Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids is investigated by analytical and numerical methods. A fermion approach for MQ spin dynamics of one-dimensional inhomogeneous systems is developed in the approximation of the dipole–dipole interactions (DDI) of nearest neighbors. It is shown that only MQ coherences of the zeroth and plus/minus second orders appear in the approximation of the DDI of the nearest neighbors even in inhomogeneous one-dimensional systems. We also investigate MQ dynamics of inhomogeneous chains numerically. Intensities of MQ NMR coherences for a linear chain consisting of 3000 spins are calculated.     

Line shapes of multiple quantum NMR coherences in one-dimensional quantum spin chains in solids

General formulae for intensities of multiple quantum (MQ) NMR coherences in systems of nuclear spins coupled by the dipole-dipole interactions are derived. The second moments of the MQ coherences of zero- and second orders are calculated for infinite linear chains in the approximation of the nearest neighbor interactions. Supercomputer simulations of intensities of MQ coherences of linear chains are performed at different times of preparation and evolution periods of MQ NMR experiments. The second moments obtained from the developed theory are compared with the results of the supercomputer analysis of MQ NMR dynamics. The linewidth information in MQ NMR experiments is discussed.     

Multiple quantum dynamics in linear chains and rings of nuclear spins in solids at low temperatures

Multiple quantum spin dynamics is studied using analytical and numerical methods for one-dimensional finite systems of nuclear spins 12 coupled by dipole-dipole interactions at low temperatures. Exact expressions for intensities of multiple quantum coherences at low temperatures were obtained in the approximation of the nearest neighbor interactions. The time growth of multiple quantum coherences was analyzed numerically when all the dipole-dipole interactions in one-dimensional systems consisting of 6/8 spins were taken into account. It is shown that the growth of multiple quantum coherences gets faster when the temperature decreases, and the intensities of multiple quantum coherences can be negative at low temperatures.     

Exact solutions in the dynamics of alternating open chains of spins s = 1/2 with the XY Hamiltonian and their application to problems of multiple-quantum dynamics and quantum information theory

A method for exactly diagonalizing the XY Hamiltonian of an alternating open chain of spins s = 1/2 has been proposed on the basis of the Jordan-Wigner transformation and analysis of the dynamics of spinless fermions. The multiple-quantum spin dynamics of alternating open chains at high temperatures has been analyzed and the intensities of multiple-quantum coherences have been calculated. The problem of the transfer of a quantum state from one end of the alternating chain to the other is studied. It has been shown that the ideal transfer of qubits is possible in alternating chains with a larger number of spins than that in homogeneous chains.     

Quantum coherence dynamics of spin pairs in many-spin cluster

We consider the dynamics of a quantum coherence of two chosen spins in systems of dipolar coupled nuclear spins s=1/2 in solid. With the purpose to study this coherence we suggest two different methods. One of them uses the partial trace technique and reduced density matrix. The second method is based on the calculation the intensity of multiple quantum coherences using two-spin operator and the density matrix of the whole spin system. Results of calculations of the multiple-quantum dynamics in spin clusters of various dimensionalities are presented. It is shown that the whole density matrix method is more informative than the method based on the reduced density matrix.     

Dynamics of multiple quantum coherences in dipole-coupling spins in solid

A perturbation method deals with dipolar coupling spins in solids is presented. As example of application the method, the multiple-quantum coherence dynamics in clusters of a linear chain of four nuclear spins and a ring of six spins coupled by dipole-dipole interaction are considered. The calculated 0Q and 2Q intensities in a linear chain of four nuclear spins and 6Q intensity in a ring of six spins vs. the duration of the preparation period agree well with the exact solutions (for linear chain of four nuclear spins) and simulation data (for linear chain of four nuclear spins and a ring of six spin).     

Integrable and nonintegrable classical spin clusters

The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.     

