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Zeeman effectThe 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. Additional recommended knowledge
IntroductionIn 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 sublevels 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 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. 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 PaschenBack effect. Theoretical presentationThe total Hamiltonian of an atom in a magnetic field is H = H_{0} + H_{M}, where H_{0} is the unperturbed Hamiltonian of the atom, and H_{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 gyromagnetic ratio (usually called 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 or (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:
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. Example: Lyman alpha transition in hydrogenThe Lyman alpha transition in hydrogen in the presence of the spinorbit interaction involves the transitions
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:
Note in particular that the size of the energy splitting is different for the different orbitals, because the g_{J} values are different.
Strong field (PaschenBack effect)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. See also
ReferencesHistorical
Modern
Categories: Atomic physics  Foundational quantum physics 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Zeeman_effect". A list of authors is available in Wikipedia. 