 Zeeman effect

The Zeeman effect ( /ˈzeɪmən/; IPA: [ˈzeːmɑn]) 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. It may also be utilized to improve accuracy in Atomic absorption 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.
Contents
Introduction
In most atoms, there exist several electron configurations with the same energy, so that transitions between these configurations and another correspond to a single spectral line. The presence of a magnetic field breaks this degeneracy, since the magnetic field interacts differently with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, they now have different energies, giving 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 (B) splits the energy levels. Therefore, a line produced by a transition from a, b or c to d, e or f will now be split into several components between different combinations of a, b, c and d, e, f. However, not all transitions will be possible (in the dipole approximation), as governed by the selection rules.
Since the distance between the Zeeman sublevels is proportional to the magnetic field, this effect can be 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 sublevels 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.
At higher magnetic fields the effect ceases to be linear. At even higher field strength, when the strength of the external field is comparable to the strength of the atom's internal field, electron coupling is disturbed and the spectral lines rearrange. This is called the PaschenBack effect.
Theoretical presentation
The total Hamiltonian of an atom in a magnetic field is
where H_{0} is the unperturbed Hamiltonian of the atom, and V_{M} is perturbation due to the magnetic field:
where 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,
where μ_{B} is the Bohr magneton, is the total electronic angular momentum, and g is the gfactor. The operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum and the spin angular momentum , with each multiplied by the appropriate gyromagnetic ratio:
where g_{l} = 1 and (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to Quantum Electrodynamics effects). In the case of the LS coupling, one can sum over all electrons in the atom:
where and 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 V_{M} is small (less than the fine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the PaschenBack effect, described below, V_{M} exceeds the LS coupling significantly (but is still small compared to H_{0}). In ultrastrong magnetic fields, the magneticfield interaction may exceed 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 spinorbit interaction dominates over the effect of the external magnetic field, and are not separately conserved, only the total angular momentum is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector . The (time)"averaged" spin vector is then the projection of the spin onto the direction of :
and for the (time)"averaged" orbital vector:
Thus,
Using and squaring both sides, we get
and: using and squaring both sides, we get
Combining everything and taking , we obtain the magnetic potential energy of the atom in the applied external magnetic field,
where the quantity in square brackets is the Lande gfactor g_{J} of the atom (g_{L} = 1 and ) and m_{j} is the zcomponent of the total angular momentum. For a single electron above filled shells s = 1 / 2 and , the Lande gfactor can be simplified into:
Example: Lyman alpha transition in hydrogen
The Lyman alpha transition in hydrogen in the presence of the spinorbit interaction involves the transitions
 and
In the presence of an external magnetic field, the weakfield Zeeman effect splits the 1S_{1/2} and 2P_{1/2} states into 2 levels each (m_{j} = 1 / 2, − 1 / 2) and the 2P_{3/2} state into 4 levels (m_{j} = 3 / 2,1 / 2, − 1 / 2, − 3 / 2). The Lande gfactors for the three levels are:
 g_{J} = 2 for 1S_{1 / 2} (j=1/2, l=0)
 g_{J} = 2 / 3 for 2P_{1 / 2} (j=1/2, l=1)
 g_{J} = 4 / 3 for 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. On the left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spinorbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.
Strong field (PaschenBack effect)
The PaschenBack 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 strongfield limit of the Zeeman effect. When s = 0, the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back.
When the magneticfield perturbation significantly exceeds the spinorbit interaction, one can safely assume [H_{0},S] = 0. This allows the expectation values of L_{z} and S_{z} to be easily evaluated for a state :
The above may be read as implying that the LScoupling is completely broken by the external field. The m_{l} and m_{s} are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the selection rule. The splitting ΔE = Bμ_{B}Δm_{l} is independent of the unperturbed energies and electronic configurations of the levels being considered. It should be noted that in general (if ), these three components are actually groups of several transitions each, due to the residual spinorbit coupling.
See also
 magnetooptic Kerr effect
 Voigt effect
 Faraday effect
 CottonMouton effect
 Polarization spectroscopy
 Zeeman Energy
References
Historical
 Condon, E. U.; G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0521092094. (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 (1424): 347. Bibcode 1897Natur..55..347Z. doi:10.1038/055347a0. http://www.nature.com/nature/journal/v55/n1424/abs/055347a0.html.
Modern
 Feynman, Richard P., Leighton, Robert B., Sands, Matthew (1965). The Feynman Lectures on Physics, Vol. 3. AddisonWesley. ISBN 0201021153.
 Forman, Paul (1970). "Alfred Landé and the anomalous Zeeman Effect, 19191921". Historical Studies in the Physical Sciences 2: 153–261.
 Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice_Hall!Prentice Hall. ISBN 013805326X.
 Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.
 Sobelman, Igor I. (2006). Theory of Atomic Spectra. Alpha Science. ISBN 1842652036.
 Foot, C. J. (2005). Atomic Physics. ISBN 0198506961.
Categories: Atomic physics
 Magnetism
 Foundational quantum physics
 Physical phenomena
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