Pauli paramagnetism

Pauli paramagnetism is the weak positive magnetic response produced when a magnetic field shifts the two spin Fermi surfaces of a degenerate electron gas. At zero temperature, if $g(E_F)$ includes both spin states, its total magnetic moment and susceptibility are

For a smooth single-spin density of states $g$ near a positive Fermi energy and fixed total particle number, the Sommerfeld expansion gives $\mu ={ϵ}_F-({\pi }^2/6)(g_F^{\prime }/g_F)(k_{B}T{)}^2+O(T^4)$. The spin contribution to the total-moment magnetic susceptibility, defined by differentiating with respect to $B$, is $2{\mu }_B^2\int g(E)[-n^{\prime }(E)]dE$. Expanding about $\mu$ and including the shift of chemical potential gives the displayed formula. The $g_F^{\prime \prime }$ term is thermal broadening; the $-g_F^{\prime 2}/g_F$ term enforces fixed particle number. A band edge or singular density of states within $k_{B}T$ of ${ϵ}_F$ requires a separate analysis. Orbital magnetic response is excluded.

For noninteracting spin-half particles with dispersion $E=cp$, the two spin populations at common chemical potential $\epsilon$ are $V(\epsilon \pm \mu B{)}^3/(6{\pi }^2{\hslash }^3{c}^3)$. Fixed total number changes $\epsilon$ only at quadratic order in $B$. Their population difference is therefore $3N\mu B/(c{p}_F)$ to first order, giving the displayed magnetic moment. Only the specified spin-energy splitting is included, not orbital magnetic effects.