# Fermi contact interaction

The Fermi contact interaction is the magnetic interaction between an electron and an atomic nucleus when the electron is inside that nucleus.

The parameter is usually described with the symbol A and the units are usually megahertz. The magnitude of A is given by this relationships

${\displaystyle A=-{\frac {8}{3}}\pi \left\langle {\boldsymbol {\mu }}_{n}\cdot {\boldsymbol {\mu }}_{e}\right\rangle |\Psi (0)|^{2}\qquad {\mbox{(cgs)}}}$

and

${\displaystyle A=-{\frac {2}{3}}\mu _{0}\left\langle {\boldsymbol {\mu }}_{n}\cdot {\boldsymbol {\mu }}_{e}\right\rangle |\Psi (0)|^{2},\qquad {\mbox{(SI)}}}$

where A is the energy of the interaction, μn is the nuclear magnetic moment, μe is the electron magnetic dipole moment, and Ψ(0) is the value of the electron wavefunction at the nucleus.[1]

It has been pointed out that it is an ill-defined problem because the standard formulation assumes that the nucleus has a magnetic dipolar moment, which is not always the case.[2]

## Use in magnetic resonance spectroscopy

Within an atom, only s-orbitals have non-zero electron density at the nucleus, so the contact interaction only occurs for s-electrons. Its major manifestation is in electron paramagnetic resonance and nuclear magnetic resonance spectroscopies, where it is responsible for the appearance of isotropic hyperfine coupling. Roughly, the magnitude of A indicates the extent to which the unpaired spin resides on the nucleus. Thus, knowledge of the A values allows one to map the singly occupied molecular orbital.[3]

## History

The interaction was first derived by Enrico Fermi in 1930.[4] A classical derivation of this term is contained in "Classical Electrodynamics" by J. D. Jackson.[5] In short, the classical energy may be written in terms of the energy of one magnetic dipole moment in the magnetic field B(r) of another dipole. This field acquires a simple expression when the distance r between the two dipoles goes to zero, since

${\displaystyle \int _{S(r)}\mathbf {B} (\mathbf {r} )\,d^{3}\mathbf {r} =-{\frac {2}{3}}\mu _{0}{\boldsymbol {\mu }}.}$

## References

1. ^ Bucher, M. (2000). "The electron inside the nucleus: An almost classical derivation of the isotropic hyperfine interaction". European Journal of Physics. 21 (1): 19. Bibcode:2000EJPh...21...19B. doi:10.1088/0143-0807/21/1/303.
2. ^ Soliverez, C. E. (1980). "The contact hyperfine interaction: An ill-defined problem". Journal of Physics C. 13 (34): L1017. Bibcode:1980JPhC...13.1017S. doi:10.1088/0022-3719/13/34/002.
3. ^ Drago, R. S. (1992). Physical Methods for Chemists (2nd ed.). Saunders College Publishing. ISBN 978-0030751769.
4. ^ Fermi, E. (1930). "Über die magnetischen Momente der Atomkerne". Zeitschrift für Physik. 60 (5–6): 320. Bibcode:1930ZPhy...60..320F. doi:10.1007/BF01339933.
5. ^ Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). Wiley. p. 184. ISBN 978-0471309321.