Inhomogeneous broadening of PAC spectra with V _zz and η joint probability distribution functions

Abstract

The perturbed angular correlation (PAC) spectrum, G ₂(t), is broadened by the presence of randomly distributed defects in crystals due to a distribution of electric field gradients (EFGs) experienced by probe nuclei. Heuristic approaches to fitting spectra that exhibit such inhomogeneous broadening (ihb) consider only the distribution of EFG magnitudes V _zz , but the physical effect actually depends on the joint probability distribution function (pdf) of V _zz and EFG asymmetry parameter η. The difficulty in determining the joint pdf leads us to more appropriate representations of the EFG coordinates, and to express the joint pdf as the product of two approximately independent pdfs describing each coordinate separately. We have pursued this case in detail using as an initial illustration of the method a simple point defect model with nuclear spin I = 5/2 in several cubic lattices, where G ₂(t) is primarily induced by a defect trapped in the first neighbor shell of a probe and broadening is due to defects distributed at random outside the first neighbor shell. Effects such as lattice relaxation are ignored in this simple test of the method. The simplicity of our model is suitable for gaining insight into ihb with more than V _zz alone. We simulate ihb in this simple case by averaging the net EFGs of 20,000 random defect arrangements, resulting in a broadened average G ₂(t). The 20,000 random cases provide a distribution of EFG components which are first transformed to Czjzek coordinates and then further into the full Czjzek half plane by conformal mapping. The topology of this transformed space yields an approximately separable joint pdf for the EFG components. We then fit the nearly independent pdfs and reconstruct G ₂(t) as a function of defect concentration. We report results for distributions of defects on simple cubic, face-centered cubic, and body-centered cubic lattices. The method explored here for analyzing ihb is applicable to more realistic cases.