Fermi contact interaction

Not to be confused with Fermi's interaction.

The Fermi contact interaction is the magnetic interaction between an electron and an atomic nucleus. Its major manifestation is in electron paramagnetic resonance and nuclear magnetic resonance spectroscopies, where it is responsible for the appearance of isotropic hyperfine coupling.

This requires that the electron occupy an s-orbital. The interaction is described with the parameter A, which takes the units megahertz. The magnitude of A is given by this relationships

<math> A = -\frac{8}{3} \pi \left \langle \boldsymbol{\mu}_n \cdot \boldsymbol{\mu}_e \right \rangle |\Psi (0)|^2\qquad \mbox{(cgs)}</math>

and

<math> A = -\frac{2}{3} \mu_0 \left \langle \boldsymbol{\mu}_n \cdot \boldsymbol{\mu}_e \right \rangle |\Psi(0)|^2, \qquad \mbox{(SI)}</math>

where A is the energy of the interaction, μ_n is the nuclear magnetic moment, μ_e is the electron magnetic dipole moment, Ψ(0) is the value of the electron wavefunction at the nucleus, and <math display="inline"> \left\langle \cdots \right\rangle </math> denotes the quantum mechanical spin coupling.<ref> 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. S2CID 250871770.</ref>

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.<ref> 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.</ref>

Simplified view of the Fermi contact interaction in the terms of nuclear (green arrow) and electron spins (blue arrow). 1: in H₂, ¹H spin polarizes electron spin antiparallel. This in turn polarizes the other electron of the σ-bond antiparallel as demanded by Pauli's exclusion principle. Electron polarizes the other ¹H. ¹H nuclei are antiparallel and ¹J_HH has a positive value.<ref name=":0">M, Balcı (2005). Basic ¹H- and ¹³C-NMR spectroscopy (1st ed.). Elsevier. pp. 103–105. ISBN 9780444518118.</ref> 2: ¹H nuclei are parallel. This form is unstable (has higher energy E) than the form 1.<ref>Macomber, R. S. (1998). A complete introduction to modern NMR spectroscopy. Wiley. pp. 135. ISBN 9780471157366.</ref> 3: vicinal ¹H J-coupling via ¹²C or ¹³C nuclei. Same as before, but electron spins on p-orbitals are parallel due to Hund's 1. rule. ¹H nuclei are antiparallel and ³J_HH has a positive value.<ref name=":0"/>

Use in magnetic resonance spectroscopy

Main article: Paramagnetic nuclear magnetic resonance spectroscopy

¹H NMR spectrum of 1,1'-dimethylnickelocene, illustrating the dramatic chemical shifts observed in some paramagnetic compounds. The sharp signals near 0 ppm are from solvent.<ref name=FK>Köhler, F. H., "Paramagnetic Complexes in Solution: The NMR Approach," in eMagRes, 2007, John Wiley. doi:10.1002/9780470034590.emrstm1229</ref>

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.<ref> Drago, R. S. (1992). Physical Methods for Chemists (2nd ed.). Saunders College Publishing. ISBN 978-0030751769.</ref>

History

The interaction was first derived by Enrico Fermi in 1930.<ref> Fermi, E. (1930). "Über die magnetischen Momente der Atomkerne". Zeitschrift für Physik. 60 (5–6): 320. Bibcode:1930ZPhy...60..320F. doi:10.1007/BF01339933. S2CID 122962691.</ref> A classical derivation of this term is contained in "Classical Electrodynamics" by J. D. Jackson.<ref> Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). Wiley. p. 184. ISBN 978-0471309321.</ref> 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

<math> \int_{S(r)} \mathbf{B}(\mathbf{r}) \, d^3\mathbf{r} = -\frac 2 3 \mu_0 \boldsymbol{\mu}. </math>