Improved Theory for the Temperature-Dependent Hyperfine Coupling Constant of Rare-EarthS-State Ions

Abstract

An improved version of the theory for the temperature dependence of the hyperfine coupling constant of rare-earth $S$-state ions in cubic crystals is developed which incorporates an exact spatial-averaging procedure valid over the entire phonon-wavelength range. The improved treatment eliminates one of the major weaknesses and points of uncertainty in the theory, viz., the ambiguity arising from the existence of separate calculations applicable only in the long- or short-wavelength limits. The question of the range of validity of these approximations is therefore removed. Contributions from both the acoustic branch, based on a Debye spectrum, and the optical branches, based on an Einstein spectrum, are determined. It is shown that the new expressions reduce exactly to those obtained in the long-wavelength limit, but not to those commonly associated with the short-wavelength approximation. The improved theory is applied to the temperature dependence of the hyperfine coupling constant of ${}_{}{}^{151}{}_{}{}^{}\mathrm{Eu}_{}^{2+}{}_{}{}^{}$ in Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$, and the results are compared with those of the long-wavelength model as well as with the experimental data. Good qualitative agreement with experiment is obtained, and possible sources of the disagreement between the observed magnitude of the temperature decrease in the hyperfine constant and that predicted by the improved theory are discussed. In addition, calculations of the rigid-lattice values of the hyperfine constant of ${\mathrm{Eu}}^{2+}$ are made, and it is found that these values are not equal in the alkaline-earth fluorides, the differences lying outside the error limits. Finally, the improved treatment shows that some of the results produced by the long-wavelength approximation are in fact spurious.