Spin-locking mechanism of spin I = 3/2 quadrupolar nuclei undergo magic angle spinning

The spin-locking mechanism of the spin I=3/2 quadrupolar nuclei under magic angle spinning (MAS) has been theoretically and experimentally investigated, and the criterion of adiabatic passage around zero-crossings of the quadrupole splitting was inferred from the time-dependent Shrödinger equation in this article. The theory, numerical simulations, and experiments conducted in this work all indicated that second-order quadrupole interaction and off-resonance play important roles in the spin-locking of the quadrupolar nuclei, and they were responsible for the great loss of the spin-locking signals. The spin-locking for a spin I=3/2 nucleus might be achieved by minimizing the effect of the second-order quadrupole interaction by using a radio frequency (RF) offset. This offset was realized by setting the RF to the opposite position of the isotropic second-order quadrupolar shift of single quantum coherences.     

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.     

CP/MAS of quadrupolar S = 3/2 nuclei.

The spin dynamics of Hartmann-Hahn cross-polarization from I = 1/2 to quadrupolar S = 3/2 nuclei is investigated. A density-matrix model applicable to cases where the quadrupole frequency vQ is much larger than the rf amplitude v1S of the S spins, predicts the time development of the spin state of an isolated I, S spin pair in static situations and in three distinct cases of magic-angle-spinning speed vR. These cases are characterized as slow, intermediate, and fast, depending on the magnitude of the parameter alpha = v1S2/vQvR relative to the intermediate value of 0.4. The model predictions are supported by numerical simulations. The polarization transfer from I to S is efficient in the limits of slow and fast sample spinning. When alpha < 1, the Hartmann-Hahn condition is shifted over once or twice vR. When the spinning rate is intermediate, poor spin-locking of the quadrupolar spins prevents the accumulation of a cross-polarization signal and, in addition, depletes the spin-locked I magnetization. Experimental CP/MAS data obtained in NaOH show that the concepts developed for isolated spin pairs are also applicable to cross-polarization in a strongly coupled multi-spin system.     

Dynamics of Spin I=3/2 under Spin-Locking Conditions in an Ordered Environment

We have derived approximate analytic solutions to the master equation describing the evolution of the spin I=3/2 density operator in the presence of a radio-frequency (RF) field and both static and fluctuating quadrupolar interactions. Spectra resulting from Fourier transformation of the evolutions of the on-resonance spin-locked magnetization into the various coherences display two satellite pairs and, in some cases, a central line. The central line is generally trimodal, consisting of a narrow component related to a slowly relaxing mode and two broad components pertaining to two faster relaxing modes. The rates of the fast modes are sensitive to slow molecular motion. Neither the amplitude nor the width of the narrow component is affected by the magnitude of the static coupling, whereas the corresponding features of the broad components depend in a rather complicated manner on the spin-lock field strength and static quadrupolar interaction. Under certain experimental conditions, the dependencies of the amplitudes on the dynamics are seen to vanish and the relaxation rates reduce to relatively simple expressions. One of the promising emerging features is the fact that the evolutions into the selectively detected quadrupolar spin polarization order and the rank-two double-quantum coherence do not exhibit a slowly relaxing mode and are particularly sensitive to slow molecular motion. Furthermore, these coherences can only be excited in the presence of a static coupling and this makes it possible to discern nuclei in anisotropic from those in isotropic environment. The feasibility of the spin-lock pulse sequences with limited RF power and a nonvanishing average electric field gradient has been demonstrated through experiments on sodium in a dense lyotropic DNA liquid crystal.     

Separation of isotropic chemical and second-order quadrupolar shifts by multiple-quantum double rotation NMR

Using a two-dimensional multiple-quantum (MQ) double rotation (DOR) experiment the contributions of the chemical shift and quadrupolar interaction to isotropic resonance shifts can be completely separated. Spectra were acquired using a three-pulse triple-quantum z-filtered pulse sequence and subsequently sheared along both the ν₁ and ν₂ dimensions. The application of this method is demonstrated for both crystalline (RbNO₃) and amorphous samples (vitreous B₂O₃). The existence of the two rubidium isotopes (⁸⁵Rb and ⁸⁷Rb) allows comparison of results for two nuclei with different spins (I = 3/2 and 5/2), as well as different dipole and quadrupole moments in a single chemical compound. Being only limited by homogeneous line broadening and sample crystallinity, linewidths of approximately 0.1 and 0.2 ppm can be measured for ⁸⁷Rb in the quadrupolar and chemical shift dimensions, enabling highly accurate determination of the isotropic chemical shift and the quadrupolar product, P_Q. For vitreous B₂O₃, the use of MQDOR allows the chemical shift and electric field gradient distributions to be directly determined—information that is difficult to obtain otherwise due to the presence of second-order quadrupolar broadening.     

