The Matrix Elements of Tensor Operators for the Electronic Configurationsfn

Abstract

It is shown that many simple relations exist between various reduced matrix elements of the form (fⁿ WUSL || U^k || fⁿW'U'SL'), where W = (w₁w₂w₃) and U = (u₁u₂) are irreducible representations of R₇ and G₂ respectively. This is done by using the fact that for k = 2, 4 and 6, the components U_q^k of the tensor operators U^k, when multiplied by (2k + 1)^1/2, transform among themselves as the irreducible representations (200) of R₇ and (20) of G₂. The number of linearly independent sets of matrix elements for a given W and W' is equal to the number of times the identity representation (000) of R₇ occurs in the reduction of the product (w₁w₂w₃) × (200) × (w₁'w₂'w₃'); similarly, the number of times the identity representation (00) of G₂ occurs in the reduction of the product (u₁u₂) × (20) × (u₁'u₂') determines the number of linearly independent sets of matrix elements that can exist with U = (u₁u₂) and U' = (u₁'u₂'). These two numbers, which are denoted by c(WW'(200)) and c(UU'(20)), are tabulated for all W, W', U and U' which occur in fⁿ, and the use of the tables in calculating the splittings induced in the levels of rare earth ions by the surrounding crystal lattice is illustrated with a number of examples.