= Dipolar recoupling
<Dipolar recoupling> uses controlled spin rotations to restore sensitivity to <NMR dipolar coupling> that would otherwise be averaged away, commonly during <magic angle spinning>. Spinning can narrow spectra by averaging an orientation-dependent interaction; recoupling deliberately preserves a selected part of that interaction so it can reveal spatial information.
<REDOR> illustrates the experimental purpose. Rotor-synchronized pulses on a second nuclear species change how the coupling contributes during the rotation cycle. A matched reference echo and a dephased echo then supply a coupling-sensitive comparison. Early <carbon-13>–<nitrogen-15> measurements on labeled <alanine> used isolated-pair calculations and a powder average to choose pulse placement.<Development of REDOR rotational-echo double-resonance NMR by Jacob Schaefer (2011)>[pp.421–422 and Figure1: recoupling motivation, pulse placement and alanine experiment]{ref}
An original commuting toy model shows how a zero average can become nonzero. Let the interaction be
$$
H(t)/\hbar=d\cos(\omega_rt)I_zS_z,\qquad T_r=2\pi/\omega_r,
$$
with dimensionless spin operators and angular-frequency scale $d$. Without control, the integrated coupling over one period vanishes. Ideal instantaneous $\pi$ rotations of the second spin at $T_r/4$ and $3T_r/4$ reverse the sign of $S_z$ between those times. The effective sign $y(t)$ then matches the sign of $\cos(\omega_rt)$, so
$$
\chi=d\int_0^{T_r}y(t)\cos(\omega_rt)\,dt
=d\int_0^{T_r}|\cos(\omega_rt)|\,dt
=\frac{2dT_r}{\pi}.
$$
Here $\chi$ is the coefficient multiplying $I_zS_z$ in the exponent of the interaction propagator $\exp(-i\chi I_zS_z)$. It is not automatically the phase of every detected coherence. The model demonstrates controlled sign reversal; it is not a reconstruction of the full <REDOR> pulse sequence or of all the angular terms in a spinning sample.
Restored sensitivity is not the same as a unique distance measurement. <NMR dipolar coupling> scales as the inverse cube of separation, but motion, additional spins and the pulse sequence affect the measured echo. <NMR pulse errors> can also change that echo, which is why reference matching and phase compensation such as <XY8> matter.<Development of REDOR rotational-echo double-resonance NMR by Jacob Schaefer (2011)>[p.422: long pulse trains and XY8 error control]{ref}
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