Abstract: The present paper presents a numerical study of the Fermi surface near and inside a Brillouin zone corner in a conducting band. Both the periodic lattice potential and mutual jnteractions among electrons are taken into account. For concreteness and illustration, the computation was carried out on the metallic aluminum on its first Brillouin zone corner. The parameters of the potential are from the calculated results of Heine. The main results obtained are as follows:The shape of the Fermi surface are far from spherical, being protruding in central part of the solid angle of the zone corner,and gradually smoothes down as the layers of electrons forming the Fermi surface are continually added; there exists a minimum of the electron energy in the continual addition of layers of extraneous electrons and thereafter the energy of electrons increases rapidly; the number of the electron density reaches about 10¹⁸/cm³.An estimation of the magnitude of exchange energy shows that.a decrease of the exchange energy about one third for the electrons from different zone corners as compared with that inside a same corner. This will give rise to a broken symmetry in the distribution of electron currents. Further investigation of the problem seems necessary in view of the direct contradiction to the Bloch's theorem of ground states without a permanent current.