Frequency moments of cubic metals and substitutional impurities: A critical review of impurity-host force-constant changes from Mössbauer data

A critical review is presented of various simplified impurity lattice models which have been used in the literature to extract effective host-impurity force-constant ratios from Mössbauer fraction and thermal-shift measurements in highly dilute alloys. It is shown that the use of some popular models, sometimes in combination with inappropriately matched dynamical parameters of the host, has led to a systematic underestimate of the range of force-constant changes for ${}_{}{}^{57}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{119}{}_{}{}^{}\mathrm{Sn}$, and ${}_{}{}^{197}{}_{}{}^{}\mathrm{Au}$ impurity systems. Mannheim's analytical impurity model provides at present the most practical and physically meaningful theoretical framework for an internally consistent method of analysis of Mössbauer lattice-dynamics experiments in cubic metals. A new and useful set of analytical relationships has been derived from the Mannheim model, expressing experimentally obtainable impurity to host moment ratios in terms of mass and force-constant ratios. Moments and their ratios have been calculated for some theoretical lattice models and 18 cubic metals and rare-gas solids for which neutron dispersion data are available. A uniform general force Born-von Kárman model was used in combination with elastic constants to assure internal consistency among the resulting data. Moment analyses of heat-capacity data for most metals used as hosts for impurity studies were performed in parallel. Generally, we found good agreement between these two independent sets of results. Remarkable uniformity and trends were found within the groups of fcc and bcc metals for the values of certain moment ratios. A summary of most reliable Mössbauer $f$ data for ${}_{}{}^{57}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{119}{}_{}{}^{}\mathrm{Sn}$, and ${}_{}{}^{197}{}_{}{}^{}\mathrm{Au}$ is presented, including some new experimental results for ${}_{}{}^{57}{}_{}{}^{}\mathrm{Fe}$ in Ir, Nb, and Rh. Using these impurity data in combination with the host data presented leads to a set of internally consistent effective host-impurity force-constant ratios which span a range $0.65<\mathrm{}\frac{A}{A^{\prime }}<2.6\mathrm{}$ for a range of mass ratios $0.3<\mathrm{}\frac{M^{\prime }}{M}<3.5\mathrm{}$. Several of the new values differ considerably from those in the literature. This study shows that no single host parameter is well correlated with the observed changes in force-constant ratio for a given impurity, despite earlier suggestions to the contrary. The observed trends must, therefore, depend upon a combination of several host parameters as yet not understood.