Generalized Cumulant Expansions and Spin-Relaxation Theory

Abstract

The generalized cumulant expansion method of Kubo is applied to an analysis of spin-relaxation theory appropriate for NMR and ESR studies of molecular systems. It leads to a general, formal solution of the equation of motion of a suitably averaged magnetization operator (and also the spin-density matrix). This solution permits a convenient perturbation expansion in the region of motional narrowing; i.e., when the random functional spin perturbation H1(t) obeys | H1(t) |τc<1, where τc is a correlation time for the random process. It is shown how this expansion valid for times t≫τc generates the time-independent R or relaxation matrix to all orders in | 1(t) |τc, and detailed expressions are given through fourth order. While the R matrix supplies the linewidth and dynamic frequency shift of the main Lorentzian resonance line, it is found that by formulating the cumulant method without the restriction t≫τc, weak subsidiary Lorentzian lines are predicted. These subsidiary lines appear as second-order perturbation corrections in |H1(t)|τc. They have widths given by τc−1 and, in general, large frequency shifts. The problem of (anisotropic) -rotational diffusion is discussed in detail, and it is shown that the spin-Hamiltonian approximation for liquids rests upon a Born–Oppenheimer-type approximation appropriate for random modulation of the nuclear coordinates, i.e., τc−1≪ωn,o, where ℏωn,o is the energy separation between the ground electronic state and the nth excited state coupled by the angular-momentum operator L. The cumulant method is then used to obtain higher-order corrections to the g-tensor line-broadening mechanism.