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9.8: Zeeman Components

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.

Example \(\PageIndex{1}\)

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:

\[\mathcal{S} (\text{C}) = 4 M^2 . \label{9.8.5}\]

Example \(\PageIndex{2}\)

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}

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