In this article, the second order perturbation energy for an ion in the crystal field due to all other ions, each being regarded as a point charge, is given as follows Considering this energy in the investigation on the elasticity theory of the alkali halide cry stals by the use of Born and Haung's method, we obtain the deviation of the Cauchy's relation in the following form As an illustration, the numerical values of the deviation are calculated and we conclude that the contribution of the quadrapole term of the second order perturbation energy to the deviation of the Cauchy's relation can not be neglected.
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