We measured Stoke's parameters P1, P2, P3, and P4 of excited Ar+ in coincidence with the scattered He for the two-electron process[Formula: see text]From the measured Stoke's parameters of the 4610 Å (1 Å = 10−10 m) radiation of the 2F(J = 7/2) to the 2D(J = 5/2) transition and the 4590 Å radiation of the 2F(J = 5/2) to 2D(J = 3/2) transition we determined the alignment and orientation parameters of the upper states using the Fano–Macek formalism. These parameters were then converted to the appropriate spherical tensor components in the J representation to obtain, in general eight values of [Formula: see text]. Since these states are best described by LS coupling, we decoupled the spherical tensor components of the density matrix of each J state in terms of the spherical components of the density matrices of the total orbital angular momentum L and total spin 5. The unknowns in the expansion are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], with [Formula: see text] because of reflection symmetry. We used Mathematica to calculate the decoupling coefficients and solve these nonlinear equations. The decoupling also allows us to optically determine the octopole moments of the total orbital momentum provided that [Formula: see text] is nonzero. When [Formula: see text], the two sets of equations are linearly dependent. In this case, the determination of a single set of [Formula: see text] specifies [Formula: see text].
Transition density matrices of Richardson–Gaudin states
