Fermi contact interaction

{{Short description|Magnetic interaction between an electron and a nucleus}}

{{distinguish|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

:$A\; =\; -\backslash frac\{8\}\{3\}\; \backslash pi\; \backslash left\; \backslash langle\; \backslash boldsymbol\{\backslash mu\}\_n\; \backslash cdot\; \backslash boldsymbol\{\backslash mu\}\_e\; \backslash right\; \backslash rangle\; |\backslash Psi\; (0)|^2\backslash qquad\; \backslash mbox\{(cgs)\}$

and

:$A\; =\; -\backslash frac\{2\}\{3\}\; \backslash mu\_0\; \backslash left\; \backslash langle\; \backslash boldsymbol\{\backslash mu\}\_n\; \backslash cdot\; \backslash boldsymbol\{\backslash mu\}\_e\; \backslash right\; \backslash rangle\; |\backslash Psi(0)|^2,\; \backslash qquad\; \backslash mbox\{(SI)\}$

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 $\backslash left\backslash langle\; \backslash cdots\; \backslash right\backslash rangle$ denotes the quantum mechanical spin coupling.

{{cite journal

|last=Bucher |first=M.

|year=2000

|title=The electron inside the nucleus: An almost classical derivation of the isotropic hyperfine interaction

|journal=European Journal of Physics

|volume=21 |issue=1 |page=19

|arxiv=

|bibcode=2000EJPh...21...19B

|doi=10.1088/0143-0807/21/1/303

|s2cid=250871770

|doi-access=free

}}

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.

{{cite journal

|last=Soliverez |first=C. E.

|year=1980

|title=The contact hyperfine interaction: An ill-defined problem

|journal=Journal of Physics C

|volume=13 |issue=34 |page=L1017

|arxiv=

|bibcode=1980JPhC...13.1017S

|doi=10.1088/0022-3719/13/34/002

}}

File:J-coupling Fermi contact mechanism.svg (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.{{Cite book|title=Basic ¹H- and ¹³C-NMR spectroscopy|last=M|first=Balcı|date=2005|publisher=Elsevier|isbn=9780444518118|edition=1st|pages=103–105}} 2: ¹H nuclei are parallel. This form is unstable (has higher energy E) than the form 1.{{Cite book|url=https://archive.org/details/ACompleteIntroductionToModernNMRSpectroscopyByRogerSMacomber|title=A complete introduction to modern NMR spectroscopy|last=Macomber|first=R. S.|date=1998|publisher=Wiley|isbn=9780471157366|pages=[https://archive.org/details/ACompleteIntroductionToModernNMRSpectroscopyByRogerSMacomber/page/n148 135]}} 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.]]