The Zeeman effect, named after the Dutch physicist Pieter Zeeman, is the effect
of splitting 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, as
governed by the selection rules. Since the distance between the Zeeman
sub-levels is a function of the magnetic field, this effect can be used to
measure the magnetic field, 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 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 that appears on transitions where
the net spin of the electrons is not 0, the number of Zeeman sub-levels being
even instead of odd if there is 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. At higher magnetic fields 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
where is the unperturbed Hamiltonian of the atom, and 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
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 . 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, 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.
In ultrastrong 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, of course, intermediate cases which
are more complex than these limit cases. Weak field
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 total angular momentum vector . The"averaged" spin
vector is then the projection of the spin onto the direction of :
and for the"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 gJ of the atom 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
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 1S1/2 and 2P1/2 levels into 2
states each and the 2P3/2 level into 4 states. The Landé g-factors for the
three levels are: for
for for .
Note in particular that the size of the energy splitting is different for the
different orbitals, because the gJ 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
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 large 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.
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. It should be noted that in
general, 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 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 and Zeeman interactions is To arrive at the Breit-Rabi formula we
will include the hyperfine structure, which is governed by the quantum number
, where is the spin angular momentum operator of the nucleus. Alternatively,
the derivation could be done with only. The constant is known as the zero field
hyperfine constant and is given in units of Hertz. is the Bohr magneton. and
are the electron and nuclear angular momentum operators. and can be found
via a classical vector coupling model or a more detailed quantum mechanical
calculation to be: As discussed, 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 large that the
Zeeman effect will dominate, and we 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.
To solve this system, we note that at all times, the total angular momentum
projection will be conserved. Furthermore, since between states will
change between only . Therefore, we can define a good basis as:
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 . Using ladder operators and We can rewrite
the Hamiltonian as Now we can determine the matrix elements
of the Hamiltonian: Solving for the eigenvalues of this
matrix, we arrive at the energy shifts: where is the splitting between two
hyperfine sublevels in the absence of magnetic field ,
is referred to as the 'field strength parameter'. 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 .
Applications = Astrophysics =
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 cooling =
The Zeeman effect is utilized in many laser cooling applications such as a
magneto-optical trap and the Zeeman slower.
See also Magneto-optic Kerr effect
Voigt effect Faraday effect
Cotton-Mouton effect Polarization spectroscopy
Zeeman energy Lamb shift
Electron configuration says at subshell p, 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, but here we have 2. l corresponding to fine structure, ml
corresponding to hyperfine structure. References
= Historical = Condon, E. U.; G. H. Shortley. The
Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.
Zeeman, P.. "On the influence of Magnetism on the Nature of the Light
emitted by a Substance". Phil. Mag. 43: 226.
Zeeman, P.. "Doubles and triplets in the spectrum produced by external magnetic
forces". Phil. Mag. 44: 55. doi:10.1080/14786449708621028.
Zeeman, P.. "The Effect of Magnetisation on the Nature of Light Emitted by a
Substance". Nature 55: 347. Bibcode:1897Natur..55..347Z.
doi:10.1038/055347a0. = Modern =
Feynman, Richard P., Leighton, Robert B., Sands, Matthew. The Feynman Lectures
on Physics 3. Addison-Wesley. ISBN 0-201-02115-3.
Forman, Paul. "Alfred Landé and the anomalous Zeeman Effect, 1919-1921".
Historical Studies in the Physical Sciences 2: 153–261.
doi:10.2307/27757307. Griffiths, David J.. Introduction to
Quantum Mechanics. Prentice Hall. ISBN 0-13-805326-X.
Liboff, Richard L.. Introductory Quantum Mechanics. Addison-Wesley. ISBN
0-8053-8714-5. Sobelman, Igor I.. Theory of Atomic
Spectra. Alpha Science. ISBN 1-84265-203-6.
Foot, C. J.. Atomic Physics. ISBN 0-19-850696-1.
External links Zeeman Effect, National MagLab
Zeeman Effect Apparatus Manufacturer