The **Zeeman effect** ( /ˈzeɪmən/ ; Dutch pronunciation: [ˈzeːmɑn] ), named after 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.

- Nomenclature
- Theoretical presentation
- Weak field (Zeeman effect)
- Example: Lyman-alpha transition in hydrogen
- Strong field (Paschen–Back effect)
- Intermediate field for j = 1/2
- Applications
- Astrophysics
- Laser cooling
- Zeeman-energy mediated coupling of spin and orbital motions
- See also
- References
- Historical
- Modern

Since the distance between the Zeeman sub-levels is a function of magnetic field strength, this effect can be used to measure magnetic field strength, e.g. that of the Sun and other stars or in laboratory plasmas. 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. A theory about the magnetic sense of birds assumes that a protein in the retina is changed due to the Zeeman effect.^{ [1] }

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

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.

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

where is the unperturbed Hamiltonian of the atom, and is the 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 here. Therefore,

where is the Bohr magneton, is the total electronic angular momentum, and 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 and the spin angular momentum , with each multiplied by the appropriate gyromagnetic ratio:

where and (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 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 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, exceeds the LS coupling significantly (but is still small compared to ). In ultra-strong magnetic fields, the magnetic-field interaction may exceed , 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, 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 Landé g-factor g_{J} of the atom ( and ) and is the z-component of the total angular momentum. For a single electron above filled shells and , the Landé g-factor can be simplified into:

Taking to be the perturbation, the Zeeman correction to the energy is

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

- and

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 () and the 2P_{3/2} level into 4 states (). The Landé g-factors for the three levels are:

- for (j=1/2, l=0)

- for (j=1/2, l=1)

- for (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.

Initial state () | Final state () | Energy perturbation |
---|---|---|

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 () and spin () angular momenta. This effect is the strong-field limit of the Zeeman effect. When , the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back.^{ [3] }

When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume . This allows the expectation values of and to be easily evaluated for a state . The energies are simply

The above may be read as implying that the LS-coupling is completely broken by the external field. However and 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 is *independent* of the unperturbed energies and electronic configurations of the levels being considered. In general (if ), 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] }

Initial state () | Initial energy Perturbation | Final state () |
---|---|---|

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

where is the hyperfine splitting (in Hz) at zero applied magnetic field, and are the Bohr magneton and nuclear magneton respectively, and are the electron and nuclear angular momentum operators and is the Landé g-factor:

- .

In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the 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 or just since and 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 and basis states. For , the Hamiltonian can be solved analytically, resulting in the **Breit–Rabi formula**. Notably, the electric quadrupole interaction is zero for (), so this formula is fairly accurate.

We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator as

These ladder operators have the property

as long as lies in the range (otherwise, they return zero). Using ladder operators and We can rewrite the Hamiltonian as

We can now see that at all times, the total angular momentum projection will be conserved. This is because both and leave states with definite and unchanged, while and either increase and decrease or vice versa, so the sum is always unaffected. Furthermore, since there are only two possible values of which are . Therefore, for every value of there are only two possible states, and we can define them as the basis:

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

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:

where is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field , is referred to as the 'field strength parameter' (Note: for the expression under the square root is an exact square, and so the last term should be replaced by ). This equation is known as the **Breit–Rabi formula** and is useful for systems with one valence electron in an () level.^{ [5] }^{ [6] }

Note that index in 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 otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different but equal (the only exceptions are ).

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.

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

Spin–orbit interaction in crystals is usually attributed to coupling of Pauli matrices to electron momentum which exists even in the absence of magnetic field . However, under the conditions of the Zeeman effect, when , a similar interaction can be achieved by coupling to the electron coordinate through the spatially inhomogeneous Zeeman Hamiltonian

- ,

where is a tensorial Landé *g*-factor and either or , or both of them, depend on the electron coordinate . Such -dependent Zeeman Hamiltonian couples electron spin to the operator representing electron's orbital motion. Inhomogeneous field 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 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 has been also demonstrated.^{ [9] }

Wikimedia Commons has media related to Zeeman effect . |

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

**Diamagnetic** materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracted by a magnetic field. Diamagnetism is a quantum mechanical effect that occurs in all materials; when it is the only contribution to the magnetism, the material is called diamagnetic. In paramagnetic and ferromagnetic substances, the weak diamagnetic force is overcome by the attractive force of magnetic dipoles in the material. The magnetic permeability of diamagnetic materials is less than the permeability of vacuum, *μ*_{0}. In most materials, diamagnetism is a weak effect which can be detected only by sensitive laboratory instruments, but a superconductor acts as a strong diamagnet because it repels a magnetic field entirely from its interior.

In atomic physics, the **fine structure** describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887, laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.

In atomic physics, the **spin quantum number** is a quantum number which describes the intrinsic angular momentum of an electron or other particle. The phrase was originally used to describe the fourth of a set of quantum numbers, which completely describe the quantum state of an electron. The name comes from a physical spinning of the electron about an axis that was proposed by Uhlenbeck and Goudsmit, and the value of m_{s} is the component of spin angular momentum parallel to a given direction, which can be either +1/2 or –1/2.

