in understanding the atom-dimer relaxation data from Refs. [@Lompe:2010; @Nakajima:2010]. A full finite temperature calculation of the $^6$Li system is beyond the scope of this work. Therefore, we perform an approximate calculation of the relaxation rate $\beta_1$ near the resonance at $B_{*}^{\rm exp}=685$ G.

![(Color online) The relaxation rate constant $\beta_1$ for a 23-dimer and an atom 1 as a function of $a/a_*$ near the resonance at $B_{*}^{\rm exp}=685$ G ($a_*=1076~a_0$). Squares and circles are data points from Ref. [@Lompe:2010] and Ref. [@Nakajima:2010], respectively. The solid (dashed) curves correspond to $\eta_*=0.016$ ($\eta_*=0.34$). The upper curves give the zero temperature result, while the lower curves give the finite temperature result for $T=60 \pm 15$ nK with the shaded area indicating the temperature uncertainty. []{data-label="fig:beta1_T"}](beta1_T3.pdf){width="15cm"}

We start from the approximate expression for the scattering length between an atom 1 and a 23-dimer near the resonance at $B_{*}$ that was extracted from a calculation of the trimer binding energy in [@Braaten:2009ey]: $$a_{1(23)} \approx \left( C_1 \cot[ s_0 \ln(a_{23}/a_*) + i \eta_*]
                         + C_2 \right) a_{23} ,
\label{aAD-a23}$$ where $a_* = a_{23}(B_*)$ and the coefficients are $C_1 = 0.67$ and $C_2 = 0.65$. Using the scattering length approximation for the S-wave atom-dimer scattering amplitude 