Abstract
A system of three dipole-coupled spins exhibits a surprisingly intricate relaxation behavior. Following Hubbard's pioneering 1958 study, many authors have investigated different aspects of this problem. Nevertheless, on revisiting this classic relaxation problem, we obtain several new results, some of which are at variance with conventional wisdom. Most notably from a fundamental point of view, we find that the odd-valued spectral density function influences longitudinal relaxation. We also show that the effective longitudinal relaxation rate for a non-isochronous three-spin system can exhibit an unusual inverted dispersion step. To clarify these and other issues, we present a comprehensive theoretical treatment of longitudinal relaxation in a three-spin system of arbitrary geometry and with arbitrary rotational dynamics. By using the Liouville-space formulation of Bloch-Wangsness-Redfield theory and a basis of irreducible spherical tensor operators, we show that the number of relaxation components in the different cases can be deduced from symmetry arguments. For the isochronous case, we present the relaxation matrix in analytical form, whereas, for the non-isochronous case, we employ a computationally efficient approach based on the stochastic Liouville equation.
Highlights
By using the Liouville-space formulation of Bloch-Wangsness-Redfield theory and a basis of irreducible spherical tensor operators, we show that the number of relaxation components in the different cases can be deduced from symmetry arguments
We find that chemical shifts can give rise to an unusual inverted step in the dispersion profile, resulting from symmetry-breaking nonsecular decoupling that principally affects distinct correlations
We establish the range of validity of the isochronous relaxation theory and we show that the effect of chemical shifts on longitudinal relaxation is always negligible outside the motional-narrowing (MN) regime
Summary
A few years after Solomon’s seminal analysis of dipolar cross relaxation in two-spin systems, Hubbard investigated longitudinal relaxation in systems of three or four dipolecoupled spins. In multi-spin systems, correlations between distinct dipole couplings (usually referred to as cross correlations) come into play, and Hubbard showed that their effect is to make the relaxation of the total longitudinal magnetization in the extreme-narrowing (EN) regime weakly bi-exponential and slightly slower. Subsequent studies confirmed Hubbard’s results and extended them to non-EN conditions, where longitudinal relaxation is tri-exponential, and to anisotropic rotation models, where distinct correlations can have a more pronounced effect. . .], just like the Liouvillian associated with the time-independent spin Hamiltonian. (Karthik and Kumar, following Jeener, refer instead to the anti-Hermitian part of R, but for all cases considered here this is identical to the imaginary part of R.) It is true that if there were no other terms in the equation of motion, the total evolution superoperator would be unitary (since H ′ is Hermitian), resulting in purely coherent evolution without dissipation. To its well-known coherent effect of inducing a second-order dynamic frequency shift.16,22,25–27 Consistent with these results, Pfeifer argued in general terms that the OSDF can affect relaxation outside the EN regime, except when the relaxation function is strictly exponential so that a unique relaxation time can be defined..
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