MATHEMATICAL-ANALYSIS OF ELECTRIC-QUADRUPOLE AND HETERONUCLEAR MAGNETIC-DIPOLE INTERACTIONS ON SPIN-5/2 HAHN-ECHO AMPLITUDES IN SOLIDS

Spin 5/2 Hahn echoes, generated by two in-phase pulses separated by a delay tau(2) greater than or equal to twice the duration of the FID, T-FID, are derived by taking into account the first-order quadrupole interaction throughout the experiment, and the heteronuclear magnetic-dipole interaction phi I-z during the evolution (tau(2)) and the detection (tau(4)) periods. Consequently, the results are valid for any ratio of the quadrupole coupling omega(Q) to the amplitude of the RF pulse omega(RF); however, phi should be smaller than omega(RF). Nine echoes are predicted: three central-transition signals occurring at tau(4) = tau(2), tau(4) = 3 tau(2), and tau(4) = 5 tau(2); two inner-satellite signals at tau(4) = tau(2) and two others at tau(4) = 3 tau(2); and two outer-satellite signals at tau(4) = tau(2). The echoes at tau(4) = tau(2), tau(4) = 3 tau(2), and tau(4) = 5 tau(2) originate from the refocusing of single-, three-, and five-quantum coherences, respectively, developed during the first pulse. This analytical approach to computing the spin-5/2 Hahn echoes can be easily extended to higher spin systems (I - 7/2 and 9/2). A numerical method for computing these echoes is also proposed. Alternating the phase of the second pulse allows us to reduce the delay tau(2) to one T-FID. The behavior of the tau(4) = tau(2) echo amplitudes versus the second-pulse flip angle is discussed for several ratios of omega(Q)/omega(RF). Some results are illustrated with Al-27 in a polycrystalline sample of AlCl3 . 6H(2)O. In this sample, the relative echo amplitudes of Al-27 depend not only on the two pulse durations, but also on the spin-spin relaxation time of each transition. (C) 1995 Academic Press, Inc.