Size Effects on the Diamagnetic Susceptibility of a Free-Electron Gas

We consider a noninteracting electron gas constrained in one dimension by a harmonic-oscillator potential as a model of a metallic film with specular surfaces. The diamagnetic response of this system to an applied field is investigated to study the effect of sample size on both the de Haas-van Alphen (dHvA) oscillations and the steady susceptibility. When the diameters of the dominant cyclotron orbits become comparable to the effective sample "thickness," we find large departures from the familiar $H^{-1\mathrm{}}$ law for the dHvA oscillations. However, the steady part of the susceptibility appears to remain independent of size.