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Zeeman Effect
The spectral lines of mercury vapor lamp at wavelength 546.1 nm, showing anomalous Zeeman effect. (A) Without magnetic field. (B) With magnetic field, spectral lines split as transverse Zeeman effect. (C) With magnetic field, split as longitudinal Zeeman effect. The spectral lines were obtained using a Fabry-Pérot interferometer.
Zeeman splitting of the 5s level of ^{87}Rb, including fine structure and hyperfine structure splitting. Here F = J + I, where I is the nuclear spin (for ^{87}Rb, I = ).
This animation shows what happens as a sunspot (or starspot) forms and the magnetic field increases in strength. The light emerging from the spot starts to demonstrate the Zeeman effect. The dark spectra lines in the spectrum of the emitted light split into three components and the strength of the circular polarisation in parts of the spectrum increases significantly. This polarisation effect is a powerful tool for astronomers to detect and measure stellar magnetic fields.
The Zeeman effect (; Dutch pronunciation: ['ze:m?n]), named after the Dutch physicist Pieter Zeeman, is the effect of 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. Also similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden (in the dipole approximation), as governed by the selection rules.
When the spectral lines are absorption lines, the effect is called inverse Zeeman effect.
Nomenclature
Historically, one distinguishes between the normal and an anomalous Zeeman effect (discovered by Thomas Preston in Dublin, Ireland^{[2]}). The anomalous effect appears on transitions where the net spin of the electrons is an odd half-integer, so that the number of Zeeman sub-levels is even. 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 field strength 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 Paschen-Back effect.
In the modern scientific literature, these terms are rarely used, with a tendency to use just the "Zeeman effect".
Theoretical presentation
The total Hamiltonian of an atom in a magnetic field is
$H=H_{0}+V_{\rm {M}},\$
where $H_{0}$ is the unperturbed Hamiltonian of the atom, and $V_{\rm {M}}$ is the perturbation due to the magnetic field:
$V_{\rm {M}}=-{\vec {\mu }}\cdot {\vec {B}},$
where ${\vec {\mu }}$ 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 here. Therefore,
where $g_{l}=1$ and $g_{s}\approx 2.0023192$ (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the effects of quantum electrodynamics). In the case of the LS coupling, one can sum over all electrons in the atom:
where ${\vec {L}}$ and ${\vec {S}}$ 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 $V_{M}$ 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, $V_{M}$ exceeds the LS coupling significantly (but is still small compared to $H_{0}$). In ultra-strong magnetic fields, the magnetic-field 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 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, $\scriptstyle {\vec {L}}$ and $\scriptstyle {\vec {S}}$ are not separately conserved, only the total angular momentum $\scriptstyle {\vec {J}}={\vec {L}}+{\vec {S}}$ is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector $\scriptstyle {\vec {J}}$. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of $\scriptstyle {\vec {J}}$:
Combining everything and taking $\scriptstyle J_{z}=\hbar m_{j}$, we obtain the magnetic potential energy of the atom in the applied external magnetic field,
where the quantity in square brackets is the Landé g-factor g_{J} of the atom ($g_{L}=1$ and $g_{S}\approx 2$) and $m_{j}$ is the z-component of the total angular momentum.
For a single electron above filled shells $s=1/2$ and $j=l\pm s$, the Landé g-factor can be simplified into:
$g_{j}=1\pm {\frac {g_{S}-1}{2l+1}}$
Taking $V_{m}$ to be the perturbation, the Zeeman correction to the energy is
$2P_{1/2}\to 1S_{1/2}$ and $2P_{3/2}\to 1S_{1/2}.$
In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S_{1/2} and 2P_{1/2} levels into 2 states each ($m_{j}=1/2,-1/2$) and the 2P_{3/2} level into 4 states ($m_{j}=3/2,1/2,-1/2,-3/2$). The Landé g-factors 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 spin-orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.
