Collision rate coefficient for an atomic transition ($q_{ij}(T)=⟨{\sigma }_{ij}v⟩$)

For a normalized electron speed distribution $f_T(v)$, the atomic transition rate coefficient is $q_{ij}(T)=\int {\sigma }_{ij}(v)v{f}_T(v)dv$. Its dimensions are volume per time, so the rate per target atom is $n_e{q}_{ij}$. Maxwellian detailed balance gives $q_{ji}=(g_i/g_j)e^{(E_j-E_i)/kT}{q}_{ij}$ for $E_j>E_i$.