the oblique configurations shown in Fig. 16, which I call right-handed,
because the directions of the projections are tangential to the threads
of a three-thread right-handed screw, and Fig. 17 (left-handed). The
prisms with their grooves and fillets will still all fit together if
they are all right-handed, or all left-handed.
[Illustration: FIG. 18.]
Fig. 18 shows the upper side of a hexagonal layer of an assemblage
thus composed of the right-handed molecule of Fig. 16. Fig. 15
unchanged, still represents a horizontal section through the centres
of the molecules. A prism built up of such layers, and finished at
each end with a pyramid according to the rule of § 48, has all the
qualities of ternary chiral symmetry required for the piezo-electricity
of quartz; for the orientational differences of the alternate pairs
of prismatic faces; for the absolute difference between the alternate
pairs of faces of each pyramid which are shown in the etching by
hydrofluoric acid; for the merely orientational difference between
the parallel faces of the two pyramids; and for the well-known
chiro-optic[21] property of quartz. Look at two contiguous faces _A_,
_B_ of our geometrical model quartz crystal now before you, with its
axis vertical. You will see a difference between them: turn it upside
down; _B_ will be undistinguishable from what _A_ was, and _A_ will be
undistinguishable from what _B_ was. Look at the two terminal pyramids,
and you will find that the face above _A_ and the face below _B_ are
identical in quality, and that they differ from the face above _B_
and below _A_. This model is composed of the right-handed constituent
molecules shown in Fig. 16. It is so placed before you that the edge of
the prismatic part of the assemblage nearest to you shows you filleted
faces of the prismatic molecules. You see two pyramidal faces; the one
to your right hand, over _B_, presents complicated projections and
hollows at the corners of the constituent molecules; and the pyramidal
face next your left hand, over _A_, presents their unmodified corners.
But it will be the face next your left hand which will present the
complex bristling corners, and the face next your right hand that
will present the simple corners, if, for the model before you, you
substitute a model composed of left-handed molecules such as those
shown in Fig. 17.
§ 50. To give all the qualities of symmetry and anti-symmetry of
the pyro-electric and piezo-electric properties of tourmaline
investigated theoretically by Voigt[22], and experimentally by himself
and Friecke[23], make a hollow in one terminal face of each of our
constituent prisms, and a corresponding projection in its other
terminal face.
§ 51. Coming back to quartz, we can now understand perfectly the
two kinds of macling which are well known to mineralogists as being
found in many natural specimens of the crystal, and which I call
respectively the orientational macling, and the chiral macling.
In the orientational macling all the crystalline molecules are
right-handed, or all left-handed; but through all of some part of the
crystal, each of our component hexagonal prisms is turned round its
axis through 60° from the position it would have if the structure
were homogeneous throughout. In each of the two parts the structure
is homogeneous, and possesses all the electric and optic properties
which any homogeneous portion of quartz crystal presents, and the
facial properties of natural uncut crystal, shown in the etching by
hydrofluoric acid; but there is a discontinuity at the interface, not
generally plane, between the two parts, which in our geometrical model
would be shown by non-fittings between the molecules on the two sides
of the interface, while all the contiguous molecules in one part, and
all the contiguous molecules in the other part, fit into one another
perfectly. In chiral macling, which is continually found in amethystine
quartz, and sometimes in ordinary clear quartz crystals, some parts
are composed of right-handed molecules, and others of left-handed
molecules. It is not known whether, in this chiral macling, there is or
there is not also the orientational macling on the two sides of each
interface; but we may say probably _not_; because we know that the
orientational macling occurs in nature without any chiral macling, and
because there does not seem reason to expect that chiral macling would
imply orientational macling on the two sides of the same interface.
I would like to have spoken to you more of this most interesting
subject; and to have pointed out to you that some of the simplest and
most natural suppositions we can make as to the chemical forces (or
electrical forces, which probably means the same thing) concerned in
a single chemical molecule of quartz, _SiO_{2}_, and acting between
it and similar neighbouring molecules, would lead essentially to
these molecules coming together in triplets, each necessarily either
right-handed or left-handed, but with as much probability of one
configuration as of the other: and to have shown you that these
triplets of silica 3(_SiO_{2}_) can form a crystalline molecule with
all the properties of ternary chiral symmetry, typified by our grooved
hexagonal prisms, and can build up a quartz crystal by the fortuitous
concourse of atoms. I should like also to have suggested and explained
the possibility that a right-handed crystalline molecule thus formed
may, in natural circumstances of high temperature, or even of great
pressure, become changed into a left-handed crystal, or _vice-versa_.
My watch, however, warns me that I must not enter on this subject.
[Illustration: FIG. 19.]
§ 52. Coming back to mere molecular tactics of crystals, remark that
our assemblage of rounded, thoroughly scalene, tetrahedrons, shown
in the stereoscopic picture (§ 36, Fig. 13 above), essentially has
chirality because each constituent tetrahedron, if wholly scalene, has
chirality[24]. I should like to have explained to you how a single or
double homogeneous assemblage of points has essentially no chirality,
and how three assemblages of single points, or a single assemblage of
triplets of points, can have chirality, though a single triplet of
points cannot have chirality. I should like indeed to have brought
somewhat thoroughly before you the geometrical theory of chirality;
and in illustration to have explained the conditions under which four
points, or two lines, or a line and two points, or a combination of
point, line and plane, can have chirality: and how a homogeneous
assemblage of non-chiral objects can have chirality; but in pity I
forbear, and I thank you for the extreme patience with which you have
listened to me.
