First-principles simulation of spin diffusion in static solids using dynamic mean-field theory

TL;DR

This work demonstrates that spin dynamic mean-field theory (spinDMFT) can accurately and efficiently simulate spectral spin diffusion and zero-quantum line shapes in static solids from first principles. By mapping dense spin ensembles to a single-site problem with a time-dependent Gaussian mean-field and employing self-consistent bath correlations, the authors reproduce experimental diffusion behavior in malonic acid (13C) and GLP (31P) without resorting to perturbative assumptions. The study proves the method's ability to predict both two-spin diffusion dynamics and ZQ lines, including situations with bath inhomogeneity, and highlights the potential of spinDMFT for large-scale NMR spin-diffusion problems and future extensions to MAS and finite temperatures.

Abstract

The dynamics of disordered nuclear spin ensembles are the subject of nuclear magnetic resonance studies. Due to the through-space long-range dipolar interaction generically many spins are involved in the time evolution, so that exact brute force calculations are impossible. The recently established spin dynamic mean-field theory (spinDMFT) represents an efficient and unbiased alternative to overcome this challenge. The approach only requires the dipolar couplings as input and the only prerequisite for its applicability is that each spin interacts with a large number of other spins. In this article, we show that spinDMFT can be used to describe spectral spin diffusion in static samples and to simulate zero-quantum line shapes which eluded an efficient quantitative simulation so far to the best of our knowledge. We perform benchmarks for two test substances that establish an excellent match with published experimental data. As spinDMFT combines low computational effort with high accuracy, we strongly suggest to use it for large-scale simulations of spin diffusion, which are important in various areas of magnetic resonance.

Figures (10)

• Figure 1: Sketch of the effective models resulting from spinDMFT. (A) SpinDMFT reduces a homogeneous spin lattice with many interacting spins (left) to a single spin interacting with a time dependent Gaussian mean-field (right). As the single spin changes its orientation over time, so does the mean-field, which is indicated here by the gray cloud. (B) SpinDMFT reduces the full lattice problem for spectral spin diffusion (left) to an effective model with explicit time-dependence (right). The coupling between the low-$\gamma$ spins (indicated by green arrows) is crucial and needs to be treated quantum-mechanically to capture spectral spin diffusion correctly. Therefore, the effective model treats both low-$\gamma$ spins explicitly. The dark gray clouds represent the mean-fields resulting from the high-$\gamma$ bath spins (indicated by gray arrows). The mean-fields are correlated, which is indicated by the light gray cloud. The blue arrows represent the chemical shifts acting on the low-$\gamma$ spins.
• Figure 2: Unit cells of the considered test substances. White, grey, red, purple and orange spheres correspond to hydrogen, carbon, oxygen, potassium and phosphor atoms, respectively. (A) Malonic acid. The unit cell contains four 13C spins, which are labelled here by black numbers. Due to inversion symmetry, two carbon pairs are each equivalent in terms of dipolar couplings and chemical shifts and therefore obtain the same number. (B) GLP. The unit cell contains two crystallographically distinct 31P spins, which are also labelled by black numbers.
• Figure 3: Universal result of the spin autocorrelations simulated by spinDMFT for a homonuclear dipolar coupling. The time axis is given in inverse units of the quadratic coupling constant $J^{\text{HH}}_{\text{Q,av}}$. The black dashed line displays the fit function provided in Equation \ref{['eqn:spindmft:longitudinalfit']}. The transverse correlation $G^{xx}$ has a Gaussian shape graes24.
• Figure 4: Spectral spin diffusion of carbon spins in malonic acid as seen in the longitudinal pair correlation $G^{zz}_{12}(t)~=~4\langle\mathbf{S}^{z}_1(t) \mathbf{S}^{z}_2(0)\rangle$ for two different crystal orientations. The solid lines correspond to the simulation results by spinDMFT; different possible crystal orientations are shown by different colors. The dashed black line is an exponential fit of the measurement result from Ref. suter85. The numerical error of the spinDMFT data is of the order or smaller than 0.02.
• Figure 5: ZQ line of carbon spins in malonic acid for two different crystal orientations. The solid lines correspond to the unbiased prediction by spinDMFT, which is the Fourier transform of Equation \ref{['eqn:spinDMFT_ZQC_analytical']}; different possible crystal orientations are shown by different colors. The dashed black line is a Lorentzian function with the experimentally determined ZQ line width suter85.