and try if it is possible to cause it to touch more than twelve of
the bodies. Attach ends of three thick flexible wires to any places
on the surface of _S_{1}_; carry the wires through interstices of the
assemblage, and attach their other ends at any three places of _A_,
_B_, _C_, respectively, these being any three of the bodies outside
the cluster of _S_{1}_ and its twelve neighbours. Cut the wires across
at any chosen positions in them; and round off the cut ends, just
leaving contact between the rounded ends, which we shall call _f′f_,
_g′g_, _h′h_. Do homogeneously for every other solid of the assemblage
what we have done for _S_{1}_. Now bend the wires slightly so as to
separate the pairs of points of contact, taking care to keep them
from touching any other bodies which they pass near on their courses
between _S_{1}_ and _A_, _B_, _C_ respectively. After having done this,
thoroughly rigidify all the wires thus altered. We may now, having
three independent variables at our disposal, so change the orientation
of the molecules, relatively to rows of the assemblage, as to bring
_f′f_, _g′g_, and _h′h_ again into contact. We have thus six fresh
points of _S_{1}_; of which three are _f′_, _g′_, _h′_; and the other
three are on the three extensions of _S_{1}_ corresponding to the
single extensions of _A_, _B_, _C_ respectively, which we have been
making. Thus we have a _real_ solution of the interesting geometrical
problem:--It is required so to form a homogeneous assemblage of solids
of any arbitrarily given shape that each solid shall be touched by
eighteen others. This problem is determinate, because the making of
the three contacts _f′f_, _g′g_, _h′h_, uses up the three independent
variables left at our disposal after we have first formed a homogeneous
assemblage with twelve points of contact on each solid. But our manner
of finding a shape for each solid which can allow the solution of the
problem to be real, proves that the solution is essentially imaginary
for every wholly convex shape.
§ 35. Pausing for a moment longer to consider afresh the geometrical
problem of putting arbitrarily given equal and similar solids together
to make a homogeneous assemblage of which each member is touched by
eighteen others, we see immediately that it is determinate (whether it
has any real solution or not), because when the shape of each body is
given we have nine disposables for fixing the assemblage: six for the
character of the assemblage of the corresponding points, and three for
the orientation of each molecule relatively to rows of the assemblage
of corresponding points. These nine disposables are determined by the
condition that each body has nine pairs of contacts with others.
Suppose now a homogeneous assemblage of the given bodies, in open
order with no contacts, to be arbitrarily made according to any nine
arbitrarily chosen values for the six distances between a point of
_S_{1}_ and the corresponding points of its six pairs of nearest and
next nearest neighbours (§ 1 above), and the three angles (§ 9 above)
specifying the orientation of each body relatively to rows of the
assemblage. We may choose in any nine rows through _S_{1}_ any nine
pairs of bodies at equal distances on the two sides of _S_{1}_ far or
near, for the eighteen bodies which are to be in contact with _S_{1}_.
Hence there is an infinite number of solutions of the problem of which
only a finite number can be real. Every solution of the problem of
eighteen contacts is imaginary when the shape is wholly convex.
[Illustration: FIG. 13.]
§ 36. Without for a moment imagining the molecules of matter to be
hard solids of convex shape, we may derive valuable lessons in the
tactics of real crystals by studying the assemblage described in §§
24 and 25 and represented in Figs. 12 and 13. I must for the present
forego the very attractive subject of the tactics presented by faces
not parallel to one or other of the four faces of the primitive
tetrahedrons which we found in § 24, and ask you only to think of the
two sides of a plate of crystal parallel to any one of them, that is to
say, an assemblage of such layers as those represented geometrically
in Fig. 12 and shown in stereoscopic view in Fig. 13. If, as is the
case with the solids[10] photographed in Fig. 13, the under side of
each solid is nearly plane but slightly convex, and the top is somewhat
sharply curved, we have the kind of difference between the upper and
under of the two parallel sides of the crystal which I have already
described to you in § 21 above. In this case the assemblage is formed
by letting the solids fall down from above and settle in the hollows to
which they come most readily, or which give them the stablest position.
It would, we may suppose, be the hollows _p′ q′ r′_, not _p q r_, (Fig.
