= Occupation of an interacting localized electron level
{title2=$\mathcal E_N=E_N-\mu N$}
A localized orbital with two <electron spin> states has <energies> $E_0=0$, $E_1=\epsilon$ and $E_2=2\epsilon+U$. Its zero-<temperature> equilibrium occupation minimizes $\mathcal E_N=E_N-\mu N$, where $\mu$ is the <chemical potential>. For repulsion $U>0$, the occupation is successively $0,1,2$ as $\mu$ crosses $\epsilon$ and $\epsilon+U$. For attraction $U<0$, the singly occupied state is never the minimum: the occupation jumps from $0$ to $2$ at $\mu=\epsilon+U/2$. Exactly at a crossing the ground-state occupation is not unique.
At positive <temperature>, its <grand canonical partition function> is $\Xi=1+2e^{-\beta(\epsilon-\mu)}+e^{-\beta(2\epsilon+U-2\mu)}$, where $\beta=1/(k_BT)$. The <expected value> of the occupation follows by differentiating: $\langle N\rangle=k_BT\,\partial_\mu\ln\Xi$. The factor two counts the two singly occupied <microstates>, not an additional pair state.
Back to article page
