Landau diamagnetism and the de Haas-van Alphen effect from a single geometric construction

Landau diamagnetism is usually derived from the grand canonical potential, a calculation that yields the correct susceptibility but little physical insight, while the rule that extremal cross sections govern the de Haas-van Alphen (dHvA) oscillations is commonly asserted rather than exhibited. We present an elementary zero-temperature construction in which the occupied states of a free-electron gas are grouped, interval by interval along the field direction, into the Landau levels onto which they condense. Within each interval the field-induced energy cost reduces to a transfer of states between two congruent triangles, and center-of-mass arithmetic yields the Landau susceptibility. The construction fails only in a narrow neighborhood of an extremal cross section of the Fermi surface; there the contribution oscillates with the dHvA period, one oscillation for each Landau level that passes through the extremal cross section. Direct zero-temperature state counting locates the oscillation: the residual is concentrated in the few intervals nearest the extremal cross section, the summed contribution of the distant ones being negligible on the scale of the peak oscillation amplitude.