Specific Heat of a Normal Fermi Liquid. I. Landau-Theory Approach

The contribution of small-momentum-transfer processes to the $T^3 \ln T$ term in the specific heat of a normal Fermi liquid is considered using Landau theory, and is evaluated exactly in terms of Landau parameters. This contribution is shown to be due to a term in the quasiparticle interaction, $f_{\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}}$, which varies as ${\left(\hat{p}·\hat{q}\right)}^2$ when $\overrightarrow{\mathrm{q}}\to 0$; the physical process responsible for this behavior is the repeated scattering of quasiparticle-quasihole pairs. It is well known that for the two-body problem the interaction energy, while not, in general, equal to the real part of the forward-scattering amplitude, may be expressed in terms of the scattering amplitude. We take over the results for the two-body problem and apply them to a quasi-particle-quasihole pair. This gives an expression for $f_{\overrightarrow{\mathrm{p}},\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}}$ in terms of the quasiparticle-quasihole scattering amplitude, which, since $\overrightarrow{\mathrm{q}}$ is small, may be expressed in terms of Landau parameters. We apply our results to liquid ${\mathrm{He}}^3$ at low pressure.