Counting echoes: Application of a complete reciprocal-space description of NMR spin dynamics

Abstract

A complete reciprocal-space formalism for describing the spatial aspects of nuclear magnetic resonance (NMR) spin dynamics in the presence of hard radiofrequency (RF) pulses and linear-refocusing inhomogeneities is reviewed. The formalism demonstrates how the magnetization in a sample can be decomposed into a linear combination of simple basis functions consisting of helical phase modulations in the transverse plane and sinusoidal amplitude modulations along the principal axis of symmetry. It is shown that plotting the evolution of the spatial Fourier variable for each basis function provides a simple way to compute both the number of echoes resulting from any multipulse experiment and when the echoes will form. The maximum number of echoes possible for a sequence of n hard RF pulses with 90°; flip angles and with arbitrary flip angles, both under the action of a time-invariant linear Iz Hamiltonian, is computed using this formalism. A simple criterion for the delay time necessary between pulses to observe the maximum number of echoes is presented. Experimental results are shown for pulse sequences of up to four pulses. ©1998 John Wiley & Sons, Inc. Concepts Magn Reson 10: 331–341, 1998