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.