Calculation of pulsed NMR signal in I=3/2 quadrupolar spin system

The rf pulse response of I=3/2 spin system experiencing first order quadrupolar splitting is studied using density matrix approach. A general expression is derived in terms of spin populations, quadrupole splitting and duration and amplitude of the rf pulse for calculating the NMR signal arising due to the centre line and satellite resonances for the situation where the impressed rf pulse excites the resonances selectively as well as non-selectively. The necessary 4×4 transformation matrix obtained analytically by diagonalyzing the Hamiltonian are used to get the expression for the centre line response. The satellite signals are obtained in the same way but by using the numerical values of the roots of the related quartics. The widths of the corresponding π/2-pulses are calculated for different initial spin populations. The variations of this pulse-width and the corresponding signal amplitude as a function of satellite splitting are studied.     

Triple quantum MQMAS spectroscopy of (I=7/2) in Na3Co(NO2)6 and trans-Co[(en2)(NO2)2]NO3 interplay between the quadrupole coupling and anisotropic shielding tensors

The purpose of this paper is to investigate the interplay between the chemical shielding anisotropy and quadrupole interaction in MQMAS spectra. in the compounds Na₃Co(NO₂)₆ and trans-Co[(en₂)(NO₂)₂]NO₃ provides model systems for such an investigation. Furthermore, only few results have been reported on the application of the MQMAS method to a spin I=7/2. The possibilities of the MQMAS spectroscopy for determining the relative orientation of the two tensors and its advantage over previous techniques are discussed. Reported experimental spectra at different spinning speeds of Na₃Co(NO₂)₆ are accurately reproduced by our theoretical simulations. The calculations are based on a recent approach, summarized in the present paper, which allows one to perform efficient simulations of MQMAS spectra including all interactions and their time-dependence throughout the experiment. This is necessary for calculating accurate MQMAS spectra including the spinning sideband pattern. In the case of trans-Co[(en₂)(NO₂)₂]NO₃ where the quadrupolar interaction and chemical shielding are stronger and their axes are non-coincident, the MQMAS spectrum is strongly distorted due to the unsufficient spinning speed and RF power. In this case, MAS at different spinning speeds is shown to provide valuable information.     

INEPT Experiments Involving Quadrupolar Nuclei in Solids

Coherence transfer from quadrupolar²⁷Al (I= ) nuclei to³¹P (I= ) via INEPT experiments is investigated.²⁷Al →³¹P INEPT experiments on a (CH₃)₃P–AlCl₃complex in zeolite NaX are performed, and the results demonstrate that the³¹P INEPT signals strongly depend on whether or not the²⁷Al pulses are applied synchronously with the rotor period, and on the length of the²⁷Al pulses. A density-matrix calculation involving the use of the spin operators for spin and nuclei has been performed to help understand the evolution behavior of the density matrix under the influence of the quadrupolar interaction, the dipolar andJ-couplings, and the pulse lengths applied to the quadrupolar nuclei. The theoretical predictions obtained from these calculations are consistent with the INEPT experimental observations.     

