= Stoner threshold for an interaction normalized per electron
{c}
{title2=$U_c=4E_F/3$}
For a three-dimensional zero-temperature parabolic <Fermi gas>, let $\zeta=(N_\uparrow-N_\downarrow)/N$ and $E_F=\hbar^2(3\pi^2N/V)^{2/3}/(2m)$. Suppose the interaction <energy> per <Electron> is $UN_\uparrow N_\downarrow/N^2=U(1-\zeta^2)/4$. The total <energy> per <Electron> is
$$
f(\zeta)=\frac{3E_F}{10}\left[(1+\zeta)^{5/3}+(1-\zeta)^{5/3}\right]+\frac U4(1-\zeta^2).
$$
Its <Taylor expansion> has quadratic coefficient $E_F/3-U/4$ and positive quartic coefficient $E_F/81$. Hence the unpolarized state loses stability at $U_c=4E_F/3\propto(N/V)^{2/3}$. This $U$ has units of energy. It differs from a contact coupling $g$ defined by interaction-energy density $g n_\uparrow n_\downarrow$, for which $g=U/n$ and $g_c=4E_F/(3n)$. Normalization must be fixed before comparing critical couplings. For $U\leq4E_F/3$, <convexity> of $x^{-1/3}$ implies $f''(\zeta)=(E_F/3)[(1+\zeta)^{-1/3}+(1-\zeta)^{-1/3}]-U/2\geq0$, so the unpolarized minimum is a <global minimum>; a first-order polarized minimum does not pre-empt this threshold in the stated model.
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