Decay of Multiple Spin Echoes in Dipolar Solids

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 $\frac{1}{T^{*}}$ in powers of the cycle time $t_c$, but there are experimental indications that this expansion is not legitimate and that more complicated $t_c$ dependences can arise. In the case when higher-order correlations decay much more rapidly than lower-order ones, our result reduces to $\frac{1}{T^{*}}=A{t}_c^4{\tau }_c^0\left(t_c\right)$, where $A$ is a quantity related to the sixth moment of the magnetization and ${\tau }_c^0\left(t_c\right)$ is a characteristic correlation time associated with decay of the lowest-order correlation function which enters the problem. The $t_c$ dependence of $T^{*}$ is then determined by the behavior of ${\tau }_c^0\left(t_c\right)$, and is in general more complex than the proportionality between $\frac{1}{T^{*}}$ and $t_c^5$ found previously. This previous result emerges in the case when ${\tau }_c^0\left(t_c\right)=t_c$. Available experimental results suggest that $\frac{1}{T^{*}}$ is in general a nonanalytic function of $t_c$, as indicated by the observed proportionality between $\frac{1}{T^{*}}$ and $t_c$ for Teflon and KAs${\mathrm{F}}_6$. Further experimental results are needed to clarify the nature of this nonanalytic behavior.