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 $Z$, $\alphaNL \propto Z \cdot T^{-1/2}$.

A Hydrogen atom can recombine directly into the ground state or first to an excited state before the electron cascades down to the ground state. To account for all the ways to recombine to the ground state ($n=1$, $l=0$ or $1s$ state), the total recombination coefficient $\alpha_A$ is $$ \alpha_A = \sum_{n=1} \sum_{l=0}^{n-1} \alphaNL$$ As will be discussed below, however, sometimes we want to omit free-to-ground transitions or recombinations directly into the ground state. For this we use $\alpha_B$: $$ \alpha_B = \sum_{n=2} \sum_{l=0}^{n-1} \alphaNL$$