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Abstract
Using Blume's stochastic theory and the approach of Winkler and Gerdau,
time-dependent effects on perturbed angular correlation (PAC) spectra
due to defect motion in solids in the case of I = 5/2 electric
quadrupole interactions have been calculated. Detailed analysis for
several models, namely, N-state + Z (N = 3, 4, 6), XYZ + S(theta, phi),
and XYZ(eta_{f }) + Z(eta_{s}) models are reported here. The influence
on the perturbation function G_2(t) of the number of fluctuating
electric field gradient (EFG) states (N), the relative static EFG
strength ( gamma), the orientation of the symmetry axis of the static
EFG versus the fluctuating EFG ( theta, phi), and the asymmetry of the
EFGs (eta_{f}, eta _{s}) are studied. A large non -Hermitian complex
matrix (Blume matrix, B) has been solved for each of the above models.
Its eigenvalues have real parts and imaginary parts corresponding to the
damping factors and the frequencies in G_2(t), respectively. Damping and
static frequencies in G _2(t) have been observed in both slow and rapid
fluctuation regimes, i.e. suitable for the low and high temperature
regions, respectively. In the intermediate fluctuation regime, complex
behaviors of the damping factors and frequencies are observed due to the
mixing of the fluctuating EFG states. Approximate forms are given for G
_2(t) in the slow and rapid fluctuation regimes which cover a wide range
of temperatures and contain the most interesting physics. These
expressions allow one to fit PAC data for a wide range of temperatures
and dopant concentrations. An application of the 4-state symmetric model
is illustrated with data from a PAC study of ceria.
... The models we have included in the study reported here are N-state analogues of the much-studied XY Z model [1,[3][4][5][6][7][8]. We have treated N = 3, 4, 6, and 12 explicitly for PAC probes with I = 5/2 in powder samples, i.e. samples made up of crystallites of random orientations. ...
... In the slow limit for the N-state symmetric models, the exact result is λ = (N − 1)r [6,10]. In the fast limit, the exact result is 504 [3,7,11]. ...
... Since more than one site was present with different ω Q s, the experimenters had to assume that the observed relaxation peak was due to a weighted average of the various site contributions. Analysis of stochastic PAC models in the rapid fluctuation limit, including models combining static and dynamic contributions [3,[5][6][7]11], shows that damping of the rapid fluctuations is proportional to ω 2 Q and that averaging several contributions to estimate ω Q should thus be carried out as ω 2 Q . This process led to a consistent physical interpretation of the defect contributions to damping in Al 11 R 3 . ...
A perturbed angular correlation (PAC) experiment that measures dynamic
damping also needs information about the fundamental quadrupole
frequency to relate the damping as a function of temperature to the EFG
fluctuation rate. When the experiment is unable to access slow electric
field gradient (EFG) fluctuations that show the fundamental quadrupole
frequency directly, one needs additional information to determine the
hyperfine field parameters and thereby the connection between observed
damping and EFG fluctuation rates. One way to solve this problem is to
estimate the hyperfine parameters from the fluctuation rate for maximum
damping (i.e. at the relaxation peak) or from the rate of maximum
damping. This work relates both the maximum damping rate and the
fluctuation rate at the relaxation peak to EFG magnitudes (or quadrupole
frequencies) for five dynamic N-state symmetric models of fluctuating
EFGs.
... In general, stochastic models and their corresponding theoretical expressions for PAC perturbation functions must be considered in detail on a case-by-case basis. There have been several detailed investigations of how damping constants and frequencies depend on fluctuation rates [49,50,51] for a variety of models. The results of these investigations can be used to analyze experimental spectra obtained from physical situations that can be described by those models. ...
This paper provides an introduction to perturbed angular correlation (PAC) spectroscopy in the context of its application in the study of point defects and diffusion. It emphasizes what we anticipate to be of interest to non-PAC specialists who are interested in understanding variations in how PAC results are presented by different research groups and in how physical quantities such as defect formation energies, association energies, and migration barriers can be extracted from analysis of PAC spectra. Numerous citations are included to emphasize the universality of the analysis methods across different classes of materials including metallic, ceramic, and semiconducting compounds.
The XYZ model describes the interaction between nuclear probes and an electric field gradient that fluctuates among three orthogonal
directions. The model presents a means to calculate the perturbation function that represents spectra obtained using perturbed
angular correlation spectroscopy. Three analytic approximations of the perturbation function have been developed previously,
and they are evaluated in the present paper in the context of Cd jumping among In-lattice sites in In3La.
A library of computational routines has been created to assist in the analysis of stochastic models of hyperfine interactions.
We call this library the stochastic hyperfine interactions modeling library (SHIML). It provides routines written in the C
programming language that (1) read a text description of a model for fluctuating hyperfine fields, (2) set up the Blume matrix,
upon which the evolution operator of the system depends, and (3) find the eigenvalues and eigenvectors of the Blume matrix
so that theoretical spectra of experimental hyperfine interaction measurements can be calculated. Example model calculations
are included in the SHIML package to illustrate its use and to generate perturbed angular correlation spectra for the special
case of polycrystalline samples when anisotropy terms of higher order than A22 can be neglected.
KeywordsPerturbed angular correlation spectroscopy–Stochastic models
The stochastic hyperfine interactions modeling library (SHIML) provides a set of routines to assist in the development and application of stochastic models of hyperfine interactions. The library provides routines written in the C programming language that (1) read a text description of a model for fluctuating hyperfine fields, (2) set up the Blume matrix, upon which the evolution operator of the system depends, and (3) find the eigenvalues and eigenvectors of the Blume matrix so that theoretical spectra of experimental techniques that measure hyperfine interactions can be calculated. The optimized vector and matrix operations of the BLAS and LAPACK libraries are utilized; however, there was a need to develop supplementary code to find an orthonormal set of (left and right) eigenvectors of complex, non-Hermitian matrices. In addition, example code is provided to illustrate the use of SHIML to generate perturbed angular correlation spectra for the special case of polycrystalline samples when anisotropy terms of higher order than A 22 can be neglected.
Atomistic simulation calculations have been used to investigate the energetics of defect clustering and migration in ceria. The defects considered are In{sup 3+}, Cd{sup 2+}, their associated oxygen vacancies, and small polarons modeled as Ce{sup 3+} ions. Thus a range of complex defect clusters is considered. The overall aim of the study is to generate a better understanding of these defects as they relate to recent experimental results obtained using perturbed-angular-correlation spectroscopy. The calculations are successful in this regard, correctly predicting both binding energies and an oxygen migration activation energy. More importantly, the calculations provide an atomistic explanation for certain of the experimental observations. As such, the synergy between calculations and experiment is an important feature of this paper.
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