Magnetization trajectory during application of adiabatic RF pulses is complicated (a) but with the unique and desirable property that beyond a minimum peak amplitude RFmax, magnetization remains in an optimally prepared state (b). Building on previous work  where adiabatic inversion pulses were parameterized using a master equation from linear plots of (RFmaxTp)2 versus Tpbwdth (c), practical considerations for extension to variable flip angle, excitation, and refocusing of spins by single and composite pulses will be presented.
The hyperbolic secant (HS1-8) [2, 3] family of pulses will be used to demonstrate construction and parameterization of calculated pulses. This class of pulse may be parameterized by the linear adiabatic master equation with minor modification of the original concept introduced by M.R. Bendall. For example, in the case of variable flip angle by a single pulse, recognising that percent inversion = 50(1+ ι0) is equivalent to θ = 90(1 + ι0) for -1 ≤ ι0 ≤ 1, three regions may be identified. These, defined as low, linear and inverted flip regions, denoted θlow, θlin and θinv, respectively also obey the master equation. Separable expressions can be derived and included in pulse sequences allowing control of parameters such as duration and power. Our motivation for ‘taming’ these pulses can be demonstrated by inspection of figure (d). The figure shows excitation profiles of a HS1 shaped pulse applied with RFmax = 0.25, 0.5, 0.75 and 1.0 kHz and degree of excitation from 6, 24, 49 and 78 degrees measured as the projection of magnetization along the z-axis of the rotating frame of reference. Broad bandwidths may be covered with a flat response over selected regions. These parameterized pulses will have immense impact in emerging techniques such as spectral-spatial imaging and spectroscopy, as well as pulsed micro-wave spectroscopy.