Irreducible spherical tensor analysis of quadrupolar nuclei

The present work introduces a new three-dimensional nuclear magnetic resonance (3D NMR) experiment for the analysis of half-integer quadrupolar nuclei in solids. The method is based on the multi-rank expansion of the high-field NMR Hamiltonian governing the central transition of these spins in terms of irreducible spherical tensor elements. This approach leads to a temporal spin evolution given by an isotropic term characteristic of each chemical site, as well as by second- and fourth-rank anisotropies depending on the principal values and relative orientations of the shielding and quadrupolar interactions. A method for extracting the 3D spectral distribution correlating these three frequency components is presented, based on the acquisition of dynamic-angle spinning NMR signals collected as a function of different initial and final spinning axes. Computational and instrumental details involved in the acquisition of these 3D dynamic-angle correlation spectroscopy (DACSY) data are discussed, and applications of the DACSY methodology to the analysis of different rubidium salts are illustrated. The new type of chemical information that this experiment can provide and its relation to other NMR techniques that have been recently developed for the analysis of quadrupolar nuclei in solids are also discussed.

Quantitative analysis of Earth’s field NMR spectra of strongly-coupled heteronuclear systems

Addition theorems can be constructed by doing three‐dimensional Taylor expansions according tof(r+r′)=exp (r′⋅∇)f(r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp (x′∂/∂x)exp (y′∂/∂y)exp (z′∂/∂z) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp (r′⋅∇) involving powers of the Laplacian ∇2and spherical tensor gradient operators\documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{Y}^{m}_{\ell}(\nabla)$\end{document}, which are irreducible spherical tensors of ranks zero and ℓ, respectively (Santos, F. D. Nucl Phys A 1973, 212, 341). In this way, it is indeed possible to derive addition theorems by doing three‐dimensional Taylor expansions (Weniger, E. J. Int J Quantum Chem 2000, 76, 280). The application of the translation operator in its spherical form is particularly simple in the case ofBfunctions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater‐type functions, bound‐state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums ofBfunctions, the addition theorems for these functions can be written down immediately. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Taking advantage of an order of magnitude in sensitivity enhancement obtained by sampling of quadrupolar-echo (QE) solid-state NMR spectra during a quadrupolar Carr−Purcell−Meiboom−Gill (QCPMG) pulse sequence, we demonstrate that the coordination environment of low-γ quadrupolar metal nuclei can be studied routinely for powders with these nuclei in natural abundance. The general applicability of the method is demonstrated by 39K, 25Mg, 67Zn, and 87Sr QCPMG solid-state NMR experiments for K2MoO4, KVO3, Mg(VO3)2, Zn(CH3COO)2·2H2O, Zn(Ala)2·H2O, Sr(NO3)2, and SrMoO4. For all samples the quadrupolar coupling parameters and the isotropic chemical shifts are extracted by numerical simulation and iterative fitting of the spin−echo sideband spectra observed in the experimental QCPMG NMR spectra. These parameters are discussed in light of the crystal structures for the compounds.

NMR spectra of polycrystalline samples of the binary nitrides ScN and VN were acquired under magic‐angle spinning. The observed nuclides 45Sc, 51V and 14N are all quadrupolar nuclei with a spin . However, due to the high symmetry of their rock‐salt type structures, the spectra of the nitrides do not exhibit effects of quadrupolar or other anisotropic interactions of significant magnitude. This allows a relatively straightforward evaluation of the acquired spectra, leading to isotropic chemical shift values (δiso) of −213 ppm (51VN), 378 ppm (V14N), 290 ppm (45ScN) and 442 ppm (Sc14N) against commonly used reference standards. In the wider context of 14N‐NMR of binary nitrides, it is shown that the distance of nitrogen to the nearest neighbour cation can be correlated to the observed chemical shift.

Feasibility of the overtone NMR for half-integer quadrupolar nuclei

Sodium ion dynamics in the nonstoichiometric layer-type oxide Na 0.67Ni 0.33Ti 0.67O 2 studied by 23Na NMR

A comparison of mesophase formation under sparging and vacuum

The transition of simple aqueous saline solutions (buffered 0.1 M NaCl) of extremely long DNA (8 kilobases ≃ 2.7 μm ≃ 54 persistence lengths) from the isotropic to anisotropic phase was characterized by solid-state 31 P NMR spectroscopy and polarized light microscopy. The anisotropic phase first appeared in equilibrium with the isotropic phase at a concentration C i * = 13 mg/mL and the isotropic phase disappeared at a concentration C a * = 67 mg/mL. The first critical concentration (C i * ) is much lower than that observed for a stiffer polyelectrolyte (xanthan) and predicted by theory incorporating the influences of chain flexibility and charge

Preparation, thermal properties, morphology, and microstructure of phosphorus-containing epoxy/SiO 2 and polyimide/SiO 2 nanocomposites

