is positive and much larger than the van der Waals length $\ell_{\rm vdW}\approx 65a_0$, the atoms $i$ and $j$ can form a dimer with binding energy $\hbar^2/(m a_{ij}^2)$. We call this dimer the ([*shallow*]{}) $ij$-dimer or simply a shallow dimer. Shallow dimers have to be distinguished from deep dimers with binding energy of order $\hbar^2/(m \ell_{\rm vdW}^2)$ or larger.

In a gas of atoms $i$ and $jk$-dimers, the atoms and dimers can undergo inelastic collisions into atoms and deeply bound dimers with a binding energy larger than that of the $jk$-dimer. This inelastic process is called [ *dimer relaxation*]{}. The difference in the binding energies of the initial and final state dimers is released as kinetic energy and the atom and dimer in the final state recoil from each other. If their kinetic energies are larger than the trapping potential, they escape the trap. The loss rate for the number density $n_i$ of atoms $i$ and number density $n_{jk}$ of $jk$-dimers is $$\frac{d\ }{dt} n_i =\frac{d\ }{dt} n_{jk} 
= - \beta_i \,n_i n_{jk},
\label{dndt}$$ where the coefficient $\beta_i$ is the [*relaxation rate constant*]{} for the $jk$-dimer and atom $i$.

In the case of identical bosons, dimer relaxation is possible only 