# 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 dipole moments that are far enough apart that they can be treated as point dipoles in calculating their interaction energy. The potential energy H of the interaction is then given by:

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

where μ0 is the magnetic constant, 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. Last term with ${\displaystyle \delta }$-function vanishes everywhere but the origin, and is necessary to ensure that ${\displaystyle \nabla \cdot \mathbf {B} }$ vanishes everywhere. 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 is a unit vector in the direction of the line joining the two spins, and |r| is the distance between them.

Finally, the interaction energy can be expressed as the dot product of the moment of either dipole into the field from the other dipole:

${\displaystyle H=-\mathbf {m} _{1}\cdot {\mathbf {B} }_{2}({\mathbf {r} }_{1})=-\mathbf {m} _{2}\cdot {\mathbf {B} }_{1}({\mathbf {r} }_{2})}$

where B2(r1) is the field that dipole 2 produces at dipole 1, and B1(r2) is the field that dipole 1 produces at dipole 2. It is not the sum of these terms.

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

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

Fourier transform of H can be calculated from the fact that

${\displaystyle {\frac {3(\mathbf {m} _{1}\cdot {\hat {\mathbf {r} }})(\mathbf {m} _{2}\cdot {\hat {\mathbf {r} }})-\mathbf {m} _{1}\cdot \mathbf {m} _{2}}{4\pi |\mathbf {r} |^{3}}}=(\mathbf {m} _{1}\cdot \mathbf {\nabla } )(\mathbf {m} _{2}\cdot \mathbf {\nabla } ){\frac {1}{4\pi |\mathbf {r} |}}}$

and is given by[source?]

${\displaystyle H=-{\mu _{0}}{\frac {(\mathbf {m} _{1}\cdot \mathbf {q} )(\mathbf {m} _{2}\cdot \mathbf {q} )-|\mathbf {q} |^{2}\mathbf {m} _{1}\cdot \mathbf {m} _{2}}{|\mathbf {q} |^{2}}}}$

## 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