Nuclear magnetic relaxation by the dipolar EMOR mechanism: General theory with applications to two-spin systems

Abstract

On the theory of the proton dipolar-correlation effect as a method for investigation of segmental displacement in polymer melts The Journal of Chemical Physics 147, 074904 (2017); 10.1063/1.4998184 THE JOURNAL OF CHEMICAL PHYSICS 144, 084202 (2016) Nuclear magnetic relaxation by the dipolar EMOR mechanism: General theory with applications to two-spin systems In aqueous systems with immobilized macromolecules, including biological tissue, the longitudinal spin relaxation of water protons is primarily induced by exchange-mediated orientational randomization (EMOR) of intra-and intermolecular magnetic dipole-dipole couplings. We have embarked on a systematic program to develop, from the stochastic Liouville equation, a general and rigorous theory that can describe relaxation by the dipolar EMOR mechanism over the full range of exchange rates, dipole coupling strengths, and Larmor frequencies. Here, we present a general theoretical framework applicable to spin systems of arbitrary size with symmetric or asymmetric exchange. So far, the dipolar EMOR theory is only available for a two-spin system with symmetric exchange. Asymmetric exchange, when the spin system is fragmented by the exchange, introduces new and unexpected phenomena. Notably, the anisotropic dipole couplings of non-exchanging spins break the axial symmetry in spin Liouville space, thereby opening up new relaxation channels in the locally anisotropic sites, including longitudinal-transverse cross relaxation. Such cross-mode relaxation operates only at low fields; at higher fields it becomes nonsecular, leading to an unusual inverted relaxation dispersion that splits the extreme-narrowing regime into two sub-regimes. The general dipolar EMOR theory is illustrated here by a detailed analysis of the asymmetric two-spin case, for which we present relaxation dispersion profiles over a wide range of conditions as well as analytical results for integral relaxation rates and time-dependent spin modes in the zero-field and motional-narrowing regimes. The general theoretical framework presented here will enable a quantitative analysis of frequency-dependent water-proton longitudinal relaxation in model systems with immobilized macromolecules and, ultimately, will provide a rigorous link between relaxation-based magnetic resonance image contrast and molecular parameters. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license

Nuclear magnetic relaxation by the dipolar EMOR mechanism: Three-spin systems

“…where the Liouvillian 5 ) and the consequent instantaneous switching of the spin Hamiltonian. For the asymmetric cases I SP-I and I SP-I S, the spin system is fragmented by the exchange.…”

“…Consequently, all multi-spin correlations that have developed as a result of dipole couplings between labile and nonlabile spins in state A are lost. 5,6 To describe both of these effects, we decompose the exchange superoperator as…”

“…These superoperators distinguish labile from nonlabile spins and they account for decorrelation of multi-spin modes by exchange fragmentation of the spin system. 5,6 Macroscopic spin observables are related to a density operator summed over all sites,…”

“…We distinguish two scenarios: symmetric exchange (m B = m A ), where the spin system exchanges as an intact unit, and asymmetric exchange (m B < m A ), where the spin system is fragmented by exchange. In asymmetric exchange, all multi-spin correlations that have developed as a result of dipole couplings between labile and nonlabile spins in the A site are lost, 5,6 leading to a qualitatively different relaxation behavior as compared to symmetric exchange. To identify different exchange cases, we use the notation "(spins in state A)-(spins in state B)."…”

“…Starting from the stochastic Liouville equation (SLE), 2,3 we have therefore developed a non-perturbative theory of relaxation induced by EMOR modulation of magnetic dipole-dipole couplings, valid without restrictions on exchange rate, dipole couplings, and Larmor frequencies. 4,5 In the dipolar EMOR theory, we consider a system of m A ≥ 2 mutually dipole-coupled spin-1/2 nuclei in the anisotropic (A) sites. This spin system comprises m B labile spins, which exchange with the isotropic bulk (B) phase, and m A − m B nonlabile spins, which reside permanently in the A site.…”