The Matrix Elements of Tensor Operators for the Electronic Configurationsfn

It is shown that many simple relations exist between various reduced matrix elements of the form ( f n WUSL || U k || f n W'U'SL' ), where W = ( w 1 w 2 w 3 ) and U = ( u 1 u 2 ) are irreducible representations of R 7 and G 2 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 (2 k + 1) 1/2 , transform among themselves as the irreducible representations (200) of R 7 and (20) of G 2 . 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 R 7 occurs in the reduction of the product ( w 1 w 2 w 3 ) × (200) × ( w 1 ' w 2 ' w 3 '); similarly, the number of times the identity representation (00) of G 2 occurs in the reduction of the product ( u 1 u 2 ) × (20) × ( u 1 ' u 2 ') determines the number of linearly independent sets of matrix elements that can exist with U = ( u 1 u 2 ) and U ' = ( u 1 ' u 2 '). 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 f n , 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.