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A simplified theoretical description of multiple-quantum excitation and mixing for nuclear magnetic resonance of half-integer quadrupolar nuclei is presented. The approach recasts the multiple-quantum nutation behavior in terms of reduced excitation and mixing curves through a scaling of the first-order offset frequency by the quadrupolar coupling constant. The two-dimensional correlation of the static first-order anisotropic line shape to the second-order anisotropic magic-angle-spinning (MAS) line shape is utilized to transform the three-dimensional integral over the three Euler angles into a single integral over the dimensionless first-order offset parameter. These transformations lead to a highly efficient algorithm for simulating the multiple-quantum (MQ)-MAS spectrum for arbitrary excitation and mixing radio frequency (RF) field strengths, pulse durations, and MAS rates within the static limit approximation, which is defined in terms of the rotation period, pulse duration, RF field strength, and quadrupolar coupling parameters. This algorithm enables a more accurate determination of the relative site populations and quadrupolar coupling parameters in a least-squares analysis of MQ-MAS spectra. Furthermore, this article examines practical considerations for eliminating experimental artifacts and employing affine transformations to improve least-squares analyses of MQ-MAS spectra. The optimum ratio of RF field strength to the quadrupolar coupling constant and the corresponding pulse durations that maximize sensitivity within experimental constraints are also examined. 

A modified shifted-echo PIETA pulse sequence is developed to acquire natural abundance²⁹Si 2DJ-resolved spectra in crystalline silicates. 

An approach is presented for simulating multipulse nuclear magnetic resonance (NMR) spectra of polycrystalline solids directly in the frequency domain. The approach integrates the symmetry pathway concept for multipulse NMR with efficient algorithms for calculating spinning sideband amplitudes and performing interpolated finite-element numerical integration over all crystallite orientations in a polycrystalline sample. The numerical efficiency is achieved through a set of assumptions used to approximate the evolution of a sparse density matrix through a pulse sequence as a set of individual transition pathway signals. The utility of this approach for simulating the spectra of complex materials, such as glasses and other structurally disordered materials, is demonstrated. 

A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman–Solum–Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique’s efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt. 

The ratio of silicon-29 nuclear magnetic resonance (NMR) coherence lifetimes for Q 4 and Q 3 sites under magic-angle spinning and a train of π pulses in a series of binary alkali silicate glasses is used to detect phase separation, even at small scales where the glass appears optically homogenous. This approach exploits the dependence of echo train coherence lifetimes on the residual heteronuclear dipolar coupling between Si-29 and the NMR active nuclei of neighboring network modifier cations. The shifted-echo phase incremented echo train acquisition NMR method is used to eliminate modulation of the echo train amplitudes due to J couplings across Si–O–Si linkages. A 2D Fourier and inverse Laplace transform of this dataset provides a correlation of the isotropic 29 Si chemical shift to echo train coherence lifetimes, giving a sensitive probe of phase separation as well as chemical composition and local structure of the different phases. The 29 Si Q 4 :Q 3 mean coherence lifetime ratios are 28.8, 23.8, and 5.8 in the phase-separated glasses, 0.05Li 2 O·0.95SiO 2 , 0.1Li 2 O·0.9SiO 2 and 0.05Na 2 O·0.95SiO 2 , respectively, while the ratio is reduced to 2.1, 1.6, and 1.6 in glasses not exhibiting signs of phase separation, 0.05K 2 O·0.95SiO 2 , 0.05Cs 2 O·0.95SiO 2 and 0.10Cs 2 O·0.90SiO 2 , respectively. Phase separation inhibition, through addition of alumina, is also verified in 0.07Li 2 O·0.02Al 2 O 3 ·0.91SiO 2 .