Zeeman effect

The Zeeman effect (pronounced /ˈzeɪmɑːn/) is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. The Zeeman effect is very important in applications such as nuclear magnetic resonance spectroscopy, electron spin resonance spectroscopy, magnetic resonance imaging (MRI) and Mössbauer spectroscopy.

When the spectral lines are absorption lines, the effect is called Inverse Zeeman effect.

The Zeeman effect is named after the Dutch physicist Pieter Zeeman.

Introduction

In most atoms, there exist several electronic configurations that have the same energy, so that transitions between different pairs of configurations correspond to a single spectral line.

The presence of a magnetic field breaks the degeneracy, since it interacts in a different way with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, now there are different energies, which give rise to several very close spectral lines.

Without a magnetic field, configurations a, b and c have the same energy, as do d, e and f. The presence of a magnetic field splits the energy levels. A line produced by a transition from a, b or c to d, e or f now will be several lines between different combinations of a, b, c and d, e, f. Not all transitions will be possible, as regulated by the transition rules.

Since the distance between the Zeeman sub-levels is proportional to the magnetic field, this effect is used by astronomers to measure the magnetic field of the Sun and other stars.

There is also an anomalous Zeeman effect that appears on transitions where the net spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of odd if there's an uneven number of electrons involved. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed the effect.

If the magnetic field strength is too high, the effect is no longer linear; at even higher field strength, electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen-Back effect.

Theoretical presentation

The total Hamiltonian of an atom in a magnetic field is

Failed to parse (Missing texvc executable; please see math/README to configure.): H = H_0 + H_M ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): H_0

is the unperturbed Hamiltonian of the atom, and Failed to parse (Missing texvc executable; please see math/README to configure.): H_M
is perturbation due to the magnetic field:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_M = -\vec{\mu} \cdot \vec{B}

,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{\mu}

is the magnetic moment of the atom. The magnetic moment consists of the electronic and nuclear parts, however, the latter is many orders of magnitude smaller and will be neglected further on. Therefore, 

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{\mu} = -\mu_B g \vec{J}

,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_B

is the Bohr magneton, Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{J}
is the total electronic angular momentum, and Failed to parse (Missing texvc executable; please see math/README to configure.): g
is the g-factor.

The operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum Failed to parse (Missing texvc executable; please see math/README to configure.): \vec l

and the spin angular momentum Failed to parse (Missing texvc executable; please see math/README to configure.): \vec s

, with each multiplied by the appropriate gyromagnetic ratio:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{\mu} = -\mu_B (g_l \vec{l} + g_s \vec{s})

,

where Failed to parse (Missing texvc executable; please see math/README to configure.): g_l = 1

or Failed to parse (Missing texvc executable; please see math/README to configure.): g_s \approx 2.0023192
(the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the relativistic effects). In the case of the LS coupling, one can sum over all electrons in the atom:

Failed to parse (Missing texvc executable; please see math/README to configure.): g \vec{J} = \left\langle\sum_i (g_l \vec{l_i} + g_s \vec{s_i})\right\rangle = \left\langle\vec{L} + g_s \vec{S}\right\rangle

,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{L}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{S}
are the total orbital momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum.

If the interaction term Failed to parse (Missing texvc executable; please see math/README to configure.): V_M

is small (less than the fine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen-Back effect, described below, Failed to parse (Missing texvc executable; please see math/README to configure.): V_M
exceeds the LS coupling significantly (but is still small compared to Failed to parse (Missing texvc executable; please see math/README to configure.): H_{0}

). In ultrastrong magnetic fields, the magnetic-field interaction may exceed Failed to parse (Missing texvc executable; please see math/README to configure.): H_0 , in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are, of course, intermediate cases which are more complex than these limit cases.

Weak field (Zeeman effect)

If the spin-orbit interaction dominates over the effect of the external magnetic field, Failed to parse (Missing texvc executable; please see math/README to configure.): \vec L

and Failed to parse (Missing texvc executable; please see math/README to configure.): \vec S
are not separately conserved, only the total angular momentum Failed to parse (Missing texvc executable; please see math/README to configure.): \vec J = \vec L + \vec S
is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector Failed to parse (Missing texvc executable; please see math/README to configure.): \vec J

. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of Failed to parse (Missing texvc executable; please see math/README to configure.): \vec J

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec S_{avg} = \frac{(\vec S \cdot \vec J)}{J^2} \vec J

.

and for the (time-)"averaged" orbital vector:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec L_{avg} = \frac{(\vec L \cdot \vec J)}{J^2} \vec J

.

Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle V_M \rangle = \frac{\mu_B}{\hbar} \vec J(g_L\frac{\vec L \cdot \vec J}{J^2} + g_S\frac{\vec S \cdot \vec J}{J^2}) \cdot \vec B

.

Using Failed to parse (Missing texvc executable; please see math/README to configure.): \vec L = \vec J - \vec S

and squaring both sides, we get 

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec S \cdot \vec J = \frac{1}{2}(J^2 + S^2 - L^2) = \frac{\hbar^2}{2}[j(j+1) - l(l+1) + s(s+1)]

, and: using Failed to parse (Missing texvc executable; please see math/README to configure.): \vec S = \vec J - \vec L

and squaring both sides, we get 

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec L \cdot \vec J = \frac{1}{2}(J^2 - S^2 + L^2) = \frac{\hbar^2}{2}[j(j+1) + l(l+1) - s(s+1)]

Combining everything and taking Failed to parse (Missing texvc executable; please see math/README to configure.): J_z = \hbar m_j , we obtain the magnetic potential energy of the atom in the applied external magnetic field,

Failed to parse (Missing texvc executable; please see math/README to configure.): V_M = \mu_B B m_j \left[ g_L\frac{j(j+1) + l(l+1) - s(s+1)}{2j(j+1)} + g_S\frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)} \right]

,

where the quantity in square brackets is the Lande g-factor g_J of the atom (Failed to parse (Missing texvc executable; please see math/README to configure.): g_L = 1

and Failed to parse (Missing texvc executable; please see math/README to configure.): g_S \approx 2

) and Failed to parse (Missing texvc executable; please see math/README to configure.): m_j

is the z-component of the total angular momentum. 

For a single electron above filled shells Failed to parse (Missing texvc executable; please see math/README to configure.): s = 1/2 .

Example: Lyman alpha transition in hydrogen

The Lyman alpha transition in hydrogen in the presence of the spin-orbit interaction involves the transitions

Failed to parse (Missing texvc executable; please see math/README to configure.): 2P_{1/2} \to 1S_{1/2}

and Failed to parse (Missing texvc executable; please see math/README to configure.): 2P_{3/2} \to 1S_{1/2}

.

In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S_1/2 and 2P_1/2 states into 2 levels each (Failed to parse (Missing texvc executable; please see math/README to configure.): m_j = 1/2, -1/2 ) and the 2P_3/2 state into 4 levels (Failed to parse (Missing texvc executable; please see math/README to configure.): m_j = 3/2, 1/2, -1/2, -3/2 ). The Lande g-factors for the three levels are:

Failed to parse (Missing texvc executable; please see math/README to configure.): g_J = 2

for Failed to parse (Missing texvc executable; please see math/README to configure.): 1S_{1/2}
(j=1/2, l=0)

Failed to parse (Missing texvc executable; please see math/README to configure.): g_J = 2/3

for Failed to parse (Missing texvc executable; please see math/README to configure.): 2P_{1/2}
(j=1/2, l=1)

Failed to parse (Missing texvc executable; please see math/README to configure.): g_J = 4/3

for Failed to parse (Missing texvc executable; please see math/README to configure.): 2P_{3/2}
(j=3/2, l=1)

Note in particular that the size of the energy splitting is different for the different orbitals, because the g_J values are different.

Image:Zeeman p s doublet.svg

Strong field (Paschen-Back effect)

The Paschen-Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently large to disrupt the coupling between orbital and spin angular momenta. This effect is the strong field generalization of the Zeeman effect. The effect was named for the German physicists Friedrich Paschen and Ernst E. A. Back.

When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume Failed to parse (Missing texvc executable; please see math/README to configure.): [H_{0}, S] = 0 . This allows the expectation values of Failed to parse (Missing texvc executable; please see math/README to configure.): L_{z}

and Failed to parse (Missing texvc executable; please see math/README to configure.): S_{z}
to be easily evaluated for a state Failed to parse (Missing texvc executable; please see math/README to configure.): |A\rangle 

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle A| \left( H_{0} + \frac{B_{z}\mu_B}{\hbar}(L_{z}+g_{s}S_z) \right) |A \rangle = E_{0} + B_z\mu_B (m_l + g_{s}m_s)

.

The above may be read as implying that the LS-coupling is completely broken by the external field. The Failed to parse (Missing texvc executable; please see math/README to configure.): m_l

and Failed to parse (Missing texvc executable; please see math/README to configure.): m_s
are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = 0, \Delta m_s = 0, \Delta L = \pm 1, \Delta m_l = 0, \pm 1
this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta m_l = 0, \pm 1
selection rule. The splitting Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta E = B \mu_B \Delta m_l
is independent of the unperturbed energies and electronic configurations of the levels being considered.

See also

• Stark effect
• magneto-optic Kerr effect
• Voigt effect
• Faraday effect
• Cotton-Mouton effect
• Polarization spectroscopy

References

Historical

• Condon, E. U.; G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.  (Chapter 16 provides a comprehensive treatment, as of 1935.)
• Zeeman, P. (1897). "On the influence of Magnetism on the Nature of the Light emitted by a Substance". Phil. Mag. 43: 226. 
• Zeeman, P. (1897). "Doubles and triplets in the spectrum produced by external magnetic forces". Phil. Mag. 44: 55. 
• Zeeman, P. (11 February 1897). "The Effect of Magnetisation on the Nature of Light Emitted by a Substance". Nature 55: 347. 

Modern

• Forman, Paul (1970). "Alfred Landé and the anomalous Zeeman Effect, 1919-1921". Historical Studies in the Physical Sciences 2: 153—261. 
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. 
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. 
• Sobelman, Igor I. (2006). Theory of Atomic Spectra. Alpha Science. ISBN 1-84265-203-6. bn:জেমান ক্রিয়া

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