in the 1(23)-channel, $$f_{1(23)}(k) = \left[ -1/a_{1(23)}-ik \right]^{-1},
\label{fAD-a23}$$ we can calculate the inelastic scattering cross section. At low temperatures, the contributions from higher partial waves can be neglected. We subtract the elastic cross section from the total cross section obtained via the optical theorem as in [@Braaten:2006nn] and find for the inelastic cross section: $$\sigma_{1(23)}^{(inelastic)}(k) = 
\frac{4\pi}{k} \frac{-{\rm Im}\, a_{1(23)}}{1-2k{\rm Im}\, a_{1(23)}
+k^2 |a_{1(23)}|^2}.
\label{sigAD-a23}$$ The total dimer-relaxation rate $\beta_1$ can then be calculated by taking a Boltzmann thermal average of the inelastic reaction rate $v_{rel}\, \sigma_{1(23)}^{(inelastic)}(k)$ where $v_{rel}=3\hbar k/(2m)$ is the relative velocity of the atom and dimer in the initial state. This leads to the expression $$\beta_1(T)=\frac{3\hbar}{2m}\langle k\sigma_{1(23)}^{(inelastic)}(k) \rangle=
\frac{3\hbar}{2m} \frac{3\lambda_T^3}{4\pi^2}\sqrt{\frac{3}{2}}
\int_0^\infty k^2 dk \;k\sigma_{1(23)}^{(inelastic)}(k)\;
e^{-\frac{3\hbar^2k^2}{4m k_BT}},
\label{betaAD-a23}$$ where $\lambda_T=\sqrt{2\pi \hbar^2/(m k_B T)}$ is the thermal de Broglie wavelength of the atoms.

Our results for the atom-dimer relaxation rate constant near the resonance at $B_{*}^{\rm exp}=685$ G are shown in Fig. \[fig:beta1\_T\]. The results for the width parameters $\eta_*=0.016$ and $\eta_*=0.34$ are given by the solid and dashed curves, respectively. In each case the upper curves give the zero temperature result while the lower curves give the finite temperature result for $T=60 \pm 15$ nK. Here, the 