NMR nutation frequency (${\omega }_1$)

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 isThe 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.^[ref]

For a resonant constant-axis rectangular pulse, the NMR flip angle is $\beta ={\omega }_1{t}_p$. An original example assigns $f_1=25\mathrm{kHz}$, giving ${\omega }_1=2\pi \times 25000rad/s$. 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 ${\omega }_1=\pi /(20\mu \mathrm{s})$ and $f_1=1/(40\mu \mathrm{s})=25\mathrm{kHz}$. Describing this as a25kHz RF field strength is frequency shorthand; the underlying magnetic field amplitude still has units of tesla.^[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.