Simulating multipulse NMR spectra of polycrystalline solids in the frequency domain.

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.

Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions.

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.

Silicon-29 echo train coherence lifetimes and geminal 2J-couplings in network modified silicate glasses.

The natural abundance 29Si echo-train coherence lifetimes in network-modified silicate glasses were examined under static and magic-angle spinning (MAS) conditions. The nuclear magnetic properties of modifier cations were found to play a major role in determining 29Si coherence lifetimes, leading to differences as large as three orders of magnitude. In compositions with abundant NMR active nuclei, such as alkali silicates, the 29Si coherence lifetimes are dominated by coherent dephasing due to residual heteronuclear dipolar couplings, whereas in compositions dilute in NMR active nuclei, such as alkaline earth silicates, the 29Si coherence lifetimes are dominated by incoherent dephasing due to paramagnetic impurities. Expressing the inverse of the coherence lifetime as a residual full width at half maximum (FWHM), we found that increasing rates of both MAS and a π-pulse train are effective in removing the residual 29Si heteronuclear broadenings, with a near-linear relationship between FWHM and MAS rotor period and π-pulse spacing. Based on these results, we conclude that accurate 29Si J coupling measurements will be the most challenging in lithium silicate glasses due to strong homonuclear dipolar couplings among 7Li nuclei, requiring MAS speeds up to 100 kHz, and be the least challenging in the alkaline earth silicate glasses. At a modest MAS speed of 14kHz, distributions of geminal J couplings across Si-O-Si linkages were measured in alkali and alkaline earth silicate glasses giving mean values of 4.2Hz and 5.1Hz in 0.4 CaO·0.6 SiO2 and 0.33 Ba2O·0.67 SiO2 glasses, respectively, and 5.2Hz and 5.3Hz in 0.33 Na2O·0.67 SiO2 and 0.33 K2O·0.67 SiO2 glasses, respectively. We also observe greater variance in the J distributions of alkaline earth silicate glasses consistent with greater structural disorder due to increased modifier cation potential, i.e., the charge-to-radius ratio, Z/r of the cation.

Statistical learning of NMR tensors from 2D isotropic/anisotropic correlation nuclear magnetic resonance spectra.

Many linear inversion problems involving Fredholm integrals of the first kind are frequently encountered in the field of magnetic resonance. One important application is the direct inversion of a solid-state nuclear magnetic resonance (NMR) spectrum containing multiple overlapping anisotropic subspectra to obtain a distribution of the tensor parameters. Because of the ill-conditioned nature of this inverse problem, we investigate the use of the truncated singular value decomposition and the smooth least absolute shrinkage and selection operator based regularization methods, which (a) stabilize the solution and (b) promote sparsity and smoothness in the solution. We also propose an unambiguous representation for the anisotropy parameters using a piecewise polar coordinate system to minimize rank deficiency in the inversion kernel. To obtain the optimum tensor parameter distribution, we implement the k-fold cross-validation, a statistical learning method, to determine the hyperparameters of the regularized inverse problem. In this article, we provide the details of the linear-inversion method along with numerous illustrative applications on purely anisotropic NMR spectra, both synthetic and experimental two-dimensional spectra correlating the isotropic and anisotropic frequencies.

Core Scientific Dataset Model: A lightweight and portable model and file format for multi-dimensional scientific data.

The Core Scientific Dataset (CSD) model with JavaScript Object Notation (JSON) serialization is presented as a lightweight, portable, and versatile standard for intra- and interdisciplinary scientific data exchange. This model supports datasets with a p-component dependent variable, {U0, …, Uq, …, Up-1}, discretely sampled at M unique points in a d-dimensional independent variable (X0, …, Xk, …, Xd-1) space. Moreover, this sampling is over an orthogonal grid, regular or rectilinear, where the principal coordinate axes of the grid are the independent variables. It can also hold correlated datasets assuming the different physical quantities (dependent variables) are sampled on the same orthogonal grid of independent variables. The model encapsulates the dependent variables' sampled data values and the minimum metadata needed to accurately represent this data in an appropriate coordinate system of independent variables. The CSD model can serve as a re-usable building block in the development of more sophisticated portable scientific dataset file standards.

Hydrogen motional disorder in crystalline iron group chloride dihydrates.

The principal components and the relative orientation of the 2H paramagnetic shift and quadrupolar coupling tensors have been measured for the MCl2·2D2O family of compounds, M = Mn, Fe, Co, Ni, and Cu, using the two-dimensional shifting-d echo nuclear magnetic resonance experiment in order to determine (1) the degree of unpaired electron delocalization and (2) the number and location of crystallographically distinct hydrogen sites around oxygen and their fractional occupancies. Expressions for the molecular susceptibility of 3d ion systems, where the spin-orbit coupling is a weak perturbation onto the crystal field, are derived using the generalized Van Vleck equation and used to predict molecular susceptibilities. These predicted molecular susceptibilities are combined with various point dipole source configurations modeling unpaired electron delocalization to predict 2H paramagnetic shift tensors at potential deuterium sites. The instantaneous deuterium quadrupolar coupling and shift tensors are then combined with parameterized motional models, developed for trigonally (M = Mn, Fe, Co, and Cu) and pyramidally (M = Ni) coordinated D2O ligands, to obtain the best fit of the experimental 2D spectra. Dipole sources placed onto metal nuclei with a small degree of delocalization onto the chlorine ligands yield good agreement with the experiment for M = Mn, Fe, Co, and Ni, while good agreement for CuCl2·2D2O is obtained with additional delocalization onto the oxygen. Our analysis of the salts with trigonally coordinated water ligands (M = Mn, Fe, Co, and Cu) confirms the presence of bisector flipping and the conclusions from neutron scattering measurements that hydrogen bonding to chlorine on two adjacent chains leads to the water molecule in the [M(D2O)2Cl4] cluster being nearly coplanar with O-M-Cl involving the shortest metal-chlorine bonds of the cluster. In the case of NiCl2·2D2O, the experimental parameters were found to be consistent with a motional model where the D2O ligands are pyramidally coordinated to the metal and undergo bisector flipping while the water ligand additionally hops between two orientations related by a 120° rotation about the Ni-O bond axis. The position of the three crystallographically distinct hydrogen sites in the unit cell was determined along with fractional occupancies. This restricted water ligand motion is likely due to van der Waals interactions and is concerted with the motion of neighboring ligands.

Modifier cation effects on (29)Si nuclear shielding anisotropies in silicate glasses.

We have examined variations in the (29)Si nuclear shielding tensor parameters of SiO4 tetrahedra in a series of seven alkali and alkaline earth silicate glass compositions, Cs2O·4.81 SiO2, Rb2O·3.96 SiO2, Rb2O·2.25 SiO2, K2O·4.48 SiO2, Na2O·4.74 SiO2, BaO·2.64 SiO2, and SrO·2.36 SiO2, using natural abundance (29)Si two-dimensional magic-angle flipping (MAF) experiments. Our analyses of these 2D spectra reveal a linear dependence of the (29)Si nuclear shielding anisotropy of Q((3)) sites on the Si-non-bridging oxygen bond length, which in turn depends on the cation potential and coordination of modifier cations to the non-bridging oxygen. We also demonstrate how a combination of Cu(2+) as a paramagnetic dopant combined with echo train acquisition can reduce the total experiment time of (29)Si 2D NMR measurements by two orders of magnitude, enabling higher throughput 2D NMR studies of glass structure.

Two-dimensional NMR measurement and point dipole model prediction of paramagnetic shift tensors in solids.

A new two-dimensional Nuclear Magnetic Resonance (NMR) experiment to separate and correlate the first-order quadrupolar and chemical/paramagnetic shift interactions is described. This experiment, which we call the shifting-d echo experiment, allows a more precise determination of tensor principal components values and their relative orientation. It is designed using the recently introduced symmetry pathway concept. A comparison of the shifting-d experiment with earlier proposed methods is presented and experimentally illustrated in the case of (2)H (I = 1) paramagnetic shift and quadrupolar tensors of CuCl2⋅2D2O. The benefits of the shifting-d echo experiment over other methods are a factor of two improvement in sensitivity and the suppression of major artifacts. From the 2D lineshape analysis of the shifting-d spectrum, the (2)H quadrupolar coupling parameters are 〈Cq〉 = 118.1 kHz and 〈ηq〉 = 0.88, and the (2)H paramagnetic shift tensor anisotropy parameters are 〈ζP〉 = - 152.5 ppm and 〈ηP〉 = 0.91. The orientation of the quadrupolar coupling principal axis system (PAS) relative to the paramagnetic shift anisotropy principal axis system is given by (α,ß,γ)=(π2,π2,0). Using a simple ligand hopping model, the tensor parameters in the absence of exchange are estimated. On the basis of this analysis, the instantaneous principal components and orientation of the quadrupolar coupling are found to be in excellent agreement with previous measurements. A new point dipole model for predicting the paramagnetic shift tensor is proposed yielding significantly better agreement than previously used models. In the new model, the dipoles are displaced from nuclei at positions associated with high electron density in the singly occupied molecular orbital predicted from ligand field theory.