To develop a framework for 3D sodium (23 Na) MR fingerprinting (MRF), based on irreducible spherical tensor operators with tailored flip angle (FA) pattern and time-efficient data acquisition for simultaneous quantification of T1 , , , and in addition to ΔB0 . 23 Na-MRF was implemented in a 3D sequence and irreducible spherical tensor operators were exploited in the simulations. Furthermore, the Cramér Rao lower bound was used to optimize the flip angle pattern. A combination of single and double echo readouts was implemented to increase the readout efficiency. A study was conducted to compare results in a multicompartment phantom acquired with MRF and reference methods. Finally, the relaxation times in the human brain were measured in four healthy volunteers. Phantom experiments revealed a mean difference of 1.0% between relaxation times acquired with MRF and results determined with the reference methods. Simultaneous quantification of the longitudinal and transverse relaxation times in the human brain was possible within 32 min using 3D 23 Na-MRF with a nominal resolution of (5 mm)3 . In vivo measurements in four volunteers yielded average relaxation times of: T1,brain = (35.0 ± 3.2) ms, = (29.3 ± 3.8) ms and = (5.5 ± 1.3) ms in brain tissue, whereas T1,CSF = (61.9 ± 2.8) ms and = (46.3 ± 4.5) ms was found in cerebrospinal fluid. The feasibility of in vivo 3D relaxometric sodium mapping within roughly ½ h is demonstrated using MRF in the human brain, moving sodium relaxometric mapping toward clinically relevant measurement times.

Simulations are essential in NMR spectroscopy for understanding and quantitatively predicting the outcomes of experiments. Dynamic NMR experiments are influenced by relaxation and chemical exchange phenomena, which complicate their interpretation. The quantum mechanical theory behind NMR simulations is well established and enables simulations under various conditions or states of the system. In particular, the representation of the spin system in terms of the density matrix, energy and pulse operators, and relaxation and chemical exchange in Liouville space is straightforward and allows for the routine simulation of dynamic NMR experiments. In fact, several advanced open-source software packages are currently available-such as Spinach, GAMMA (pyGAMMA), Simpson, ChemEx, and NMR TITAN-which users can choose from based on their simulation needs. Many of these software packages are optimized for computational speed and make use of sophisticated bases, such as the irreducible spherical tensor basis. Here, we present simulations using the direct method (spin product basis), as implemented in GAMMA, which we believe will help users better understand the underlying theory and thereby explore the other available software resources more effectively. We chose two experiments-CPMG-RD and CEST-as a case study to understand the conformational changes in proteins or macromolecules occurring in the fast and slow exchange regimes, which are frequently investigated by structural biologists.

The Full Rotation Group and Irreducible Spherical Tensors

The anisotropic character of the chemical shift can be measured by experiments that provide shift tensor values and comparing these experimental components, obtained from microcrystalline powders, with 3D nuclear shielding tensor components, calculated with quantum chemistry, yields structural models of the analyzed molecules. The use of a metric tensor for evaluating the mean squared deviations, d2, between two or more tensors provides a statistical way to verify the molecular structure governing the theoretical shielding components. The sensitivity of the method is comparable with diffraction methods for the heavier organic atoms (i.e., C, O, N, etc.) but considerably better for the positions of H atoms. Thus, the method is especially powerful for H‐bond structure, the position of water molecules in biomolecular species, and other proton important structural features, etc. Unfortunately, the traditional Cartesian tensor components appear as reducible metric representations and lack the orthogonality of irreducible icosahedral and irreducible spherical tensors, both of which are also easy to normalize. Metrics give weighting factors that carry important statistical significance in a structure determination. Details of the mathematical analysis are presented and examples given to illustrate the reason nuclear magnetic resonance are rapidly assuming an important synergistic relationship with diffraction methods (X‐ray, neutron scattering, and high energy synchrotron irradiation). © 2009 Wiley Periodicals, Inc.Concepts Magn Reson Part A 34A: 217–237, 2009.

All Hamiltonians of spin interactions, which are based on second-rank tensors (see Tables 46.1 to 46.5), can be expressed in the general form $$ \mathcal{H}_{i}=a_{i}\sum_{l=0,2} \sum_{m=-2}^{l} \left ( -1 \right)^{m}A_{l},-mT_{l,m} $$ (49.1) where ai is a constant specific for the interaction. The quantities A l ,_ m are functions of the spatial orientation and distances (usually given in polar coordinates (r, ϕ, ϑ) as defined in Chap. 48), whereas T l,m represents functions of spin operators. The analytical form of these expressions are chosen in such a way that the Hamiltonians transform under rotations like spherical harmonics of rank l. They are therefore referred to as irreducible spherical tensor (IST) operators of the spatial and the spin coordinates, respectively. “Irreducible” means that no decomposition in terms transforming under rotations other than spherical harmonics of the same rank l is possible.1 The subscript m indicates the order of multiple-quantum transition induced by T l,m .