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 strong to disrupt the coupling between orbital (${\vec {L}}$) and spin (${\vec {S}}$) angular momenta. This effect is the strong-field limit of the Zeeman effect. When $s=0$, the two effects are equivalent. The effect was named after the GermanphysicistsFriedrich Paschen and Ernst E. A. Back.^{[3]}
When the magnetic-field perturbation significantly exceeds the spin-orbit 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 $|\psi \rangle$. The energies are simply
The above may be read as implying that the LS-coupling is completely broken by the external field. However $m_{l}$ and $m_{s}$ are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., $\Delta s=0,\Delta m_{s}=0,\Delta l=\pm 1,\Delta m_{l}=0,\pm 1$ this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the $\Delta m_{l}=0,\pm 1$ selection rule. The splitting $\Delta E=B\mu _{\rm {B}}\Delta m_{l}$ is independent of the unperturbed energies and electronic configurations of the levels being considered. In general (if $s\neq 0$), these three components are actually groups of several transitions each, due to the residual spin-orbit coupling.
In general, one must now add spin-orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure') as a perturbation to these 'unperturbed' levels. First order perturbation theory with these fine-structure corrections yields the following formula for the Hydrogen atom in the Paschen–Back limit:^{[4]}
where $A$ is the hyperfine splitting (in Hz) at zero applied magnetic field, $\mu _{\rm {B}}$ and $\mu _{\rm {N}}$ are the Bohr magneton and nuclear magneton respectively, ${\vec {J}}$ and ${\vec {I}}$ are the electron and nuclear angular momentum operators and $g_{J}$ is the Landé g-factor:
In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the $|F,m_{f}\rangle$ basis. In the high field regime, the magnetic field becomes so strong that the Zeeman effect will dominate, and one must use a more complete basis of $|I,J,m_{I},m_{J}\rangle$ or just $|m_{I},m_{J}\rangle$ since $I$ and $J$ will be constant within a given level.
To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the $|F,m_{F}\rangle$ and $|m_{I},m_{J}\rangle$ basis states. For $J=1/2$, the Hamiltonian can be solved analytically, resulting in the Breit-Rabi formula. Notably, the electric quadrupole interaction is zero for $L=0$ ($J=1/2$), so this formula is fairly accurate.
To solve this system, we note that at all times, the total angular momentum projection $m_{F}=m_{J}+m_{I}$ will be conserved. Furthermore, since $J=1/2$ between states $m_{J}$ will change between only $\pm 1/2$. Therefore, we can define a good basis as:
as long as $m_{L}$ lies in the range ${-L,\dots ...,L}$ (otherwise, they return zero). Using ladder operators $J_{\pm }$ and $I_{\pm }$
We can rewrite the Hamiltonian as
where $\Delta W$ is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field $B$,
$x$ is referred to as the 'field strength parameter' (Note: for $m=-(I+1/2)$ the square root is an exact square, and should be interpreted as $+(1-x)$). This equation is known as the Breit-Rabi formula and is useful for systems with one valence electron in an $s$ ($J=1/2$) level.^{[5]}^{[6]}
Note that index $F$ in $\Delta E_{F=I\pm 1/2}$ should be considered not as total angular momentum of the atom but as asymptotic total angular momentum. It is equal to total angular momentum only if $B=0$
otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different $F$ but equal $m_{F}$ (the only exceptions are $|F=I+1/2,m_{F}=\pm F\rangle$).
Applications
Astrophysics
Zeeman effect on a sunspot spectral line
George Ellery Hale was the first to notice the Zeeman effect in the solar spectra, indicating the existence of strong magnetic fields in sunspots. Such fields can be quite high, on the order of 0.1 tesla or higher. Today, the Zeeman effect is used to produce magnetograms showing the variation of magnetic field on the sun.
