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3: Magnetic Interaction Energy (Anomalous Zeeman Effect)

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    66538
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    The interaction energy of the magnetic moments of the atom, \scriptstyle \boldsymbol{\mu}_\mathbf{j}\, with the external field \scriptstyle \mathbf{H}\, is given by

    \scriptstyle \Delta E = \mathbf{H}\boldsymbol{.}\,\boldsymbol{\mu}_\mathbf{j} =H\,\mu_j\,cos\,(H,\,\mu_j)=H\,\mu_j\,cos\,(H,\,j) ---------------[45]

    With the help of relation \scriptstyle \frac{\mu_j}{\mathbf{j}}=g\frac{e}{2m_e}, where \scriptstyle \mathbf{j} =j\hbar, the equation (45) is simplified to

    \scriptstyle \Delta E = g\frac{eH}{2m_e}j\hbar\,cos\,(H,\,j) ---------------[45']

    With projection of \scriptstyle j\, on \scriptstyle H\,, i.e. component of \scriptstyle j\, along \scriptstyle H\,, relation (45') reduces to

    \scriptstyle \Delta E =g\frac{eH}{2m_e}j\hbar\frac{m_j}{j}= gh\frac{eH}{4\pi m_e}m_j = gh\nu_L.\, m_j (in mks units)
    \scriptstyle = gh\frac{eH}{4\pi m_e c}m_j = gh\nu_L.\, m_j (in cgs units) with \scriptstyle \nu_L=\frac{eH}{4\pi m_e c} ---------------[46]

    Finally, in terms of Bohr magneton,

    \scriptstyle \Delta \mathbf{E} = \mathbf{gh}\frac{\mathbf{eH}}{\mathbf{4}\boldsymbol{\pi} \mathbf{m_e}} \mathbf{m_j} = \mathbf{gH}\boldsymbol{\mu}_\mathbf{B}\mathbf{m_j}
---------------[47]

    Or \scriptstyle \Delta \mathbf{T(cm^{-1})} =-\frac{\Delta \mathbf{E}}{\mathbf{ch}}=- \mathbf{g}\frac{\mathbf{eH}}{\mathbf{4}\boldsymbol{\pi} \mathbf{m_e} \mathbf{c^2}}\mathbf{m_j}\,\,\mathbf{or}\,\,-\Delta \mathbf{T} = \mathbf{g}\, \mathbf{L(cm^{-1}) m_j} ---------------[47']

    \scriptstyle L (=\frac{eH}{4\pi m_e c^2}cm^{-1}) is Lorentz unit ---------------[48]

    \scriptstyle \Delta T\, gives the change in energy for each of \scriptstyle m_j\, from the original line. Obviously, apart from the dependence on \scriptstyle H\,, the Landeʹg factor plays an important role to find the relative separations of the Zeeman levels. The value of the g-factor depends on \scriptstyle l,\,s, and \scriptstyle j\,; consequently, value of \scriptstyle g\, and therefore the energy differences, is different for different terms i.e. different \scriptstyle j\, values. This results in the appearance of the several lines in accordance with the selection rule \scriptstyle \Delta m_j = 0,\,\pm 1. Further, when spin is disregarded \scriptstyle (s = 0)\,, value of \scriptstyle g\, turns out to be one; in such a case, the differences between the Zeeman levels again reduce to \scriptstyle L\,(cm^{-1}), that is the equation reduces to Lorentz’s classical formula. \scriptstyle \underline{It\, implies\, that\, the\, difference\, between\, normal\, and\, anomalous\, Zeeman\, Effect\, lies\, in\, g-\, factor.}

    Levels in Anomalous Zeeman levels split into equidistant \scriptstyle (2j + 1)\, levels for a given \scriptstyle j\, but with a magnitude \scriptstyle g\, times \scriptstyle h\nu_L\,, the separation observed in the normal Zeeman Effect. In the absence of anomalous behaviour of the spin, \scriptstyle \frac{\mu_l}{l}\, is expected to be equal to \scriptstyle \frac{\mu_s}{S};\,\mu_{ls} would have been in line with resultant mechanical moment; consequently