12) that would be chosen; and thus, of the two formations described in
§ 25, we should have that in which the hollows above _p′ q′ r′_ are
occupied by the comparatively flat under sides of the molecules of
the layer above, and the hollows below the apertures _p q r_ by the
comparatively sharp tops of the molecules of the layers below.
§ 37. For many cases of natural crystals of the wholly asymmetric
character, the true forces between the crystalline molecules will
determine precisely the same tactics of crystallization as would be
determined by the influence of gravity and fluid viscosity in the
settlement from water, of sand composed of uniform molecules of the
wholly unsymmetrical convex shape represented in Figs. 12 and 13.
Thus we can readily believe that a real crystal which is growing
by additions to the face seen in Fig. 12, would give layer after
layer regularly as I have just described. But if by some change of
circumstances the plate, already grown to a thickness of many layers in
this way, should come to have the side facing _from_ us in the diagram
exposed to the mother-liquor, or mother-gas, and begin to grow from
that face, the tactics might probably be that each molecule would find
its resting-place with its most nearly plane side in the wider hollows
under _p′ q′ r′_, instead of with its sharpest corner in the narrower
and steeper hollows under _p q r_, as are the molecules in the layer
below that shown in the diagram in the first formation. The result
would be a compound crystal consisting of two parts, of different
crystalline quality, cohering perfectly together on the two sides of
an interfacial plane. It seems probable that this double structure may
be found in nature, presented by crystals of the wholly unsymmetric
class, though it may not hitherto have been observed or described in
crystallographic treatises.
[Illustration: FIG. 14.]
§ 38. This asymmetric double crystal becomes simply the well-known
symmetrical ‘twin-crystal’[11] in the particular case in which each of
the constituent molecules is symmetrical on the two sides of a plane
through it parallel to the plane of our diagrams, and also on the two
sides of some plane perpendicular to this plane. We see, in fact, that
in this case if we cut in two the double crystal by the plane of Fig.
14, and turn one part ideally through 180° round the intersection of
these two planes, we bring it into perfect coincidence with the other
part.
This we readily understand by looking at Fig. 14, in which the solid
shown in outline may be either an egg-shaped figure of revolution, or
may be such a figure flattened by compression perpendicular to the
plane of the diagram. The most readily chosen and the most stable
resting-places for the constituents of each successive layer might be
the wider hollows _p′ q′ r′_: and therefore if, from a single layer to
begin with, the assemblage were to grow by layer after layer added to
it on each side, it might probably grow as a twin-crystal. But it might
also be that the presence of a molecule in the wider hollow _p′ q′ r′_
on one side, might render the occupation of the corresponding hollow on
the other side by another molecule less probable, or even impossible.
Hence, according to the configuration and the molecular forces of the
particular crystalline molecule in natural crystallization, there may
be necessarily, or almost necessarily, the twin, when growth proceeds
simultaneously on the two sides: or the twin growth may be impossible,
because the first occupation of the wider hollows on one side, may
compel the continuity of the crystalline quality throughout, by leaving
only the narrower hollows _p q r_ free for occupation by molecules
attaching themselves on the other side.
§ 39. Or the character of the crystalline molecule may be such
that when the assemblage grows by the addition of layer after layer
on one side only, with a not very strongly decided preference to the
wider hollows _p′ q′ r′_, some change of circumstances may cause
the molecules of one layer to place themselves in a hollow _p q r_.
The molecules in the next layer after this would find the hollows
_p′ q′ r′_ occupied on the far side, and would thus have a bias in
favour of the hollows _p q r_. Thus layer after layer might be added,
constituting a twinned portion of the growth, growing, however,
with less strong security for continued homogeneousness than when
the crystal was growing, as at first, by occupation of the wider
hollows _p′ q′ r′_. A slight disturbance might again occur, causing
the molecules of a fresh layer to settle, not in the narrow hollows
_p q r_, but in the wider hollows _p′ q′ r′_, notwithstanding the
nearness of molecules already occupying the wider hollows on the
other side. Disturbances such as these occurring irregularly during
the growth of a crystal, might produce a large number of successive
twinnings at parallel planes with irregular intervals between them,
or a large number of twinnings in planes at equal intervals might be
produced by some regular periodic disturbance occurring for a certain
number of periods, and then ceasing. Whether regular and periodic, or
irregular, the tendency would be that the number of twinnings should
be even, and that after the disturbances cease the crystal should
go on growing in the first manner, because of the permanent bias in
favour of the wider hollows _p′ q′ r′_. These changes of molecular
tactics, which we have been necessarily led to by the consideration
of the fortuitous concourse of molecules, are no doubt exemplified in
a large variety of twinnings and counter-twinnings found in natural
minerals. In the artificial crystallization of chlorate of potash they
are of frequent occurrence, as is proved, not only by the twinnings and
counter-twinnings readily seen in the crystalline forms, but also by
the brilliant iridescence observed in many of the crystals found among
a large multitude, which was investigated scientifically by Sir George
Stokes ten years ago, and described in a communication to the Royal
Society ‘On a remarkable phenomenon of crystalline reflection’ (_Proc.
R.S._, vol. xxxviii, 1885, p. 174).
§ 40. A very interesting phenomenon, presented by what was originally
a clear homogeneous crystal of chlorate of potash, and was altered by
heating to about 245°-248° Cent., which I am able to show you through
the kindness of Lord Rayleigh, and of its discoverer, Mr. Madan,
presents another very wonderful case of changing molecular tactics,
most instructive in respect of the molecular constitution of elastic
solids. When I hold this plate before you with the perpendicular to
its plane inclined at 10° or more to your line of vision, you see a
tinsel-like appearance, almost as bright as if it were a plate of
polished silver, on this little area, which is a thin plate of chlorate
of potash cemented for preservation between two pieces of glass; and,
when I hold a light behind, you see that the little plate is almost
perfectly opaque like metal foil. But now when I hold it nearly
perpendicular to your line of vision the tinsel-like appearance is
lost. You can see clearly through the plate, and you also see that very
little light is reflected from it. As a result both of Mr. Madan’s own
investigations, and further observations by himself, Lord Rayleigh came
to the conclusion that the almost total reflection of white light which
you see is due to the reflection of light at many interfacial planes
between successive layers of twinned and counter-twinned crystal of
small irregular thicknesses, and not to any splits or cavities or any
other deviation from homogeneousness than that presented by homogeneous
portions of oppositely twinned-crystals in thorough molecular contact
at the interfaces.
§ 41. When the primitive clear crystal was first heated very gradually
by Madan to near its melting-point (359° according to Carnelly), it
remained clear, and only acquired the tinsel appearance after it had
cooled to about 245° or 248°[12]. Rayleigh found that if a crystal
thus altered was again and again heated it always lost the tinsel
appearance, and became perfectly clear at some temperature considerably
below the melting-point, and regained it at about the same temperature
in cooling. It seems, therefore, certain that at temperatures above
248°, and below the melting-point, the molecules had so much of thermal
motions as to keep them hovering about the positions of _p q r_, _p′ q′
r′_, of our diagrams, but not enough to do away with the rigidity of
the solid; and that when cooled below 248° the molecules were allowed
to settle in one or other of the two configurations, but with little
of bias for one in preference to the other. It is certainly a very
remarkable fact in Natural History, discovered by these observations,
that, when the molecules come together to form a crystal out of the
watery solution, there should be so much more decided a bias in favour
of continued homogeneousness of the assemblage than when, by cooling,
they are allowed to settle from their agitations in a rigid, but nearly
melting, solid.
§ 42. But even in crystallization from watery solution of chlorate of
potash the bias in favour of thorough homogeneousness is not in every
contingency decisive. In the first place, beginning, as the formation
seems to begin, from a single molecular plane layer such as that
ideally shown in Fig. 14, it goes on, not to make a homogeneous crystal
on the two sides of this layer, but probably always so as to form a
twin-crystal on its two sides, exactly as described in § 38, and, if
so, certainly for the reason there stated. This is what Madan calls
the ‘inveterate tendency to produce twins (such as would assuredly
drive a Malthus to despair)[13]’; and it is to this that he alludes as
‘the inevitable twin-plate’ in the passage from his paper given in the
foot-note to § 41 above.
