Definitions

# Hyperfine structure

[hahy-per-fahyn, hahy-]
In atomic physics, hyperfine coupling is the weak magnetic interaction between electrons and nuclei. Hyperfine coupling causes the hyperfine splitting of atomic or molecular energy levels. The totality of energy levels spawned by hyperfine splitting is called the hyperfine structure of the atom's or molecule's spectrum.

## Terminology

The following terminology has evolved to describe atomic and/or molecular spectra:

• The gross structure is due to the energy difference of electronic orbitals with different principal quantum number n.
• The fine structure occurs only for n>0; it is due to the spin-orbit coupling (the energy difference between the electron spin being parallel or antiparallel to the electron's orbital moment).
• The hyperfine structure is due to an unpaired electron interacting with a nucleus having nuclear spin quantum number I $ne$ 0. The electron and nucleus (nuclei) are on the same atom or within the same molecule.
• The superhyperfine structure is due to an unpaired electron interacting with a nucleus having I $ne$ 0. The electron and nucleus (nuclei) are on different atoms or different molecules.
• The spin-spin structure is due to interactions among nuclei having I $ne$ 0. This phenomenon is especially important in NMR spectra

## Theory

In first order, hyperfine coupling is a magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic moment with the magnetic field of the electron.

According to classical thinking, the electron moving around the nucleus has a magnetic dipole moment, because it is charged. The interaction of this magnetic dipole moment with the magnetic moment of the nucleus (due to its spin) leads to hyperfine splitting.

However, due to the electron's spin, there is also hyperfine splitting for s-shell electrons, which have zero orbital angular momentum. In this case, the magnetic dipole interaction is even stronger, as the electron probability density does not vanish inside the nucleus ($~r=0~$).

The amount of correction to the Bohr energy levels due to hyperfine splitting of the hydrogen atom is of the order

$frac\left\{m\right\}\left\{m_\left\{rm p\right\}\right\} alpha^4 m c^2$

where

$~m~$ is the mass of an electron,
$~m_\left\{rm p\right\}~$ is the mass of a proton,
$~alpha~$ is the fine structure constant $~\left(~alpha approx 1/137.036~\right)~$, and
$~c~$ is the speed of light.

For atoms other than hydrogen, the nuclear spin $vec\left\{I\right\}$ and the total electron angular momentum $vec\left\{J\right\} = vec\left\{L\right\} + vec\left\{S\right\}$ get coupled, giving rise to the total angular momentum $vec\left\{F\right\} = vec\left\{J\right\} + vec\left\{I\right\}$. The hyperfine splitting is then

$Delta E_\left\{rm hfs\right\} = - \left(vec\left\{mu\right\}_I cdot vec\left\{B\right\}_J\right) = frac\left\{a\right\}\left\{2\right\} \left[F\left(F+1\right) - I\left(I+1\right) - J\left(J+1\right)\right],$
where
$a = frac\left\{g_I \left(vec\left\{mu\right\}_\left\{rm N\right\} cdot vec\left\{B\right\}_J\right)\right\}\left\{sqrt\left\{J\left(J+1\right)\right\}\right\},$
$vec\left\{mu\right\}_\left\{rm N\right\}$ the magnetic dipole moment of the nucleus, and
$vec\left\{B\right\}_J$ is the atomic magnetic field.

This interaction obeys the Lande interval rule: The energy level is split into $~\left(J+I\right) - |J-I|+ 1~$ energy levels, where $~J~$ denotes the total electron angular momentum and $~I~$ denotes the nuclear spin.

Usually, $~Delta E_\left\{rm hfs\right\}/hbar~$ is of order of GHz; the hyperfine splitting is orders of magnitude smaller perturbation than the fine structure.

In a more advanced treatment, one also has to take the nuclear magnetic quadrupole moment into account. This is sometimes (?) referred to as "hyperfine structure anomaly".

## History

The optical hyperfine structure was already observed in 1881 by Albert Abraham Michelson. It could, however, only be explained in terms of quantum mechanics in the 1920s. Wolfgang Pauli proposed the existence of a small nuclear magnetic moment in 1924.

In 1935, M. Schiiler and T. Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure.

## Measurements

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in electron paramagnetic resonance spectra of free radicals and transition-metal ions.

## Applications

### Astrophysics

As the hyperfine splitting is very small, the transition frequencies usually are not optical, but in the range of radio- or microwave frequencies.

Hyperfine structure gives the 21 cm line observed in HI region in interstellar medium.

Carl Sagan and Frank Drake considered the hyperfine transition of hydrogen to be a sufficiently universal phenomenon so as to be used as a base unit of time and length on the Pioneer plaque and later Voyager Golden Record.

In radio astronomy, heterodyne receivers are widely used in detection of the electromagnetic signals from celestial objects. The separations among various components of a hyperfine structure are usually small enough to fit into the receiver's IF band. Because optical depth varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic intensities. From this we can derive the object's physical parameters.

### Nuclear technology

The AVLIS process uses the hyperfine splitting of between optical transitions in uranium-235 and uranium-238 to selectively photoionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned dye lasers are used as the sources of the necessary exact wavelength radiation.

### Use in defining the SI second and meter

The hyperfine structure transition can be used to make a microwave notch filter with very high stability, repeatability and Q factor, which can thus be used as a basis for very precise atomic clocks. Typically, the hyperfine structure transition frequency of a particular isotope of caesium or rubidium atoms is used as a basis for these clocks.

Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One second is now defined to be exactly 9,192,631,770 cycles of the hyperfine structure transition frequency of caesium-133 atoms.

Since 1983, the meter is defined by declaring the speed of light in a vacuum to be exactly 299,792,458 metres per second. Thus:

The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

### Precision tests of quantum electrodynamics

The hyperfine splitting in hydrogen and in muonium have been used to measure the value of the fine structure constant α. Comparison with measurements of α in other physical systems provides a stringent test of QED.

### Qubit in ion-trap quantum computing

The hyperfine states of a trapped ion are commonly used for storing qubits in ion-trap quantum computing. They have the advantage of having very long lifetimes, experimentally exceeding ~10 min (compared to ~1 s for metastable electronic levels).

The frequency associated with the states' energy separation is in the microwave region, making it possible to drive hyperfine transitions using microwave radiation. However, at present no emitter is available that can be focused to address a particular ion from a sequence. Instead, a pair of laser pulses can be used to drive the transition, by having their frequency difference (detuning) equal to the required transition's frequency. This is essentially a stimulated Raman transition.