XXXIV. On the Dispersion of Diamond. By L. SILBERSTEIN, Ph.D., Lecturer in Mathematical Physics at the University of Rome".

THE HE object of the present paper is to apply the concept of electrical interaction of atoms, developed in previous papers †, to the refractive properties of diamond considered as a known assemblage of fixed "atomic centres," each containing a single dispersive electron and becoming a doublet in presence of an external electric field. The assumption of the mutual immobility of the "centres themselves, i. e. of the whole atoms, need not be given up until one comes to contemplate the infra-red free frequencies; the ultra-violet ones, which chiefly interest us for the present, can all be thrown upon the electrons.

Consider a space lattice of points, indefinitely extended in all directions. Let all these points be occupied by the centres of equal atoms, each containing a single electron whose charge and mass are e, m. Write Bem, and denote by the vector p; the electric moment of the doublet produced at the i-th centre by the electric force E of the incident light-wave (and by the co-operation of all the remaining doublets). Then, for monochromatic light of

* Communicated by the Author.

For what follows it will suffice to consult the second paper on "Refractivity and Atomic Interaction," Phil. Mag. vol. xxxiii. p. 521 (1917), more especially the general formulæ (2), (3) on p. 522.

squared frequency y, the equation of motion of the i-th electron gives, for any arrangement of equal centres,

where is the squared free frequency, common to all the electrons, u the unit vector drawn from the i-th towards the j-th centre (or vice versa), r the mutual distance of these centres, and G the force on the i-th electron, per unit charge, due to the external field E and due to all those doublets not already taken account of under the sign of summation.

If the sum, in our case a triply infinite series, embraces all the centres, that is to say, all "neighbours" of i, no matter how distant, then the force G reduces simply to E. For, according to the opinion generally held, electromagnetic phenomena are influenced by particles of matter only so far as these contain electric charges. Now, in the case of our present problem it will be easy to exhaust all centres.

Thus,

According to Bragg, diamond consists of the superposition of two face-centred cubic lattices, one of which is obtained from the other by translating it rigidly along a cube diagonal one-quarter of the length of the diagonal*. Each point of both lattices is occupied by a carbon atom. The result of this superposition is that each carbon atom occupies the centre of a regular tetrahedron whose four corners are occupied by the four nearest neighbours of that atom. The relation of this tetrahedron to the original face-centred lattice is well-represented in fig. 1, drawn in perspective (for which I am indebted to Mr. E. Hatschek). The lattice points are the centres of the five spheres drawn so as to be in contact with one another. The rough sketch, fig. 2, corresponding to Bragg's photograph on Plate III. of his work, represents a number of centres linked up by straight lines to their four nearest neighbours. (The actual distance of nearest neighbours, i. e. centres, or in usual notation d is 1.54. 10-8 cm. This, however, is irrelevant for our present

* W. H. and W. L. Bragg, 'X-Rays and Crystal Structure,' London, Bell (1915), pp. 105 et seq. A beautiful construction of the "fundamental domains" of Bragg's diamond lattice was given by L. Föppl in Phys. Zeitschrift, vol. xv. p. 191 (1914).

purpose, owing to the ultimate cancelling of all interactions.) Details showing the very strong experimental evidence in

Fig. 1.

Fig. 2.

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support of the view that such is precisely the structure of diamond will be found in Bragg's work.

In order to evaluate the vector sum appearing in formula (1), fix as the i-th centre any of the lattice points, say O in fig. 2. Call 1, 2, 3, 4 its four nearest neighbours. The assemblage being unlimited and homogeneous, all the atomic centres are in equal conditions. Disregarding the free oscillations of the system, which would soon die away, let us take account only of the vibrations of p; forced by the external field. Then, in a homogeneous * a homogeneous field E, the moments p; will either be all equal in size and direction, or, at the utmost, split into two classes, p' and p". In order to see this we can proceed as follows. Let, for instance, E be parallel to and concurrent with the vector 01. Then every atomic centre will be either such as O, in which E points. away from it towards its next neighbour (viz. 1), or such as 1, or 2, etc., where E is directed towards them, away from their nearest neighbours. Let p' correspond to the first, and p' to the second class of centres. Then, denoting by N the linear vector operator in formula (1), i. e. writing it, as a dyadic,

Ως

1 4π

1

[3u.u-1],

(2)

the vector sum, say P, representing the total action of all doublets upon the electron in O will be

P='p'+Q"p",

where ' is as in (2) with u=u', r=r', and with Σ extended to the first class of points, and similarly for ".

Now, E being along 01, the vectors p', p' can, by reasons of symmetry, have only the same or opposite directions. Thus, denoting the unit vector from 0 to 1 by i,

p'p'i, p' sp'i,

where s is a scalar (undetermined thus far), and therefore, P=p'('+s')i.

But if the field E is reversed, the rôles of the two classes of centres will be exchanged: the moment for each member of the first class will be sp'i, and for the second class -p'i; at the same time the total P will be reversed, so that (the

*This will also hold with sufficient accuracy in the case of luminous oscillations, since their wave-length is very great as compared with the mutual distance of neighbour centres. The latter being of the order of 10 cm., a wave-length cube of visible, or of not too remote ultraviolet light, will contain 101 or 1010 atomic doublets.

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