 NMR dipolar coupling

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 $\mathbf{r}=r\hat{\mathbf{r}}$, the interaction energy isHere ${\mu }_0$ is the vacuum permeability. Write ${\mathbf{\mu }}_I={\gamma }_I\hslash \mathbf{I}$ and ${\mathbf{\mu }}_S={\gamma }_S\hslash \mathbf{S}$, 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 thenThis 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 Hamiltonianwhere $\theta$ is the angle between the internuclear vector and the field. The angular factor vanishes at $\theta =\arccos (1/\sqrt{3})$. 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.^[ref]