Intensities of multiple-quantum coherences in a dilute spin system

The temporal evolution of the intensities of multiple-quantum coherences in a magnetically dilute system of dipole-dipole coupled spins is calculated. It is shown that not only the temporal scale but also the form of the time dependence itself are different from those in concentrated systems. Fiz. Tverd. Tela (St. Petersburg) 39, 510–512 (March 1997)     

Second moment of multiple-quantum NMR and a time-dependent growth of the number of multispin correlations in solids

The time evolution of multispin correlations (the growth of the number of correlated spins as a function of time) can be observed directly using the multiple-quantum nuclear magnetic resonance spectroscopy of solids. A quantity related to this number, namely, the second moment 〈n ²(t)〉 of the intensity distribution of coherences of different orders in the multiple-quantum spectrum can be calculated using the theory proposed in this work. An approach to the calculation of the four-spin time correlation function through which this moment is expressed is developed. The main sequences of contributions in the expansion of this function into a time power series are summed using the approximation of a large number of neighbors both for systems with a secular dipole-dipole interaction and for systems with a nonsecular effective interaction. An exponential dependence of 〈n ²(t)〉 is obtained. The value of 〈n ²(t)〉 is additionally calculated using an expansion in terms of orthogonal operators for three model examples corresponding to different limiting realizations of spin systems. It is shown that the results of the microscopic theory at least qualitatively agree with both the results obtained for model examples and experimental results obtained recently for adamantane.     

Triple-quantum dynamics in multiple-spin systems undergoing magic-angle spinning: application to 13C homonuclear correlation spectroscopy

We analyze the multiple-quantum dynamics governed by a new homonuclear recoupling strategy effecting an average dipolar Hamiltonian comprising three-spin triple-quantum operators (e.g., S(p)+S(q)+S(r)+) under magic-angle spinning conditions. Analytical expressions are presented for polarization transfer processes in systems of three and four coupled spins-1/2 subject to triple-quantum filtration (3QF), and high-order multiple-quantum excitation is investigated numerically in moderately large clusters, comprising up to seven spins. This recoupling approach gives highly efficient excitation of triple-quantum coherences: ideally, up to 67% of the initial polarization may be recovered by 3QF in three-spin systems in polycrystalline powders. Two homonuclear 2D correlation strategies are demonstrated experimentally on powders of uniformly 13C-labeled alanine and tyrosine: the first correlates the single-quantum spectrum in the first dimension with the corresponding 3QF spectrum along the other. The second protocol correlates triple-quantum coherences with their corresponding single-quantum coherences within triplets of coupled spins.     

Heisenberg-Dirac-van Vleck vector model for a 1D antiferromagnetic chain of localized spins <Emphasis Type="Italic">S</Emphasis> = 1

Magnetic properties of anisotropic crystals with localized spins S = 1 are investigated; for these crystals, the Hamiltonian is derived in the Heisenberg-Dirac-van Vleck form, which includes biquadratic contributions apart from bilinear terms. The ground-state energy of the antiferromagnetic chain of spins S = 1 is calculated in the model of nearest neighbors, and the interaction constant is renormalized using the renorm group method in the case of coarsening of the system. The temperature criterion for the formation of long-range order in the system is obtained. The excitations of this chain in the linear approximation have a dispersion relation differing from that for antiferromagnets with spin S = 1/2 and are separated by an energy gap from the ground state. Allowance for nonlinear contribution leads to the formation of a solitary wave in the form of a dark-bright soliton.     

Influences of Quantum and Disorder Effects on Solitons Excited in Protein Molecules in Improved Model

Utilizing the improved model with quasi-coherent two-quantum state and new Hamiltonian containing an additional interaction term [Phys. Rev. E62 (2000) 6989 and Euro. Phys. J. B19 (2001) 297] we study numerically the influences of the quantum and disorder effects including distortion of the sequences of masses of amino acid molecules and fluctuations of force constant of molecular chains, and of exciton-phonon coupled constants and of the dipole-dipole interaction constant and of the ground state energy on the properties of the solitons transported the bio-energy in the protein molecules by Runge-Kutta method. The results obtained show that the new soliton is robust against these structure disorders, especially for stronger disorders in the sequence of masses spring constants and coupling constants, except for quite larger fluctuations of the ground state energy and dipole-dipole interaction constant. This means that the new soliton in the improved model is very stable in normal cases and is possibly a carrier of bio-energy transport in the protein molecules.     

The multiple quantum NMR dynamics in systems of equivalent spins with a dipolar ordered initial state

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.     

