### Magnetic dipole-dipole interaction

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

- $H\; =\; -\; \backslash frac\{\; \backslash mu\_0\; \}\; \{4\; \backslash pi\; r\_\{jk\}^3\; \}\; \backslash left(\; 3\; (\backslash bold\{m\}\_j\; \backslash cdot\; \backslash bold\{e\}\_\{jk\})\; (\backslash bold\{m\}\_k\; \backslash cdot\; \backslash bold\{e\}\_\{jk\})\; -\; \backslash bold\{m\}\_j\; \backslash cdot\; \backslash bold\{m\}\_k\; \backslash right)$

where **e**_{jk} is a unit vector parallel to the line joining the centers of the two dipoles. r_{jk} is the distance between two dipoles, **m**_{k} and **m**_{j}.

For two interacting nuclear spins

- $H\; =\; -\; \backslash frac\{\; \backslash mu\_0\; \}\{\; 4\; \backslash pi\; \}\; \backslash frac\{\; \backslash gamma\_j\; \backslash gamma\_k\; \backslash hbar^2\}\{\; r\_\{jk\}^3\; \}\; \backslash left(\; 3\; (\backslash bold\{I\}\_j\; \backslash cdot\; \backslash bold\{e\}\_\{jk\})\; (\backslash bold\{I\}\_k\; \backslash cdot\; \backslash bold\{e\}\_\{jk\})\; -\; \backslash bold\{I\}\_j\; \backslash cdot\; \backslash bold\{I\}\_k\; \backslash right)$

where $\backslash mu\_0$ is the magnetic constant, $\backslash gamma\_j$, $\backslash gamma\_k$ are gyromagnetic ratios of two spins, and r_{jk} is the distance between the two spins.

Force between two magnetic dipoles:

- $$

\vec{F}_{ab}= \frac {3 \mu_0} {4 \pi |r|^4} [ (\hat r \times \vec{m}_a) \times \vec{m}_b + (\hat r \times \vec{m}_b) \times \vec{m}_a - 2 \hat r(\vec{m}_a \cdot \vec{m}_b) + 5 \hat r ((\hat r \times \vec{m}_a) \cdot (\hat r \times \vec{m}_b)) ]

where $\backslash hat\{r\}$ is unit vector pointing from magnetic moment $m\_a$ to $m\_b$, and $|r|$ is the distance between those two magnetic dipole moments.

## 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. 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.