    \scriptstyle \mu_{ls} = \mu_j =\hbar\frac{e}{2m_e} j =\frac{e}{2m_e} j

    A relation similar to the one obtained for orbital motion except that \scriptstyle l\, is replaced by \scriptstyle j\,. Magnetic interaction energy (in \scriptstyle cm^{-1}\,) then is given by

    \scriptstyle \Delta T =-\frac{\Delta E}{ch}=-\frac{eH}{4\pi m_e c^2}m_j or \scriptstyle -\Delta T = L(cm^{-1})\, m_j ---------------[49]

    Separations between Zeeman lines (because \scriptstyle \Delta m_j = \pm1) would have again been same like in the case of normal Zeeman Effect. \scriptstyle \underline{In\, other\, words\, without\, the\, introduction\, of\, spin\, and\, more\, particularly\, its\, anomalous\, behavior\, and} \scriptstyle \underline{hence\, the\, Lande^'\, g\, factor,\, Anomalous\, Zeeman\, Effect\, could\, not\, have\, understood.}

    Zeeman patterns for yellow sodium \scriptstyle D\, lines (first member of the Principal series) are derived as an \scriptstyle \underline{example.}

    \scriptstyle g-\, factor for \scriptstyle ^2P_{1/2}\,\And\,^2P_{3/2} are

    \scriptstyle ^2P_{1/2} = \left\{1 + (\mathbf{j^2 + s^2 - l^2)/ 2j^2}\right\} = \left\{1+ [j(j + 1) + s(s + 1) - l(l + 1)/ 2j(j + 1)\right\}]
    \scriptstyle = \left\{1+ [1/2(1/2 + 1) + 1/2(1/2 + 1) - 1(l + 1)/ 2\times 1/2(1/2 + 1)\right\}]
    \scriptstyle  = 2/3\,

    Similarly, g- factor for \scriptstyle ^2P_{3/2}\, and \scriptstyle ^2S_{1/2}\, are \scriptstyle 4/3\, and 2 respectively. Self explanatory figure (8) shows splitting of Zeeman levels, transitions in accordance with the selection rules \scriptstyle \Delta m_j = 0,\,\pm1, and finally pattern of expected anomalous Zeeman spectrum. Dotted horizontal line on the left side represents the centre of gravity of the levels, dotted vertical transitions indicate forbidden transitions and the heights of the expected spectral lines (shown at the bottom of the Fig. 8) indicate relative intensities.

    Fig.8 Anomalous Zeeman pattern of Sodium \scriptstyle D\, lines.

    Electromagnetic waves that have undergone Zeeman Effect get polarized; nature of polarization as mentioned in the beginning of the section 3 depends on the direction (with respect to the magnetic field) of the field of view. Polarization rules can be derived both from the classical theory as well as from the quantum mechanics. These are summarised below:

    Transverse View (viewed perpendicular to the direction of magnetic field, \scriptstyle H\,)

    1. Transitions with \scriptstyle \Delta m_j = \pm1\, yield plane polarized light having its electric vector oscillating perpendicular to \scriptstyle H\,, termed as \scriptstyle S\,(\sigma)- components.

    2. Transitions with \scriptstyle \Delta m_j = 0\, yield plane polarized light having its electric vector oscillating parallel to \scriptstyle H\,, termed as \scriptstyle P\,(\pi)- components.

    Longitudinal View (viewed parallel to the direction of magnetic field, \scriptstyle H\,)

    1. Transitions with \scriptstyle \Delta m_j = \pm1\, yield circularly polarized light, termed as \scriptstyle S\,(\sigma)- components.

    2. Transitions with \scriptstyle \Delta m_j = 0\, are forbidden.

    Problem: Discuss the Anomalous Zeeman spectra of the green \scriptstyle (546.1nm,\,^3P_2 - ^3S_1) and violet \scriptstyle (435.8nm,\,^3P_1 - ^3S_1) lines of mercury.


    3: Magnetic Interaction Energy (Anomalous Zeeman Effect) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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