= Contact-interaction Hartree-Fock spin energy
{title2=$\mathcal E=A(n_\uparrow^{5/3}+n_\downarrow^{5/3})-\gamma(n_\uparrow^2+n_\downarrow^2)/2$}
In a uniform three-dimensional <Hartree-Fock electron gas> with repulsive contact coupling $\gamma$, a spin-independent neutralizing background can cancel the <Hartree energy>. The remaining same-spin exchange term is negative and quadratic in each <spin> <density>. Integrating the filled single-spin <Fermi sphere> gives $A=3\hbar^2(6\pi^2)^{2/3}/(10m)$. At fixed total <density> the exchange term favors <spin polarization>, whereas the kinetic term opposes it.
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