ABSTRACT
The diagonal matrix elements of the Zeeman Hamiltonian, Hmag for a state |αSL J M〉 are given by (2-53),
〈αSL J M |Hmag|αSL J M〉 = Bzµ0 Mg(SL J ) In intermediate coupling, we have from (5-1)
〈a J M |Hmag|a J M〉 = ∑ α,S,L
〈a J |αSL J 〉〈αSL J M |Hmag|αSL J M〉〈αSL J |a J 〉
= Bzµ0 M ∑ α,S,L
〈a J |αSL J 〉g(SL J )〈αSL J |a J 〉 (5-3)
Thus, using the linear combination from (5-2) we have for the ground state of Pr3+
〈a4M |Hmag|a4M〉 = Bzµ0 M × [(−0.0282)2g(3 F4) + (0.1523)2g(1G4) + (0.9879)2g(3 H4)]
= Bzµ0 M(0.8045) (5-4)
where g(3 F4) = 1.2506 g(1G4) = 1.0000 g(3 H4) = 0.79954 (5-5)
As a result of intermediate coupling the value of g−factor is changed from 0.7995 for the pure 3 H4 to 0.8045 for the intermediate coupling functions. Here the correction is rather small. As a second example consider the level calculated at 10, 004cm−1 with the eigenstate
|b4〉 = 0.5037|3 F4〉 + 0.8558|1G4〉 − 0.1175|3 H4〉 (5-6)
with the dominant component of 1G symmetry. The g−factor evaluated with this function has the value of 1.0607, compared with that of the pure 1G4 g−factor of 1.0000. Finally, for the third J = 4 eigenstate
|c4〉 = −0.8634|3 F4〉 + 0.4943|1G4〉 − 0.1009|3 H4〉 (5-7)
which in its major part is of the 3 F symmetry. The intermediate coupling corrected g−factor is 1.1847 compared with that for a pure 3 F4 g−factor of 1.2506. The sum of intermediate coupling corrected g−factors gives
gi = 0.8045 + 1.0607 + 1.1847 = 3.0499
whereas the sum of the three L S g−factors gives 3∑
i=1 gi = 0.7995 + 1.0000 + 1.2506 = 3.0501
Although these two results are the same to within the precision of the calculation, they illustrate the g-sum rule.
