9.8: Zeeman Components
- Page ID
- 9049
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this section I give \(\mathcal{S}(\text{C})\), the relative strengths of Zeeman components within a line.
I consider first lines for which \(J\) changes by \(1\), and then lines for which \(J\) does not change.
\(\underline{\text{Lines connecting } J \text{ to } J-1.}\)
Components connecting \(M\) to \(M − 1\):
\[\mathcal{S}(\text{C}) = (J+M_>)(J+M_<) \label{9.8.1}\]
Components connecting \(M\) to \(M+1\):
\[\mathcal{S}(\text{C}) = (J-M_<)(J-M_>) . \label{9.8.2}\]
Components in which \(M\) does not change:
\[\mathcal{S}(\text{C}) = 4 (J+M) (J-M) . \label{9.8.3}\]
In these equations \(J\) is the larger of the two \(J\)-values involved in the line; \(M_>\) and \(M_<\) are, respectively, the larger and the smaller of the two \(M\)-values involved in the component. Note that these formulas are not normalized to a sum of unity. In order to do so, the strength of each component should be divided by the sum of the strengths of all the components – i.e. by the strength of the line.
Consider the Zeeman pattern of figure \(\text{VII.1}\). The strength factors for each of the nine components, reading from left to right in the figure, will be found to be
0 2 6 12 16 12 6 2 0
Normalized to unity, these are
0.0000 0.0357 0.1071 0.2143 0.2857 0.2143 0.1071 0.0357 0.0000
As described in section 7.27 in connection with figure \(\text{VII.1}\), the components within each group of three are unresolved, so the relative strengths of the three groups are \(\frac{1}{7}\) \(\frac{5}{7}\) \(\frac{1}{7}\).
Consider also the Zeeman pattern of figure \(\text{VII.2}\). The strength factors for each of the six components, reading from left to right in the figure, will be found to be
2 6 8 8 6 2
or, normalized to unity,
\(\frac{1}{16}\) \(\frac{3}{16}\) \(\frac{4}{16}\) \(\frac{4}{16}\) \(\frac{3}{16}\) \(\frac{1}{16}\).
\(\underline{\text{Lines for which } J \text{ does not change}.}\)
Components for which \(M\) changes by \(\pm 1\)
\[\mathcal{S}(\text{C}) = (J+ M_<)(J-M_>) . \label{9.8.4}\]
Components for which \(M\) does not change:
\[\mathcal{S} (\text{C}) = 4 M^2 . \label{9.8.5}\]
For a line \(J − J = 2 − 2\), the relative strengths of the components are
\begin{array}{c c c}
M^\prime & M^{\prime \prime} & \mathcal{S}(\text{C}) \\
\\
-2 & -2 & 16 \\
-2 & -1 & 4 \\
-1 & -2 & 4 \\
-1 & -1 & 4 \\
-1 & 0 & 6 \\
0 & -1 & 6 \\
0 & 0 & 0 \\
0 & 1 & 6 \\
1 & 0 & 6 \\
1 & 1 & 4 \\
1 & 2 & 4 \\
2 & 1 & 4 \\
2 & 2 & 16 \\
\end{array}