Note on the internal consistency of the signs in the matrix of S1/2

Abstract

A discrepancy in the evaluation of the sign of the quantity S1/2 is stated. The Shore and Menzel(1) phases differ from the Condon and Shortley(2) phases for transitions from J to J + 1. This result is a consequence of the fact that the first pair of authors consider S1/2 as transforming like a reduced matrix element, while the second take S1/2 (αJ, αJ) as transforming like the quantity (αJ P αJ). The discrepancy in signs is removed if one looks at the relation between the reduced matrix element and the matrix element and the quantity (αJ P αJ). The Condon and Shortley(2) determination of the phase is more trustworthy, since for every breakdown of the LS-coupling from αJ to ΓJ the value of the line strength of the inverse transition equals the value of the line strength of the transition involved, which is not necessarily the case with the Shore and Menzel treatment. It is pointed out that in the case that the quantum numbers J do not get mixed up, it does not matter by which method the sign of S1/2 is determined.