= Solution
<Electrons> contribute magnetic response through both <spin> and orbital motion. In an ordinary weakly interacting metal, a field splits the two spin energies. Only states near the <Fermi energy> can change occupancy because of the <Pauli exclusion principle>, producing <Pauli paramagnetism>. For a total two-spin density of states $g(E_F)$, the simple result is $\chi_P=\mu_0\mu_B^2g(E_F)$. Orbital cyclotron motion gives <Landau diamagnetism>; for an isotropic parabolic free-electron band it contributes $-\chi_P/3$. Filled shells can also supply a diamagnetic background.
Interactions can favor parallel spin occupancy. In an itinerant description, the exchange-energy gain competes with the kinetic-energy cost of redistributing electrons between spin bands. The <Stoner criterion> describes when an unpolarized state becomes unstable, with the precise factor in $Ig(E_F)$ depending on whether the density of states counts one spin or both. This mechanism can yield spontaneous <ferromagnetism> and magnetic domains. Alternatively, sufficiently localized moments in a metal can interact through conduction electrons. A high density of states alone is therefore not a complete microscopic explanation of a material's magnetism; band structure, electron interactions and temperature matter.
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