FOOTNOTES:
[1] See foot-note on § 22 below.
[2] The holes in the cylinders are bored obliquely, as shown in Fig.
4, which causes them to remain at any desired position on the cord and
allows them to be freed to move up and down by slackening the cord for
a moment.
[3] ‘On the Homogeneous Division of Space,’ by Lord Kelvin, _Royal
Society Proceedings_, vol. lv, Jan. 18, 1894.
[4] Similar curves are said to be parallel when the tangents to them at
corresponding points are parallel.
[5] See foot-note to § 12 above.
[6] ‘On the Division of Space with Minimum Partitional Area,’
_Philosophical Magazine_, vol. xxiv, 1887, p. 502, and _Acta
Mathematica_ of the same year.
[7] A. Levy, _Edinburgh Philosophical Journal_, April, 1822;
Whewell, _Phil. Trans. Royal Society_, 1825; Miller, _Treatise on
Crystallography_.
[8] I call any geometrical figure, or group of points, _chiral_, and
say that it has chirality, if its image in a plane mirror, ideally
realized, cannot be brought to coincide with itself. Two equal and
similar right hands are homochirally similar. Equal and similar right
and left hands are heterochirally similar or ‘allochirally’ similar
(but heterochirally is better). These are also called ‘enantiomorphs,’
after a usage introduced, I believe, by German writers. Any chiral
object and its image in a plane mirror are heterochirally similar.
[9] _Philosophical Magazine_, vol. xx, 1885, second half year, p. 469,
and _British Association Report_, 1885, Aberdeen, p. 896.
[10] The solids of the photograph are castings in fine plaster of Paris
from a scalene tetrahedron of paraffin wax, with its corners and edges
rounded, used as a pattern.
[11] ‘A twin-crystal is composed of two crystals joined together in
such a manner that one would come into the position of the other by
revolving through two right angles round an axis which is perpendicular
to a plane which either is, or may be, a face of either crystal.
The axis will be called the twin-axis, and the plane to which it is
perpendicular the twin-plane.’ Miller’s _Treatise on Crystallography_,
p. 103. In the text the word ‘twin-plane,’ quoted from the writings
of Stokes and Rayleigh, is used to signify the plane common to the
two crystals in each of the cases referred to: and not the plane
perpendicular to this plane, in which one part of the crystal must be
rotated to bring it into coincidence with the other, and which is the
twin-plane as defined by Miller.
[12] ‘A clear transparent crystal of potassium chlorate, from which
the inevitable twin-plate had been ground away so as to reduce it to
a single crystal film about 1 mm. in thickness, was placed between
pieces of mica and laid on a thick iron plate. About 3 cm. from it
was laid a small bit of potassium chlorate, and the heat of a Bunsen
burner was applied below this latter, so as to obtain an indication
when the temperature of the plate was approaching the fusing-point of
the substance (359° _C_ according to Prof. Carnelly). The crystal plate
was carefully watched during the heating, but no depreciation took
place, and no visible alteration was observed, up to the point at which
the small sentinel crystal immediately over the burner began to fuse.
The lamp was now withdrawn, and when the temperature had sunk a few
degrees a remarkable change spread quickly and quietly over the crystal
plate, causing it to reflect light almost as brilliantly as if a film
of silver had been deposited upon it. No further alteration occurred
during the cooling; and the plate, after being ground and polished on
both sides, was mounted with Canada balsam between glass plates for
examination. Many crystals have been similarly treated with precisely
similar results; and the temperature at which the change takes place,
has been determined to lie between 245° and 248°, by heating the
plates upon a bath of melted tin in which a thermometer was immersed.
With single crystal plates no decrepitation has ever been observed,
while with the ordinary twinned-plates it always occurs more or less
violently, each fragment showing the brilliant reflective power above
noticed.’--_Nature_, May 20, 1886.
[13] _Nature_, May 20, 1886.
[14] _Philosophical Magazine_, 1888, second half year, p. 260.
[15] See foot-note to § 22 above.
[16] Widmanstätten, 1807. Leydolt (1855, Wien. Akad. Ber. 15, 59, T.
9, 10. Baumhauer, Pogg. Ann. 138, 563 (1869); 140, 271; 142, 324; 145,
460; 150, 619.) For an account of these investigations, see Mallard,
_Traité de Crystallographie_ (Paris, 1884), Tome II, chapitre xvi.
[17] More exactly .9525, being 3/4 × cot 38° 13′; see p. 53.
[18] J. and P. Curie and C. Friedel, _Comptes Rendus_, 1882, 1883,
1886, 1892.
[19] Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen
an Krystallen. W. Voigt, Königl. Gesellschaft der Wissenschaften zu
Göttingen, August 2, 1890.
[20] Wiedemann, _Annalen_, 1892, xlv, p. 923.
[21] Generally miscalled ‘rotational.’
[22] See foot-note (2) to p. 54 above.
[23] See foot-note (3) to p. 54 above.
[24] See foot-note to § 22 above.
THE END
Oxford
PRINTED AT THE CLARENDON PRESS
BY HORACE HART, PRINTER TO THE UNIVERSITY
End of Project Gutenberg's The Molecular Tactics of a Crystal, by Lord Kelvin