Orbital magnetic susceptibility of electrons confined in a rectangular box

The orbital magnetic susceptibility of a gas of noninteracting electrons confined by hard walls to a rectangular box has been calculated in the low-magnetic-field limit. The result for the size-corrected susceptibility in the high-temperature limit is $\ensuremath{\chi}={\ensuremath{\chi}}_{L}[1\ensuremath{-}\frac{{\ensuremath{\lambda}}_{T}({L}_{x}^{\ensuremath{-}1}+{L}_{y}^{\ensuremath{-}1})}{16{\ensuremath{\pi}}^{\frac{1}{2}}}]$ where ${\ensuremath{\chi}}_{L}$ is the Landau diamagnetism, ${\ensuremath{\lambda}}_{T}$ is the thermal de Broglie wavelength, and ${L}_{x}$ and ${L}_{y}$ are the dimensions of the box perpendicular to the direction of the magnetic field. A slightly more complicated formula for the case of Boltzmann statistics is accurate for the range ${\ensuremath{\lambda}}_{T}\ensuremath{\le}2L$. A similar expression is found for Fermi-Dirac statistics valid for high values of the Fermi energy. The result corrects an error by Papapetrou, and agrees with the recent results of Angelescu, Nenciu, and Bundaru. When applied to the case of infinite slabs (${L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}$), it agrees with the results of several previous authors. The result, however, indicates that calculations by Dingle and Osborne for spherical and cylindrical volumes are in error. Numerical calculations for the case of Fermi-Dirac statistics at temperatures a few percent of the Fermi level give results which agree reasonably well with the size-corrected susceptibility formula even for a very few electrons in the box (${\ensuremath{\lambda}}_{F}\ensuremath{\lesssim}2L$, where ${\ensuremath{\lambda}}_{F}$ is the de Broglie wavelength at the Fermi level).

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