The Zeeman effect (; ) is the effect of splitting of a
spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
into several components in the presence of a static
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
. It is named after the
Dutch physicist
Pieter Zeeman, who discovered it in 1896 and received a Nobel prize for this discovery. 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.
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
star
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...
s or in laboratory
plasmas.
The Zeeman effect is very important in applications such as
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
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.
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). The anomalous effect appears on transitions where the net
spin of the
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary partic ...
s 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?".
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".
Theoretical presentation
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
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
, 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 particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
). In the case of the
LS coupling
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact th ...
, 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
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact th ...
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.
Weak field (Zeeman effect)
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
:
Example: Lyman-alpha transition in hydrogen
The
Lyman-alpha transition in
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
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.
Strong field (Paschen–Back effect)
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
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate ca ...
s
Friedrich Paschen and
Ernst E. A. Back.
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:
:
:
:
Intermediate field for j = 1/2
In the magnetic dipole approximation, the Hamiltonian which includes both the
hyperfine
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the ...
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
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
. 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
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
, 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.
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
).
Applications
Astrophysics
George Ellery Hale
George Ellery Hale (June 29, 1868 – February 21, 1938) was an American solar astronomer, best known for his discovery of magnetic fields in sunspots, and as the leader or key figure in the planning or construction of several world-lead ...
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 cooling
The Zeeman effect is utilized in many
laser cooling
Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) ...
applications such as a
magneto-optical trap
A magneto-optical trap (MOT) is an apparatus which uses laser cooling and a spatially-varying magnetic field to create a trap which can produce samples of cold, trapped, neutral atoms. Temperatures achieved in a MOT can be as low as several micro ...
and the
Zeeman slower.
Zeeman-energy mediated coupling of spin and orbital motions
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. 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, and driving spins by electric field due to inhomogeneous
has been also demonstrated.
See also
*
Magneto-optic Kerr effect
*
Voigt effect
*
Faraday effect
The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the ...
*
Cotton–Mouton effect
*
Polarization spectroscopy Polarization spectroscopy comprises a set of spectroscopic techniques based on polarization properties of light (not necessarily visible one; UV, X-ray, infrared, or in any other frequency range of the electromagnetic radiation). By analyzing the ...
*
Zeeman energy
*
Stark effect
*
Lamb shift
In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which ...
References
Historical
* ''(Chapter 16 provides a comprehensive treatment, as of 1935.)''
*
*
*
*
*
Modern
*
*
*
*
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