Tensors and Rotations in NMR

The transformation of second-rank Cartesian tensors under rotation plays a fundamental role in the theoretical description of nuclear magnetic resonance experiments, providing the framework for describing anisotropic phenomena such as single crystal rotation patterns, tensor powder patterns, sideband intensities under magic-angle sample spinning, and as input for relaxation theory. Here, two equivalent procedures for effecting this transformation—direct rotation in Cartesian space and the decomposition of the Cartesian tensor into irreducible spherical tensors that rotate in subgroups of rank 0, 1, and 2—are reviewed. In a departure from the standard formulation, the explicit use of the spherical tensor basis for the decomposition of a spatial Cartesian tensor is introduced, helping to delineate the rotational properties of the basis states from those of the matrix elements. The result is a uniform approach to the rotation of a physical system and the corresponding transformation of the spatial components of the NMR Hamiltonian, expressed as either Cartesian or spherical tensors. This clears up an apparent inconsistency in the NMR literature, where the rotation of a spatial tensor in spherical tensor form has typically been partnered with the inverse rotation in Cartesian form to produce equivalent transformations. 2011 Wiley Periodicals, Inc. Concepts Magn Reson Part A 38: 221–235, 2011.