In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule.

Hyperfine structure contrasts with fine structure, which results from the interaction between the magnetic moments associated with electron spin and the electrons' orbital angular momentum. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the nucleus (or nuclei, in molecules) with internally generated electric and magnetic fields.

History

The first theory of atomic hyperfine structure was given in 1930 by Enrico Fermi ^[1] for an atom containing a single valence electron with an arbitrary angular momentum. The Zeeman splitting of this structure was discussed by S. A. Goudsmit and R. F. Bacher later that year. In 1935, H. Schüler and Theodor Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure of Europium, Cassiopium(older name for Lutetium), Indium, Antimony and Mercury.^[2]

Theory

The theory of hyperfine structure comes directly from electromagnetism, consisting of the interaction of the nuclear multipole moments (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to each nucleus in a molecule. Following this there is a discussion of the additional effects unique to the molecular case.

Atomic hyperfine structure

Magnetic dipole

The dominant term in the hyperfine Hamiltonian is typically the magnetic dipole term. Atomic nuclei with a non-zero nuclear spin ${\displaystyle \mathbf {I} }$ have a magnetic dipole moment, given by:

$${\displaystyle {\boldsymbol {\mu }}_{\text{I}}=g_{\text{I}}\mu _{\text{N}}\mathbf {I} ,}$$

${\displaystyle g_{\text{I}}}$

${\displaystyle \mu _{\text{N}}}$

There is an energy associated with a magnetic dipole moment in the presence of a magnetic field. For a nuclear magnetic dipole moment, μ_I, placed in a magnetic field, B, the relevant term in the Hamiltonian is given by:^[3]

$${\displaystyle {\hat {H}}_{\text{D}}=-{\boldsymbol {\mu }}_{\text{I}}\cdot \mathbf {B} .}$$

In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital (ℓ) and spin (s) angular momentum of the electrons:

$${\displaystyle \mathbf {B} \equiv \mathbf {B} _{\text{el}}=\mathbf {B} _{\text{el}}^{\ell }+\mathbf {B} _{\text{el}}^{s}.}$$

Electron orbital magnetic field

Electron orbital angular momentum results from the motion of the electron about some fixed external point that we shall take to be the location of the nucleus. The magnetic field at the nucleus due to the motion of a single electron, with charge –e at a position r relative to the nucleus, is given by:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }={\frac {\mu _{0}}{4\pi }}{\frac {-e\mathbf {v} \times -\mathbf {r} }{r^{3}}},}$$

where −r gives the position of the nucleus relative to the electron. Written in terms of the Bohr magneton, this gives:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}{\frac {\mathbf {r} \times m_{\text{e}}\mathbf {v} }{\hbar }}.}$$

Recognizing that m_ev is the electron momentum, p, and that r×p/ħ is the orbital angular momentum in units of ħ, ℓ, we can write:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}\mathbf {\ell } .}$$

For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, ${\displaystyle \mathbf {L} }$, by summing over the electrons and using the projection operator, ${\displaystyle \varphi _{i}^{\ell }}$, where ${\textstyle \sum _{i}\mathbf {\ell } _{i}=\sum _{i}\varphi _{i}^{\ell }\mathbf {L} }$. For states with a well defined projection of the orbital angular momentum, L_z, we can write ${\displaystyle \varphi _{i}^{\ell }={\hat {\ell }}_{z_{i}}/L_{z}}$, giving: