Collision strength (${\Omega }_{ji}$)

The dimensionless collision strength ${\Omega }_{ij}(E)$ describes electron-impact transitions between two atomic levels before thermal averaging. It can vary rapidly with collision energy, especially near excitation thresholds. The quantity entering a thermal collision rate coefficient for an atomic transition is its effective collision strength, usually denoted ${{\rm Y}}_{ij}(T_e)$, rather than one value of ${\Omega }_{ij}$.^[ref]

For levels $E_j>E_i$, write $\Delta E=E_j-E_i$ and measure the scattering variable $E\ge 0$ above the upper-level threshold. With a Maxwellian electron population,Let $g_i,g_j$ be the statistical weights of atomic levels, and set $T_K=T_e/(1\mathrm{K})$. The excitation and de-excitation rate coefficients areTheir ratio obeys detailed balance. The numerical coefficient is $8.63\times 10^{-6}$ when the rate coefficient is expressed in $\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$.^[ref]

If ${\Omega }_{ij}(E)$ is constant over the sampled energy range, the normalized integral gives ${{\rm Y}}_{ij}={\Omega }_{ij}$. Otherwise low-energy resonances can strongly alter the thermal result. For a non-Maxwellian velocity distribution, compute the appropriate cross-section–speed average; a tabulated Maxwellian effective collision strength is not automatically transferable. Multiplying $q_{ij}$ by electron number density gives the excitation rate per target chemical atom.

The effective collision strength is the dimensionless thermal average of an electron-impact collision strength. For a Maxwellian electron population, with collision energy $E$ measured above the upper-level threshold,It is a function of temperature, rather than a cross section evaluated at one representative energy.

If $\Omega$ is constant, the normalized exponential weight gives ${\rm Y}=\Omega$. Near-threshold resonances can instead strongly change the average, particularly at low temperature. Atomic excitation and de-excitation rate coefficients use this average together with statistical weights and, for excitation, the appropriate Boltzmann threshold factor.^[ref]