On the statistical theory of the shape of multiple quantum nmr spectra in solids

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Abstract

The statistical model developed in this work allows us to calculate the shape of the multiple quantum (MQ) NMR spectra (the dependence of the amplitudes of the corresponding multiple quantum coherences on their orders) by decomposing the desired time-correlation functions (TCF) over the infinite set of orthogonal operators and by using some well-known facts from the physics of traditional model systems. The resulting expression contains series with terms depending on the gradually growing up with the time number of spins in clusters of correlated spins. The influence of the possible degradation of these clusters on the shape of the spectra is taken into account. Analytical and numerical calculations were performed for various parameter values included in the final expressions. The developed theory adequately describes the results of numerical calculations of the MQ spectra performed by us and experiments: the transformation of the Gaussian profile into an exponential one, the asymptotics (wings) of the spectrum depending on the coherence order M, the dependence of the relaxation rate of the MQ spectrum on M, as well as the narrowing and stabilization of the MQ spectrum under the influence of perturbation.

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About the authors

V. E. Zobov

Kirensky Institute of Physics, Federal Research Center KSC SB RAS

Author for correspondence.
Email: rsa@iph.krasn.ru
Russian Federation, Krasnoyarsk

A. A. Lundin

Semenov Federal Research Center for Chemical Physics, Russian Academy of Sciences

Email: ya-andylun2012@yandex.ru
Russian Federation, Moscow

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Supplementary files

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2. Fig. 1. Comparison of the experimental MC spectrum [53] (filled circles) at p = 0 and T = 11τ0 (τ0 = 57.6 ps) with the results of theoretical calculations: calculation using the formula GM (T) = (1 + 2^2 M2/K) exp (-2^2 M2/K)/42K at K— = 428 (1), 550 (2) and 700 (3); curve 4 – calculation using the Gauss formula GM (T) = exp(-M 2/K)/4PK at K = 428. Only the right halves of the spectra are shown.

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3. Fig. 2. Comparison of four experimental MC spectra at p = 0 and different durations of preparatory periods [53]: T = 5 (filled circles), 8 (squares), 11 (triangles) and 14τ0 (open circles), where τ0 = 57.6 ps, with theoretical MC spectra calculated using the formula GM (T) = (1 + 2—2 M2/ K) exp (-2^2 M2/ K)/LK, respectively, at: K— = 50 (1), 150 (2),

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4. Fig. 3. Multiquantum spectra (logarithmic values ​​of the right halves) for T = 5 (squares) and 8 (crosses), calculated at C = 0.001 and A = 0 both by formula (36) (solid lines) and by numerical integration of expression (21) taking into account formula (27). The dashed line is the calculation by the simplified formula (36), obtained after removing θ(′1) in expressions (36) and (34).

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5. Fig. 4. Multiquantum spectra (logarithmic values ​​of the right halves) for T = 8, calculated at C = 0.0001 both by formula (36) (solid lines) and by numerical integration of expression (21) taking into account formula (27): α = 1 (squares) and α = 0 (crosses). In both cases, integration over K was carried out from M to 1,000,000.

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6. Fig. 5. Emitted at the disturbance parameter p = 0.108 and two different preparatory periods: T = 5т0 (open circles) and T = 9т0 (filled circles) from [53], and the corresponding MK spectra calculated by formula (36) at T = 5т0(m2)1/2 = 1.89 (solid line) and T = 9т0( m 2)1/2 = 3.41 (dashed line). Other parameters used in the calculations: C = p2B2/m2 = 0.029, a = A2/B2 = 0.5, as well as at T = 5т0 N1 = 1/1.7, K0 = 46.9, at T = 9т0 N1 = 1/29, K0 = 403.2.

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