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 non-zero. 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. Wolfgang Pauli recalls that when asked by a colleague why he looks unhappy he replied "How can one look happy when he is thinking about the anomalous Zeeman effect?".^[3]

At higher magnetic field strength the effect ceases to be linear. At even higher field strengths, comparable to the strength of the atom's internal field, the 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".

The total Hamiltonian of an atom in a magnetic field is

${\displaystyle H=H_{0}+V_{\rm {M}},\ }$ 

where ${\displaystyle H_{0}}$  is the unperturbed Hamiltonian of the atom, and ${\displaystyle V_{\rm {M}}}$  is the perturbation due to the magnetic field:

${\displaystyle V_{\rm {M}}=-{\vec {\mu }}\cdot {\vec {B}},}$ 

where ${\displaystyle {\vec {\mu }}}$  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,

${\displaystyle {\vec {\mu }}\approx -{\frac {\mu _{\rm {B}}g{\vec {J}}}{\hbar }},}$ 

where ${\displaystyle \mu _{\rm {B}}}$  is the Bohr magneton, ${\displaystyle {\vec {J}}}$  is the total electronic angular momentum, and ${\displaystyle g}$  is the Landé g-factor. A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum ${\displaystyle {\vec {L}}}$  and the spin angular momentum ${\displaystyle {\vec {S}}}$ , with each multiplied by the appropriate gyromagnetic ratio:

${\displaystyle {\vec {\mu }}=-{\frac {\mu _{\rm {B}}(g_{l}{\vec {L}}+g_{s}{\vec {S}})}{\hbar }},}$ 

where ${\displaystyle g_{l}=1}$  and ${\displaystyle g_{s}\approx 2.0023192}$  (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:

${\displaystyle g{\vec {J}}=\left\langle \sum _{i}(g_{l}{\vec {l_{i}}}+g_{s}{\vec {s_{i}}})\right\rangle =\left\langle (g_{l}{\vec {L}}+g_{s}{\vec {S}})\right\rangle ,}$ 

where ${\displaystyle {\vec {L}}}$  and ${\displaystyle {\vec {S}}}$  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 ${\displaystyle V_{M}}$  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, ${\displaystyle V_{M}}$  exceeds the LS coupling significantly (but is still small compared to ${\displaystyle H_{0}}$ ). In ultra-strong magnetic fields, the magnetic-field interaction may exceed ${\displaystyle H_{0}}$ , 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.

If the spin–orbit interaction dominates over the effect of the external magnetic field, ${\displaystyle {\vec {L}}}$  and ${\displaystyle {\vec {S}}}$  are not separately conserved, only the total angular momentum ${\displaystyle {\vec {J}}={\vec {L}}+{\vec {S}}}$  is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector ${\displaystyle {\vec {J}}}$ . The (time-)"averaged" spin vector is then the projection of the spin onto the direction of ${\displaystyle {\vec {J}}}$ :

${\displaystyle {\vec {S}}_{\rm {avg}}={\frac {({\vec {S}}\cdot {\vec {J}})}{J^{2}}}{\vec {J}}}$ 

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

${\displaystyle {\vec {L}}_{\rm {avg}}={\frac {({\vec {L}}\cdot {\vec {J}})}{J^{2}}}{\vec {J}}.}$ 

Thus,

${\displaystyle \langle V_{\rm {M}}\rangle ={\frac {\mu _{\rm {B}}}{\hbar }}{\vec {J}}\left(g_{L}{\frac {{\vec {L}}\cdot {\vec {J}}}{J^{2}}}+g_{S}{\frac {{\vec {S}}\cdot {\vec {J}}}{J^{2}}}\right)\cdot {\vec {B}}.}$ 

Using ${\displaystyle {\vec {L}}={\vec {J}}-{\vec {S}}}$  and squaring both sides, we get

${\displaystyle {\vec {S}}\cdot {\vec {J}}={\frac {1}{2}}(J^{2}+S^{2}-L^{2})={\frac {\hbar ^{2}}{2}}[j(j+1)-l(l+1)+s(s+1)],}$ 

and: using ${\displaystyle {\vec {S}}={\vec {J}}-{\vec {L}}}$  and squaring both sides, we get

${\displaystyle {\vec {L}}\cdot {\vec {J}}={\frac {1}{2}}(J^{2}-S^{2}+L^{2})={\frac {\hbar ^{2}}{2}}[j(j+1)+l(l+1)-s(s+1)].}$ 

Combining everything and taking ${\displaystyle J_{z}=\hbar m_{j}}$ , we obtain the magnetic potential energy of the atom in the applied external magnetic field,

