Magnetic dipole–dipole interaction

Magnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. Roughly speaking, the magnetic field of a dipole goes as the inverse cube of the distance, and the force of its magnetic field on another dipole goes as the first derivative of the magnetic field. It follows that the dipole-dipole interaction goes as the inverse fourth power of the distance.

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, 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. 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 ${\displaystyle {\hat {\mathbf {r} }}}$ 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:

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

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

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