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*{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 $\textrm{v}$ and $T$ in the expression above. Since recombination involves positively charged ions and electrons, $\sigfb$ accounts for Coulomb Focusing and scales as $\textrm{v}^{-2}$. 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 $\alpha_A$ with units $\textrm{cm}^3 \textrm{s}^{-1}$ 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 which are recombinations that are direct into the ground state. For these cases we use $\alpha_B$: $$\alpha_B = \sum_{n=2} \sum_{l=0}^{n-1} \alphaNL$$

\subsection*{Back of the Envelope Approximation and True Values} For Hydrogen, the rate of recombination per unit volume is approximately equal to the probability of finding an electron and proton near one another in 3-dimensional space multiplied by the rate of an electron decaying from a free to a bound state. For an electron and proton to be about a few Bohr radii apart, their proximity traces out a volume $\sim 10^{-23} \textrm{cm}^3$. The atomic decay rate is $\sim 10^9 \textrm{s}^{-1}$. Let's include an extra order of magnitude because of coulomb focusing. Multiplying these together, we would predict that $\alpha_A \sim 10^{-13} \textrm{cm}^3 \textrm{s}^{-1}$ which is quite close to Hydrogen's recombination coefficients at $T=10^4 \textrm{K}$, a typical nebula temperature:

$$\alpha_A=4\e{-13}{cm^3\over s}$$ $$\alpha_B=2\e{-13}{cm^3\over s}$$

\subsection*{Timescales} For a typical nebula at $T=10^4 \textrm{K}$, the recombination time is: $$t_{rec} \approx {1 \over n_e * \alpha_A} \approx 3 \times 10^{12} n_e^{-1} \textrm{sec} \approx 10^{5} n_e^{-1} \textrm{yrs}$$

The recombination time is longer than the timescales of the other relevant processes, namely photoionization and electron-electron scattering/collisions. This is good verification that it was appropriate to use a Maxwellian distribution of velocities in the derivation above. After being photoionized, the electrons thermalize so their velocities settle into a Maxwellian on timescales shorter than those associated with recombination or ionization. (The $e^--e^-$ cross-section is larger than the photoionization cross-section by several orders of magnitude.)

\subsection*{Case A and B Recombination} For optically thin nebulae, $\alpha_A$ is appropriate and we include recombinations that go directly from free to the ground bound state. But when a nebula is optically thick, no ionizing photons ($h \nu > 13.6 eV$ for an H nebula) escape the nebula--each photoionizing photon released from a direct free-to-ground recombination immediately photoionizes a neutral Hydrogen atom nearby, having no impact on the net recombination. In this case we want to ignore free-to-ground recombination and the appropriate recombination coefficient is $\alpha_B$.

Instead of recombining directly to the \emph{ground state}, sometimes a proton and an electron can recombine to form Hydrogen in an \emph{excited state} followed by a cascading down to lower energy levels. This process releases multiple lower energy photons that are not able to subsequently photoionize other bound Hydrogen atoms because they have $h \nu < 13.6 eV$. Essentially one photoionizing photon (the one that produced the proton-electron pair in quesion) is exchanged for several lower energy photons. Eventually as you get further away from the source of the photoionizing photons, the Hydrogen nebula will be entirely neutral because this exchange happened enough times that no photoionizing photons remain. This is the premise of Str\"{o}mgren spheres--bubbles of ionized Hydrogen that form around young, hot stars in nebulae of neutral hydrogen. These bubbles have definitive radii and thin edges (the boundary between the ionized and neutral Hydrogen regions) that.