Sideband separation experiments in NMR with phase incremented echo train acquisition.

A general approach for enhancing sensitivity of nuclear magnetic resonance sideband separation experiments, such as Two-Dimensional One Pulse (TOP), Magic-Angle Turning (MAT), and Phase Adjust Spinning Sidebands (PASS) experiments, with phase incremented echo-train acquisition (PIETA) is described. This approach is applicable whenever strong inhomogeneous broadenings dominate the unmodulated frequency resonances, such as in non-crystalline solids or in samples with large residual frequency anisotropy. PIETA provides significant sensitivity enhancements while also eliminating spectral artifacts would normally be present with Carr-Purcell-Meiboom-Gill acquisition. Additionally, an intuitive approach is presented for designing and processing echo train acquisition magnetic resonance experiments on rotating samples. Affine transformations are used to relate the two-dimensional signals acquired in TOP, MAT, and PASS experiments to a common coordinate system. Depending on sequence design and acquisition conditions two significant artifacts can arise from truncated acquisition time and discontinuous damping in the T2 decay. Here we show that the former artifact can always be eliminated through selection of a suitable affine transformation, and give the conditions in which the latter can be minimized or removed entirely.

Communication: Phase incremented echo train acquisition in NMR spectroscopy.

We present an improved and general approach for implementing echo train acquisition (ETA) in magnetic resonance spectroscopy, particularly where the conventional approach of Carr-Purcell-Meiboom-Gill (CPMG) acquisition would produce numerous artifacts. Generally, adding ETA to any N-dimensional experiment creates an N + 1 dimensional experiment, with an additional dimension associated with the echo count, n, or an evolution time that is an integer multiple of the spacing between echo maxima. Here we present a modified approach, called phase incremented echo train acquisition (PIETA), where the phase of the mixing pulse and every other refocusing pulse, φ(P), is incremented as a single variable, creating an additional phase dimension in what becomes an N + 2 dimensional experiment. A Fourier transform with respect to the PIETA phase, φ(P), converts the φ(P) dimension into a Δp dimension where desired signals can be easily separated from undesired coherence transfer pathway signals, thereby avoiding cumbersome or intractable phase cycling schemes where the receiver phase must follow a master equation. This simple modification eliminates numerous artifacts present in NMR experiments employing CPMG acquisition and allows "single-scan" measurements of transverse relaxation and J-couplings. Additionally, unlike CPMG, we show how PIETA can be appended to experiments with phase modulated signals after the mixing pulse.

TOP-PASS: a processing algorithm to reduce 2D PASS acquisition time.

A slow speed MAS spectrum contains a pattern of spinning sideband resonances separated by integer multiples of the rotor frequency and centered about an isotropic frequency. The 2D signal acquired in a two-dimensional Phase Adjusted Spinning Sideband (PASS) experiment correlates this slow speed MAS spectrum, obtained in the direct dimension, to an indirect dimension spectrum containing the same pattern of spinning sideband resonances centered about a frequency of zero. An affine transformation is used to convert the acquired 2D PASS signal into a 2D signal that correlates a spectrum of pure isotropic frequencies to a spectrum of spinning sideband resonances with no isotropic frequency contributions. The conventional affine transform applied to 2D PASS consists of an active shear of the signal parallel to the indirect time domain coordinate followed by a passive scaling of the indirect time domain coordinate. Here we show that an alternative affine transform, previously employed in the Two-dimensional One Pulse (TOP) experiment, can be employed to create the same 2D signal correlation with an enhanced spectral width in the anisotropic (spinning sideband) dimension. This enhancement can provide a significant reduction in the minimum experiment time required for a 2D PASS experiment, particularly for spectra where the individual spinning sideband patterns are dispersed over a wider spectral range than the isotropic resonance frequencies. The TOP processing consists of an active shear of the signal parallel to the direct time domain, followed by an active shear of the signal parallel to the new indirect time domain coordinate followed by a passive scaling of the new direct time domain coordinate. A theoretical description of the affine transformation in the context of 2D PASS is given along with illustrative examples of (29)Si in Clinoenstatite and (13)C in l-Histidine.

Broadband inversion for MAS NMR with single-sideband-selective adiabatic pulses.

