= NMR nutation frequency
{c}
{title2=$\omega_1$}
The resonant <NMR nutation frequency> is the rate at which a driven spin rotates about the effective transverse <radio frequency> field in the <rotating frame>. With $B_1$ the co-rotating field amplitude and $\gamma$ the angular <gyromagnetic ratio>, its magnitude is
$$
\omega_1=|\gamma|B_1,\qquad f_1=\frac{\omega_1}{2\pi}.
$$
The angular rate $\omega_1$ is in radians per second, while $f_1$ is in hertz. It differs from the resonance carrier frequency set mainly by the static field.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter3, On-resonance pulses, pp.3-8–3-9; section3.7, p.3-12: angular and hertz field-strength conventions]{ref}
For a resonant constant-axis rectangular pulse, the <NMR flip angle> is $\beta=\omega_1t_p$. An original example assigns $f_1=25\,\mathrm{kHz}$, giving $\omega_1=2\pi\times25000\,\mathrm{rad/s}$. A10microsecond pulse then rotates by $\pi/2$. Reducing $f_1$ to12.5kHz and extending the duration to20microseconds gives the same ideal angle. Amplitude and duration enter as a product in this model.
A measured inversion duration supplies a way to infer the rate. If a calibration gives a20microsecond $180^{\circ}$ pulse, then $\omega_1=\pi/(20\,\mathrm{\mu s})$ and $f_1=1/(40\,\mathrm{\mu s})=25\,\mathrm{kHz}$. Describing this as a25kHz RF field strength is frequency shorthand; the underlying magnetic field amplitude still has units of tesla.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section3.7, pp.3-11–3-12: inference from inversion time and frequency-unit shorthand]{ref}
The simple interpretation assumes resonance and a fixed pulse axis. With an offset, the effective field also has a longitudinal component, so the spin rotates about a tilted axis. Increasing a nominal pulse duration then need not reproduce the resonant trajectory. A linearly oscillating laboratory field must also be decomposed before identifying its co-rotating amplitude with $B_1$.
<NMR pulse calibration> connects the rate to an observed response, rather than relying only on a transmitter setting. Spatial field variation and pulse imperfections can give different nuclei different rotations even when one nominal rate is assigned to the experiment.
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