Hahn Spin Echoes in Large Static Gradients Following a Series of 90° Pulses

The intensities of the Hahn spin echoes produced by a series of 90° pulses applied to a sample in large static field gradients are calculated. Static gradients are important because they can be very much larger than pulsed field gradients and echoes can be collected without recovery time problems. One application is for imaging using the stray field of a magnet (STRAFI). The high strength of the gradients in the stray field allows the imaging of solids and other systems with very broad lines. In an imaging pulse sequence consisting of pulses with phases 90(°)x-(90(°)y)n, the echoes (calculated in the absence of spin relaxation) have relative intensities of 1:${{3}\over{2}}$:${{3}\over{2}}$:1(1/8):1(1/8), etc., whereas a 90(°)x-(90(°)x)n pulse sequence produces echoes which have relative intensities of 1:½:-½:-(3/8):(3/8), etc. A simple way of calculating these intensities is presented, using an irreducible tensor approach that has been termed the superspin formalism. These numbers are for infinitely short pulses on resonance and without relaxation or diffusion, but the extension to real systems is also discussed. In particular, it is shown that the attenuation down the echo train is due to a varying contribution from both T1 and T2 processes, as well as diffusion. With long T1's and slow diffusion, the decay of the echoes approximates T2.