Nuclear magnetic relaxation in spin-1 systems: the response to three-pulse sequences

Abstract

The response of a system of spin-1 nuclei to sequences of three r.f. pulses is calculated using the Redfield density matrix equation. The system is taken to be in a large homogeneous B ₀ field and subject to both weak secular and time-dependent quadrupole couplings. Two sequences of pulses are considered, these being represented as 90° _x -τ-θ° _y -T-Θ° _y -t →, where for sequence I, θ° _y = Θ° _y = 45° _y ; whilst for sequence II, θ° _y = Θ° _y = 90° _y ; the subscripts x, y referring to axes in the usual rotating reference frame. Expressions are obtained for the signals following the third pulse in terms of the time intervals, τ, T and t and the parameters characterizing the quadrupolar interactions. In the case of sequence I, often referred to as the Jeener-Broekaert sequence, it is shown that when the second pulse is applied at a zero-crossing of the on-resonance free induction decay, the 90° _x -τ-45° _y pulse pair is adiabatic in its effect, i.e. the total entropy of the spin system is conserved. It is shown that three-quarters of the Zeeman polarization in the laboratory frame is transformed by this sequence into order in the zero-field quadrupolar levels in the rotating frame there producing a very low effective spin-temperature. The remaining quarter of the Zeeman polarization appears as coherence in the double-quantum states. These results are compared with the predictions of conventional spin-temperature theory, as normally used in discussion of solids, and it is found that the spin-temperature theory agrees quantitatively with the Redfield theory in so far as the transfer of order to the zero-field rotating-frame quadrupole levels is concerned but does not account for the double-quantum coherence. It is shown that this is because conventional spin-temperature theory assumes a diagonal spin-density matrix. Experimental verification of the main features of the theoretical results is presented in the form of measurements on the ²D nuclei in the magnetically orientable lamellar phase of the caesium perfluoro-octanoate/D₂O system.