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9.8: Zeeman Components

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    9049

    • 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}