The **Kerr–Newman metric** is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.

The **Curie–Weiss law** describes the magnetic susceptibility `χ` of a ferromagnet in the paramagnetic region above the Curie point:

In physics, the **Landé g-factor** is a particular example of a

In atomic physics, the **electron magnetic moment**, or more specifically the **electron magnetic dipole moment**, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10^{−24} J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10^{13}.

The **nuclear magnetic moment** is the magnetic moment of an atomic nucleus and arises from the spin of the protons and neutrons. It is mainly a magnetic dipole moment; the quadrupole moment does cause some small shifts in the hyperfine structure as well. All nuclei that have nonzero spin also possess a nonzero magnetic moment and vice versa, although the connection between the two quantities is not straightforward or easy to calculate.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

A **spin wave** is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only μeV in keeping with typical Curie points at room temperature and below.

An **electromagnetic reverberation chamber** is an environment for electromagnetic compatibility (EMC) testing and other electromagnetic investigations. Electromagnetic reverberation chambers have been introduced first by H.A. Mendes in 1968. A reverberation chamber is screened room with a minimum of absorption of electromagnetic energy. Due to the low absorption very high field strength can be achieved with moderate input power. A reverberation chamber is a cavity resonator with a high Q factor. Thus, the spatial distribution of the electrical and magnetic field strengths is strongly inhomogeneous. To reduce this inhomogeneity, one or more *tuners* (*stirrers*) are used. A tuner is a construction with large metallic reflectors that can be moved to different orientations in order to achieve different boundary conditions. The *Lowest Usable Frequency* (LUF) of a reverberation chamber depends on the size of the chamber and the design of the tuner. Small chambers have a higher LUF than large chambers.

In chemistry and physics, the **exchange interaction** is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an **exchange force** in an analogy to classical force, it is not a true force as it lacks a force carrier.

In quantum mechanics, **Landau quantization** refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.

In quantum mechanics, the **Pauli equation** or **Schrödinger–Pauli equation** is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

A **Zeeman slower** is a scientific apparatus that is commonly used in quantum optics to cool a beam of atoms from room temperature or above to a few kelvins. At the entrance of the Zeeman slower the average speed of atoms is on the order of a few hundred m/s. The spread of velocity is also in the order of a few hundred m/s. Final speed at the exit of the slower is few 10 m/s with an even smaller spread.

For many paramagnetic materials, the magnetization of the material is directly proportional to an applied magnetic field, for sufficiently high temperatures and small fields. However, if the material is heated, this proportionality is reduced. For a fixed value of the field, the magnetic susceptibility is inversely proportional to temperature, that is

**Magnetochemistry** is concerned with the magnetic properties of chemical compounds. Magnetic properties arise from the spin and orbital angular momentum of the electrons contained in a compound. Compounds are diamagnetic when they contain no unpaired electrons. Molecular compounds that contain one or more unpaired electrons are paramagnetic. The magnitude of the paramagnetism is expressed as an effective magnetic moment, μ_{eff}. For first-row transition metals the magnitude of μ_{eff} is, to a first approximation, a simple function of the number of unpaired electrons, the spin-only formula. In general, spin-orbit coupling causes μ_{eff} to deviate from the spin-only formula. For the heavier transition metals, lanthanides and actinides, spin-orbit coupling cannot be ignored. Exchange interaction can occur in clusters and infinite lattices, resulting in ferromagnetism, antiferromagnetism or ferrimagnetism depending on the relative orientations of the individual spins.

**Electric dipole spin resonance** (**EDSR**) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.

In mathematical physics, the **Gordon decomposition** of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

In condensed matter and atomic physics, **Van Vleck paramagnetism** refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells.

- ↑ Thalau, Peter; Ritz, Thorsten; Burda, Hynek; Wegner, Regina E.; Wiltschko, Roswitha (18 April 2006). "The magnetic compass mechanisms of birds and rodents are based on different physical principles".
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*Physica*(in German).**1**: 261–273. Available at: Leiden University (Netherlands) - ↑ Griffiths, David J. (2004).
*Introduction to Quantum Mechanics*(2nd ed.). Prentice Hall. p. 247. ISBN 0-13-111892-7. OCLC 40251748. - ↑ Woodgate, Gordon Kemble (1980).
*Elementary Atomic Structure*(2nd ed.). Oxford, England: Oxford University Press. pp. 193–194. - ↑ First appeared in: Breit, G.; Rabi, I.I. (1931). "Measurement of nuclear spin".
*Physical Review*.**38**(11): 2082–2083. Bibcode:1931PhRv...38.2082B. doi:10.1103/PhysRev.38.2082.2. - ↑ S. I. Pekar and E. I. Rashba, Combined resonance in crystals in inhomogeneous magnetic fields, Sov. Phys. - JETP
**20**, 1295 (1965) http://www.jetp.ac.ru/cgi-bin/dn/e_020_05_1295.pdf - ↑ 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.
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- 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. - 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.

- Feynman, Richard P., Leighton, Robert B., Sands, Matthew (1965).
*The Feynman Lectures on Physics*.**3**. Addison-Wesley. ISBN 0-201-02115-3.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).
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