Recombination Coefficients

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle #1\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\qot{q_{12}} \def\qto{q_{21}} \def\ehvkt{e^{-h\nu_{21}\over2kT}} \def\hf{\frac12} \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document}

\subsection*{THIS PAGE IS STILL UNDER CONSTRUCTION} \subsection*{Derivation}

\def\sigbf{\sigma_{bf}} \def\alphaNL{\alpha_{n,l}} \def\sigfb{\sigma_{fb}}

The recombination coefficient $\alphaNL$ is analogous to $q_{12}$, the collisional rate coefficient, and is derived in a similar way. The rate of recombinations per unit volume is:

$${\textrm{\#\ of\ recombinations}\over \textrm{volume} \cdot \textrm{time} }= n_+n_e\mean{\sigfb v} = n_+n_e\int_0^{\infty}\sigfb(v)v f(v)dv = n_+n_e \alphaNL$$ Therefore the recombination coefficient for recombination to level \emph{n, l} (\emph{n} is the principal quantum number and \emph{l} is the orbital quantum number) is defined below where $f(v)$, the distribution of velocities, is a Maxwellian. $$\alphaNL \equiv \int_0^{\infty}\sigfb(v)v f(v)dv = 4\pi \int_0^{\infty}\sigfb(v) \left(\frac{m}{ 2 \pi kT} \right)^{3/2} v^3 \exp\left(-\frac{mv^2}{2kT}\right) dv$$

To find how $\alphaNL$ scales with temperature, it is necessary to track all the factors of $v$ and $T$ in the expression above. Note that since recombination involves positively charged ions and electrons, $\sigfb$ accounts for Coulomb Focusing and scales as $v^{-2}$. It follows that $\alphaNL \propto T^{-1/2}$. For Hydrogen-like ions with nuclear charge $Z$, $\alphaNL \propto Z \cdot T^{-1/2}$.