14.2: Relation between the Einstein, Mass Absorption, and Atomic Absorption Coefficients
- Page ID
- 141694
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Since the Einstein coefficient \(\mathrm{B_{ik}}\) is basically the probability that an atom will make a transition from the ith state to the kth state in a given time interval, the relationship to the mass absorption coefficient can be found by relating all upward transitions to the total absorption of photons that must take place. From the definition of the Einstein coefficient of absorption, the total number of transitions that take place per unit time is \[\mathrm{N}_{\mathrm{i} 6 \mathrm{k}}=\mathrm{n}_{\mathrm{i}} \mathrm{B}_{\mathrm{ik}} \mathrm{I}_{\nu} \mathrm{dt}\label{14.1.1}\]
where \(\mathrm{n_i}\) is the number density of atoms in the ith state. Since the number of photons available for absorption at a particular frequency is \(\left(\mathrm{I}_\nu / \mathrm{h} \nu\right) \mathrm{d} \nu\), the total number of upward transitions is also \[N_{i \rightarrow k}=4 \pi n_i \int \kappa_\nu \rho \frac{I_\nu}{h \nu} d \nu d t\label{14.1.2}\]
If we assume that the radiation field seen by the atom is relatively independent of frequency throughout the spectral line, then the integral of the mass absorption coefficient over the line is \[\int \kappa_\nu d \nu=\frac{1}{4 \pi} \frac{h \nu_0}{\rho} B_{i k}\label{14.1.3}\]
where \(\nu_0\) is the frequency of the center of the line.
For the remainder of this chapter, we will be concerned with the determination of the frequency dependence of the line absorption coefficient. Thus, we will be calculating the absorption coefficient of a single atom at various frequencies. We will call this absorption coefficient the atomic line absorption coefficient, which is related to the mass absorption coefficient by \[S_\nu=\frac{S_\omega}{2 \pi}=\kappa_\nu \rho / \mathrm{n}_{\mathrm{i}}\label{14.1.4}\]
Note that we will occasionally use the circular frequency \(\omega\) instead of the frequency \(\nu\), where \(\omega=2\pi \nu\).