 NMR pulse error

An NMR pulse error is a difference between an intended spin rotation and the rotation actually produced by a radio frequency pulse. A pulse can have the wrong NMR flip angle, the wrong rotation-axis phase, or a tilted effective axis because the irradiation is off resonance. These discrepancies differ from changes occurring during free evolution or spin relaxation.

For an ideal resonant rectangular pulse, the magnitude of its NMR flip angle is $\theta =|\gamma |B_1{t}_p$, where $\gamma$ is the angular gyromagnetic ratio, $B_1$ is the co-rotating transverse field amplitude and $t_p$ is pulse duration. A fixed amplitude miscalibration can therefore produce a reproducible angle error. Using the co-rotating amplitude explicitly avoids an ambiguous factor of two for a linearly oscillating field.

An original pulse-only calculation illustrates accumulation. Start with a unit magnetization vector along $z$, and apply $N$ instantaneous rotations about the same $x$ axis, each with angle $\pi +\epsilon$. Ignore free evolution, relaxation and every other error. Their product is a rotation by $N\pi +N\epsilon$, givingFor even $N$, ideal pulses return $M_z$ to one. With an invented systematic error of $1^{\circ }$ per pulse, ten pulses give $\cos 10^{\circ }\approx 0.985$, whereas one hundred give $\cos 100^{\circ }\approx -0.174$. Small systematic errors can accumulate coherently and even reverse the component being measured. These are not measured decay data, nor does $N\epsilon$ describe arbitrary independently varying errors.

Alternating rotation-axis phases can control accumulation. A 1990 study experimentally compared compensated pulse trains designed to protect all three magnetization components; its applications included REDOR.^[ref] The XY8 pattern is one important example. An ideal product equal to the identity verifies its nominal net rotation, but a robustness claim requires calculating imperfect pulses in a stated error model.

A useful teaching experiment would compare phase patterns while varying a calibrated pulse-amplitude offset and keeping preparation, readout and nominal evolution window controlled. The REDOR retrospective supplies historical motivation: XY8 reduced off-resonance amplitude and phase distortions in long pulse trains.^[ref]