Relaxation Effects in a System of a Spin-1/2 Nucleus Coupled to a Quadrupolar Spin Subjected to RF Irradiation: Evaluation of Broadband Decoupling Schemes

We have investigated the suitability and performance of various decoupling methods on systems in which an observed spin-1/2 nucleusI(¹³C or¹⁵N) is scalar-coupled to a quadrupolar spinS(²H). Simulations and experiments have been conducted by varying the strength of the irradiating radiofrequency (RF) field, RF offset, relaxation times, and decoupling schemes applied in the vicinity of theS-spin resonance. TheT₁relaxation of the quadrupolar spin has previously been shown to influence the efficiency of continuous wave (CW) decoupling applied on resonance in such spin systems. Similarly, the performance of broadband decoupling sequences should also be affected by relaxation. However, virtually all of the more commonly used broadband decoupling schemes have been developed without consideration of relaxation effects. As a consequence, it is not obvious how one selects a suitable sequence for decoupling quadrupolar nuclei with exotic relaxation behavior. Herein we demonstrate that, despite its simplicity, WALTZ-16 decoupling is relatively robust under a wide range of conditions. In these systems it performs as well as the more recently developed decoupling schemes for wide bandwidth applications such as GARP-1 and CHIRP-95. It is suggested that in macromolecular motional regimes, broadband deuterium decoupling can be achieved with relatively low RF amplitudes (500–700 Hz) using WALTZ-16 multiple pulse decoupling.     

Complete description of the interactions of a quadrupolar nucleus with a radiofrequency field. Implications for data fitting

We present a theory, with experimental tests, that treats exactly the effect of radiofrequency (RF) fields on quadrupolar nuclei, yet retains the symbolic expressions as much as possible. This provides a mathematical model of these interactions that can be easily connected to state-of-the-art optimization methods, so that chemically-important parameters can be extracted from fits to experimental data. Nuclei with spins >1/2 typically experience a Zeeman interaction with the (possibly anisotropic) local static field, a quadrupole interaction and are manipulated with RF fields. Since RF fields are limited by hardware, they seldom dominate the other interactions of these nuclei and so the spectra show unusual dependence on the pulse width used. The theory is tested with ²³Na NMR nutation spectra of a single crystal of sodium nitrate, in which the RF is comparable with the quadrupole coupling and is not necessarily on resonance with any of the transitions. Both the intensity and phase of all three transitions are followed as a function of flip angle. This provides a more rigorous trial than a powder sample where many of the details are averaged out. The formalism is based on a symbolic approach which encompasses all the published results, yet is easily implemented numerically, since no explicit spin operators or their commutators are needed. The classic perturbation results are also easily derived. There are no restrictions or assumptions on the spin of the nucleus or the relative sizes of the interactions, so the results are completely general, going beyond the standard first-order treatments in the literature.     

Multiple quadrupolar spin echoes in the La-filled skutterudite LaOs4As12

We report experimental results of ¹³⁹La pulse NMR studies in LaOs₄As₁₂. Measurements have been performed on a powder sample obtained from high quality single crystals. For the first time the pattern of quadrupole echoes for ¹³⁹La nuclei (I=7/2) was obtained. All the allowed quadrupolar echoes expected for spin I=7/2 were observed at times t=(4/3)τ, (3/2)τ, (5/3)τ, 2τ, (5/2)τ, 3τ, 4τ. The presence of quadrupolar echoes is the fingerprint of the deviation from perfect cubic symmetry of the structure and can be used as a simple and fast test of the sample quality.     

Critical factors in obtaining meaningful fast MAS NMR spectra of non-integral spin quadrupolar nuclei. A review with particular emphasis on27Al MAS NMR of catalysts and minerals

In the last decade, magic angle spinning (MAS) NMR has become an extremely important method for studying the structure of inorganic solids. Advances in NMR technology have greatly aided in understanding the structure of catalysts, minerals, clays, ceramics, glasses, etc. Obtaining meaningful MAS spectra of spin-1/2 nuclei such as²⁹Si and³¹P is relatively straightforward and well understood. In contrast, obtaining meaningful MAS spectra is far from simple with non-integral spin quadrupolar nuclei such as¹¹B (I=3/2),¹⁷O (I=5/2),²³Na (I=3/2),²⁷Al (I=5/2),⁶⁹Ga (I=3/2), and⁷¹Ga (I=3/2)?to name some of the most commonly studied nuclei. Many additional factors have to be considered. This paper will deal with these factors and the utility of very fast MAS for studying non-integral spin quadrupolar nuclei in inorganic solids.     

