Racah seniority number

The Racah seniority number (seniority quantum number) ${\displaystyle \nu }$ was introduced by Giulio Racah for the classification of electrons in an atomic configuration.[1] The "seniority number", in a loosing statement, is quantum number additional to the total angular momentum ${\displaystyle L}$ and total spin ${\displaystyle S}$, which gives the degree of unpaired particles.

A spin-independent interaction ${\displaystyle {\hat {V}}}$ is assumed with the property[2]

${\displaystyle <l^{2};LM_{L}|{\hat {V}}|l^{2};LM_{L}>=g(2l+1)\delta _{L0}}$,

where ${\displaystyle L}$ is the combined angular momentum, ${\displaystyle M_{L}}$ magnetic quantum number, ${\displaystyle l}$ is electrons' orbital angular momenta, and ${\displaystyle g}$ is the dimensionless magnetic moment. The equation above shows there is no interaction unless the two electrons' orbital angular momenta are coupled to ${\displaystyle L=0}$. The eigenvalue is the "seniority number" ${\displaystyle \nu }$.

References

1. Racah, Giulio (1943). "Theory of Complex Spectra. III". Phys. Rev. 63: 367. doi:10.1103/physrev.63.367.
2. Isacker, P. Van (2010). "Seniority in quantum many-body systems". Symmetries in Nature: Symposium in Memoriam Marcos Moshinsky: 141.