Theory of symmetry in nuclear magnetic relaxation

Publisher Website
Google Scholar
Abstract
Recently introduced methods of discussing and simplifying problems concerning magnetic relaxation in liquids, which involve investigation of the transformation properties of the Liouville operator, are further developed. It is shown that the previous results can be obtained by a simpler method and that the Liouville operator can be expressed as a sum of super-operators describing the time evolution of the density operator caused by individual nuclear Zeeman interactions, scalar couplings between pairs of nuclei and individual relaxation processes whose effects are introduced by means of Redfield relaxation theory. These methods prove useful when dealing with multispin systems (being also applicable to single-spin systems) because the super-matrix elements of these Liouville operators between operators, which are irreducible tensor operators with respect to rotations in the Liouville space appropriate to the entire spin system, constructed by vector coupling the irreducible tensor operators pertaining to single nuclei, are given by the Wigner-Eckart theorem. Methods for expressing the reduced super-matrix elements between vector-coupled irreducible tensor operators in terms of reduced super-matrix elements expressed in sub-spaces (of the total Liouville space) pertaining to individual nuclei, which obviate the need to construct such tensor operators explicitly, are developed and explicit expressions for these latter reduced super-matrix elements are given. It is shown that such reduced super-matrix elements of Liouville operators for nuclear Zeeman interactions, scalar couplings and straight or cross relaxation in the extreme narrowing approximation caused by a relaxation hamiltonian, which in space-fixed axes takes the form of a sum of scalar contractions of irreducible tensors in which each product of spin operators contains only operators pertaining to one nucleus, can be written as one single product of 6j symbols and reduced super-matrix elements pertaining to individual nuclei. It is also shown that such expressions can be evaluated on inspection; the system of two spin-1 nuclei subject to quadrupole relaxation is treated explicitly. Expressions are presented which decompose the reduced super-matrix elements of the super-operators describing the time evolution of the density operator, caused by intramolecular dipole-dipole relaxation in the extreme narrowing approximation, into sums of products of 6j, 9j and 18j symbols and reduced super-matrix elements pertaining to individual nuclei.