Theory of symmetry in nuclear magnetic relaxation including applications to high resolution N.M.R. line shapes

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Abstract
It is shown in discussing problems involving magnetic relaxation in liquids that, whilst the usual Hilbert space spanned by all the eigenkets of the spin hamiltonian does not reflect any symmetry inherent in the spin system, the vector space (Liouville space) comprising all operators of the spin system does so. The transformation properties of the Liouville operator, as reflected by those of the high resolution spin hamiltonian and relaxation operators whose effects are introduced by means of Redfield relaxation theory, with respect to arbitrary rotations of the coordinate system are investigated. The use of irreducible tensor operators as a set of basis operators spanning Liouville space is stressed, since it is shown that their super-matrix elements of the Liouville operator are given by the Wigner-Eckart theorem provided that relaxation by anisotropy of the chemical shift or anisotropic random fields is absent. These arguments are independent of the fine details of molecular reorientation in the extreme narrowing approximation, since use is made of Hubbard's symmetry arguments to evaluate correlation functions. A further symmetry operation in Liouville space, that of spin inversion conjugation, is defined. These arguments are used to prove the existence of and equality of the time constants (T 1 and T 2) describing the decays of the longitudinal and transverse magnetizations, in the extreme narrowing approximation for systems comprising any number of isochronous nuclei of any spin subject to relaxation mechanisms which are scalar contractions of irreducible spherical tensors of which the spin components operate on one nucleus only. The degree of correlation of the relaxation mechanisms at different nuclei does not enter the calculation. The inequality of T 1 and T 2 for relaxation by anisotropy of the chemical shift is shown to arise from lack of symmetry in the hamiltonian. Non-exponential relaxation in the presence of the dipole-dipole mechanism in systems of isochronous spin-½ nuclei is discussed, the number of exponentials needed to describe the relaxation being symmetry determined. The case of three spin-½ nuclei located at the corners of an equilateral triangle is treated in detail, a previous result being corrected. Relaxation outside the extreme narrowing approximation is discussed for relaxation by chemical shift anisotropy and the quadrupole interaction. The utility of the symmetry arguments in the problem of calculating the band shapes of the resonances of spin-½ nuclei which are scalar coupled to nuclei of spin > ½ which are undergoing rapid quadrupole relaxation is discussed.