 NMR flip angle

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,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.^[ref]

An original numerical example uses $f_1={\omega }_1/(2\pi )=25\mathrm{kHz}$. The durations for $90^{\circ }$ and $180^{\circ }$ rotations areSubstitution of the angular rate $2\pi \times 25000rad/s$ 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.^[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.^[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.