What Are the Conditions for Exponential Time-Cubed Echo Decays?

Abstract

Diffusion of precessing spins through a constant field gradient is well-known to produce two distinctive features: an exp(−bt3) decay of the echo amplitude in response to two pulses and a much slower decay of the Carr–Purcell echo train. These features will appear whenever the spin frequency is described by a continuous random-walk. The present work shows that this may also occur in the presence of motions with long correlation times τc—continuous Gaussian frequency noise with an exponential autocorrelation has the correct properties over time durations smaller than τc. Thus, time-cubed echo decays will occur in situations other than physical diffusion. The decay rate of the Carr–Purcell echo train is shown to vary with the pulse spacing τ whenever the correlation time τc is long; the slower Carr–Purcell decay compared to the two-pulse echo decay is not unique to diffusion. Simulations are presented that display time-cubed decays. The simulations confirm two important criteria: the echo time must be less than τc and the frequency noise must consist of nearly continuous variations, as opposed to step-like changes. These criteria define the range of physical parameters for which time-cubed decays will be observable.