Analytical Solution to Solomon Equations for Three-Spin Groupings

Abstract

Analytical solutions are provided for a set of three simultaneous first-order differential equations which describe either cross relaxation among three groupings of spin-$\text{1}\text{2}$ nuclei (regardless of the number of spins within each grouping) or the complete longitudinal relaxation of a system of two spins-$\text{1}\text{2}$, including CSA-dipolar interference terms (which couple the longitudinal spin order to conventional longitudinal magnetizations). In spite of their complexity, the expressions so obtained afford a time savings by a factor of 50 when used in a computer program. The efficiency of the method is illustrated by the fit of experimental data, exhibiting an unusual evolution due to both intra- and intermolecular dipolar couplings.