Effect of rf field on spin diffusion

A study was made of the effect of an rf field on spin diffusion. The interaction of spins with the rf field is described quantum mechanically. It is shown that the effect of the rf field on the system of spins can, in some approximation, be interpreted as the effect of a change in the Larmor frequency of the spin and a decrease in the magnitude of the dipole-dipole interaction between spins. These conclusions were obtained on the basis of a unitary transformation which eliminates the explicit form of the spin-photon interaction operator in the Hamiltonian of the system considered. An expression is derived for the spin-diffusion coefficient under saturation. The presence of the rf field results in a decrease in the diffusion coefficient.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 7–11, September, 1979.     

Constants of motion in NMR spectroscopy

We present a general method for constructing a subset of the constants of motion in terms of products of spin operators. These operators are then used to give insight into the multi-spin orders comprising the quasi-equilibrium state formed under a Jeener-Broekaert sequence in small, dipolar-coupled, spin systems. We further show that constants of motion that represent single-quantum coherences are present due to the symmetry of the dipolar Hamiltonian under 180 degrees spin rotations, and that such coherences contribute a DC component to the FID which vanishes in the absence of the flip-flop terms and is only present for spin clusters with an odd number of spins.     

Spin-vibrational interactions in geminate radical pairs: Effective spin Hamiltonian including anharmonicity

An equation for the effective spin Hamiltonian including the interaction of radical pair spins with vibrations in both linear and higher approximations was obtained. Even in the linear approximation, the Hamiltonian was found to be non-Heisenberg. The influence of a sound field on the probability of geminate radical pair recombination is considered in the Appendix. The results obtained can be of use for solving certain nanochemistry problems.     

ARPES measurements of SnAs electronic band structure

A change in the time dependence of the second moment of the distribution of intensities of coherences with various orders in the spectrum of multiple-quantum NMR in a solid at the inclusion of an inhomogeneous magnetic field in the effective interaction is studied. Both the secular dipole–dipole and nonspecular twoquantum interactions are considered as nucleus–nucleus interactions, which correspond to traditional experimental realizations. It is shown that, with an increase in the magnitude of the inhomogeneous field, an exponential increase in the second moment of multiple-quantum NMR with time changes to a power-law increase. The results obtained in this work indicate that this second moment, which determines the average number of dynamically correlated spins, can be used as a convenient characteristic for studying a transition to a many-body localized state.     

Multiple-quantum cross-polarization and two-dimensional MQMAS NMR of quadrupolar nuclei

Cross-polarization from (1)H to the multiple-quantum coherences of a quadrupolar nucleus is used in combination with the two-dimensional multiple-quantum magic angle spinning (MQMAS) NMR experiment in order to extract high-resolution CPMAS NMR spectra. The technique is demonstrated on (23)Na (S = 3/2), (17)O, (27)Al (both S = 5/2), and (45)Sc (S = 7/2) nuclei, showing the applicability of multiple-quantum cross-polarization to systems with differing spin quantum number, gyromagnetic ratio, and relative nuclide abundance. The utility of this two-dimensional MAS NMR experiment for spectral editing and site-specific measurement of cross-polarization intensities is demonstrated. The possibility of direct cross-polarization to higher order multiple-quantum coherences is also considered and three-, five-, and seven-quantum cross-polarized (45)Sc MAS NMR spectra are presented.     

Low temperature MQ NMR dynamics in dipolar ordered state

We investigate analytically and numerically the Multiple Quantum (MQ) NMR dynamics in dipolar ordered spin systems of nuclear spins 1/2 at low temperatures. We consider two different methods of MQ NMR. One of them is based on the measurement of the dipolar energy. The other method uses an additional resonance (π/4)_y-pulse after the preparation period of the standard MQ NMR experiment in solids and allows one to measure the Zeeman energy. Both considered methods are sensitive to the contribution of remote spins in the interaction and to the spin system structure. The QS method is sensitive to the spin number in the molecule while the PS method gives very similar time dependencies of the intensities of MQ coherences for different spin numbers. It is shown that the use of the dipolar ordered initial state has the advantage of exciting the highest order MQ coherences in clusters of 4m identical spins, where m=1,2,3,…, that is impossible to do with the standard MQ method. MQ NMR methods based on the dipolar ordered initial states at low temperatures complement the standard NMR spectroscopy for better studying structures and dynamic processes in solids.