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 f₀(Ω) of order 3N×3N, where Ω is the mass-reduced force-constant matrix. Our approach is to use a Cauchy singular integral representation for f₀(Ω) involving an integration along the real frequency, ω, axis. Matrix partitioning techniques are used to reduce our problem to one of evaluating the 3×{}3 impurity atom dynamic response matrix, G₁₁=(1+ε)[I₃+τεA₁₁]^-1A₁₁, where τ=ω²-iδ. Here, δ is an arbitrarily small number, and ε+1=ratioofimpurityatomtohostatommass, (MIMH). For an arbitrary physical arrangement of the atoms, A₁₁=[I3(z+1)-Dz+1(ΔFMH)]^-1Dz+1₁₁, where the subscript, 1, refers to the impurity atom coordinates, ΔF is the perturbation in force-constant matrix, and z is the number of sites over which the perturbation extends. The Dz+1 matrix has matrix elements obtained from the elements of the pure host matrix DH=[τI3N-FHMH^-1]^-1, FH is the pure host force-constant matrix. Iₖ is a k×k unit matrix.The general approach is used to study the dynamic response of an impurity atom substituted in the aluminum lattice with arbitrary ε and nearest neighbor ΔF. The A matrix is block diagonalized by introducing the molecular vibration symmetry coordinates and A₁₁ is characterized by a 4×{}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 DH, the pure aluminum lattice Green's function matrix. Similar studies are carried out for a Sn¹¹⁹ 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 G₁₁. Typical K functions are presented for Fe⁵⁷ in Al for various changes in ΔF and for Sn¹¹⁹ in Ge with ΔF=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 γ rays absorbed by the impurity nucleus, f, the Lamb-M\"ossbauer coefficient, $2W$, and mean-square velocity, (vI²)ₐᵥ, of Fe⁵⁷-Al are tabulated for several ΔF changes as a function of temperature. Our results are extrapolated to study the temperature dependence of $2W$ and f for Fe⁵⁷-Cu and Fe⁵⁷-Pt. From the results derived in this paper, it is possible to determine K, $2W$, and (vI²)ₐᵥ for any ε and ΔF for Al as a host lattice.