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6: Paschen–Back Effect

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    66544
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    In describing the anomalous Zeeman effect, it is assumed that external magnetic field is weak compared to the internal fields; consequently the interaction between \)J\) and \(H\) is weak compared to the interaction of orbital and spin magnetic fields. If the strength of the external magnetic field is increased to the extent that it starts competing with the internal field, the interaction between \(l\) and \(s\) starts loosening or gets broken at sufficiently large value. Under this condition \scriptstyle J\, starts losing its significance and \scriptstyle l\, & \scriptstyle s\, interact independently with the external magnetic field resulting in the independent precessions of \scriptstyle l\, & \scriptstyle s\, about \scriptstyle H\,; their precession much faster than the precession of residual interaction of \scriptstyle l\, & \scriptstyle s\, about \scriptstyle j\,. This is known as Paschen-Back Effect. It may be recalled that, as emphasized earlier, the separation between the spin components or between the Zeeman components is a measure of the corresponding precessional frequency.

    \scriptstyle \underline{fairly\, strong\, for\, hydrogen\, atom.}
    Figure11).

    \scriptstyle H\, and their space quantization.
    Figure 11. When the ls-interaction is significantly loosened or broken, major part of the total energy of the atom consists of the energies due to the precession of \scriptstyle l\, around \scriptstyle H\, plus energy due to precession of \scriptstyle s\, about \scriptstyle H\,. The major energy shift, therefore, is given as
    \scriptstyle \Delta E_{mag.int.} = \Delta E_{l,\,H} + \Delta E_{s,\,H}  = \mathbf{H}\boldsymbol{.}\boldsymbol{\mu}_\mathbf{l} + \mathbf{H}\boldsymbol{.}\boldsymbol{\mu}_\mathbf{s}  = H\,\mu_l\,cos\,(l,\,H) + H\,\mu_s\,cos\,(s,\,H) ------------[50]

    Using equations \scriptstyle I-45^'\,\And\,I-49^' for \scriptstyle \mu_l\,\And\,\mu_s respectively and substituting the values of \scriptstyle cos\,(l,\,H)\,\And\,\,cos\,(s,\,H), the equation (50) reduces to

    \scriptstyle \Delta E_{mag.int.} =\hbar\,(m_l+2m_s)\frac{eH}{2mc}\,\,\,=H\,\mu_B(m_l+2m_s),\,\mu_B is Bohr magneton ---------------[51]

    In terms of wave numbers, \scriptstyle -\Delta T =(m_l+2m_s)\,L\,cm^{-1} ---------------[51']

    Although \scriptstyle l\,\And\,s precess independently, yet each produces magnetic field on the electron causing some perturbation to other’s motion. In other words, a small \scriptstyle ls-\, interaction is inherent that one has to take into considerations despite it being small compared to the effect of the external magnetic field. The separation due to this residual interaction is of the same order of magnitude as the field free fine structure doublet separations. Therefore, using equation \scriptstyle I.76\, and that the \scriptstyle \mathbf{j}\, is vector sum of \scriptstyle \mathbf{l}\,\mathbf{\And\,s}, its contribution can be written as

    \scriptstyle -\Delta T_{l,\,s} =a\,\,\,ls\,cos\,(l,\,s)\,\,\,\,\,\,\,\,\,\,\,where\,\,\,\,a =\frac{R\alpha^2 z^4}{n^3 l(l+ 1/2)(l+1)} ---------------[52]

    In field free \scriptstyle l,\,s- interaction (sec 9.2, chapter I), \scriptstyle l\, and \scriptstyle s\, rotate together as a rigid system about \scriptstyle j\, thereby the angle between \scriptstyle l\, and \scriptstyle s\, is constant rendering easy evaluation of \scriptstyle cos\,(l,\,s). In the present case angle between \scriptstyle l\, and \scriptstyle s\, is varying continuously because of the anomalous behavior of spin \scriptstyle (s\, precess faster than \scriptstyle l\,). This necessitate the use of average value of \scriptstyle cos\,(l,s) that can be evaluated using trigonometry theorem, \scriptstyle viz\,\overline{cos\,(l,s)}= cos\,(l,H)\times\,cos\,(s,H). Using this theorem along with the values of \scriptstyle cos\,(l,H) and \scriptstyle cos\,(s,H) from the figure (11), one finds

