Effective Spinndash;Orbit Hamiltonian

摘要

Deviations from the Landeacute; interval rule governing separations between theJmultiplets of atomic terms are known to stem from term mixing by the spinndash;orbit interaction. A phenomenological interaction of the formHEthinsp;equals;thinsp;Sgr;nlgr;nlpar;Lmiddot;Srpar;nwith1thinsp;le;thinsp;nthinsp;le;thinsp;pcan account for any such separations ifpthinsp;equals;thinsp;2S, one less than the multiplicity of the term. The general theory of effective operators is applied to the spinndash;orbit interaction for electron configurations of the formlN, and it is shown that, throughpth order in the perturbation expansion of the spinndash;orbit Hamiltonianhthinsp;equals;thinsp;eegr;Sgr;i1imiddot;si, the above form forHEobtains wherelgr;nthinsp;equals;thinsp;eegr;Sgr;mlpar;eegr;thinsp;sol;thinsp;F2rpar;mminus;1lgr;nm, nthinsp;le;thinsp;mthinsp;le;thinsp;p. Assuming hydrogenlike wavefunctions forfelectrons, thelgr;nmare shown to be constants depending only on the term being considered and the number of equivalent electronsN. Usingeegr;andF2values given in the literature, thelgr;nare calculated for the ground terms of the triply ionized rare earths. Values for thelgr;nhave been reported for Nd3plus;, Tb3plus;, Ho3plus;, and Er3plus;in CaWO4where the Starkhyphen;split energy levels of the ground terms were analyzed by treating thelgr;nand the crystalhyphen;field parameters inHXthinsp;equals;thinsp;Sgr;kmBkmdagger;Ckmas adjustable. The theoretical and experimental values forlgr;1, lgr;2andlgr;3agree to within 2.6percnt;, 7.0percnt;, and 23percnt;, respectively. The calculation is then inverted andeegr;andF2values for the four ions are determined by requiring the theoreticallgr;nparameters to equal those obtained empirically. Theeegr;andF2values obtained in this manner differ from those reported by 2.3percnt; and 5.1percnt;, respectively. It is therefore feasible, through the use of the effective spinndash;orbit HamiltonianHE, to determineeegr;andF2values for the ground configuration of an ion by considering the ground term alone.