Modulation of Spin Echoes in Multihyphen;Halfhyphen;Spin Systems. I. Closed Formulas of Carrndash;Purcell Spin Echoes in Several AaBXxSystems

摘要

A densityhyphen;matrix approach to derive the effects of chemical shift and both homohyphen; and heteronuclear spin coupling in a molecule on the Carrndash;Purcell spinhyphen;echo (CPSE) train is developed. Closed formulas are derived for several multihyphen;halfhyphen;spin systems of theldquo;AaBXxrdquo;type whereaandxare arbitrary numbers of spins, andXis either the same nuclear species asAandBor different. As a special case of this type, the formulas are also obtained for both homohyphen; and heteronuclearAAprime;XXprime;systems. In heteronuclear multihyphen;halfhyphen;spin systems, the general equation is derived as a function of all the coupling constants. Features of the modulation patterns inldquo;AaBXxrdquo;systems are discussed and the following relation is obtained: By introducing ldquo;effective internal chemical shiftsrdquo; the modulation pattern ofAaBresonance in a heteronuclearAaBXxsystem can be related to that in the homonuclearAaBsystem. Modulation patterns in homonuclearldquo;AaBXxrdquo;systems are not summations of those in simpler spin systems and these facts arise from the effect of the 180deg; pulse on homonuclear systems. In heteronuclearAXxsystems, there are only unmodulated terms. When the coupling constants between two magnetically equivalent sets are equal inAaXxsystems, the results show that the heteronuclear system has no modulation and only one modulation frequency exists in the homonuclear system.