Dynamical Motion and Gamma-Ray Cross Section of an Impurity Nucleus in a Crystal. I. Isolated Impurities in Germanium and Aluminum

The general theory of the dynamical motion and gamma-ray cross section for a single impurity nucleus harmonically coupled to an arbitrary collection of $N$ atoms is developed in supermatrix representation. The relevant properties of the system are expressed in terms of a functional matrix $f_0\left(\Omega \right)$ of order $3N\times 3N$, where $\Omega$ is the mass-reduced force-constant matrix. Our approach is to use a Cauchy singular integral representation for $f_0\left(\Omega \right)$ involving an integration along the real frequency, $\omega$, axis. Matrix partitioning techniques are used to reduce our problem to one of evaluating the 3×3 impurity atom dynamic response matrix, ${\left\{\mathrm{G}\right\}}_{11}=(1+\epsilon ){[{\mathrm{I}}_3+\tau \epsilon {\mathrm{A}}_{11}]}^{-1\mathrm{}}{\mathrm{A}}_{11}$, where $\tau ={\omega }^2-i\delta$. Here, $\delta$ is an arbitrarily small number, and $\epsilon +1=\mathrm{ratio}\mathrm{of}\mathrm{impurity}\mathrm{atom}\mathrm{to}\mathrm{host}\mathrm{atom}\mathrm{mass}$, ($\frac{M_I}{M_H}$). For an arbitrary physical arrangement of the atoms, ${\mathrm{A}}_{11}={\left\{{[{\mathrm{I}}_{3(z+1\mathrm{})\mathrm{}}-{\mathrm{D}}_{z+1\mathrm{}}(\frac{\Delta \mathrm{F}}{M_H}\left)\right]}^{-1\mathrm{}}{\mathrm{D}}_{z+1\mathrm{}}\right\}}_{11}$, where the subscript, 1, refers to the impurity atom coordinates, $\Delta \mathrm{F}$ is the perturbation in force-constant matrix, and $z$ is the number of sites over which the perturbation extends. The ${\mathrm{D}}_{z+1\mathrm{}}$ matrix has matrix elements obtained from the elements of the pure host matrix ${\mathrm{D}}_H={[\tau {\mathrm{I}}_{3N}-{\mathrm{F}}_H{M_H}^{-1\mathrm{}}]}^{-1\mathrm{}}$, ${\mathrm{F}}_H$ is the pure host force-constant matrix. ${\mathrm{I}}_k$ is a $k\times k$ unit matrix.