${\displaystyle {\begin{aligned}V_{\rm {M}}&=\mu _{\rm {B}}Bm_{j}\left[g_{L}{\frac {j(j+1)+l(l+1)-s(s+1)}{2j(j+1)}}+g_{S}{\frac {j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}}\right]\\&=\mu _{\rm {B}}Bm_{j}\left[1+(g_{S}-1){\frac {j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}}\right],\\&=\mu _{\rm {B}}Bm_{j}g_{j}\end{aligned}}}$ 

where the quantity in square brackets is the Landé g-factor g_J of the atom (${\displaystyle g_{L}=1}$  and ${\displaystyle g_{S}\approx 2}$ ) and ${\displaystyle m_{j}}$  is the z-component of the total angular momentum. For a single electron above filled shells ${\displaystyle s=1/2}$  and ${\displaystyle j=l\pm s}$ , the Landé g-factor can be simplified into:

${\displaystyle g_{j}=1\pm {\frac {g_{S}-1}{2l+1}}}$ 

Taking ${\displaystyle V_{m}}$  to be the perturbation, the Zeeman correction to the energy is

${\displaystyle {\begin{aligned}E_{\rm {Z}}^{(1)}=\langle nljm_{j}|H_{\rm {Z}}^{'}|nljm_{j}\rangle =\langle V_{M}\rangle _{\Psi }=\mu _{\rm {B}}g_{J}B_{\rm {ext}}m_{j}\end{aligned}}}$ 

Example: Lyman-alpha transition in hydrogenEdit

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

${\displaystyle 2P_{1/2}\to 1S_{1/2}}$  and ${\displaystyle 2P_{3/2}\to 1S_{1/2}.}$ 

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 (${\displaystyle m_{j}=1/2,-1/2}$ ) and the 2P_3/2 level into 4 states (${\displaystyle m_{j}=3/2,1/2,-1/2,-3/2}$ ). The Landé g-factors for the three levels are:

${\displaystyle g_{J}=2}$  for ${\displaystyle 1S_{1/2}}$  (j=1/2, l=0)

${\displaystyle g_{J}=2/3}$  for ${\displaystyle 2P_{1/2}}$  (j=1/2, l=1)

${\displaystyle g_{J}=4/3}$  for ${\displaystyle 2P_{3/2}}$  (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 (${\displaystyle {\vec {L}}}$ ) and spin (${\displaystyle {\vec {S}}}$ ) angular momenta. This effect is the strong-field limit of the Zeeman effect. When ${\displaystyle s=0}$ , the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back.^[4]

When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume ${\displaystyle [H_{0},S]=0}$ . This allows the expectation values of ${\displaystyle L_{z}}$  and ${\displaystyle S_{z}}$  to be easily evaluated for a state ${\displaystyle |\psi \rangle }$ . The energies are simply

${\displaystyle E_{z}=\left\langle \psi \left|H_{0}+{\frac {B_{z}\mu _{\rm {B}}}{\hbar }}(L_{z}+g_{s}S_{z})\right|\psi \right\rangle =E_{0}+B_{z}\mu _{\rm {B}}(m_{l}+g_{s}m_{s}).}$ 

The above may be read as implying that the LS-coupling is completely broken by the external field. However ${\displaystyle m_{l}}$  and ${\displaystyle m_{s}}$  are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., ${\displaystyle \Delta s=0,\Delta m_{s}=0,\Delta l=\pm 1,\Delta m_{l}=0,\pm 1}$  this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the ${\displaystyle \Delta m_{l}=0,\pm 1}$  selection rule. The splitting ${\displaystyle \Delta E=B\mu _{\rm {B}}\Delta m_{l}}$  is independent of the unperturbed energies and electronic configurations of the levels being considered. In general (if ${\displaystyle s\neq 0}$ ), 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:^[5]

${\displaystyle E_{z+fs}=E_{z}+{\frac {m_{e}c^{2}\alpha ^{4}}{2n^{3}}}\left\{{\frac {3}{4n}}-\left[{\frac {l(l+1)-m_{l}m_{s}}{l(l+1/2)(l+1)}}\right]\right\}.}$ 

In the magnetic dipole approximation, the Hamiltonian which includes both the hyperfine and Zeeman interactions is

