# Crystal fields at rare-earth ions

## Abstract

The object of this paper is to find out why the calculated rare-earth crystal-field parameters do not agree with those determined experimentally. As is well known, in explaining the 4f spectra in the rare-earth compounds, one needs relatively few crystal-field constants, Vnm's. These Vnm's have been measured for many rare-earth ions in LaCl3, ethylsulfates and other lattices. One can study the V nm's in a given lattice as the nuclear charge is varied (starting from Ce, Z=58 to Yb, Z =70). For a given n and m one finds that the Vnm's vs Z vary fairly smoothly. However, when the crystal fields are calculated using an ionic model one cannot get good agreement with the Vnm's. In general, neither the magnitude nor the Z dependence of the Vnm's can be reproduced. The calculated crystal-field parameters are written as: (1-αn) 〈r n〉Anm, where αn is the shielding factor (the outer 5s2p6 electrons to some extent shield the inner 4f electrons from the crystal field), 〈r n〉 is the expectation value of rn over the 4f electrons, and Am is a lattice sum over the charges in the lattice that produce the crystal field. To try to determine why the agreement is bad, the effect of the lattice in altering the values of α and 〈r n〉 was calculated. However, the principle trouble appears to come from the Am values. Anm∝1/R n+1, where R is the distance between the rare-earth ion and the charges in the lattice that produce the crystal fields. For the important n=4 and 6 lattice sums one gets very rapid convergence. The Anm sums are usually calculated assuming the lattice can be replaced by point charges, point dipoles, etc., at the nucleus of the ions surrounding the rare-earth ions. Thus, R is taken as the various internuclear distances. However, for lattice sums that depend on such high powers of R, one must consider the extended nature of the charge distribution of the ions surrounding the rare earths. Another way of stating the trouble is: Since the distance between the observer (rare-earth ion) and the charge distribution is not large compared to the extent of the charge distribution it is not a good approximation to replace the charge distribution by point moments. From numerical examples it is easy to see, for n=4 and 6, that most of the contribution to the A nm sums comes from the parts of the surrounding ions that are closest to the rare-earth ions. Thus it becomes difficult to calculate the Anm's accurately. However, using this idea, agreement is obtained with the dependence of the Fnm's on Z and some of the interrelationships between the different n terms. For the lattices usually used as hosts one cannot calculate the A2° sum because of its sensitivity to the x-ray data and charge distribution parameters. However, by using nuclear quadrupole resonance data to eliminate calculating the actual lattice sum, one gets good agreement between the measured V20's and (1-α2)〈r2〉 A20.