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]

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

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" ${\displaystyle \nu }$.