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Magnetic interaction between an electron and a nucleus
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
=
−
8
3
π
⟨
μ
n
⋅
μ
e
⟩
|
Ψ
(
0
)
|
2
(cgs)
{\displaystyle A=-{\frac {8}{3}}\pi \left\langle {\boldsymbol {\mu }}_{n}\cdot {\boldsymbol {\mu }}_{e}\right\rangle |\Psi (0)|^{2}\qquad {\mbox{(cgs)}}}
and
A
=
−
2
3
μ
0
⟨
μ
n
⋅
μ
e
⟩
|
Ψ
(
0
)
|
2
,
(SI)
{\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, 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
⟨
⋯
⟩
{\textstyle \left\langle \cdots \right\rangle }
denotes the quantum mechanical spin coupling.
[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]
Simplified view of the Fermi contact interaction in the terms of nuclear (green arrow) and electron
spins (blue arrow). 1 : in
H2 ,
1 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 1 H. 1 H nuclei are antiparallel and 1 JHH has a positive value.
[3] 2 : 1 H nuclei are parallel. This form is unstable (has higher energy E) than the form 1.
[4] 3 : vicinal 1 H J-coupling via
12 C or
13 C nuclei. Same as before, but electron spins on
p-orbitals are parallel due to
Hund's 1. rule . 1 H nuclei are antiparallel and 3 JHH has a positive value.
[3]
Use in magnetic resonance spectroscopy
1 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.
[5]
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 .
[6]
History
The interaction was first derived by
Enrico Fermi in 1930.
[7] A classical derivation of this term is contained in "Classical Electrodynamics" by
J. D. Jackson .
[8] 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
∫
S
(
r
)
B
(
r
)
d
3
r
=
−
2
3
μ
0
μ
.
{\displaystyle \int _{S(r)}\mathbf {B} (\mathbf {r} )\,d^{3}\mathbf {r} =-{\frac {2}{3}}\mu _{0}{\boldsymbol {\mu }}.}
References
^
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 .
^
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 .
^
a
b M, Balcı (2005). Basic ¹H- and ¹³C-NMR spectroscopy (1st ed.). Elsevier. pp. 103–105.
ISBN
9780444518118 .
^ Macomber, R. S. (1998).
A complete introduction to modern NMR spectroscopy . Wiley. pp.
135 .
ISBN
9780471157366 .
^ Köhler, F. H., "Paramagnetic Complexes in Solution: The NMR Approach," in eMagRes, 2007, John Wiley.
doi :
10.1002/9780470034590.emrstm1229
^
Drago, R. S. (1992). Physical Methods for Chemists (2nd ed.).
Saunders College Publishing .
ISBN
978-0030751769 .
^
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 .
^
Jackson, J. D. (1998).
Classical Electrodynamics (3rd ed.).
Wiley . p.
184 .
ISBN
978-0471309321 .