Numerical analysis of Solomon echo amplitudes in static solids

Abstract

For the first time a numerical procedure for computing the Solomon echo amplitudes of half-integer quadrupole spins (I=3/2, 5/2, 7/2, and 9/2) has been derived from a detailed analysis of the evolution of the spin system using the density operator formalism. As the first-order quadrupole interaction is taken into account throughout the experiment, consisting in exciting the spin system with two pulses either in phase or in quadrature phase and separated by a delay τ2, the results are valid for any ratio of the quadrupole coupling, ωQ, to the amplitude of the pulses, ωrf. These results are applicable to light nuclei at high magnetic field, for which the second-order quadrupole interaction and the chemical shift anisotropy are negligible. We predict (4I2−1)(2I−1)/16 echoes and their positions in the detection period τ4, (I−1/2)2 of which are allowed echoes, the others being forbidden echoes. All of these echoes are satellite-transition signals, which are superimposed on the free induction decay (FID) of the central transition following the second pulse. This means that τ2 must be short when compared to the duration of the central-transition FID. In other words, during τ2 the magnetic dipole interactions have no time to affect the evolution of the spin system. The behavior of the τ4=τ2 echoes versus the second-pulse flip angle ωrft3 is analyzed: in the hard-pulse excitation condition (ωQ/ωrf≪1), the (±1/2↔±3/2) echoes generated by two pulses in quadrature phase are twice as large as those generated by two in-phase pulses; the echoes of the other transitions have comparable amplitudes. The echo amplitudes in the soft-pulse excitation condition (ωQ/ωrf=1) are also presented.