Geometrical properties of the Fermi energy

The Fermi energy at 0°K is evaluated for electrons confined to cubical and spherical rigid-walled boxes of equal volume, respectively, in the Sommerfeld approximation. Due primarily to large differences in single-particle degeneracies, Fermi energies compared for equal numbers of particles in these two configurations are found to be unequal. Approximate expressions of the Fermi energy in the large particle-number limit for the spherical case reveal that it agrees in form with the Fermi energy for the cubical configuration. The finite cylindrical box is also examined and it is found that for fixed number of particles and fixed volume, the Fermi energy varies as the two characteristic lengths of the box are changed. Experimental corroboration of the theory is suggested through measurement of the work function for equal-volume micrometallic samples in the different geometries.