    \scriptstyle -\Delta T_{l,s}=a\,l\frac{m_l}{l}.s\frac{m_s}{s} ---------------[52']

    Adding this to equation (51'), total energy shift becomes

    \scriptstyle -\Delta T= [(m_l+2m_s)L+a\,m_l\,m_s]\,cm^{-1} ---------------[53]

    A general relation for the term values may, thus, be written as

    \scriptstyle T =T_0-[(m_l+2m_s)\,L-a\,m_l\,m_s]\,cm^{-1} ---------------[54]

    \scriptstyle T_0\, is the term value of the hypothetical center of gravity of the fine structure doublet. It may be reminded that \scriptstyle J\, is no longer well defined quantum number; therefore, Paschen Back effect is described in terms of quantum number \scriptstyle (m_l+2m_s)\,. As an example, Paschen Back effect of sodium \scriptstyle D_1\And\,D_2 lines: quantum numbers for the states involved in the transitions exhibiting Paschen Back effect are given below.

                     \scriptstyle State  \quad \quad \quad \, l  \quad \quad \quad s  \quad \quad \quad \,\,\,\,\,\, m_l  \quad \quad m_s  \quad \quad \quad \quad (m_l + m_s)  \quad \quad \quad \,\,\,\, (m_l + 2m_s) 
    \scriptstyle Excited\, (Upper)\,States
                    \scriptstyle P_{3/2} \quad \quad \quad \quad 1 \,\, \quad \quad +1/2 \quad \quad \,\,\,\,  1  \quad \quad +1/2 \quad \quad \,\,\,\, +3/2  \quad\quad \quad \quad \quad  +2  
                                  \scriptstyle  1\,\, \quad\quad +1/2 \quad \quad \,\,\,\, 0 \quad \quad +1/2 \quad \quad \,\,\,\, +1/2 \quad \quad \quad \quad \quad +1
                                  \scriptstyle  1\,\, \quad \quad +1/2 \,\,\,\,\,\,\,\,\, -1 \,\,\,\,\,\quad +1/2 \,\,\,\,\,\,\quad \,\,\,\, -1/2 \,\,\,\,\,\,\,\,\,\,\, \quad \quad \quad \,\,\, 0
                    \scriptstyle P_{1/2}\quad \quad \quad \quad 1 \,\, \quad \quad -1/2 \quad \quad \,\,\,\, 1 \quad \quad -1/2 \quad \quad \,\,\,\, +1/2 \quad \quad \quad \quad \quad\,\, 0
                                  \scriptstyle 1\,\, \quad \quad -1/2 \,\,\,\,\,\,\,\,\,\quad \,  0 \,\,\,\,\,\quad \, -1/2 \quad \quad \,\,\,\, -1/2 \quad \quad \quad \quad \,\,\,\,\,  -1
                                  \scriptstyle 1\,\, \quad \quad -1/2 \,\,\,\quad  -1 \,\,\,\,\,\quad \, +1/2 \quad \quad \,\,\,\, -3/2 \quad \quad \quad \quad \,\,\,\,\, -2 
    \scriptstyle Ground\, State
                    \scriptstyle S_{1/2}\quad \quad \quad \quad 0 \,\, \quad \quad +1/2 \quad \quad \,\,\,\, 0 \quad \quad +1/2 \quad \quad \,\,\,\, +1/2 \quad\quad \quad \quad \quad +1
                                  \scriptstyle 0 \,\, \quad \quad -1/2 \,\,\,\quad \quad \, 0 \,\,\,\,\,\quad \, -1/2 \quad \quad \,\,\,\, -1/2 \quad \quad \quad \quad \,\,\,\,\, -1  
    

