Implementation equations for HS n RF pulses

Implementation equations for the family of stretched hyperbolic secant (HS(n)) pulses are derived in the linear adiabatic range for inversion of spins. These master equations provide convenience relations for relating the peak amplitude RF(max) of the pulse to the frequency sweep (bwdth) range of the pulse and its duration T(p). The bandwidth of the pulse can also be related to the effective coverage (bw(eff)) of the pulse to a defined or chosen spectral region. The choice of pulse determined by the use of these derived expressions guarantees uniform inversion to a prescribed efficiency across the selected spectral region. The performance of HS(n) pulses in determining the cut-off region between spectral regions was also examined. It is found that beyond a unique T(p)bwdth product no additional gain may be obtained by extending pulse durations for a chosen bwdth of pulse. An example of practical implementation of the inversion pulses is presented for adiabatic decoupling using HS(7) and HS(8) pulses. It is shown that despite added B(1) inhomogeneity in the form of additional amplifier power to 400% from optimal, these pulses can still yield reproducible decoupled spectra.