Spin precession: A spin-1 case study using irreducible tensor operators

Abstract

Using a Cartesian operator basis set, precession equations have previously been derived for spin-1 systems using some 23 Cartesian operator commutators. We avoid the explicit evaluation of these commutators, and use instead fundamental properties of irreducible tensor operators (ITO) to obtain these precession equations. First, advantage is taken of the angle-axis parametrization of the rotation matrices that transform second-rank ITO under rotation to define the unitarily equivalent rotation matrix that transforms second-rank Cartesian tensors. From this latter transformation, and using simple matrix analysis techniques, all the equations that describe spin-1 precession in the presence of radiofrequency fields and resonance offsets are obtained. Second, information on the ITO commutation relations can be encoded in angular momentum coupling coefficients in a generalized spin precession equation. In the case of spin-1, this leads to a set of coupled differential equations for the statistical tensor components . After transformation of these components to their Cartesian counterparts, the corresponding vector differential equations that define the time evolution of the Cartesian operator expectation values are easily solved, again using simple matrix analysis. This solution yields all the equations that describe spin-1 precession in the presence of radiofrequency fields, resonance offsets, and the quadrupolar interaction.