Theory of the vibrations of the sodium chloride lattice

Abstract

The interest in the frequency spectrum of the thermal vibrations in a crystal arose chiefly in connexion with the problem of the specific heat of crystals at low temperatures. Debye’s theory of the specific heat, however, has been so successful that the actual determination of the frequency spectrum according to Born and v. Karman (1912) has been pushed into the background. But recent investigations, especially those of Blackman (1935,1937 a,b , 1938), have shown that appreciable deviations from Debye’s theory should occur according to the correct atomistic treatment. These deviations appear to be most pronounced near the absolute zero of temperature. It, therefore, seemed desirable to calculate the exact frequency spectrum of a crystal. The first attempt to calculate the frequency spectrum of a crystal was made by Born and v. Karman in their original paper. They assumed only quasi-elastic forces between neighbouring particles. Later calculations have been made for ionic lattices, for which we have a fair knowledge of the real forces which determine the equilibrium positions and the vibrations about them. The chief difficulty in that calculation has always been the long range of the Coulomb force which makes a direct summation over all lattice points impossible. Born and Thompson (1934), using a method developed by Ewald (1921), suggested a way of transforming these sums into more rapidly convergent expressions, and Thompson (1935) has given the final formulae for the coupling coefficients due to the Coulomb force in the equation of motion, but in his paper a slight mistake occurred in the definition of the coefficients, and so far no numerical results of these calculations have been published. Broch (1937) has given formulae for the case of a one-dimensional lattice making use of Epstein’s Zeta functions. Lyddane and Herzfeld (1938) have used an extension of Madelung’s method (1918) and they have given some numerical results, but their formulae are rather complicated, so that one cannot expect to compute the whole frequency spectrum by this method. Moreover, the problem of the thermal oscillations of an ionic lattice is not a purely electrostatic problem, and this point has not been made sufficiently clear by Lyddane and Herzfeld. Their treatment of the case of the residual rays is open to objection, and the question whether the potential, from which the coupling coefficients are obtained, satisfies the Laplace equation or Poisson’s equation is not clear.