Nuclear spin -- Mathematics, Electron spin echoes, Mathematical physics
In this paper we derive a general expression describing the evolution of the transverse nuclear-spin magnetization for the Ostroff-Waugh multiple-spin-echo experiment in dipolar solids. Our approach consists of expressing the formula for the magnetization at even echoes in a form resembling an ordinary time-correlation function, and then evaluating this quantity by means of Zwanzig's projection-operator technique. For long times, we show that under certain conditions the echo envelope decays exponentially, in agreement with experiment. A general expression is obtained for the time constant T* associated with the decay. This result may be used to generate an expansion of 1T* in powers of the cycle time tc, but there are experimental indications that this expansion is not legitimate and that more complicated tc dependences can arise. In the case when higher-order correlations decay much more rapidly than lower-order ones, our result reduces to 1T*=At4cτ0c(tc), where A is a quantity related to the sixth moment of the magnetization and τ0c(tc) is a characteristic correlation time associated with decay of the lowest-order correlation function which enters the problem. The tc dependence of T* is then determined by the behavior of τ0c(tc), and is in general more complex than the proportionality between 1T* and t5c found previously. This previous result emerges in the case when τ0c(tc) = tc. Available experimental results suggest that 1T* is in general a nonanalytic function of tc, as indicated by the observed proportionality between 1T* and tc for Teflon and KAsF6. Further experimental results are needed to clarify the nature of this nonanalytic behavior.
C.H. Wang and J.D. Ramshaw, "Decay of Multiple Spin Echoes in Dipolar Solids," Phys. Rev. B 6, 3253 (1972)