The spherical tensor formalism applied to relaxation in magnetic resonance

Abstract

Spherical tensor operators are used to simplify the relaxation terms of the Redfield equation. A method for treating relaxation is developed using trace-normalized spherical tensor operators. The Redfield equation is recast as a set of coupled linear differential equations for the expectation values of a complete set of spherical tensor operators. A single spin-1/2 particle is treated, and the correspondence of the equation of motion with the Bloch equation is demonstrated. The advantage of the spherical tensor approach is that both the coherent and relaxation terms of the equation of motion are written as matrices that operate on a common vector consisting of the expectation values of spin variables. The process used to construct the equation of motion helps to organize the various contributions to spin relaxation and can be applied uniformly regardless of spin system size. As an example, the relaxation of a two-spin system by the chemical shift anisotropy and dipolar Hamiltonians is developed using the spherical tensor formalism. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 270–290, 2006