Magnetic dipole–dipole interaction

Magnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles.

Suppose m₁ and m₂ are two magnetic moments in space. The potential energy H of the interaction is then given by:

${\displaystyle H=-{\frac {\mu _{0}}{4\pi |{\mathbf {r}}|^{3}}}\left(3({\mathbf {m}}_{1}\cdot {\hat {\mathbf {r}}})({\mathbf {m}}_{2}\cdot {\hat {\mathbf {r}}}){\dfrac {1}{|{\mathbf {r}}|^{2}}}-{\mathbf {m}}_{1}\cdot {\mathbf {m}}_{2}\right)}$

where μ₀ is the magnetic constant, r̂ is a unit vector parallel to the line joining the centers of the two dipoles, and |r| is the distance between the centers of m₁ and m₂. Alternatively, suppose γ₁ and γ₂ are gyromagnetic ratios of two particles with spin quanta S₁ and S₂. (Each such quantum is some integral multiple of 1/2.) Then:

${\displaystyle H=-{\frac {\mu _{0}\gamma _{1}\gamma _{2}\hbar ^{2}}{4\pi |{\mathbf {r}}|^{3}}}\left(3({\mathbf {S}}_{1}\cdot {\hat {\mathbf {r}}})({\mathbf {S}}_{2}\cdot {\hat {\mathbf {r}}})-{\mathbf {S}}_{1}\cdot {\mathbf {S}}_{2}\right)}$

where r̂ is a unit vector in the direction of the line joining the two spins, and |r| is the distance between them.

The force F arising from the interaction between m₁ and m₂ is given by:

${\mathbf F}={\frac {3\mu _{0}}{4\pi |{\mathbf r}|^{4}}}(({\hat {\mathbf r}}\times {\mathbf m}_{1})\times {\mathbf m}_{2}+({\hat {\mathbf r}}\times {\mathbf m}_{2})\times {\mathbf m}_{1}-2{\hat {\mathbf r}}({\mathbf m}_{1}\cdot {\mathbf m}_{2})+5{\hat {\mathbf r}}(({\hat {\mathbf r}}\times {\mathbf m}_{1})\cdot ({\hat {\mathbf r}}\times {\mathbf m}_{2})))$

Dipolar coupling and NMR spectroscopy

The direct dipole-dipole coupling is very useful for molecular structural studies, since it depends only on known physical constants and the inverse cube of internuclear distance. Estimation of this coupling provides a direct spectroscopic route to the distance between nuclei and hence the geometrical form of the molecule, or additionally also on intermolecular distances in the solid state leading to NMR crystallography notably in amorphous materials.

For example, in water, NMR spectra of hydrogen atoms of water molecules are narrow lines because dipole coupling is averaged due to chaotic molecular motion.^[1] In solids, where water molecules are fixed in their positions and do not participate in the diffusion mobility, the corresponding NMR spectra have the form of the Pake doublet. In solids with vacant positions, dipole coupling is averaged partially due to water diffusion which proceeds according to the symmetry of the solids and the probability distribution of molecules between the vacancies.^[2]

Although internuclear magnetic dipole couplings contain a great deal of structural information, in isotropic solution, they average to zero as a result of diffusion. However, their effect on nuclear spin relaxation results in measurable nuclear Overhauser effects (NOEs).

The residual dipolar coupling (RDC) occurs if the molecules in solution exhibit a partial alignment leading to an incomplete averaging of spatially anisotropic magnetic interactions i.e. dipolar couplings. RDC measurement provides information on the global folding of the protein-long distance structural information. It also provides information about "slow" dynamics in molecules

Relevance to current research

While the theory of magnetic dipole-dipole interactions has deep roots, it is not a dead subject by any means. For instance, the dipole-dipole interaction is critical to the understanding of magnetic dipoles in optical lattices.^[3]

See also

• J-coupling
• Magic angle
• Residual dipolar coupling
• Nuclear Overhauser effect
• Magnetic moment

References

• Malcolm H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance. ISBN 0-471-48922-0.

1. Abraham, A. (1961) The Principles of Nuclear Magnetism. Oxford University Press, Oxford.
2. Gabuda, S.P.; Lundin, A.G.(1969) Diffusion of Water Molecules in Hydrates and NMR Spectra. JETP, 28 (3), 555. http://www.jetp.ac.ru/cgi-bin/dn/e_028_03_0555.pdf
3. Wall, M. L.; Carr, L. D. (2013). "Dipole-dipole interactions in optical lattices do not follow an inverse cube power law". arXiv:1303.1230 [cond-mat.quant-gas].