Differential Line Broadening in the Presence of Quadrupolar–CSA Interference

The formalism for calculating the lineshape of a spin 1/2J-coupled to a high-spin nucleus undergoing quadrupolar and chemical shift anisotropy (CSA) relaxations is derived in the case where the tensors of both interactions are noncoincident and nonaxial. The expressions show that the CSA–quadrupolar interference term which is responsible for the asymmetry of lines involves a term depending on tensorial parameters. The effect of this term on the lineshapes is discussed with respect to three cases, namely coincident–axially symmetric, noncoincident–axially symmetric, and general noncoincident quadrupolar and CSA tensors. These cases are considered in the analysis of the lineshape of the¹H-decoupled spectra of the³¹P nucleusJ-coupled to the⁵⁹Co nucleus encountered in the tetrahedral cluster HFeCo₃(CO)₁₁PPh₂H.     

Simulations of molecular dynamics in solid-state NMR spectra of spin-1 nuclei including effects of CSA- and EFG-terms up to second order

By numerical simulations MAS and QCPMG methods for acquiring spectra of spin-1 nuclei were compared in order to determine the most sensitive experiment for analysis of molecular dynamics. To comply with the large quadrupolar constants for 14N and the CSA reported for 6Li both of these interactions are included up to second order. For 2H and 6Li both QCPMG and single-pulse MAS experiments were suitable for dynamics studies whereas the single-pulse MAS experiment were the method of choice for investigation of 14N dynamics for C(Q)'s larger than 750kHz at 14.1T. This property prohibits excitation of the 14N lineshape using either single hard or softer composite rf-pulses. Focusing on 14N it was demonstrated that the centerband lineshape is sensitive toward both off-MAS and CSA effects. In addition, excitation by real-time pulses showed that proper lineshapes corresponding to a site with a C(Q) of 3MHz may be excited by a very short pulse.     

An adaptive line-by-line—statistical model for fast and accurate spectral simulations in low-pressure plasmas

An adaptive grid method is proposed for the simulation of low-pressure plasma radiation. The method relies on two complementary approaches which significantly reduce calculation times and the size of the obtained grids: Weak lines are calculated as a so-called “pseudo-continuum”, hence reducing the number of calculated lines, and a numerical algorithm has been developed for accurately calculating Voigt lineshapes using a minimum number of points. The method is fully user-parametric, allowing the choice of privileging calculation efficiency, or alternatively privileging the accuracy of the computed spectra. Sample radiative transfer calculations are presented, which show the efficiency of the method, also providing some guidelines on how to define lineshape calculation parameters, depending on the problem to be solved.     

Two dimensional NQR separation of inhomogeneous and homogeneous lineshapes in disordered solids

A two-dimensional (2D) zero field NQR separation of inhomogeneous and homogeneous lineshapes technique is described. The nuclear spin Hamiltonian for spinsI>1/2 in zero magnetic field consists to a good approximation only of the electric quadrupole term. The 2D separation technique enables a separate spectroscopic observation of a static and a randomly time-fluctuating dynamic part of the quadrupole interaction. The separation is based on the fact that nuclear spin precession under a static quadrupolar Hamiltonian can be time-reversed whereas this can not be achieved under the action of a randomly time-fluctuaing Hamiltonian. The 2D spectrum displays in theω ₂-domain the inhomogeneously broadened lineshape, which is a convolution of the inhomogeneous static frequency distribution function and the homogeneous (adiabatic) lineshape. Theω ₁-domain shows the pure homogeneous lineshape. A deconvolution of the inhomogeneous lineshape with the homogeneous one yields a pure static inhomogeneous frequency distribution function which is characteristic and theoretically known for many different models of disordered solids like glasses and incommensurate systems. This technique is important in studies where both lineshapes have comparable widths. The 2D NQR separation technique has been applied to⁷⁵As in a proton glass Rb_0.98(NH₄)_0.02H₂AsO₄.     

Iterative lineshape analysis of quadrupolar echo spectra of a damped CD(3) quantum rotor: preliminary evidence of a novel mechanism of stochastic spin exchange

It is demonstrated that the wealth of information about damped quantum rotation of CD(3) groups, contained in quadrupolar echo spectra, can be fully explored in a broad temperature range using a method of iterative analysis of the spectral lineshapes. The recently reported lineshape equation which, apart from the quantum tunneling and the dissipative Alexander--Binsch terms, contains an additional dissipative term having no classical analog is shown to be capable of describing even subtle details of the spectra of a crystal of acetylsalicylic acid--CD(3) oriented specifically in the magnetic field. Preliminary evidence of the occurrence of this novel dissipative mechanism in the system studied is reported. The results obtained seem to suggest that there is no "classical limit" in the dissipative behavior of this system.     

