= Stoner criterion for an opposite-spin contact repulsion
{c}
{title2=$U g_s(E_F)>1$}
For an interaction-energy density $U n_\uparrow n_\downarrow$, let $n_\uparrow=n/2+\delta$, $n_\downarrow=n/2-\delta$ at fixed total <carrier density>. If $g_s(E_F)$ is the unpolarized <single-spin density of states> per unit volume, the kinetic-energy cost of the two occupation shifts is $\delta^2/g_s(E_F)$, whereas the interaction changes by $-U\delta^2$. Therefore
$$
\Delta\mathcal E=\left[\frac1{g_s(E_F)}-U\right]\delta^2+O(\delta^4),\qquad
U_c=\frac1{g_s(E_F)}=\frac2{g_{\rm tot}(E_F)}.
$$
For the three-dimensional <free electron model>, $g_{\rm tot}(E_F)=3n/(2E_F)$ and $U_c=4E_F/(3n)$. Both the interaction normalization and the spin counting in the <density of states> must be specified when quoting a <Stoner criterion>.
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