    Two highlighted states have same value of Paschen Back quantum number \scriptstyle (m_l+2m_s=0)\, and will, therefore, constitute a single component of the state \scriptstyle P\,. States corresponding to these quantum numbers are shown in the figure 12. Selection rules \scriptstyle \Delta (m_l+2m_s)=0,\,\pm 1 allow only six out of possible ten transitions. If resolution of the detecting system is small enough to neglect the \scriptstyle ls-\, interaction, the Paschen Back will constitute three lines; each is a coinciding pair (a coincides with \scriptstyle A,\,b with \scriptstyle B\, and \scriptstyle c\, with \scriptstyle C\,) since the values of \scriptstyle \Delta (m_l+2m_s)\, for each of the components of a pair of lines are \scriptstyle +1\,, 0 and \scriptstyle -1\,respectively (fig 12). Dotted lines represent the forbidden transitions. Three pair of lines is obtained under the assumption that ls- interaction is zero.

    Figure 12: Transitions showing Paschen Back effect in the absence of ls-interaction.

    The effect of the external magnetic field (weak to strong with respect to internal magnetic field) on the spectrum can be summarized as: as long as the external magnetic field is unable to perturb the inner precession, one gets anomalous Zeeman spectrum. On the other side, if the strength of the external field yields the magnetic resolution

    Figure 13. Transition from anomalous Zeeman effect to Paschen Back effect.

    more than the spin –orbit fine structure, one ends up with a normal Zeeman triplet, shown in the Figure 13 (transition of interaction for WEAK to STRONG field). In a way, these are the two extreme situations; what about the intermediate fields, i.e. during the transition from relatively weak to strong external field? Usually this transitional zone is referred to as the Paschen Back effect. We will see below that the Paschen Back spectrum is not as simple as discussed and shown in figure 12; it is somewhat complicated and separation between the various magnetic field components is a function of external magnetic field.

    The situation can be addressed by using the classical laws of the conservation of angular momentum. The conservation law demands that \scriptstyle m_j\,, projection of \scriptstyle j\, on \scriptstyle H\,, that describes the internal magnetic field & is well defined under weak field, must be equal to the total angular momentum \scriptstyle (m_l+m_s)\, when the field is relatively strong. That is, conservation law says that \scriptstyle m_j=(m_l+m_s)\,. In addition, in keeping with the quantum mechanics, levels with same \scriptstyle m_j\, must not cross in the correlation between the two extreme conditions. Correlation between the magnetic levels of the transitions \scriptstyle ^2P_{3/2,\,1/2}\rightarrow ^2S_{1/2} are shown in the fig 14.

    Figure 14 Correlation between the magnetic levels of the transitions \scriptstyle ^2P_{3/2,1/2}\,\rightarrow ^2S_{1/2}

    In a relatively strong field, as discussed above, the extreme limit (spin is not coupled to orbital motion) of the Zeeman effect leads to normal Zeeman triplet. Quantum numbers \scriptstyle m_l\, and \scriptstyle m_s\,, since \scriptstyle l\,\And\,s are independently & strongly coupled to \scriptstyle H\,, are well defined and, therefore, keep their physical significance. Conservation of angular momentum accounts for the selection rules: \scriptstyle \Delta l = 0,\,\pm 1,\,\Delta m_l = 0,\,\pm 1,\,\Delta s =0,\,\And \Delta m_s = 0 for the electric dipole radiation. Since the orientation of the spin does not alter during any radiation process (emission or absorption), the selection rules \scriptstyle \Delta m_l = 0,\,\pm 1 and \scriptstyle \Delta m_s = 0\, for dipole radiation hold well. Together with these, selection rules \scriptstyle \Delta l =0\,\pm 1 and \scriptstyle \Delta s=0\, allow us to ignore spin. Or one may argue that on substituting \scriptstyle \Delta m_s =0\, in the Equation 51, that is magnetic energy shift \scriptstyle =(m_l+2m_s)\,L\,cm^{-1}, the shift turns out to be proportional to \scriptstyle H\,\Delta m_l; that is spin is completely dropped out. Consequently, only three lines corresponding to \scriptstyle \Delta m_l =-1,\,0\And\,=1 are observed. Implication of the \scriptstyle \Delta m_s = 0\, selection rule is that the polarization remains unaltered throughout all the magnetic field strengths. It may be remarked that should \scriptstyle s \neq 0\,, there will be residual spin-orbit coupling yielding group of several transitions around each of the three components.


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