The two-pulse spin echo revisited

Abstract

Abstract A formulation of the two-pulse spin echo problem appropriate to inhomogeneously broadened NMR lines in solids is presented and discussed. The classical formulation is discussed in the text, with the (equivalent) quantum case outlined in an appendix. The in-phase and quadrature components of the echo signal are given by integrals over the lineshape functiong(w);these integrals can be evaluated numerically in a straightforward fashion. An exact expression is given for the area under the echo waveform, relating it to the areas under the two r.f. pulse field profiles H1(t). The echo area is directly proportional to the lineshape amplitude at the applied r.f. frequencyg(wrr),Thus, one can in principle resolve structure of arbitrary complexity by recording the area under the echo. Numerical calculations are presented to show the dependence of the echo waveforms on pulse shape and length, and on the r.f. frequency relative to the peaks of symmetric and asymmetric NMR lines. The exact phase symmetries of spin echoes are shown to be useful in eliminating unwanted electrical transients from experimental echo signals. A practical example is given.