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# Hyperfine structure

Hyperfine structure is a small perturbation in the energy levels (or spectrum) of atoms or molecules due to the magnetic dipole-dipole interaction, arrising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron.

 Contents

## Theory

(This section should be expanded and get more professional!)

Thinking classically, the electron moving around the nucleus has a magnetic dipole moment, as it is charged. However, there is also hyperfine splitting for s-shell electrons which do not have an angular momentum, and then it is even stronger, as the electron probability distribution does not vanish in the nucleus.

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

$\frac{m}{m_p} \alpha^4 m c^2$

where

m is the mass of an electron
mp is the mass of a proton
α is the fine structure constant (1/137.036)
c is the speed of light.

This is a much smaller perturbation than the fine structure or Lamb shift.

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

In a more advanced treatment, one also has to take the nuclear magnetic quadrupole moment into account. This is sometimes (?) refered 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. Wolfang Pauli porposed the existence of a small nuclear magnetic momentum 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.

## Applications

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.

### 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.