§ 43. In the second place, I must tell you that many of the crystals
produced from the watery solution by the ordinary process of slow
evaporation and crystallization, show twinnings and counter-twinnings
at irregular intervals in the otherwise homogeneous crystal on either
one or both sides of the main central twin-plane, which henceforth,
for brevity, I shall call (adopting the hypothesis already explained,
which seems to me undoubtedly true) the ‘initial plane.’ Each twinning
is followed, I believe, by a counter-twinning at a very short distance
from it; at all events Lord Rayleigh’s observations[14] prove that the
whole number of twinnings and counter-twinnings in a thin disturbed
stratum of the crystal on one side of the main central twin-plane
is generally, perhaps always, even; so that, except through some
comparatively very small part or parts of the whole thickness, the
crystal on either side of the middle or initial plane is homogeneous.
This is exactly the generally regular growth which I have described
to you (§ 39) as interrupted occasionally or accidentally by some
unexplained disturbing cause, but with an essential bias to the
homogeneous continuance of the more easy or natural one of the two
configurations.
§ 44. I have now great pleasure in showing you a most interesting
collection of the iridescent crystals of chlorate of potash, each
carefully mounted for preservation between two glass plates, which have
been kindly lent to us for this evening by Mr. Madan. In March, 1854,
Dr. W. Bird Herapath sent to Prof. Stokes some crystals of chlorate
of potash showing the brilliant and beautiful colours you now see,
and, thirty years later, Prof. E. J. Mills recalled his attention to
the subject by sending him ‘a fine collection of splendidly coloured
crystals of chlorate of potash of considerable size, several of the
plates having an area of a square inch or more, and all of them
being thick enough to handle without difficulty.’ The consequence
was that Stokes made a searching examination into the character of
the phenomenon, and gave the short, but splendidly interesting,
communication to the Royal Society of which I have already told you.
The existence of these beautifully coloured crystals had been well
known to chemical manufacturers for a long time, but it does not appear
that any mention of them was to be found in any scientific journal or
treatise prior to Stokes’ paper of 1885. He found that the colour was
due to twinnings and counter-twinnings in a very thin disturbed stratum
of the crystal showing itself by a very fine line, dark or glistening,
according to the direction of the incident light when a transverse
section of the plate of crystal was examined in a microscope. By
comparison with a spore of lycopodium he estimated that the breadth
of this line, and therefore the thickness of the disturbed stratum of
the crystal, ranged somewhere about the one-thousandth of an inch. He
found that the stratum was visibly thicker in those crystals which
showed red colour than in those which showed blue. He concluded that
‘the seat of the coloration is certainly a thin twinned stratum’ (that
is to say, a homogeneous portion of crystal between a twinning and a
counter-twinning), and found that ‘a single twin-plane does not show
anything of the kind.’
§ 45. A year or two later Lord Rayleigh entered on the subject with
an exhaustive mathematical investigation of the reflection of light at
a twin-plane of a crystal (_Philosophical Magazine_, September, 1888),
by the application of which, in a second paper ‘On the remarkable
phenomenon of Crystalline Reflection described by Prof. Stokes,’
published in the same number of the _Philosophical Magazine_, he gave
what seems certainly the true explanation of the results of Sir George
Stokes’ experimental analysis of these beautiful phenomena. He came
very decidedly to the conclusion that the selective quality of the
iridescent portion of the crystal, in virtue of which it reflects
almost totally light nearly of one particular wave-length for one
particular direction of incidence (on which the brilliance of the
coloration depends), cannot be due to merely a single twin-stratum,
but that it essentially is due to a considerable number of parallel
twin-strata at nearly equal distances. The light reflected by this
complex stratum is, for any particular direction of incident and
reflected ray, chiefly that of which the wave-length is equal to twice
the length of the period of the twinning and counter-twinning, on a
line drawn through the stratum in the direction of either the incident
or the reflected ray.
