Dynamics of Cubic Crystals with a Local Change of Mass

The effects of a local change of mass on lattice vibrations of cubic systems are discussed for simple Bravais lattices, on the basis of the harmonic theory of crystal dynamics. It is shown that the change of any extensive property, as well as the values of the discrete frequencies, are given in terms of simple expressions involving only the distribution function $G_0\left({\omega }^2\right)$ of the square frequency for the unperturbed lattice; the theoretical expressions are found to be equivalent to those holding for linear chains, provided that the distribution function is normalized to one. Perturbative techniques are discussed in some detail; it is found that a perturbative expansion of the change of any extensive property converges to the right value, provided the fractional change of mass, $\epsilon =-\frac{(M^{\prime }-M)}{M}$, lies inside the range from $-\infty$ to +½. Some applications are carried out starting from Overton's distribution function for a fcc lattice: a discrete frequency, threefold degenerate, is found to occur for $\epsilon >~0.215$, and the self-entropy of a substitutional impurity (neglecting the influence of the elastic distortion) is shown to be related only to the structure of the distribution function, in the high-temperature limit; this change of entropy is evaluated for several light impurities. Finally, arguments are presented which suggest that the contribution to the self-entropy of a vacant lattice site, arising from the loss of coupling with the neighbors, may be higher than was estimated by Huntington and coworkers.