The Theory of the Raman Effect in Crystals, in Particular Rock-Salt
Proceedings of the Royal Society A
The R am an effect in crystals is treated in this paper with the help of Placzek's approxim ation. I t consists of contributions of different orders with respect to th e am plitudes of the vibrations; the first-order effect is a line spectrum depending only on the vibrations of infinite w ave length, the second-order effect is a continuous spectrum depending on com bination fre quencies of all pairs of branches of the lattice vibrations, each pair taken for the same wave vector. In highly symm
... trical crystals like rock-salt the first-order effect is zero. The secondorder effect can be,calculated for rock-salt w ith the help of the tables of the lattice frequencies published by K ellerm ann. I t consists of thirty-six peak's, each belonging to a combination frequency. The superposition of these allows us to determine w ithout any arb itrary assum p tion about the coupling constants, the frequency of the observable m axim a in fair agreement w ith K rishnan's m easurem ents. By adapting three coupling constants one can also determine the relative intensities of the m ost prom inent peaks and obtain a curve which in its m ain features agrees w ith the observed one. The results show th a t lattice dynamics can account quantitatively for the R am an effect in crystals and th a t R am an's attacks against the theory are unfounded. Vol. 188. A [ 161 ] II on July 19, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from Raman for demonstrating this statement: specific heat, diffuse X-ray scattering, optical effects. The temperature dependence of the specific heat is, of course, not sensitive enough to supply a sharp criterion; it is, for example, well known th a t Lindemann's formula, which uses only two frequencies, can hardly be distinguished from Debye's formula, which uses a continuous spectrum. In the case of diffuse X-ray scattering the situation is now even less favourable for R am an's ideas; for it turns out th at the diffuse thermal X-ray spots, when correctly interpreted, are a direct image of the continuous vibrational spectrum, which could in principle be derived from measurements of the intensity distribution of the scattering. There remains the optical effects, which, according to Raman, show directly the existence of only a small number of characteristic frequencies in crystals. R am an's pupil Krishnan (1943) has written a paper on the Raman effect in rock-salt, discussing and reinterpreting the observations of Fermi & Rasetti. He contends th a t their result has to be regarded as a line spectrum (of 9 Stokes and 9 anti-Stokes lines). Now the photograph of the spectrum may possibly be interpreted as a system of faint lines, though the continuous background seems obvious; Fermi & Rasetti have, however, published above the photograph the microphotometric curve of the intensity distribution which shows without the slightest doubt th at the Italian authors are right in describing it as a continuum with small peaks. Krishnan says th at the 9 lines, which he counts rather arbitrarily on each side of the incident line, are exactly what Ram an's theory predicts; but he makes no attem pt to calculate the position of the lines from his theory nor to discuss the intensities. Recently, Krishnan (1945 a) has repeated the observations and obtained a spec trum and microphotometer curve which is a confirmation of R asetti's work and an improvement on it in so far as the little peaks are much sharper. Krishnan declares again th at they constitute a line spectrum, but he counts only 6 strong and 2 weak lines, some of which may be double. So it seems th a t he has given up the claim th a t these peaks confirm the existence of 9 lines as predicted by Ram an's theory. Under these circumstances it seemed to us desirable to develop the correct lattice theory of the Raman effect for crystals and apply it quantitatively to rock-salt. This substance is particularly suited for this purpose, since Kellermann (1940) has published the theory of the lattice vibrations of rock-salt containing tables of the coefficients of the equations of motion and of the six frequency branches for a fairly narrow distribution of wave vectors.