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# Magnetic dipole-dipole interaction

Magnetic dipole-dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. The energy of the interaction is as follows:

$\bold{H} = - \frac{ \hbar \mu_0 } {4 \pi r_{ij}^3 } \left( 3 (\bold{m}_j \cdot \bold{e}_{jk}) (\bold{m}_k \cdot \bold{e}_{jk}) - \bold{m}_j \cdot \bold{m}_k \right)$

### Additional recommended knowledge

where ejk is a unit vector parallel to the line joining the centers of the two dipoles. rij is the distance between two dipoles, mk and mj.

For two interacting nuclear spins:

$\bold{H} = - \frac{ \mu_0 }{ 4 \pi } \frac{ \gamma_j \gamma_k }{ r_{ij}^3 } \left( 3 (\bold{I}_j \cdot \bold{e}_{jk}) (\bold{I}_k \cdot \bold{e}_{jk}) - \bold{I}_j \cdot \bold{I}_k \right)$

γj, γk and rij are gyromagnetic ratios of two spins and spin-spin distance respectively.

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

The residual dipolar coupling (RDC) occur 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.

## References

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