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Journal of Experimental and Theoretical Physics Letters

Dipolar temperature and multiple-quantum NMR dynamics in dipolar ordered-spin systems

Serge I. Doronin, Edward B. Fel’dman, E. I. Kuznetsova, G. B. Furman, S. D. Goren

Published Sep 1, 2007• 15 citations
Publisher

Abstract

We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins 1/2 coupled by dipole-dipole interactions in the case of the dipolar-ordered initial state. We suggest a new method of MQ NMR based on the measurement of the dipolar temperature in the quasi-equilibrium state, which establishes after the time of order ω loc −1 (ωloc is the dipolar local field) after the MQ NMR experiment. Manyspin clusters and correlations are created faster in such an experiment than in usual MQ NMR experiments and can be used for the investigation of the many-spin dynamics of nuclear spins in solids.

DOI
10.1134/s0021364007130061

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We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins 1/2 coupled by dipole-dipole interactions in the case of the dipolar-ordered initial state. We suggest a new method of MQ NMR based on the measurement of the dipolar temperature in the quasi-equilibrium state, which establishes after the time of order ω loc −1 (ωloc is the dipolar local field) after the MQ NMR experiment. Manyspin clusters and correlations are created faster in such an experiment than in usual MQ NMR experiments and can be used for the investigation of the many-spin dynamics of nuclear spins in solids.
We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins 1∕21∕21∕2 coupled by the dipole-dipole interactions in the case of the dipolar ordered initial state. We suggest two different methods of MQ NMR. One of them is based on the measurement of the dipolar temperature in the quasiequilibrium state which is established after the time of order ${\ensuremath{\omega}}_{\mathit{loc}}^{\ensuremath{-}1}$ (${\ensuremath{\omega}}_{\mathit{loc}}$ is the dipolar local field) after the MQ NMR experiment. The other method uses an additional resonance ${(\ensuremath{\pi}∕4)}_{y}$ pulse after the preparation period of the standard MQ NMR experiment in solids. Many-spin clusters and correlations are created faster in such experiments than in the usual MQ NMR experiments and can be used for the investigation of many-spin dynamics of nuclear spins in solids.
We investigate numerically the multiple quantum (MQ) NMR dynamics in systems of nuclear spins 1/2 coupled by dipole-dipole interactions in the case of the pseudopure initial state. Simulations of the MQ NMR with real molecular structures such as six dipolar-coupled proton spins of benzene, hydroxyl proton chains in calcium hydroxyapatite, and fluorine chains in calcium fluorapatite open the way to experimental NMR testing of the obtained results. It was found that multiple-spin correlations are created faster in such experiments than in the usual MQ NMR experiments and can be used for the investigation of many-spin dynamics of nuclear spins in solids.
The multiple quantum (MQ) NMR dynamics in the system of equivalent spins with the dipolar ordered initial state is considered. The high symmetry of the Hamiltonian responsible for the MQ NMR dynamics (the MQ Hamiltonian) is used to develop analytic and numerical methods for the investigation of the MQ NMR dynamics in systems consisting of hundreds of spins from the “first principles.” We obtain the dependence of the intensities of the MQ NMR coherences on their orders (profiles of the MQ NMR coherences) for systems of 200–600 spins. It is shown that these profiles may be well approximated by exponential distribution functions. We also compare the MQ NMR dynamics in the systems of equivalent spins having two different initial states, the dipolar ordered state and the thermal equilibrium state in a strong external magnetic field.
Multiple quantum (MQ) nuclear magnetic resonance (NMR) spin dynamics are investigated analytically in infinite one-dimensional (1D) chains of spins 1/2. The representation of spin 1/2 operators with fermion field operators allows to calculate exactly the spin density operator, and hence NMR observables, under a variety of different conditions for 1D spin systems. The exact expressions are valid for all times and for a macroscopic number of coupled spins. The calculations for a 1D spin system initially at thermal equilibrium, and evolving under a 2-quantum/2-spin average dipolar Hamiltonian, in the presence of nearest-neighbor dipolar interactions yield MQ NMR spectra with 0- and 2-quantum coherences only. For a nonequilibrium initial condition with transverse magnetization, the analogous spin dynamics calculations produce MQ NMR spectra with all possible coherences of odd orders. Calculations at the level of perturbation theory, which include next-nearest-neighbor dipolar interactions, generate MQ spectra with higher even order coherences for equilibrium initial condition and evolution under a 2-quantum/2-spin propagator. Consideration of multiple spin correlations, 0-quantum coherences, and rf pulse imperfections are also presented. The relevance and implications of these theoretical results for comparison with the recent MQ NMR experiments of Yesinowski et al. on materials with quasi-one-dimensional distributions of spins, and for MQ NMR of solids in general are discussed.
We investigated dynamics of entangled states on the basis of multiple-quantum (MQ) NMR methods. A time evolution of MQ coherences of nuclear spins coupled by the dipole-dipole interactions (DDI) in solids is simply connected with dynamics of quantum entangled states. We studied analytically dynamics of the entangled states for two- and three-spin systems. In this case dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one and vice versa.
The dynamics of entanglement is investigated on the basis of exactly solvable models of multiple quantum (MQ) NMR spin dynamics. It is shown that the time evolution of MQ coherences of systems of coupled nuclear spins in solids is directly connected with dynamics of the quantum entanglement. We studied analytically the dynamics of entangled states for two- and three-spin systems coupled by the dipole-dipole interaction. In this case the dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that the dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one, and vice versa. A pure state which generalizes the Greenberger-Horne-Zeilinger (GHZ) and W states is found. Different measures of the entanglement of this state are analyzed for tripartite systems.