Hyperfine structure

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 magnitude smaller than those of a fine-structure shift, results from the interactions of the nucleus with internally generated electric and magnetic fields.
File:Fine hyperfine levels.svg|thumb|right|Schematic illustration of fine and hyperfine structure in a neutral hydrogen atom

History

The first theory of atomic hyperfine structure was given in 1930 by Enrico Fermi 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, indium, antimony, and mercury.

Theory

The theory of hyperfine structure comes directly from electromagnetism, consisting of the interaction of the nuclear multipole moments 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 have a magnetic dipole moment, given by:
where is the g-factor and is the nuclear magneton.
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:
In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital and spin angular momentum of the electrons:

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:
where −r gives the position of the nucleus relative to the electron. Written in terms of the Bohr magneton, this gives:
Recognizing that m_ev is the electron momentum, p, and that is the orbital angular momentum in units of ħ, ℓ, we can write:
For a many-electron atom this expression is generally written in terms of the total orbital angular momentum,, by summing over the electrons and using the projection operator,, where. For states with a well defined projection of the orbital angular momentum,, we can write, giving:

Electron spin magnetic field

The electron spin angular momentum is a fundamentally different property that is intrinsic to the particle and therefore does not depend on the motion of the electron. Nonetheless, it is angular momentum and any angular momentum associated with a charged particle results in a magnetic dipole moment, which is the source of a magnetic field. An electron with spin angular momentum, s, has a magnetic moment, μ_s, given by:
where g_s is the electron spin g-factor and the negative sign is because the electron is negatively charged.
The magnetic field of a point dipole moment, μ_s, is given by:

Electron total magnetic field and contribution

The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by:
The first term gives the energy of the nuclear dipole in the field due to the electronic orbital angular momentum. The second term gives the energy of the "finite distance" interaction of the nuclear dipole with the field due to the electron spin magnetic moments. The final term, often known as the Fermi contact term relates to the direct interaction of the nuclear dipole with the spin dipoles and is only non-zero for states with a finite electron spin density at the position of the nucleus. It has been argued that one may get a different expression when taking into account the detailed nuclear magnetic moment distribution. The inclusion of the delta function is an admission that the singularity in the magnetic induction B owing to a magnetic dipole moment at a point is not integrable. It is B which mediates the interaction between the Pauli spinors in non-relativistic quantum mechanics. Fermi avoided the difficulty by working with the relativistic Dirac wave equation, according to which the mediating field for the Dirac spinors is the four-vector potential. The component V is the Coulomb potential. The component A is the three-vector magnetic potential, which for the point dipole is integrable.
For states with this can be expressed in the form
where:
If hyperfine structure is small compared with the fine structure, I and J are good quantum numbers and matrix elements of can be approximated as diagonal in I and J. In this case, we can project N onto J and we have:
This is commonly written as
with being the hyperfine-structure constant which is determined by experiment. Since , this gives an energy of:
In this case the hyperfine interaction satisfies the Landé interval rule.

Electric quadrupole

Atomic nuclei with spin have an electric quadrupole moment. In the general case this is represented by a rank-2 tensor,, with components given by:
where i and j are the tensor indices running from 1 to 3, x_i and x_j are the spatial variables x, y and z depending on the values of i and j respectively, δ_ij is the Kronecker delta and ρ is the charge density. Being a 3-dimensional rank-2 tensor, the quadrupole moment has 3² = 9 components. From the definition of the components it is clear that the quadrupole tensor is a symmetric matrix that is also traceless, giving only five components in the irreducible representation. Expressed using the notation of irreducible spherical tensors we have:
The energy associated with an electric quadrupole moment in an electric field depends not on the field strength, but on the electric field gradient, confusingly labelled, another rank-2 tensor given by the outer product of the del operator with the electric field vector:
with components given by:
Again it is clear this is a symmetric matrix and, because the source of the electric field at the nucleus is a charge distribution entirely outside the nucleus, this can be expressed as a 5-component spherical tensor,, with:
where:
The quadrupolar term in the Hamiltonian is thus given by:
A typical atomic nucleus closely approximates cylindrical symmetry and therefore all off-diagonal elements are close to zero. For this reason the nuclear electric quadrupole moment is often represented by.

Molecular hyperfine structure

The molecular hyperfine Hamiltonian includes those terms already derived for the atomic case with a magnetic dipole term for each nucleus with and an electric quadrupole term for each nucleus with. The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley, and the resulting hyperfine parameters are often called the Frosch and Foley parameters.
In addition to the effects described above, there are a number of effects specific to the molecular case.

Direct nuclear spin–spin

Each nucleus with has a non-zero magnetic moment that is both the source of a magnetic field and has an associated energy due to the presence of the combined field of all of the other nuclear magnetic moments. A summation over each magnetic moment dotted with the field due to each other magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian,.
where α and α are indices representing the nucleus contributing to the energy and the nucleus that is the source of the field respectively. Substituting in the expressions for the dipole moment in terms of the nuclear angular momentum and the magnetic field of a dipole, both given above, we have

Nuclear spin–rotation

The nuclear magnetic moments in a molecule exist in a magnetic field due to the angular momentum, T, associated with the bulk rotation of the molecule, thus

Small molecule hyperfine structure

A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions of hydrogen cyanide in its ground vibrational state. Here, the electric quadrupole interaction is due to the ¹⁴N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen, ¹⁴N, and hydrogen, ¹H, and a hydrogen spin-rotation interaction due to the ¹H-nucleus. These contributing interactions to the hyperfine structure in the molecule are listed here in descending order of influence. Sub-doppler techniques have been used to discern the hyperfine structure in HCN rotational transitions.
The dipole selection rules for HCN hyperfine structure transitions are,, where is the rotational quantum number and is the total rotational quantum number inclusive of nuclear spin, respectively. The lowest transition splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components carries only 0.6% of the rotational transition intensity in the case of. This contribution drops for increasing J. So, from upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components together with two widely spaced components; one on the low frequency side and one on the high frequency side relative to the central hyperfine triplet. Each of these outliers carry ~ the intensity of the entire transition. For consecutively higher- transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.