The configuration l ^{N} has more than one allowed term of certain LS types if l > 1 and 2 < N < 4l (d ^{3} - d ^{7}, f ^{3} - f ^{11}, etc.). The recurring terms of a particular LS term type from d ^{N} and f ^{N} configurations are assigned sequential index numbers in the tables of Nielson and Koster [7]; the index numbers stand for additional numbers having group-theoretical significance that serve to differentiate the recurring terms, except for a few terms of f ^{5} and f ^{9}, f ^{6} and f ^{8}, and f ^{7}. These remaining terms, which occur only in pairs, are further labeled A or B to indicate Racah's separation of the two terms.
The index numbers of Nielson and Koster are in practice the most frequently used labels for the recurring terms of f ^{N} configurations. Use of their index numbers for the recurring terms of d ^{N} configurations has perhaps the disadvantage of substituting an arbitrary number for a quantum number (the seniority) that itself distinguishes the recurring terms in all cases. The actual value of the seniority number is rarely needed, however, and a consistent notation for the d ^{N} and f ^{N} configurations is desirable. A table of the allowed LS terms of the l^{ N} electrons for l ≤ 3 is given in Ref. [8], with all recurring terms having the index numbers of Nielson and Koster as a following on-line integer. The theoretical group labels are also listed. Thus the d^{ 3} ^{2}D term having seniority 3 is designated ^{2}D2, instead of D, in this scheme; and the level having J = ^{3}/_{2} is designated ^{2}D_{3/2}2.
l_{j}^{N} | Allowed J values |
---|---|
l_{1/2} | ^{1}/_{2} |
l ^{2}_{1/2} | 0 |
l_{3/2} and l ^{3}_{3/2} | ^{3}/_{2} |
l ^{2}_{3/2} | 0, 2 |
l ^{4}_{3/2} | 0 |
l_{5/2} and l ^{5}_{5/2} | ^{5}/_{2} |
l ^{2}_{5/2} and l ^{4}_{5/2} | 0, 2, 4 |
l ^{3}_{5/2} | ^{3}/_{2}, ^{5}/_{2}, ^{9}/_{2} |
l ^{6}_{5/2} | 0 |
l_{7/2} and l ^{7}_{7/2} | ^{7}/_{2} |
l ^{2}_{7/2} and l ^{6}_{7/2} | 0, 2, 4, 6 |
l ^{3}_{7/2} and l ^{5}_{7/2} | ^{3}/_{2}, ^{5}/_{2}, ^{7}/_{2}, ^{9}/_{2}, ^{11}/_{2}, ^{15}/_{2} |
l ^{4}_{7/2} | 0, _{2}2, _{4}2, _{2}4, _{4}4, 5, 6, 8 |
l ^{8}_{7/2} | 0 |
The allowed levels of the configuration nl ^{N} may be obtained by dividing the electrons into sets of two groups , Q + R = N. The possible sets run from Q = N - 2l (or zero if N > 2l) up to Q = N or Q = 2l + 2, whichever is smaller. The (degenerate) levels for a set with both Q and R nonzero have wave functions defined by the quantum numbers (αJ_{1}, βJ_{2}) J, with J_{1} and J_{2} deriving from the Q and R groups, respectively. The symbols α and β represent any additional quantum numbers required to identify levels. The J values of the allowed levels for each (αJ_{1}, βJ_{2}) subset are obtained by combining J_{1} and J_{2} in the usual way.