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2: Anomalous Zeeman Effect (Vector Model)

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    66537
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    To determine the number of components (Zeeman lines) one has to find the magnitude of separations among Zeeman levels (split energy levels under Zeeman Effect); and to know the separations, knowledge of the total magnetic moment of the atom is a prime requirement. Let us take simple case of a single valence electron atom in which lone electron is moving around the core (stationary nucleus plus completely filled orbit(s)) whose net moments is zero. That means whatever magnetic moments an atom has is all exclusively due to the valence electron.

    Whatever magnetic moments an atom has is all exclusively due to the valence electron.

    We have already discussed that, because of the orbital motion,

    \scriptstyle \frac{\mu_l}{l}=-\frac{e}{2m_e}=-g\frac{\mu_B}{\hbar} ---------------[I.45']

    and for the spin motion,

    \scriptstyle \frac{\mu_s}{S}=-2\frac{e}{2m_e}=-g\frac{\mu_B}{\hbar} ---------------[I.49']

    \scriptstyle \mu_B\left(=\frac{e\hbar}{2m_e}\right)\, is Bohr Magneton and \(g\) is Lande’ \(g\) factor and has value 1 for pure orbital motion and 2 for pure spin motion. Since, \(l\) and \scriptstyle s\, precess about their resultant \scriptstyle j,\,\boldsymbol{\mu}_\mathbf{l} and \scriptstyle \boldsymbol{\mu}_\mathbf{s} must also precess about \scriptstyle j\,. However, the total magnetic moment, \scriptstyle \boldsymbol{\mu}_\mathbf{{ls}} (= \boldsymbol{\mu}_\mathbf{l} + \boldsymbol{\mu}_\mathbf{s)},\, because \scriptstyle \frac\boldsymbol{{\mu}_\mathbf{l}}\mathbf{{l}}\neq\frac\boldsymbol{{\mu}_\mathbf{s}}\mathbf{{S}},\, does not coincide with \scriptstyle j\, (Figure \(\PageIndex{1}\)).

    Figure \(\PageIndex{1}\): Precessions of various moments about the resultant

    As resultant mechanical moment, \scriptstyle j\, is invariant, all the moment vectors \scriptstyle (l,\,s,\,\mu_l,\,\mu_s and \scriptstyle \mu_{ls})\, precess around it (Figure \(\PageIndex{1}\)). Recalling that mechanical and magnetic moments (for orbital as well as for spin motion) lie along the same line but in opposite direction, the total magnetic moment of the atom is, likewise expected to be in line with its resultant mechanical moments though directed in opposite directions. To meet this requirement, let us resolve \scriptstyle \mu_{ls}\, into components, one component parallel to \scriptstyle j\, and the second perpendicular to it. The perpendicular component will constantly be changing its direction; consequently its time average over a period is zero. Under this assumption, net effective magnetic moments of the atom, \scriptstyle \mu_j\,, is the component parallel to \scriptstyle j\, whose magnitude will be vector sum of the components of \scriptstyle \mu_l\, and \scriptstyle \mu_s\, along \scriptstyle j\,.

    Therefore, \scriptstyle \mu_j =\boldsymbol{(\mu}_\mathbf{l} +\boldsymbol{\mu}_\mathbf{s)}\, //components \scriptstyle =l\hbar\frac{e}{2m_e}\,cos(l,\,j) + 2\,s\hbar\frac{e}{2m_e}\,cos(s,\,j)

    \scriptstyle =\hbar\frac{e}{2m_e}\left\{l \,cos(l,\,j) + 2s\,cos(s,\,j)\right\} ---------------[39]

    From the figure (7) it is clear that \scriptstyle s^2 = l^2 + j^2- 2lj\,cos\,(l,\,j) leading to \scriptstyle l\,cos\,(l,\,j) = (l^2 + j^2- s^2)/ 2j and similarly \scriptstyle 2s\,cos\,(s,\,j)= (s^2 + j^2-l^2)/ j. Substitution of the values for \scriptstyle l\,cos\,(l,\,j)\,\And\,2s\,cos\,(s,\,j) reduce the relation for \scriptstyle \mu_j\,, to

    \scriptstyle \mu_j =\hbar\frac{e}{2m_e}\left\{(3j^2 + s^2-l^2)/2j\right\}\,
    \scriptstyle =\hbar\frac{e}{2m_e}\,j\left\{1+(j^2 + s^2 - l^2)/ 2j^2\right\} ---------------[40]

