= Collision strength
{title2=$\Omega_{ji}$}
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 $\Upsilon_{ij}(T_e)$, rather than one value of $\Omega_{ij}$.<The CHIANTI upsilon files (ups and scups) by Peter Young>[PDF p. 2, introductory definition: Maxwellian-integrated strengths depend on temperature]{ref}
For levels $E_j>E_i$, write $\Delta E=E_j-E_i$ and measure the scattering variable $E\geq0$ above the upper-level threshold. With a <Maxwellian> electron population,
$$
\Upsilon_{ij}(T_e)=\int_0^\infty\Omega_{ij}(E)e^{-E/(k_BT_e)}\,d\left(\frac{E}{k_BT_e}\right).
$$
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 are
$$
q_{ij}=\frac{8.63\times10^{-12}}{g_i\sqrt{T_K}}\Upsilon_{ij}(T_e)e^{-\Delta E/(k_BT_e)}\,\mathrm{m^3\,s^{-1}},\qquad
q_{ji}=\frac{8.63\times10^{-12}}{g_j\sqrt{T_K}}\Upsilon_{ij}(T_e)\,\mathrm{m^3\,s^{-1}}.
$$
Their ratio obeys <detailed balance>. The numerical coefficient is $8.63\times10^{-6}$ when the rate coefficient is expressed in $\mathrm{cm^3\,s^{-1}}$.<Fine-structure collision strengths and line ratios for Ne V in infrared and optical sources by Michael Dance and others>[Section 2.3, equations (3)–(4): Maxwellian averaging and excitation coefficient]{ref}
If $\Omega_{ij}(E)$ is constant over the sampled <energy> range, the normalized integral gives $\Upsilon_{ij}=\Omega_{ij}$. 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>.
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