Many-body effects on the magnetic susceptibilities of metals

The paramagnetic and diamagnetic susceptibilities of an electron fluid at low but finite temperatures are determined by taking into consideration the free electron, first- and second-order exchange, and ring diagrams, without using the variational method adopted frequently in the past, but rather using a grand-ensemble method. In this method, the grand partition function is evaluated as a function of the magnetic field. The contribution from the ring diagrams is treated accurately with the finite momentum corrections to order $q^4$ and with the finite temperature corrections to order ${\mathrm{}\left(\mathrm{kT}\right)\mathrm{}}^2$ to the eigenvalues. The paramagnetic susceptibility, when compared with the data on alkali metals, is found satisfactory. Both the $r_s$ and temperature dependences of the para- and diamagnetic susceptibilities are determined. When plotted against $r_s$, the paramagnetic susceptibility increases over the Pauli susceptibility. The diamagnetic susceptibility is also somewhat increased by interaction, although this depends on $r_s$. The Coulomb energy and the thermal energy are found to counteract in the manifestation of the magnetism. The energy associated with the external magnetic field is evaluated as a function of $r_s$. In comparison with the case without interaction, the field energy varies more slowly with $r_s$. The correlation part of the field energy is also determined.