We explain how and under which conditions it is possible to obtain an efficient inversion of an entire sideband family of several hundred kHz using low-power, sideband-selective adiabatic pulses, and we illustrate with some experimental results how this framework opens new avenues in solid-state NMR for manipulating spin systems with wide spinning-sideband (SSB) manifolds. This is achieved through the definition of the criteria of phase and amplitude modulation for designing an adiabatic inversion pulse for rotating solids. In turn, this is based on a framework for representing the Hamiltonian of the spin system in an NMR experiment under magic angle spinning (MAS). Following earlier ideas from Caravatti et al. [J. Magn. Reson. 55, 88 (1983)], the so-called "jolting frame" is used, which is the interaction frame of the anisotropic interaction giving rise to the SSB manifold. In the jolting frame, the shift modulation affecting the nuclear spin is removed, while the Hamiltonian corresponding to the RF field is frequency modulated and acquires a spinning-sideband pattern, specific for each crystallite orientation.

Trading sensitivity for information: Carr-Purcell-Meiboom-Gill acquisition in solid-state NMR.

The Carr-Purcell-Meiboom-Gill (CPMG) experiment has gained popularity in solid-state NMR as a method for enhancing sensitivity for anisotropically broadened spectra of both spin 1/2 and half integer quadrupolar nuclei. Most commonly, the train of CPMG echoes is Fourier transformed directly, which causes the NMR powder pattern to break up into a series of sidebands, sometimes called "spikelets." Larger sensitivity enhancements are observed as the delay between the pi pulses is shortened. As the duration between the pi pulses is shortened, however, the echoes become truncated and information about the nuclear spin interactions is lost. We explored the relationship between enhanced sensitivity and loss of information as a function of the product Omega 2tau, where Omega is the span of the anisotropic lineshape and 2tau is the pi pulse spacing. For a lineshape dominated by the nuclear shielding anisotropy, we found that the minimum uncertainty in the tensor values is obtained using Omega 2tau values in the range Omega 2tau approximately 12(-1)(+6) and Omega 2tau approximately 9(-3)(+3) for eta(s)=0 and eta(s)=1, respectively. For an anisotropic second-order quadrupolar central transition lineshape under magic-angle spinning (MAS), the optimum range of Omega 2tau approximately 9(-2)(+3) was found. Additionally, we show how the Two-dimensional One Pulse (TOP) like processing approach can be used to eliminate the cumbersome sideband pattern lineshape and recover a more familiar lineshape that is easily analyzed with conventional lineshape simulation algorithms.

Q(n) species distribution in K2O.2SiO2 glass by 29Si magic angle flipping NMR.

Two-dimensional magic angle flipping (MAF) was employed to measure the Q((n)) distribution in a (29)Si-enriched potassium disilicate glass (K(2)O.2SiO(2)). Relative concentrations of [Q((4))] = 7.2 +/- 0.3%, [Q((3))] = 82.9 +/- 0.1%, and [Q((2))] = 9.8 +/- 0.6% were obtained. Using the thermodynamic model for Q((n)) species disproportionation, these relative concentrations yield an equilibrium constant k(3) = 0.0103 +/- 0.0008, indicating, as expected, that the Q((n)) species distribution is close to binary in the potassium disilicate glass. A Gaussian distribution of isotropic chemical shifts was observed for each Q((n)) species with mean values of -82.74 +/- 0.03, -91.32 +/- 0.01, and -101.67 +/- 0.02 ppm and standard deviations of 3.27 +/- 0.03, 4.19 +/- 0.01, and 5.09 +/- 0.03 ppm for Q((2)), Q((3)), and Q((4)), respectively. Additionally, nuclear shielding anisotropy values of zeta =-85.0 +/- 1.3 ppm, eta = 0.48 +/- 0.02 for Q((2)) and zeta = -74.9 +/- 0.2 ppm, eta = 0.03 +/- 0.01 for Q((3)) were observed in the potassium disilicate glass.

Optimum excitation of "enhanced" central transition populations.

Central transition (CT) sensitivity enhancement schemes that transfer polarization from satellites to the CT through selective saturation or inversion of neighboring satellite transitions have provided a welcome improvement for magic-angle spinning spectra of half-integer quadrupole nuclei. While many researchers have investigated and developed different methods of creating enhanced CT populations, here we investigate the conversion of these enhanced CT populations into observable CT coherence. We show a somewhat unexpected result that a conversion pulse length optimized for maximum sensitivity on equilibrium populations may not be optimum for an enhanced (non-equilibrium) polarization. Furthermore, CT enhancements can be lost if excessive rf field strength is used to convert this enhanced polarization into CT coherence. While a maximally enhanced CT signal is expected when using a perfectly selective CT conversion pulse, we have found that significant sensitivity loss can occur when using surprisingly low rf field strengths, even for sites with relatively large quadrupole coupling constants. We have systematically investigated these issues, and present some general guidelines and expectations when optimizing the conversion of enhanced (non-equilibrium) CT populations into observable CT coherence.

