233.

Footnote 16:

  _Ibid._, 1891, 59, 96.

Footnote 17:

  _Phil. Trans. Roy. Soc._, 1895, A, 185, 913.

Footnote 18:

  _Mineralogical Magazine_, 1892, 10, 123.

Footnote 19:

  _Journ. Chem. Soc._, 1906, 89, 413; _Proc. Roy. Soc., A_, 1907, 79,
  322.

Footnote 20:

  “Lehrbuch der Allgemeinen Chemice,” vol. 2, part 2, p. 780.

Footnote 21:

  _Phil. Trans._, 1903, A, 202, 459.

Footnote 22:

  _Phil. Trans._, 1903, A, 202, 519.

Footnote 23:

  _Zeitschr. für Kryst._, 1901, 34, 449.

Footnote 24:

  _Loc. cit._, p. 531.

Footnote 25:

  _Sitzungsber. d. Physik. Med. Soc._, Erlangen, 1878, 10, 59.

Footnote 26:

  _Neues Jahrbuch_, 1887, 138.

Footnote 27:

  _Ber. der deutsch. Chem. Ges._, 1907, 40, 1970.

Footnote 28:

  _Zeitschr. f. Phys. Chemie._, 1907, 57, 357.

Footnote 29:

  This apparent contradiction in terms the author takes to mean that,
  whatever compression is produced by pressure on one part of the
  sphere, is counterbalanced by a corresponding protuberance produced in
  the part not confined under pressure, the total volume being
  incompressible.

Footnote 30:

  _Proc. Roy. Soc._, 1898, 63, 270, 286, and 296; 1901, 67, 493.

------------------------------------------------------------------------




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