Generalization of the lineshape useful in magnetic resonance spectroscopy

A new lineshape function is derived from the Tsallis distribution to describe electron paramagnetic resonance (EPR) spectra, and possibly nuclear magnetic resonance (NMR) spectra as well. This lineshape generalizes the Gaussian and Lorentzian lineshapes that are widely used in simulations. The main features of this lineshape function are presented: the normalization, moments, and first derivative. A number of experimental EPR spectra are compared with the results of simulations employing the new lineshape function. The results show that the new lineshape often provides a better approximation of the experimental spectrum. It is also shown that the new parameter of the lineshape function can be used to quantify the intermolecular spin-spin interactions.     

Quadrupolar and Chemical Shift Tensors Characterized by 2D Multiple-Quantum NMR Spectroscopy

The present work discusses a new 2D NMR method for characterizing the principal values and relative orientations of the electric field gradient and the chemical shift tensors of half-integer quadrupolar sites. The technique exploits the different contributions that quadrupolar and shielding interactions impart on the evolution of multiple-quantum and of single-quantum coherences, in order to obtain 2D powder lineshapes that are highly sensitive to these nuclear spin coupling parameters. Different spinning variants of this experiment were assayed, but it was concluded that a static version can yield the highest sensitivity to the values of the principal components and to the relative geometries of the local coupling tensors. It was found that correlating the central transition evolution with the highest available order of the spin coherence was also helpful for maximizing this spectral information. Good agreement between data obtained on ⁸⁷Rb (S = ) and ⁵⁹Co (S = ) samples and ideal theoretical lineshape predictions of this experiment was obtained, provided that heterogeneities in the multiple-quantum excitation and conversion processes were suitably accounted by procedures similar to those described in the spin- multiple-quantum NMR literature.     

Geometry of kinked protein helices from NMR data

Residual dipolar couplings between spin-1/2 and quadrupolar nuclei are often observed and exploited in the magic-angle spinning (MAS) NMR spectra of spin-1/2 nuclei. These orientation-dependent splittings contain information on the dipolar interaction, which can be translated into structural information. The same type of splittings may also be observed for pairs of quadrupolar nuclei, although information is often difficult to extract from the quadrupolar-broadened lineshapes. Here, the complete theory for describing the dipolar coupling between two quadrupolar nuclei in the frequency domain by Hamiltonian diagonalization is given. The theory is developed under MAS and double-rotation (DOR) conditions, and is valid for any spin quantum numbers, quadrupolar coupling constants, asymmetry parameters, and tensor orientations at both nuclei. All terms in the dipolar Hamiltonian become partially secular and contribute to the NMR spectrum. The theory is validated using experimental 11B and 35/37Cl NMR experiments carried out on powdered B-chlorocatecholborane, where both MAS and DOR are used to help separate effects of the quadrupolar interaction from those of the dipolar interaction. It is shown that the lineshapes are sensitive to the quadrupolar coupling constant of both nuclei and to the J coupling (including its sign). From these experiments, the dipolar coupling constant for a heteronuclear spin pair of quadrupolar nuclei may be obtained as well as the sign of the quadrupolar coupling constant of the perturbing nucleus; these are two parameters that are difficult to obtain experimentally otherwise.     

MAS NMR spectra of quadrupolar nuclei in disordered solids: the Czjzek model

Structural disorder at the scale of two to three atomic positions around the probe nucleus results in variations of the EFG and thus in a distribution of the quadrupolar interaction. This distribution is at the origin of the lineshape tailing toward high fields which is often observed in the MAS NMR spectra of quadrupolar nuclei in disordered solids. The Czjzek model provides an analytical expression for the joint distribution of the NMR quadrupolar parameters upsilon(Q) and eta from which a lineshape can be predicted. This model is derived from the Central Limit Theorem and the statistical isotropy inherent to disorder. It is thus applicable to a wide range of materials as we have illustrated for 27Al spectra on selected examples of glasses (slag), spinels (alumina), and hydrates (cement aluminum hydrates). In particular, when relevant, the use of the Czjzek model allows a quantitative decomposition of the spectra and an accurate extraction of the second moment of the quadrupolar product. In this respect, it is important to realize that only rotational invariants such as the quadrupolar product can make sense to describe the quadrupolar interaction in disordered solids.     

