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

$$\begin{array}{c c c} M^\prime & M^{\prime \prime} & \mathcal{S}(\text{C}) \\ \\ -2 & -2 & 16 \\ -2 & -1 & 4 \\ -1 & -2 & 4 \\ -1 & -1 & 4 \\ -1 & 0 & 6 \\ 0 & -1 & 6 \\ 0 & 0 & 0 \\ 0 & 1 & 6 \\ 1 & 0 & 6 \\ 1 & 1 & 4 \\ 1 & 2 & 4 \\ 2 & 1 & 4 \\ 2 & 2 & 16 \\ \end{array}$$