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

$$\sigbf\sim{\lambda^2\over 8\pi}{A_{21}\over \Delta\nu}$$ It turns out that $\Delta\nu$ is about $\nu$. $\lambda\approx 912\AA$, so scaling from Lyman-alpha: $$\sigbf\sim{(912\AA)^3\over c\cdot8\pi}A_{21,Ly\alpha} \left({1216\AA\over912\AA}\right)^3\sim 10^{-18}cm^2$$ It turns out that the real answer is $\sigbf\sim 6\cdot 10^{-18}cm^2$. In general: $$\sigbf=\sigma\eval{edge}\left({E_{photon\,in}\over E_{edge}}\right)^{-3}$$ That exponent (-3) is actually $-\frac83$ near the edge and goes to $-\frac72$ far from it. So you see $\sigbf$ spike up as the photon reaches the ionization energy, and then decrease exponentially as energy increases. However, you can see new spikes from ionizing electrons in inner shells.

\subsection*{ Radiative Recombination}

\def\sigfb{\sigma_{fb}} This is the inverse process of photoionization, so $\sigfb$ is the cross-section for an ion recapturing its electron and emitting a photon. We'll relate $\sigfb$ to $\sigbf$. This is called the Milne Relation. In this derivation, we'll start by assuming complete thermal equilibrium and derive a result which will end up being independent of thermal equilibrium. Let's start calculating the rate of radiative recombinations. Thermal equilibrium dictates that this must equal the rate of photoionization. For radiative recombination: $$rate\ of\ recombination =n_+n_e\sigfb(v)v[f(v)dv]={\#\ of\ recombinations\over volume\ time}$$ We'll set this equal to the rate of photoionization. This rate is: $$rate\ of\ photoionization ={B_\nu4\pi d\nu\over h\nu}n_0\sigbf \overbrace{\left(1-{g_0\over g_+}{n_+\over n_0}\right)}^{{correction\ for\atop stimulated}\atop recombination}$$ where $n_0$ is the \# density of neutrals. Note this has units of \# flux.