= NMR flip angle
{c}
The <NMR flip angle> is the angle of a spin rotation produced by a <radio frequency> pulse under specified resonance and axis conditions. For an ideal resonant rectangular pulse with constant effective transverse axis,
$$
\beta=\omega_1t_p,\qquad |\beta|=|\gamma|B_1t_p.
$$
Here $\omega_1$ is the angular <NMR nutation frequency>, $t_p$ is duration and $B_1$ is the co-rotating field amplitude. The angular <gyromagnetic ratio> is in radians per second per tesla. A linearly oscillating laboratory field and its co-rotating component must not be assigned the same amplitude without checking conventions.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter3, On-resonance pulses, pp.3-8–3-9: flip-angle definition]{ref}
An original numerical example uses $f_1=\omega_1/(2\pi)=25\,\mathrm{kHz}$. The durations for $90^{\circ}$ and $180^{\circ}$ rotations are
$$
t_{90}=\frac1{4f_1}=10\,\mathrm{\mu s},\qquad
t_{180}=\frac1{2f_1}=20\,\mathrm{\mu s}.
$$
Substitution of the angular rate $2\pi\times25000\,\mathrm{rad/s}$ gives $\pi/2$ and $\pi$. Confusing the rate in hertz with the angular rate would change the result by a factor of $2\pi$.
For a vector initially along $+z$, choose the positive-$x$ rotation convention. Its final components are $M_z=M_0\cos\beta$ and $M_y=-M_0\sin\beta$. An ideal $180^{\circ}$ pulse therefore gives no transverse component while leaving the vector inverted along $-z$; a transverse-signal null does not mean that the <magnetization> vanished.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Figures3.12–3.13, pp.3-8–3-9: vector motion and projections]{ref}
<NMR pulse calibration> can vary duration and follow the resulting sinusoidal signal. Finding the inversion null, then halving its duration at unchanged amplitude, supplies a nominal $90^{\circ}$ pulse in the ideal model.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section3.7, pp.3-11–3-12, Figure3.18: calibration through pulse duration]{ref} In the invented25kHz example, a10-percent amplitude increase with the10microsecond duration held fixed produces $99^{\circ}$ and $M_z/M_0\approx-0.156$. This illustrates an <NMR pulse error>, not a measured instrument performance.
The simple pulse-area formula assumes resonance and a fixed axis. Off-resonance or phase-modulated pulses generally require the full effective-field rotation or propagator, rather than interpreting one integrated amplitude as a complete description.
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