${\displaystyle H=hA{\vec {I}}\cdot {\vec {J}}-{\vec {\mu }}\cdot {\vec {B}}}$ 

${\displaystyle H=hA{\vec {I}}\cdot {\vec {J}}+(\mu _{\rm {B}}g_{J}{\vec {J}}+\mu _{\rm {N}}g_{I}{\vec {I}})\cdot {\vec {\rm {B}}}}$ 

where ${\displaystyle A}$  is the hyperfine splitting (in Hz) at zero applied magnetic field, ${\displaystyle \mu _{\rm {B}}}$  and ${\displaystyle \mu _{\rm {N}}}$  are the Bohr magneton and nuclear magneton respectively, ${\displaystyle {\vec {J}}}$  and ${\displaystyle {\vec {I}}}$  are the electron and nuclear angular momentum operators and ${\displaystyle g_{J}}$  is the Landé g-factor:

In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the ${\displaystyle |F,m_{f}\rangle }$  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 ${\displaystyle |I,J,m_{I},m_{J}\rangle }$  or just ${\displaystyle |m_{I},m_{J}\rangle }$  since ${\displaystyle I}$  and ${\displaystyle J}$  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 ${\displaystyle |F,m_{F}\rangle }$  and ${\displaystyle |m_{I},m_{J}\rangle }$  basis states. For ${\displaystyle J=1/2}$ , the Hamiltonian can be solved analytically, resulting in the Breit–Rabi formula. Notably, the electric quadrupole interaction is zero for ${\displaystyle L=0}$  (${\displaystyle J=1/2}$ ), so this formula is fairly accurate.

We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator ${\displaystyle L}$  as

${\displaystyle L_{\pm }\equiv L_{x}\pm iL_{y}}$ 

These ladder operators have the property

${\displaystyle L_{\pm }|L_{,}m_{L}\rangle ={\sqrt {(L\mp m_{L})(L\pm m_{L}+1)}}|L,m_{L}\pm 1\rangle }$ 

as long as ${\displaystyle m_{L}}$  lies in the range ${\displaystyle {-L,\dots ...,L}}$  (otherwise, they return zero). Using ladder operators ${\displaystyle J_{\pm }}$  and ${\displaystyle I_{\pm }}$  We can rewrite the Hamiltonian as

${\displaystyle H=hAI_{z}J_{z}+{\frac {hA}{2}}(J_{+}I_{-}+J_{-}I_{+})+\mu _{\rm {B}}Bg_{J}J_{z}+\mu _{\rm {N}}Bg_{I}I_{z}}$ 

We can now see that at all times, the total angular momentum projection ${\displaystyle m_{F}=m_{J}+m_{I}}$  will be conserved. This is because both ${\displaystyle J_{z}}$  and ${\displaystyle I_{z}}$  leave states with definite ${\displaystyle m_{J}}$  and ${\displaystyle m_{I}}$  unchanged, while ${\displaystyle J_{+}I_{-}}$  and ${\displaystyle J_{-}I_{+}}$  either increase ${\displaystyle m_{J}}$  and decrease ${\displaystyle m_{I}}$  or vice versa, so the sum is always unaffected. Furthermore, since ${\displaystyle J=1/2}$  there are only two possible values of ${\displaystyle m_{J}}$  which are ${\displaystyle \pm 1/2}$ . Therefore, for every value of ${\displaystyle m_{F}}$  there are only two possible states, and we can define them as the basis:

${\displaystyle |\pm \rangle \equiv |m_{J}=\pm 1/2,m_{I}=m_{F}\mp 1/2\rangle }$ 

This pair of states is a Two-level quantum mechanical system. Now we can determine the matrix elements of the Hamiltonian:

${\displaystyle \langle \pm |H|\pm \rangle =-{\frac {1}{4}}hA+\mu _{\rm {N}}Bg_{I}m_{F}\pm {\frac {1}{2}}(hAm_{F}+\mu _{\rm {B}}Bg_{J}-\mu _{\rm {N}}Bg_{I}))}$ 

${\displaystyle \langle \pm |H|\mp \rangle ={\frac {1}{2}}hA{\sqrt {(I+1/2)^{2}-m_{F}^{2}}}}$ 

Solving for the eigenvalues of this matrix, (as can be done by hand - see Two-level quantum mechanical system, or more easily, with a computer algebra system) we arrive at the energy shifts:

${\displaystyle \Delta E_{F=I\pm 1/2}=-{\frac {h\Delta W}{2(2I+1)}}+\mu _{\rm {N}}g_{I}m_{F}B\pm {\frac {h\Delta W}{2}}{\sqrt {1+{\frac {2m_{F}x}{I+1/2}}+x^{2}}}}$ 

