Shielding of Crystal Fields at Rare-Earth Ions

In connection with the interpretation of the spectra of the salts of rare-earth ions, the appropriate shielding factors ₂, ₄, and ₆ have been calculated for Pr^3+ (Z=59) and Tm^3+ (Z=69). The shielding factor ₊ measures the effective reduction of the A₊^_^qr^kY₊ₐ term of the crystal potential at the location of the 4f shell which is produced by the more external 5s and 5p electrons of the rare earths (and also to a smaller extent by the inner core electrons). The values of ₄ and ₆ were found to be small, i. e. , of order 0. 1. Thus ₄ is weakly shielding (₄=0. 091 for Pr^3+), while ₆ gives a small antishielding (₆=-0. 040 for Pr^3+). A previous calculation of ₂ has been improved by including the effect of the 4s, 4p, and 4d subshells. The resulting values of ₂ are in reasonable agreement with experiment. Finally, an angular factor which arises in the evaluation of the direct and exchange terms of ₊ has been expressed in terms of 3-j and 6-j symbols involving the various angular momenta of the problem.