Fast and accurate characterization of biological membranes by EPR spectral simulations of nitroxides

A method by which it is possible to characterize the membranes of biological samples on the basis of the EPR spectral lineshape simulation of membrane-dissolved nitroxide spin probes is described. The presented simulation procedure allows the determination of the heterogeneous structure of biological membranes and fluidity characteristics of individual membrane domains. The method can deal with isotropic and anisotropic orientations of nitroxides introduced into the biological samples described by restricted fast motion with a correlation time between 0.01 and 10 ns. The linewidths of the Lorentzian lineshapes are calculated in a restricted fast-motion approximation. In the special case of samples with high concentrations of nitroxides or in the presence of paramagnetic ions, the lineshapes are calculated directly from the exchange-coupled Bloch equations. The parameters describing ordering, relaxation, polarity, and the portions of the individual spectral components are extracted by optimizing the simulated spectra to the experimental spectrum with either a Simplex or a Monte Carlo algorithm. To improve the algorithm's efficiency, a new way of characterizing the goodness of fits is introduced. The new criterion is based on the standard least-squares function, but with special weighting of the partial sums. Its benefits are confirmed with membrane spectral simulation. Two classes of examples-simulation and optimizations of synthetic spectra to evaluate the accuracy of the optimization algorithms and simulation and optimization of EPR spectra of nitroxides in liposome suspensions in the presence of a broadening agent and in human leukocytes are shown.     

Singularities in the lineshape of a second-order perturbed quadrupolar nucleus and their use in data fitting

Even for large quadrupolar interactions, the powder spectrum of the central transition for a half-integral spin is relatively narrow, because it is unperturbed to first order. However, the second-order perturbation is still orientation dependent, so it generates a characteristic lineshape. This lineshape has both finite step discontinuities and singularities where the spectrum is infinite, in theory. The relative positions of these features are well-known and they play an important role in fitting experimental data. However, there has been relatively little discussion of how high the steps are, so we present explicit formulae for these heights. This gives a full characterization of the features in this lineshape which can lead to an analysis of the spectrum without the usual laborious powder average.The transition frequency, as a function of the orientation angles, shows critical points: maxima, minima and saddle points. The maxima and minima correspond to the step discontinuities and the saddle points generate the singularities. Near a maximum, the contours are ellipses, whose dimensions are determined by the second derivatives of the frequency with respect to the polar and azimuthal angles. The density of points is smooth as the contour levels move up and down, but then drops to zero when a maximum is passed, giving a step. The height of the step is determined by the Hessian matrix—the matrix of all partial second derivatives. The points near the poles and the saddle points require a more detailed analysis, but this can still be done analytically. The resulting formulae are then compared to numerical simulations of the lineshape.We expand this calculation to include a relatively simple case where there is chemical shielding anisotropy and use this to fit experimental ¹³⁹La spectra of La₂O₃.     

Direct determination of gas velocity and gas temperature in an atmospheric-pressure argon-hydrogen plasma jet

Direct gas temperature and gas velocity measurements made in the exit plane of a subsonic argon-hydrogen thermal plasma jet from high-resolution lineshape analysis of laser light scattered by the plasma are reported. The lineshapes are in general a superposition of the ion feature of the Thomson-scattered light and the lineshape of Rayleigh scattered light. In the center of the jet Thomson scattering dominates while at larger radii Rayleigh scattering dominates. Because of the complexity of the lineshapes of light scattered by multicomponent plasmas, only those that are predominantly due to Thomson scattering can in practice be analyzed for gas temperature. Gas velocity can be determined from the Doppler shift of the lineshapes relative to the frequency of the incident laser if the velocity is greater than about 50 m s⁻¹. The maximum gas temperature measured was 14,500 K±5%. The maximum gas velocity measured was 1100 m s⁻¹±3%. Temperature values and the radial velocity profile are compared with those previously obtained from a subsonic all-argon plasma jet operated under similar conditions.     

A new method for suppressing the central transition in I=3/2 NMR spectra with a demonstration for 23Na in bovine articular cartilage

The splitting and the lineshape of the satellite transitions of 23Na are measures of the residual quadrupolar interaction and its distribution, which are related to the degrees of order and binding of sodium in biological tissues. However, these transitions are often masked by the stronger signals of the central transition and the isotropic sodium ions. A way to suppress the central signals, while preserving the lineshape and the intensity of the satellites, is suggested and tested on a liquid crystal and on bovine articular cartilage.     

Influence of atomic recoil on power broadened lineshapes in 2-level atoms

The power broadened absorption lineshapes of 2-level sodium atoms were carefully studied, using a frequency stabilized cw dye laser and a highly collimated atomic beam. At low driving fields the absorption lineshape was lorentzian as expected. However, at field intensities greater than 50 mW/cm², the lineshape exhibited an asymmetry which was attributed to atomic recoil effects.     