${\displaystyle x\equiv {\frac {B(\mu _{\rm {B}}g_{J}-\mu _{\rm {N}}g_{I})}{h\Delta W}}\quad \quad \Delta W=A\left(I+{\frac {1}{2}}\right)}$ 

where ${\displaystyle \Delta W}$  is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field ${\displaystyle B}$ , ${\displaystyle x}$  is referred to as the 'field strength parameter' (Note: for ${\displaystyle m_{F}=\pm (I+1/2)}$  the expression under the square root is an exact square, and so the last term should be replaced by ${\displaystyle +{\frac {h\Delta W}{2}}(1\pm x)}$ ). This equation is known as the Breit–Rabi formula and is useful for systems with one valence electron in an ${\displaystyle s}$  (${\displaystyle J=1/2}$ ) level.^[6][7]

Note that index ${\displaystyle F}$  in ${\displaystyle \Delta E_{F=I\pm 1/2}}$  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 ${\displaystyle B=0}$  otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different ${\displaystyle F}$  but equal ${\displaystyle m_{F}}$  (the only exceptions are ${\displaystyle |F=I+1/2,m_{F}=\pm F\rangle }$ ).

AstrophysicsEdit

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.

Laser coolingEdit

The Zeeman effect is utilized in many laser cooling applications such as a magneto-optical trap and the Zeeman slower.

Zeeman-energy mediated coupling of spin and orbital motionsEdit

Spin–orbit interaction in crystals is usually attributed to coupling of Pauli matrices ${\displaystyle {\vec {\sigma }}}$  to electron momentum ${\displaystyle {\vec {k}}}$  which exists even in the absence of magnetic field ${\displaystyle {\vec {B}}}$ . However, under the conditions of the Zeeman effect, when ${\displaystyle {\vec {B}}\neq 0}$ , a similar interaction can be achieved by coupling ${\displaystyle {\vec {\sigma }}}$  to the electron coordinate ${\displaystyle {\vec {r}}}$  through the spatially inhomogeneous Zeeman Hamiltonian

${\displaystyle H_{\rm {Z}}={\frac {1}{2}}({\vec {B}}{\hat {g}}{\vec {\sigma }})}$ ,

where ${\displaystyle {\hat {g}}}$  is a tensorial Landé g-factor and either ${\displaystyle {\vec {B}}={\vec {B}}({\vec {r}})}$  or ${\displaystyle {\hat {g}}={\hat {g}}({\vec {r}})}$ , or both of them, depend on the electron coordinate ${\displaystyle {\vec {r}}}$ . Such ${\displaystyle {\vec {r}}}$ -dependent Zeeman Hamiltonian ${\displaystyle H_{\rm {Z}}({\vec {r}})}$  couples electron spin ${\displaystyle {\vec {\sigma }}}$  to the operator ${\displaystyle {\vec {r}}}$  representing electron's orbital motion. Inhomogeneous field ${\displaystyle {\vec {B}}({\vec {r}})}$  may be either a smooth field of external sources or fast-oscillating microscopic magnetic field in antiferromagnets.^[8] Spin–orbit coupling through macroscopically inhomogeneous field ${\displaystyle {\vec {B}}({\vec {r}})}$  of nanomagnets is used for electrical operation of electron spins in quantum dots through electric dipole spin resonance,^[9] and driving spins by electric field due to inhomogeneous ${\displaystyle {\hat {g}}({\vec {r}})}$  has been also demonstrated.^[10]

OtherEdit

Old high-precision frequency standards, i.e. hyperfine structure transition-based atomic clocks, may require periodic fine-tuning due to exposure to magnetic fields. This is carried out by measuring the Zeeman effect on specific hyperfine structure transition levels of the source element (cesium) and applying a uniformly precise, low-strenght magnetic field to said source, in a process known as degaussing.^[11]

• Magneto-optic Kerr effect
• Voigt effect
• Faraday effect
• Cotton–Mouton effect
• Polarization spectroscopy
• Zeeman energy
• Stark effect
• Lamb shift

HistoricalEdit

• 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. (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. Bibcode:1896VMKAN...5..181Z.
• Zeeman, P. (1897). "On the influence of magnetism on the nature of the light emitted by a substance". Philosophical Magazine. 5th series. 43 (262): 226–239. doi:10.1080/14786449708620985.
• 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.
• Zeeman, P. (1897). "Doublets and triplets in the spectrum produced by external magnetic forces". Philosophical Magazine. 5th series. 44 (266): 55–60. doi:10.1080/14786449708621028.

ModernEdit

• Feynman, Richard P., Leighton, Robert B., Sands, Matthew (1965). The Feynman Lectures on Physics. Vol. 3. Addison-Wesley. ISBN 0-201-02115-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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. JSTOR 27757307.
• 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.
• Foot, C. J. (2005). Atomic Physics. ISBN 0-19-850696-1.