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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.


This is the publisher's final pdf. Article appears in Physical Review B ( and is copyrighted by APS Journals (



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