    In quantum mechanics, the equation is expressed in terms of the operator \scriptstyle \mathbf{j^2,\,l^2,\,s^2} and is expressed as

    \scriptstyle \mu_j =\hbar\frac{e}{2m_e}\,{j}\left\{1+(\mathbf{j^2} + \mathbf{s^2}-\mathbf{l^2})/2\mathbf{j^2}\right\}\, ---------------[40']

    These operators commute with each other. Using the properties of the commuting operators and the digitalization of their matrix, operators can be replaced by the corresponding eigenvalues. Therefore, the relation is rewritten as

    \scriptstyle \mu_j =\hbar\frac{e}{2m_e}j\left\{1 + [j(j + I) + s(s + 1) - l(l + 1)]/ 2j(j + 1)\right\} ---------------[41]

    or

    \scriptstyle \boldsymbol{\mu}_\mathbf{j} =g\,\hbar\frac{e}{2m_e}j = \mathbf{g}\frac{e\hbar}{2m_e}j\frac{\hbar}{\hbar}=g\frac{\mu_B}{\hbar} \mathbf{j} ---------------[42]

    \[ \vec{mu}_j = g \dfrac{\hbar}{2 m_e} j = g \dfrac{e \hbar}{2m_e} j \dfrac{\mu_B}{\hbar}j \label{42}\]

    where \scriptstyle \mathbf{j} = j\hbar\,\And\,\boldsymbol{\mu}_\mathbf{B}\left(=\frac{e\hbar}{2m_e}\right) is Bohr Magneton

    and

    \[g = \dfrac{1+[j(j+1) + s(2+1)- l(l+1)}{2j(j+1)} \label{43}\]

    In case of a pure orbital motion which means \(s=0\); value of \(g\) turns out to be 1. For pure spin motion, \(l=0\); consequently value of \(g\) is 2.

    Equation \(\ref{42}\) is similar to the ones we obtained earlier for orbital and spin motion of the electron (equations I.45' and I.49'). The precession of \scriptstyle j\, around the external field is the result of the torque acting on both \(l\) and \(s\). When the external field is not very strong, \(j\) retains its significance and therefore precesses with a compromise angular velocity (as given by Larmor’s theorem)

    \[ \omega_L = g \dfrac{eH}{2M_e} \label{44}\]

    Equation \(\ref{44}\) is same as for the pure orbital motion except for the Landeʹ g-factor which, in any case, is 1 for the orbital motion.

    Note that, in the above treatment, no where external magnetic field is considered. \scriptstyle \boldsymbol{\mu}_\mathbf{j}\, is the resultant magnetic moment in an atom arising out of the internal motions of the electron and hence refers to as ‘INNER PRECESSION’.

    When the atom is subjected to external magnetic field, \(H\), it will start precessing around the field direction; the precession is termed as ‘OUTER PRECESSION’. Although the atom precesses as a whole around \(H\), yet the nature of the internal precessions (and hence the magnetic interactions) depend largely on the relative strength of the external magnetic field, \(H\).

    1. If field strength of \(H\) is weak compared to that of internal magnetic moment, \scriptstyle \boldsymbol{\mu}_\mathbf{j}\,, spin –orbit coupling remains intact and therefore, continues to have physical meaning. That means \scriptstyle j (= l \pm s)\, is well defined and precess around \scriptstyle H\, when atom is subjected to it. \scriptstyle l-s\, coupling is intact means precessions of \scriptstyle l\, and \scriptstyle s\, as a rigid system around \scriptstyle j\, is more rapid compared to that of \scriptstyle j\, around \scriptstyle H\,; and under this condition one can legitimately assume that perpendicular components of \scriptstyle l\,\And\,s average out to zero over a period of rotation. We will see below that weak external field compared to internal leads to the anomalous Zeeman Effect.
    2. If field strength of \(H\) is fairly strong compared to internal magnetic moment, \scriptstyle \boldsymbol{\mu}_\mathbf{j}\,, spin –orbit coupling is weakened or even broken. Under this condition \scriptstyle j\, starts losing its physical significance thereby \scriptstyle l\And\,sstart precessing independently around \scriptstyle H\,. This means precessions of \scriptstyle l\, and \scriptstyle s\, about \scriptstyle H\, is more rapid compared to their precession about \scriptstyle j\,, a must requirement for Paschen-Back effect, which is discussed later.

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

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