Superadiabaticity in magnetic resonance.

Adiabaticity plays a central role in modern magnetic resonance experiments, as excitations with adiabatic Hamiltonians allow precise control of the dynamics of the spin states during the course of an experiment. Surprisingly, many commonly used adiabatic processes in magnetic resonance perform well even though the adiabatic approximation does not appear to hold throughout the process. Here we show that this discrepancy can now be explained through the use of Berry's superadiabatic formalism, which provides a framework for including the finite duration of the process in the theoretical and numerical treatments. In this approach, a slow, but finite time-dependent Hamiltonian is iteratively transformed into time-dependent diagonal frames until the most accurate adiabatic approximation is obtained. In the case of magnetic resonance, the magnetization during an adiabatic process of finite duration is not locked to the effective Hamiltonian in the conventional adiabatic frame, but rather to an effective Hamiltonian in a superadiabatic frame. Only in the superadiabatic frame can the true validity of the adiabatic approximation be evaluated, as the inertial forces acting in this frame are the true cause for deviation from adiabaticity and loss of control during the process. Here we present a brief theoretical background of superadiabaticity and illustrate the concept in the context of magnetic resonance with commonly used shaped radio-frequency pulses.

Separating chemical shift and quadrupolar anisotropies via multiple-quantum NMR spectroscopy.

Chemical shift anisotropy (CSA) has been an invaluable probe of structure and dynamics for a variety of systems in NMR spectroscopy. Unfortunately, the presence of strong quadrupolar couplings has severely limited the ability to measure CSA in nuclei with spins I > 1/2. Here we show that these two interactions can be refocused at different times in a 2D multiple-quantum NMR experiment on polycrystalline samples. Combining this experiment with appropriate affine transformations allows these interactions to be cleanly separated into orthogonal dimensions. The 1D projection onto each axis can be fit to extract the respective principal tensor components. These components can then be used to fit the 2D spectrum for the relative orientation between the CSA and quadrupolar-coupling tensors. The necessary affine transformation parameters are given for all possible I values. Illustrative examples of spectra and analyses are given for 63Cu in K3[Cu(CN)4], 59Co in K3[Co(CN)6], and 87Rb in RbCrO4.

Solid-state nuclear magnetic resonance in the rotating tilted frame.

Recent methodological advances have made it possible to measure fine structure on the order of a few hertz in the nuclear magnetic resonance (NMR) spectra of quadrupolar nuclei in polycrystalline samples. Since quadrupolar couplings are often a significant fraction of the Zeeman coupling, a complete analysis of such experimental spectra requires a theoretical treatment beyond first-order. For multiple pulse NMR experiments, which may include sample rotation, the traditional density matrix approaches for treating higher-order effects suffer from the constraint that undesired fast oscillations (i.e., multiples of the Zeeman frequency), which arise from allowed overtone transitions, can only be eliminated in numerical simulations by employing sampling rates greater than 2I times the Zeeman frequency. Here, we present a general theoretical approach for arbitrary spin I that implements an analytical "filtering" of undesired fast oscillations in the rotating tilted frame, while still performing an exact diagonalization. Alternatively, this approach can be applied using a perturbation expansion for the eigenvalues and eigenstates, such that arbitrary levels of theory can be explored. The only constraint in this approach is that the Zeeman interaction remains the dominant interaction. Using this theoretical framework, numerical simulations can be implemented without the need for a high sampling rate of observables and with significantly reduced computation times. Additionally, this approach provides a general procedure for focusing on the excitation and detection of both fundamental and overtone transitions. Using this approach we explore higher-order effects on a number of sensitivity and resolution issues with NMR of quadrupolar nuclei.

A computational investigation of 17O quadrupolar coupling parameters and structure in alpha-quartz phase GeO2.

Ab initio band-structure calculations based on density functional theory have been completed for alpha-quartz phase GeO2 to obtain electric-field gradients (efg) for oxygen atoms, including those for GeO2 at elevated pressure and temperature. To interpret the resulting efg values and examine correlations between structure and 17O quadrupolar coupling parameters, additional ab initio self-consistent Hartree-Fock molecular orbital calculations were completed. The quadrupolar coupling constant was found to have a strong dependence on Ge-O distance and angleGe-O-Ge, with the quadrupolar asymmetry parameter being primarily dependent on angleGe-O-Ge. Analytical expressions describing these dependencies consistent with earlier investigations of analogous silicate compounds are also reported.