Charge penetration effects in rare-earth crystal fields

The theory of charge penetration by a rare-earth ion into the charge distribution on a ligand has been developed. The results are cast into a form reminiscent of the Sternheimer shielding factors, sigma n, which relate the observed A prime nm < r to the nth power > to the lattice sum A nm < r to the nth power > in the form A prime nm < r to the nth power > = A prime nm < r to the nth power > (1 - sigma n). The corresponding form developed here is A prime nm < r to the nth power > = A prime nm < r to the nth power > (1 - delta n(x,y)/< r to the nth power >) with explicit expressions given for the delta n(x,y). The delta n(x,y) is then expanded for small x, and a form convenient for calculation is given. The results obtained here can be used with the three-parameter theory of crystal fields to obtain crystal field parameters Bnm (tau;q eta gamma), which give a better representation of the phenomenological Bnm than we obtained previously.