Magnetic resonance lineshape in the presence of slow motion

The effect on the magnetic resonance lineshape of slow molecular motion is considered. The molecules are assumed to undergo diffusive jumps of angle ε described by the distribution function W(ε)=(N/τ) exp (λ cos ε), where τ is the mean lifetime between jumps and λ a parameter describing the width of the distribution. As λ is varied from zero to infinity this distribution describes diffusion models that change continuously from the strong collision limit to Brownian rotational diffusion. Magnetic resonance lineshapes are calculated by Freed's method using the asymptotic expansion technique, and results are presented for an axially symmetric (secular) Zeeman hamiltonian.

An exact expression for the magnetic resonance lineshape is derived in the strong collision limit for any secular hamiltonian and a few examples are presented. It is also shown how this technique can be used as a basis for lineshape calculations in the presence of moderately large jumps.     

Single- and multiple-quantum cross-polarization in NMR of quadrupolar nuclei in static samples

Cross-polarization from a spin I=1/2 nucleus (e.g., ¹H) to a spin S = 3/2 nucleus (e.g., ²³Na) or a spin S = 5/2 nucleus (e.g., ²⁷A1 or ⁿO) in static powder samples is investigated. The results of conventional (single-quantum), three-quantum, and five-quantum cross-polarization experiments are presented and discussed. Based on a generalization of an existing theory of cross-polarization to quadrupolar nuclei, computer simulations are used to model the intensity and lineshape variations observed in cross-polarized NMR spectra as a function of the radio-frequency field strengths of the two simultaneous spin-locking pulses. These intensity and lineshape variations can also be understood in terms of the spin S = 3/2 or 5/2 nutation rates determined from experimental quadrupolar nutation spectra. The results of this study are intended as a preliminary step towards understanding single- and multiple-quantum cross-polarization to quadrupolar nuclei under MAS conditions and the application of these techniques to the MQMAS NMR experiment.     

Molecular dynamics of half-integer quadrupolar nuclei studied by QCPMG solid-state NMR experiments on static and rotating samples. Theory and simulations

Simulations of QCPMG NMR type experiments have been used to explore dynamic processes of half-integer quadrupolar nuclei in solids. By setting up a theoretical approach that is well suited for efficient numerical simulations the QCPMG type experiments have been analyzed regarding the effect of the magnitude of the EFG- and CSA-tensors, the spin-quantum number, different dynamical processes and MAS. Compared to the QE experiment the QCPMG experiment offers not only intensity gain by an order of magnitude and changes in overall lineshape as a function of the kinetic rate constant but the lineshape of the individual spin-echo sidebands is also very sensitive towards dynamics. Hereby a visual identification of the dynamics is obtained. In common for all the simulations the spin-echo sidebands are narrow in the slow (k< or =10(2) Hz) and the fast (k> or =10(7) Hz) dynamic regime whereas they are broadened in the intermediate regime 10(3)< or =k< or =10(7) Hz. The maximum intensity of the spin-echo sidebands for two-site jumps is highly dependent on the type of anisotropic interactions involved and the type of QCPMG experiment. Hence, in the fast limit the maximum intensity was 140% of the initial intensity when significant CSA was present or under the QCPMG-MAS experiment compared to 89 or 71% for the static experiment influenced by the quadrupolar interaction only. For 3-, 4-, and 6-site jumps the maximum intensity in the fast limit reached up to 339% of the intensity in the static limit.     

Solomon echoes for spin 7/2 by soft pulse excitation

Solomon echoes are calculated for spin 7/2 in solids taking into account the first-order quadrupolar interaction while the pulses are on. The computation is performed using the algebraic computer program ‘MAPLE’. Fifteen echoes are predicted and the amplitude of each echo is calculated. Each satellite transition produces five echoes whereas no echo is detected for the central transition. Among these echoes, six are ‘forbidden’ which are a result of the refocusing of exclusively multiple quantum coherences which are developed during the first pulse. These echoes cannot be predicted by a calculation based upon ‘hard’ pulse excitation. The results are valid for any ratio of the quadrupolar coupling to the frequency of the RF field (ω_Q/ω₁).     

Hot luminescence in polyacetylene

We propose that the unusual frequency dependent lineshapes observed in Raman scattering experiments on polyacetylene are due to hot luminescence in very long polyene chains. Employing a generalized equation of motion technique we find that a photoexcited polyene is unstable with respect to a variety of symmetry lowering distortions which vary as a function of the frequency of the exciting radiation. Radiative decay during this relaxation is observably strong in the inelastically scattered photon spectrum and would explain the dominant observed lineshape variations.