Unitarity-Limited Elastic Collision Rate in a Harmonically-Trapped Fermi Gas

We derive the elastic collision rate for a harmonically-trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is $\sigma (k)=4\pi \mathrm{/}{k}^2$, with $\mathrm{\hslash }k$ the relative momentum. The collision rate is given in the form $\mathrm{\Gamma }=\gamma I(T\mathrm{/}{T}_F)$–the product of a universal collision rate $\gamma =k_B{T}_F\mathrm{/}(6\pi \mathrm{\hslash })$ and a dimensionless function of the ratio of the temperature $T$ to the Fermi temperature $T_F$. We find $I$ has a peak value of $\simeq 4.6$ at $T\mathrm{/}{T}_F\simeq 0.4$, $I\simeq 82(T\mathrm{/}{T}_F{)}^2$ for $T\mathrm{/}{T}_F\le 0.15$, and $I\simeq 2(T_F\mathrm{/}T{)}^2$ for $T\mathrm{/}{T}_F>1.5$. We estimate the collision rate for recent experiments on a strongly-interacting degenerate Fermi gas of atoms.