Line Profile Functions

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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\hf{\frac12} \def\^{\hat } \def\.{\dot } \def\tnTemplate:\tilde\nu \def\eboltz{e^Template:-h\nu 0 \over kT} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}}

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Line Profile Function}

$\phi(\nu)\equiv$ is called the line profile function.  It 

describes some (maybe gaussian) distribution of absorption around $\nu_0$ (the absorption frequency), and is subject to the requirement that: $$\int_0^\infty{\phi(\nu)d\nu}=1$$ Therefore, the line profile function represents the probability of getting a photon around $\nu_0$. Say that $\Delta\nu$ is the width of the distribution around $\nu_0$. $\Delta \nu $ is affected by three factors: $\ato$ (the natural, uncertainty-based broadening of at atom in isolation), $\nu_0 V_T/ c$ (the thermal, Doppler-based broadening), and $n_{coll}\sigma_{coll}v_{rel}$ (collisional broadening, a.k.a. pressure broadening). So really, the transition probability per unit time is: $$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\bar J$$

\subsection*{ Zeeman Effect}

The Zeeman Effect concerns the splitting of electronic levels in a magnetic field. We've already talked a little bit about this in hyperfine splitting, which was caused by magnetic fields intrinsic to an atom. We find that a single line can split into $\sim3-27$ components, depending on the number of combinations of $\vec L$ and $\vec S$ there are in the atom. The strengths of these various lines depend on viewing geometry, and can be polarized. The change in energy between previously degenerate states set up by an external $B$ field is: $$\begin{aligned}\Delta E&\sim\mu B\sim{e\hbar\over m_ec}B\\ &\sim{hc\over\lambda_1}-{hc\over\lambda_2}\\ &\sim{hc\over\lambda}{\Delta\lambda\over\lambda}\sim{e\hbar B\over m_ec}\\ \end{aligned}$$ Thus the fractional change in wavelength is: $$\boxed{{\Delta\lambda\over\lambda}\sim\lambda{eB\over2\pi m_ec^2}}$$ In practice, we find that the various split components of the original absorption lines are hard to resolve, and we see the effect expressed mostly as a broadening of the original line.

\end{document} <\latex>