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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}