Modulation of Spin Echoes in Multi-Half-Spin Systems. I. Closed Formulas of Carr–Purcell Spin Echoes in Several AaBXx Systems

Abstract

A density-matrix approach to derive the effects of chemical shift and both homo- and heteronuclear spin coupling in a molecule on the Carr–Purcell spin-echo (CPSE) train is developed. Closed formulas are derived for several multi-half-spin systems of the “AaBXx” type where a and x are arbitrary numbers of spins, and X is either the same nuclear species as A and B or different. As a special case of this type, the formulas are also obtained for both homo- and heteronuclear AA′XX′ systems. In heteronuclear multi-half-spin systems, the general equation is derived as a function of all the coupling constants. Features of the modulation patterns in “AaBXx” systems are discussed and the following relation is obtained: By introducing “effective internal chemical shifts” the modulation pattern of AaB resonance in a heteronuclear AaBXx system can be related to that in the homonuclear AaB system. Modulation patterns in homonuclear “AaBXx” systems are not summations of those in simpler spin systems and these facts arise from the effect of the 180° pulse on homonuclear systems. In heteronuclear AXx systems, there are only unmodulated terms. When the coupling constants between two magnetically equivalent sets are equal in AaXx systems, the results show that the heteronuclear system has no modulation and only one modulation frequency exists in the homonuclear system.