Part 4

§ 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.

§ 49. Change now the directions of the grooves and fillets to either of 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.

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.

§ 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