Behaviour of a coupled two-spin-1/2 system in the presence of a spin-locking radio-frequency field. Relaxation and Hartmann-Hahn transfers

Abstract

The theory of the evolution of a J-coupled homonuclear system of two nuclei of spin 1/2 (in the liquid state) subjected to a spin-locking field is explicitly reformulated in terms of simultaneous differential equations. Quantities destroyed by the inhomogeneity of the spin-locking field are delineated; from these considerations it is found that only four quantities interfere: the two transverse magnetizations of concern (supposed to be in absorption), one observable mode corresponding to the difference of antiphase dispersive doublets and one mode (unobservable) involving zero/double quantum coherences. Transfers of Hartmann-Hahn type occur necessarily via the antiphase doublet mode. Analytical solutions could be derived for these transfers, which occur in an oscillatory fashion. The frequency of the Hartmann-Hahn oscillations is shown to depend not only upon J but also upon the characteristics of the spin-locking field (amplitude and offset with respect to the resonance frequencies of the two nuclei). When relaxation phenomena are included, the resulting differential equations are no longer analytically solvable. Approximate treatments, as well as an exact numerical analysis, lead to the conclusion that Hartmann-Hahn oscillations always exist as long as the J coupling is not too small with regard to relaxation parameters, their amplitude decreasing when the amplitude of the spin-locking field decreases. The numerical analysis allows extraction, from experimental data, of the relaxation parameters and particularly the transverse cross-relaxation term. Experimental one-dimensional results confirm the validity of the present theory. Implications for the intensities of cross-peaks in two-dimensional ROESY experiments are outlined.