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{{Short description|Magnetic interaction between an electron and a nucleus}}
{{distinguish|Fermi's interaction}}
The '''Fermi contact interaction''' is the [[magnetism|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 [[electron paramagnetic resonance#Hyperfine coupling|hyperfine coupling]].

This<!-- seemingly impossible feat--> 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''<sub>''n''</sub> is the [[nuclear magnetic moment]], ''μ''<sub>''e''</sub> 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>
{{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
 }}</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>
{{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
}}</ref>

[[File:J-coupling Fermi contact mechanism.svg|center|thumb|450px|Simplified view of the Fermi contact interaction in the terms of nuclear (green arrow) and electron [[Spin (physics)|spins]] (blue arrow). '''1''': in [[Hydrogen|H<sub>2</sub>]], [[Isotopes of hydrogen|<sup>1</sup>H]] 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 <sup>1</sup>H. <sup>1</sup>H nuclei are antiparallel and <sup>1</sup>J<sub>HH</sub> has a positive value.<ref name=":0">{{Cite book|title=Basic ¹H- and ¹³C-NMR spectroscopy|last=M|first=Balcı|date=2005|publisher=Elsevier|isbn=9780444518118|edition=1st|pages=103–105}}</ref> '''2''': <sup>1</sup>H nuclei are parallel. This form is unstable (has higher energy E) than the form 1.<ref>{{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]}}</ref> '''3''': vicinal <sup>1</sup>H J-coupling via [[Carbon-12|<sup>12</sup>C]] or [[13C|<sup>13</sup>C]] nuclei. Same as before, but electron spins on [[p-orbitals]] are parallel due to [[Hund's rule of maximum multiplicity|Hund's 1. rule]]. <sup>1</sup>H nuclei are antiparallel and <sup>3</sup>J<sub>HH</sub> has a positive value.<ref name=":0"/>]]

==Use in magnetic resonance spectroscopy==
{{main|Paramagnetic nuclear magnetic resonance spectroscopy}}
[[File:NMRMeCp2Ni.png|thumb|right|284px|<sup>1</sup>H NMR spectrum of 1,1'-dimethyl[[nickelocene]], 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>
{{cite book
 |last=Drago |first=R. S.
 |year=1992
 |title=Physical Methods for Chemists
 |edition=2nd
 |publisher=[[Saunders College Publishing]]
 |pages=
 |isbn=978-0030751769
}}</ref>

==History==
The interaction was first derived by [[Enrico Fermi]] in 1930.<ref>
{{cite journal
 |last=Fermi |first=E.
 |year=1930
 |title=Über die magnetischen Momente der Atomkerne
 |journal=[[Zeitschrift für Physik]]
 |volume=60 |issue=5–6 |page=320
 |arxiv=
 |bibcode= 1930ZPhy...60..320F
 |doi= 10.1007/BF01339933
|s2cid=122962691
 }}</ref> A classical derivation of this term is contained in "Classical Electrodynamics" by [[John David Jackson (physicist)|J. D. Jackson]].<ref>
{{cite book
 |last=Jackson |first=J. D.
 |year=1998
 |title=Classical Electrodynamics
 |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |page=[https://archive.org/details/classicalelectro00jack_449/page/n183 184] |edition=3rd
 |publisher=[[John Wiley & Sons|Wiley]]
 |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>
{{clear}}

==References==
{{Reflist}}

[[Category:Magnetostatics]]
[[Category:Magnetism]]
[[Category:Electric and magnetic fields in matter]]