4: Intensity of Zeeman components

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Intensity sum rules of the alkali spectra are equally applicable to provide relative intensities of the Zeeman components also. In this case the initial and the final states are Zeeman levels. This may be justified on the basis of the correspondence principle. Assuming that the magnetic field strength is such that the degeneracy is removed; the populations, at equilibrium, of the various Zeeman levels, are proportional to the respective statistical weights. Further, taking into account the relation between the Einstein emission and absorption coefficients (statement to be verified), i.e.

$\scriptstyle B_{21} = (c^3\,/ 8\pi h\nu_{21}^3)g_2/g_1.\,A_{21}$ ,

the sum rules may be stated as: $\scriptstyle \underline{the\, sum\, of\, intensities\, of\, all\, the\, transitions\, from\, an\, initial\, Zeeman\, level\,}$ $\scriptstyle \underline{is\, equal\, to\, the\, sum\, of\, intensities\, of\, all\, the\, transitions\, originating\,from\, any\, other\, level\, of\, the\,}$ $\scriptstyle \underline{same\, field\, free\, line\, (i.e.\, same\, n\, \And\, l).\,The\, same\, sum\, rule\, is\, true\, for\, the\, final\,(terminating)}$ $\scriptstyle \underline{Zeeman\, level.}$

The arguments, on which the above sum rules are based, can be extended to more general case of transitions: $\scriptstyle J\,\rightarrow J,\,J\,\rightarrow\,J + 1$ . Calculations are somewhat tedious and therefore the resulting formulae for the intensity, $\scriptstyle I\,$ , are given below.

$\scriptstyle J\,\rightarrow\,J,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, m_j\,\rightarrow\,m_j\pm 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\sigma\,\,or\,\,s)\,\,\,\,\,\,\,\,\,\,\,\,\,I\,\,\,\,\,\,\,\,\,\,\,\,\,= A(J \pm m + 1)(J \mp 1)$

$\scriptstyle m_j\,\rightarrow\,m_j\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\pi\,\,or\,\,p)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= 4Amj^2$

$\scriptstyle J\,\rightarrow\,J + 1,\,\,\,\,\,\,m_j\,\rightarrow\,m_j\pm 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\sigma\,\,or\,\,s)\,\,\,\,\,\,\,\,\,\,\,I\,\,\,\,\,\,\,\,\,\,\,\,\,= B(J\,\pm\,m + 1)(J\,\pm\,m + 2)$

$\scriptstyle m_j\,\rightarrow\,m_j\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\pi\,\,or\,\,p)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= B(J\,+ m + 1)(J\,- m + 2)$

$\scriptstyle A\,\And\,B$ are constants; their values need not be determined for the relative intensity considerations within a Zeeman pattern. Since final and initial states can be interchanged, transitions $\scriptstyle J\,\rightarrow\,J-1$ are included in these relations.

It is important to note that half of the circularly polarized light is seen in the transverse direction and the second half of the circularly polarized is observed in the longitudinal direction. On the other hand, total intensity of the linearly polarized light is seen in the transverse direction and zero intensity (no light) in the longitudinal direction. These considerations have been taken into account in the above equations.

A careful observation of the figure (8) showing Zeeman transitions and of the equation (47), reveal that the magnitude of separation between the consecutive Zeeman levels for a given values of $\scriptstyle J\,$ and of the magnetic field, is same; the quantum, of course, depends on $\scriptstyle g\,$ – factor. This is true for given $\scriptstyle J\,\And\,H$ values irrespective of the electronic state. This fact coupled with selection rule $\scriptstyle \Delta m_j = 0,\,\pm1 (\Delta m_j = 0$ yields $\scriptstyle \sigma\,$ or $\scriptstyle s\,$ –components and $\scriptstyle \Delta m_j = \pm1\,\pi$ or $\scriptstyle p\,$ –components) lead to a quick method to obtain the Zeeman pattern. The method is explained taking example of Zeeman pattern of sodium $\scriptstyle D_2\,(^2P_{3/2}\,\rightarrow\,^2S_{1/2})$ line.

Write $\scriptstyle m_j\,$ values in a row. Below this are the separation factors $\scriptstyle (m_jg)\,$ for both the initial and final states in two rows such that equal values of $\scriptstyle m_j\,$ lie directly one above the other (Fig. 9).

Fig. 9 Zeeman components

Vertical arrows indicate $\scriptstyle \pi\,$ (p)-components $\scriptstyle (m_j = 0)\,$ and the diagonal arrows $\scriptstyle \sigma\, (s)-$ components $\scriptstyle (m_j = \pm\, 1)$ . The differences between the separation factors (i.e. the difference between $\scriptstyle m_jg\,$ values of the start and the end of the arrows) with a least common denominators are: