= NMR dipolar coupling
{c}
{title2=$d_{IS}\propto r^{-3}$}
<NMR dipolar coupling> is the interaction between the <magnetic moments> of <atomic nuclei> through their magnetic fields. It depends on internuclear separation and orientation, allowing <nuclear magnetic resonance> measurements to constrain spatial structure. It differs from <NMR scalar coupling>, whose mechanism involves intervening electrons.
For two point <magnetic dipoles> separated by $\mathbf r=r\widehat{\mathbf r}$, the interaction energy is
$$
H_{\mathrm{dd}}=\frac{\mu_0}{4\pi r^3}\left[\boldsymbol\mu_I\cdot\boldsymbol\mu_S-3(\boldsymbol\mu_I\cdot\widehat{\mathbf r})(\boldsymbol\mu_S\cdot\widehat{\mathbf r})\right].
$$
Here $\mu_0$ is the <vacuum permeability>. Write $\boldsymbol\mu_I=\gamma_I\hbar\mathbf I$ and $\boldsymbol\mu_S=\gamma_S\hbar\mathbf S$, where $\mathbf I,\mathbf S$ are dimensionless <spin angular momentum> operators and each $\gamma$ is an angular <gyromagnetic ratio> in radians per second per tesla. A signed angular-frequency coupling scale is then
$$
d_{IS}=\frac{\mu_0\gamma_I\gamma_S\hbar}{4\pi r^3}.
$$
This convention puts the geometrical factor separately; some texts absorb an extra minus sign or convert to cycles per second. Comparing numerical coupling constants requires matching those conventions.
For an unlike-spin pair with well-separated resonance frequencies in a strong field, neglecting rapidly oscillating terms gives the ideal secular <Quantum Hamiltonian>
$$
\frac{H_{\mathrm{sec}}}{\hbar}=d_{IS}(1-3\cos^2\theta)I_zS_z,
$$
where $\theta$ is the angle between the internuclear vector and the field. The angular factor vanishes at $\theta=\arccos(1/\sqrt3)$. <Magic angle spinning> exploits related orientation averaging, while <dipolar recoupling> deliberately restores sensitivity to interactions suppressed by spinning.
An original scaling comparison is especially useful: doubling $r$ reduces $|d_{IS}|$ by a factor of eight. Thus a coupling strength can constrain distance, but orientation averaging, motion and additional spins must be modeled before turning a measured signal change into a unique distance. <REDOR> supplies an experimental route to that information. Its early <carbon-13>–<nitrogen-15> <alanine> measurements compared a full rotational echo with a dephased echo under timed pulses.<Development of REDOR rotational-echo double-resonance NMR by Jacob Schaefer (2011)>[p.422, Figure1 and discussion: alanine echo comparison, isolated-pair calculation and powder average]{ref}
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