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

Abstract

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 ${f}_{0}(\ensuremath{\Omega})$ of order $3N\ifmmode\times\else\texttimes\fi{}3N$, where $\ensuremath{\Omega}$ is the mass-reduced force-constant matrix. Our approach is to use a Cauchy singular integral representation for ${f}_{0}(\ensuremath{\Omega})$ involving an integration along the real frequency, $\ensuremath{\omega}$, axis. Matrix partitioning techniques are used to reduce our problem to one of evaluating the 3\ifmmode\times\else\texttimes\fi{}3 impurity atom dynamic response matrix, ${{\mathrm{G}}}_{11}=(1+\ensuremath{\epsilon}){[{\mathrm{I}}_{3}+\ensuremath{\tau}\ensuremath{\epsilon}{\mathrm{A}}_{11}]}^{\ensuremath{-}1}{\mathrm{A}}_{11}$, where $\ensuremath{\tau}={\ensuremath{\omega}}^{2}\ensuremath{-}i\ensuremath{\delta}$. Here, $\ensuremath{\delta}$ is an arbitrarily small number, and $\ensuremath{\epsilon}+1=\mathrm{ratio}\mathrm{of}\mathrm{impurity}\mathrm{atom}\mathrm{to}\mathrm{host}\mathrm{atom}\mathrm{mass}$, ($\frac{{M}_{I}}{{M}_{H}}$). For an arbitrary physical arrangement of the atoms, ${\mathrm{A}}_{11}={{{[{\mathrm{I}}_{3(z+1)}\ensuremath{-}{\mathrm{D}}_{z+1}(\frac{\ensuremath{\Delta}\mathrm{F}}{{M}_{H}})]}^{\ensuremath{-}1}{\mathrm{D}}_{z+1}}}_{11}$, where the subscript, 1, refers to the impurity atom coordinates, $\ensuremath{\Delta}\mathrm{F}$ is the perturbation in force-constant matrix, and $z$ is the number of sites over which the perturbation extends. The ${\mathrm{D}}_{z+1}$ matrix has matrix elements obtained from the elements of the pure host matrix ${\mathrm{D}}_{H}={[\ensuremath{\tau}{\mathrm{I}}_{3N}\ensuremath{-}{\mathrm{F}}_{H}{{M}_{H}}^{\ensuremath{-}1}]}^{\ensuremath{-}1}$, ${\mathrm{F}}_{H}$ is the pure host force-constant matrix. ${\mathrm{I}}_{k}$ is a $k\ifmmode\times\else\texttimes\fi{}k$ unit matrix.The general approach is used to study the dynamic response of an impurity atom substituted in the aluminum lattice with arbitrary $\ensuremath{\epsilon}$ and nearest neighbor $\ensuremath{\Delta}\mathrm{F}$. The A matrix is block diagonalized by introducing the molecular vibration symmetry coordinates and ${\mathbf{A}}_{11}$ is characterized by a 4\ifmmode\times\else\texttimes\fi{}4 symmetry adapted Green's function matrix whose elements have been tabulated. A generalized tensor force-constant model is used with Walker's force constants characterizing ${\mathrm{D}}_{H}$, the pure aluminum lattice Green's function matrix. Similar studies are carried out for a ${\mathrm{Sn}}^{119}$ atom isotopically substituted in Ge, where the relevant Green's functions are derived from Phillip's frequency spectrum.The dynamical motion and gamma-ray cross section of impurity nuclei are characterized by a dynamic response function, $K$, which is related to the imaginary part of ${{\mathrm{G}}}_{11}$. Typical $K$ functions are presented for ${\mathrm{Fe}}^{57}$ in Al for various changes in $\ensuremath{\Delta}\mathrm{F}$ and for ${\mathrm{Sn}}^{119}$ in Ge with $\ensuremath{\Delta}\mathrm{F}=0$. Our results show that the dynamical behavior of impurity atoms in real lattices is quite sensitive to the vibrational properties of the host lattice. The resonant fraction of $\ensuremath{\gamma}$ rays absorbed by the impurity nucleus, $f$, the Lamb-M\"ossbauer coefficient, $2W$, and mean-square velocity, ${({{v}_{I}}^{2})}_{\mathrm{av}}$, of ${\mathrm{Fe}}^{57}$-Al are tabulated for several $\ensuremath{\Delta}\mathrm{F}$ changes as a function of temperature. Our results are extrapolated to study the temperature dependence of $2W$ and $f$ for ${\mathrm{Fe}}^{57}$-Cu and ${\mathrm{Fe}}^{57}$-Pt. From the results derived in this paper, it is possible to determine $K$, $2W$, and ${({{v}_{I}}^{2})}_{\mathrm{av}}$ for any $\ensuremath{\epsilon}$ and $\ensuremath{\Delta}\mathrm{F}$ for Al as a host lattice.