Magnetic dipole–dipole interaction

Consider two classical point-like magnetic dipole moment m₁ and m₂ in Cartesian coordinate system. In our model we will consider a dipole interaction in the follow way. Due to the long range interaction of the magnetic field produced by the dipole, one dipole can interact with the field produced by the other such that the potential energy landscape is given by ${\displaystyle -\mathbf {m} _{i}\cdot {\mathbf {B} }_{j}}$, where the indices label the dipole moments. The interaction Hamiltonian of this system calculated to be of the following form:

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

where μ₀ is the magnetic constant, ${\displaystyle {\hat {\mathbf {r} }}}$ is a unit vector parallel center line passing through both 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.

This previous Hamiltonian can be simply adapted for a quantum mechanical picture. Consider two spin-1/2 particles S₁ and S₂ with gyromagnetic ratios γ₁ and γ₂. The Hamiltonian can now be written as

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

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

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

The 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^{[citation needed]}

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

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.

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