NMR pulse error
= NMR pulse error
{c}
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 $\theta=|\gamma|B_1t_p$, 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 $\pi+\varepsilon$. Ignore free evolution, relaxation and every other error. Their product is a rotation by $N\pi+N\varepsilon$, giving
$$
M_z=(-1)^N\cos(N\varepsilon).
$$
For even $N$, ideal pulses return $M_z$ to one. With an invented systematic error of $1^{\circ}$ per pulse, ten pulses give $\cos10^{\circ}\approx0.985$, whereas one hundred give $\cos100^{\circ}\approx-0.174$. Small systematic errors can accumulate coherently and even reverse the component being measured. These are not measured decay data, nor does $N\varepsilon$ 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>.<New, compensated Carr-Purcell sequences by Terry Gullion and others>[Original publisher Abstract: x/y phase alternation, cumulative error compensation and experimental comparison]{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.<Development of REDOR rotational-echo double-resonance NMR by Jacob Schaefer (2011)>[p.422, XY8 phase-routing discussion]{ref}