Specific Heat of a Degenerate Electron Gas at High Density

The methods of the preceding paper of Gell-Mann and Brueckner are generalized so that not only the ground state but also the low excited states of an electron gas can be discussed. The energy levels relative to the ground state are the same as those of a gas of independent particles where the energy of each particle (in rydbergs) is a certain function $W\left(p\right)\mathrm{}$ of its momentum (expressed in units of the Fermi momentum). The specific heat of the gas at low temperature is proportional to the density of single particle levels at the surface of the Fermi sea, or inversely proportional to ${\left(\frac{\mathrm{dW}}{\mathrm{dp}}\right)}_{p=1\mathrm{}}$. This last quantity is calculated for high density (small $r_s$, where density is proportional to ${r_s}^{-3\mathrm{}}$) and compared to the corresponding quantity for a free electron gas. The ratio is found to be $1+0.083r_s[-\ln r_s-0.203]+\mathrm{higher}\mathrm{terms}$ in $r_s$. The expansion is exact and may be compared with the approximate result of Pines, who finds $1+0.083r_s[-\ln r_s+1.47]+\cdots$.