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The Zeeman effect (/ˈzeɪmən/; [ˈzeːmɑn]), 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 (in the dipole approximation), as governed by the selection rules.
Contents
- Nomenclature
- Theoretical presentation
- Weak field Zeeman effect
- Example Lyman alpha transition in hydrogen
- Strong field Paschen Back effect
- Intermediate field for j 12
- Astrophysics
- Laser cooling
- References
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 (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 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 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
where
where
where
where
If the interaction term
Weak field (Zeeman effect)
If the spin-orbit interaction dominates over the effect of the external magnetic field,
and for the (time-)"averaged" orbital vector:
Thus,
Using
and: using
Combining everything and taking
where the quantity in square brackets is the Landé g-factor gJ of the atom (
Taking
Example: Lyman alpha transition in hydrogen
The Lyman alpha transition in hydrogen in the presence of the spin-orbit interaction involves the transitions
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 (
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 (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 large to disrupt the coupling between orbital (
When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume
The above may be read as implying that the LS-coupling is completely broken by the external field. However
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 and Zeeman interactions is
To arrive at the Breit-Rabi formula we will include the hyperfine structure (interaction between the electron's spin and the magnetic moment of the nucleus), which is governed by the quantum number
As discussed, in the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the
To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the
To solve this system, we note that at all times, the total angular momentum projection
We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator
These ladder operators have the property
as long as
Now we can determine the matrix elements of the Hamiltonian:
Solving for the eigenvalues of this matrix, (as can be done by hand, or more easily, with a computer algebra system) we arrive at the energy shifts:
where
Note that index
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.