= NMR pulse calibration
{c}
<NMR pulse calibration> determines pulse settings that produce intended spin rotations under specified experimental conditions. For a resonant rectangular pulse, varying duration at fixed amplitude changes the <NMR flip angle> as $\beta=\omega_1t_p$, with $\omega_1$ the angular <NMR nutation frequency>. A pulse-acquire measurement follows how the resulting transverse <magnetization> changes.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter3, section3.7, pp.3-11–3-12: pulse calibration and duration scan]{ref}
Assume that each acquisition begins from the same $+z$ <magnetization>, relaxation during the pulse is negligible, and the receiver phase reads the signed $y$ component after a positive-$x$ rotation. The ideal response is
$$
M_y(t_p)=-M_0\sin(\omega_1t_p),\qquad M_z(t_p)=M_0\cos(\omega_1t_p).
$$
A duration scan can locate an inversion null and give a nominal $90^{\circ}$ duration by halving the corresponding $180^{\circ}$ duration at unchanged amplitude.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Figures3.13 and3.18: vector projections and inversion-null calibration]{ref}
For an original ideal data set with $f_1=25\,\mathrm{kHz}$,
$$
\begin{array}{c|rrrrrr}
t_p\;(\mathrm{\mu s})&0&5&10&15&20&25\\
M_y/M_0&0&-1/\sqrt2&-1&-1/\sqrt2&0&1/\sqrt2
\end{array}
$$
The10microsecond pulse gives a transverse extremum, while the20microsecond null corresponds to inverted longitudinal <magnetization>. The null at zero duration is a different point of the same curve, so a single zero signal cannot identify the rotation. These values are constructed from the model, not recorded calibration data.
The phase convention matters: displaying only magnitude would hide the sign change beyond inversion. Reproducible initial conditions also matter; a sequence of differently timed pulses without consistent recovery would not test this one-pulse model. Sample-dependent RF loading and a distribution of effective field amplitudes can alter the measured response.
Calibration establishes a chosen pulse response in the tested setting. It does not by itself establish every offset response, cancel all <NMR pulse errors>, or validate a complete multi-pulse experiment. Recording the calibration conditions and the quantity being fitted makes later pulse-sequence comparisons interpretable.
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