Unitary thermodynamics from thermodynamic geometry

Degenerate Fermi gases of atoms near a Feshbach resonance show universal thermodynamic properties, which are here calculated with the geometry of thermodynamics, and the thermodynamic curvature $R$. Unitary thermodynamics is expressed as the solution to a pair of ordinary differential equations, a "superfluid" one valid for small entropy per atom $z\equiv S/N k_B$, and a "normal" one valid for high $z$. These two solutions are joined at a second-order phase transition at $z=z_c$. Define the internal energy per atom in units of the Fermi energy as $Y=Y(z)$. For small $z$, $Y(z)=y_0+y_1 z^{\alpha}+y_2 z^{2 \alpha}+\cdots,$ where $\alpha$ is a constant exponent, $y_0$ and $y_1$ are scaling factors, and the series coefficients $y_i$ ($i\ge 2$) are determined uniquely in terms of $(\alpha, y_0, y_1)$. For large $z$ the solution follows if we also specify $z_c$, with $Y(z)$ diverging as $z^{5/3}$ for high $z$. The four undetermined parameters $(\alpha,y_0,y_1,z_c)$ were determined by fitting the theory to experimental data taken by a Duke University group on $^6$Li in an optical trap with a Gaussian potential. The very best fit of this theory to the data had $\alpha=2.1$, $z_c=4.7$, $y_0=0.277$, and $y_1=0.0735$, with $\chi^2=0.95$. The corresponding Bertsch parameter is $\xi_B=0.462(40)$.