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 ${\mathrm{“A}}_a{\mathrm{BX}}_x”$ 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 $\mathrm{AA\prime XX\prime }$ 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 ${\mathrm{“A}}_a{\mathrm{BX}}_x”$ systems are discussed and the following relation is obtained: By introducing “effective internal chemical shifts” the modulation pattern of $A_{a}B$ resonance in a heteronuclear $A_a{\mathrm{BX}}_x$ system can be related to that in the homonuclear $A_{a}B$ system. Modulation patterns in homonuclear ${\mathrm{“A}}_a{\mathrm{BX}}_x”$ 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 ${\mathrm{AX}}_x$ systems, there are only unmodulated terms. When the coupling constants between two magnetically equivalent sets are equal in $A_a{X}_x$ systems, the results show that the heteronuclear system has no modulation and only one modulation frequency exists in the homonuclear system.