Zeeman-energy mediated coupling of spin and orbital motions
Spin-orbit interaction in crystals is usually attributed to coupling of Pauli matrices ${\boldsymbol {\sigma }}$ to electron momentum ${\boldsymbol {k}}$ which exists even in the absence of magnetic field ${\boldsymbol {B}}$. However, under the conditions of the Zeeman effect, when ${\boldsymbol {B}}\neq 0$, a similar interaction can be achieved by coupling ${\boldsymbol {\sigma }}$ to the electron coordinate ${\boldsymbol {r}}$ through the spatially inhomogeneous Zeeman Hamiltonian
where ${\hat {g}}$ is a tensorial Landé g-factor and either ${\boldsymbol {B}}={\boldsymbol {B}}({\boldsymbol {r}})$ or ${\hat {g}}={\hat {g}}({\boldsymbol {r}})$, or both of them, depend on the electron coordinate ${\boldsymbol {r}}$. Such ${\boldsymbol {r}}$-dependent Zeeman Hamiltonian $H_{\rm {Z}}({\boldsymbol {r}})$ couples electron spin ${\boldsymbol {\sigma }}$ to the operator ${\boldsymbol {r}}$ representing electron's orbital motion. Inhomogeneous field ${\boldsymbol {B}}({\boldsymbol {r}})$ may be either a smooth field of external sources or fast-oscillating microscopic magnetic field in antiferromagnets.^{[7]} Spin-orbit coupling through macroscopically inhomogeneous field ${\boldsymbol {B}}({\boldsymbol {r}})$ of nanomagnets is used for electrical operation of electron spins in quantum dots through electric dipole spin resonance,^{[8]} and driving spins by electric field due to inhomogeneous ${\hat {g}}({\boldsymbol {r}})$ has been also demonstrated.^{[9]}
Electron configuration says at subshell p (l=1), there are 3 energy level ml=-1,0,1, but we see only two p1/2 and p3/2. for subshell s(l=0), there is only 1 energy level (ml=0), but here we have 2. l corresponding to fine structure, ml corresponding to hyperfine structure.
^Paschen, F.; Back, E. (1921). "Liniengruppen magnetisch vervollständigt" [Line groups magnetically completed [i.e., completely resolved]]. Physica (in German). 1: 261-273. Available at: Leiden University (Netherlands)
^Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Coherent single electron spin control in a slanting Zeeman field, Phys. Rev. Lett. 96, 047202 (2006)
^Salis G, Kato Y, Ensslin K, Driscoll DC, Gossard AC, Awschalom DD (2001). "Electrical control of spin coherence in semiconductor nanostructures". Nature. 414 (6864): 619-622. doi:10.1038/414619a.CS1 maint: uses authors parameter (link)
Historical
Condon, E. U.; G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN0-521-09209-4.(Chapter 16 provides a comprehensive treatment, as of 1935.)
Zeeman, P. (1896). "Over de invloed eener magnetisatie op den aard van het door een stof uitgezonden licht" [On the influence of magnetism on the nature of the light emitted by a substance]. Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling (Koninklijk Akademie van Wetenschappen te Amsterdam) [Reports of the Ordinary Sessions of the Mathematical and Physical Section (Royal Academy of Sciences in Amsterdam)] (in Dutch). 5: 181-184 and 242-248.
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.
Zeeman, P. (1897). "Over doubletten en tripletten in het spectrum, teweeggebracht door uitwendige magnetische krachten" [On doublets and triplets in the spectrum, caused by external magnetic forces]. Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling (Koninklijk Akademie van Wetenschappen te Amsterdam) [Reports of the Ordinary Sessions of the Mathematical and Physical Section (Royal Academy of Sciences in Amsterdam)] (in Dutch). 6: 13-18, 99-102, and 260-262.
Forman, Paul (1970). "Alfred Landé and the anomalous Zeeman Effect, 1919-1921". Historical Studies in the Physical Sciences. 2: 153-261. doi:10.2307/27757307. JSTOR27757307.