Crystal Field for Yb3+ in Garnets
Authors: Hutchings M. T. , Wolf W. P. 
Published: April 2005

Nine parameters Anm〈rn〉 are necessary to define the crystal field acting on the rare-earth ions in garnets, and a priori none of these may be considered negligible. As there are in general insufficient data to determine these from experiment alone, we have attempted to reduce the number of unknowns by estimating the ratio of terms of the same degree from an accurate point charge calculation, leaving only the effective values of 〈r2〉, 〈r4〉, and 〈r6〉 to be determined from observed electron spin resonance and optical data. Calculations for a number of garnet lattices show that all nine parameters may be quite important. There are significant contributions from ions more distant than the eight nearest oxygen neighbors usually considered, and the frequently used approximation of near cubic symmetry is in general poor. There is a large uncertainty in the two second degree terms and it is therefore necessary to treat both A20〈r2〉 and A22〈r2〉 as unknowns.
Ytterbium is the simplest of the rare earths to consider theoretically as it has a large spin-orbit coupling constant, and L and S, as well as J, are good quantum numbers. Most experimental work has been done on Yb3+ in yttrium gallium garnet for which there are seven essentially independent pieces of information. Using a computer program we have fitted to these data the four overdetermined adjustable parameters A20〈r2〉, A22〈r2〉, A40〈r4〉, and A60〈r6〉, the remaining five parameters being determined from the ratio of calculated potentials. The final fit, to within 0.2 for all six g values and 2 cm−1 for the two J=52 splittings, gives some indication of the validity of the approximations made.
Choosing a coordinate system (ξ, η, ζ), in which the ζ axis is a ``fourfold'' axis of the distorted ``cube'' formed by the nearest oxygen neighbors and the ξ axis lies along a unit cell [001] axis, the fitted parameters are A20〈r2〉=−86 cm−1, A22〈r2〉=297 cm−1, A40〈r4〉=−193 cm−1, A42〈r4〉=159 cm−1, A44〈r4〉=535 cm−1, A60〈r6〉=72 cm−1, A62〈r6〉=−229 cm−1, A64〈r6〉=1315 cm−1, A66〈r6〉=−233 cm−1. The corresponding energy levels for the J=72 state are at 0, 517, 697, and 796 cm−1, in good agreement with indirect evidence from susceptibility data. A comparison of the wavefunction of the lowest state shows it to be quite close to that of a pure cubic Γ7 doublet, and combined with the large energy separation to the next level explains why the near cubic approximation is so successful for ytterbium. It has not been possible to extend these calculations to other lattices because of the lack of experimental data, and no definite predictions for the iron garnet lattices can be made.
Several severe approximations have been made in this analysis, and it is certain that our crystal field parameters are not completely accurate. We have neglected the effect of the finite nonspherical extent of the ionic charges, possible covalency, both between the magnetic and the nonmagnetic ions, and we have assumed the complex shielding and antishielding effects of closed electron shells to depend on the degree n only. When further experimental data become available to determine the crystal field parameters uniquely, a comparison with our results will provide an interesting measure of the validity of the point charge approximation.
For the present we believe that our set of crystal field parameters represents the best approximation that can be obtained from the data now available for Yb3+ in the garnets.
A full account of this work is being published elsewhere.