Numerical Example: ⁴F_J States of Nd³⁺ (4f³)

Abstract

As a numerical example of the calculation of the crystal-field splitting, we calculate the splitting for Nd³⁺ (4f³) in a field of S₄ point symmetry. We assume that the levels are pure ⁴F_J. We assume that L, S, and J are all good quantum numbers; then we consider matrix elements of H₃ in equation (9.9) with J’ = J and L’ = L. Thus,

$$<JM'LS|{H_3}|JMLS>=\sum\limits_{kq}{B_{kq}^*<J(M)k(q)|J(M')>}<J|{U^k}|J><\ell(0)k(0)|\ell(0)>$$

((11.1))

. The values of the reduced matrix elements <J | U^k| J> <ℓ | C_k |ℓ> for the ⁴F state of Nd³⁺ are as in table 11.1. In obtaining these values we have used Nielson and Koster’s results (1963) in equations (9.9) and (9.10) for the reduced matrix elements, (LS|U^k|LS), for L = 3 and S = 3/2; the Racah coefficients (6-j symbols) are found in Rotenberg et al (1969).