§ 46. It seems to me probable that each twinning is essentially
followed closely by a counter-twinning. Probably three or four of these
twin-strata might suffice to give colour; but in any of the brilliant
specimens as many as twenty or thirty, or more, might probably be
necessary to give so nearly monochromatic light as was proved by
Stokes’ prismatic analysis of the colours observed in many of his
specimens. The disturbed stratum of about a one-thousandth of an inch
thickness, seen by him in the microscope, amply suffices for the 5,
10, or 100 half wave-lengths required by Rayleigh’s theory to account
for perceptible or brilliant coloration. But what _can_ be the cause
of any approach to regular periodicity in the structure sufficiently
good to give the colours actually observed? Periodical motion of
the mother-liquor relatively to the growing crystal might possibly
account for it. But Lord Rayleigh tells us that he tried rocking the
pan containing the solution without result. Influence of light has
been suggested, and I believe tried, also without result, by several
enquirers. We know, by the beautiful discovery of Edmond Becquerel,
of the prismatic colours photographed on a prepared silver plate by
the solar spectrum, that ‘standing waves’ (that is to say, vibrations
with stationary nodes and stationary places of maximum vibration),
due to co-existence of incident and reflected waves, do produce such
a periodic structure as that which Rayleigh’s theory shows capable
of giving a corresponding tint when illuminated by white light. It
is difficult, therefore, not to think that light may be effective in
producing the periodic structure in the crystallization of chlorate of
potash, to which the iridescence is due. Still, experimental evidence
seems against this tempting theory, and we must perforce be content
with the question unanswered:--What can be the cause of 5, or 10, or
100 pairs of twinning and counter-twinning following one another in
the crystallization with sufficient regularity to give the colour: and
why, if there are twinnings and counter-twinnings, are they not at
irregular intervals, as those produced by Madan’s process, and giving
the observed white tinsel-like appearance with no coloration?
§ 47. And now I have sadly taxed your patience: and I fear I have
exhausted it and not exhausted my subject! I feel I have not got
halfway through what I hoped I might be able to put before you this
evening regarding the molecular structure of crystals. I particularly
desired to speak to you of quartz crystal with its ternary symmetry
and its chirality[15]; and to have told you of the etching[16] by
hydrofluoric acid which, as it were, commences to unbuild the crystal
by taking away molecule after molecule, but not in the reverse order of
the primary up-building; and which thus reveals differences of tactics
in the alternate faces of the six-sided pyramid which terminates at
either end, sometimes at both ends, the six-sided prism constituting
generally the main bulk of the crystal. I must confine myself to giving
you a geometrical symbol for the ternary symmetry of the prism and its
terminal pyramid.
[Illustration: FIG. 15.]
§ 48. Make an equilateral equiangular hexagonal prism, with
its diagonal from edge to edge ninety-five hundredths[17] of its
length. Place a number of these close together, so as to make up
a hexagonal plane layer with its sides perpendicular to the sides
of the constituent hexagonal prisms: see Fig. 15 and imagine the
semicircles replaced by their diameters. You see in each side of the
hexagonal assemblage, edges of the constituent prisms, and you see
at each corner of the assemblage a face (not an edge) of _one_ of
the constituent prisms. Build up a hexagonal prismatic assemblage
by placing layer after layer over it with the constituent prisms of
each layer vertically over those in the layer below; and finish the
assemblage with a six-sided pyramid by building upon the upper end
of the prism, layer after layer of diminishing hexagonal groups,
each less by one circumferential row than the layer below it. You
thus have a crystal of precisely the shape of a symmetrical specimen
of rock crystal, with the faces of its terminal pyramid inclined at
38° 13′ to the faces of the prism from which they spring. But the
assemblage thus constituted has ‘senary’ (or six-rayed symmetry). To
reduce this to ternary symmetry, cut a groove through the middle of
each alternate face of the prismatic molecule, making this groove
in the first place parallel to the edges: and add a corresponding
projection, or fillet, to the middles of the other three faces, so
that two of the cylinders similarly oriented would fit together, with
the projecting fillet on one side of one of them entering the groove
in the anti-corresponding side of the other. The prismatic portion
of the assemblage thus formed shows (see Fig. 15), on its alternate
edges, faces of molecules with projections and faces of molecules with
grooves; and shows only orientational differences between alternate
faces, whether of the pyramid or of the prism. Having gone only so far
from ‘senary’ symmetry, we have exactly the triple, or three-pair,
anti-symmetry required for the piezo-electricity of quartz investigated
so admirably by the brothers Curie[18], who found that a thin plate of
quartz crystal cut from any position perpendicular to a pair of faces
of a symmetrical crystal, becomes positively electrified on one side
and negatively on the other when pulled in a direction perpendicular
to those faces. But this assemblage has not the chiral piezo-electric
quality discovered theoretically by Voigt[19], and experimentally in
quartz and in tourmaline by himself and Riecke[20], nor the well-known
optic chirality of quartz.
[Illustration: FIG. 16.]
[Illustration: FIG. 17.]