Exact calculation,using angular momentum,of combined Zeeman and quadrupolar interactions in NMR

The NMR of nuclei with spins greater than ½ is often strongly influenced by the quadrupole interaction. This combination of Zeeman and quadrupole terms can usually be treated using perturbation theory, but an exact calculation is also needed. We explain an exact approach that eliminates the evaluation of commutators of complicated operators. Instead, the calculation is based on matrix elements of the Liouvillian, the commutator with the Hamiltonian. The spectrum can then be calculated directly from the eigenvectors and eigenvalues of the Liouvillian. With the aid of angular momentum methods, it can be shown that the quadupole interaction for spin I is fully determined by only (2I ?1) reduced matrix elements—for spin 3/2, this means only two quantities. The exact nature of the various basis operators is not needed, since the calculation only needs the angular momentum quantum numbers. The full Liouvillian matrix can be calculated from selection rules and the Wigner-Eckart theorem. Furthermore, we present an expression for these reduced matrix elements which is valid for any spin. This theory covers the whole range from quadrupole-perturbed NMR spectra to Zeeman-perturbed nuclear quadrupole resonance.     

Nonsecular contribution to the decay of spin-echo signals of quadrupolar nuclei in magnetically ordered materials

We analyze the influence of fluctuations of the nonsecular part of the spin Hamiltonian on the decay of ordinary and multiquantum signals of the two-pulse spin echo in a quadrupole spin system with an inhomogeneously broadened spectral line. Expressions are obtained for the rate of decay of an echo in the case of selective excitation of a signal from quadrupole nuclei with arbitrary spin. These expressions are then used to analyze the experimentally observed ordinary and multiquantum echo signals from quadrupole nuclei with spin I=3/2 (⁵³Cr, ⁶³Cu, and ⁶⁵Cu) in ferromagnetic chromium chalcogenide spinels. Zh. éksp. Teor. Fiz. 116, 204–216 (July 1999)     

NMR relaxation in multipolar AMX systems under spin-locking conditions

A relaxation network has been calculated for multipolar AMX systems under application of a spin-locking RF field. Systems of this type are of interest in the study of proteins with fractional ²H enrichment. All possible auto- and cross-correlation terms involving dipolar, quadrupolar, and CSA interactions have been taken into account. The results show the presence of spectral densities at zero frequency for interactions associated with the locked nuclei, which are nonvanishing in the absence of fast motions. In addition, the application of a spin-locking field blocks certain cross-correlation interactions, thereby considerably simplifying the relaxation network.     

Electron spin-echo envelope modulation at S-band. 2: Hyperfine and weak nuclear quadrupole interaction effects forI>1/2

The analysis of the nuclear modulation often observed in the electron spin-echo signals gives information on the structural arrangement around the paramagnetic center and on the interactions undergone by the spin system. Most applications to date have been carried out at X-band (≈9 GHz). However, to run experiments at different operating frequencies yields more accurate results and increases the number of experimental situations that can be investigated. Here we analyze the two- and three-pulse modulation patterns simulated at S-band (≈3 GHz) where the modulation is due to two different nuclei of experimental interest at different electron-nucleus distances, and comparison is done with the corresponding X-band patterns.²H (I=1) and⁷Li (I=3/2) nuclei are chosen due to their low quadrupole moment, so that comparison can be done between the effects due to weak nuclear quadrupole interaction at the two bands. It is shown that at short distances the complicated patterns due to hyperfine couplings are successfully interpreted by Fourier transform in the frequency domain, and that the use of S-band is really of some advantage with respect to X-band in detecting the presence of a weak nuclear quadrupole interaction.     

Efficient Time Propagation Technique for MAS NMR Simulation: Application to Quadrupolar Nuclei

The quantum mechanical Floquet theory is investigated in order to derive an efficient way of performing numerical calculations of the dynamics of nuclear spin systems in MAS NMR experiments. Here, we take advantage of time domain integration of the quantum evolution over one period as proposed by Edenet al.(1). But a full investigation of the propagatorU(t,t₀), and especially its dependence with respect totandt₀within a formalized approach, leads to further simplifications and to a substantial reduction in computation time when performing powder averaging for any complex sequence. Such an approximation is suitable for quadrupolar nuclei (I> 1/2) and can be applied to the simulation of the RIACT (rotational induced adiabatic coherence transfer) phenomenon that occurs under special experimental conditions in spin locking experiments (2–4). The present method is also compared to the usual infinite dimensional Floquet space approach (5, 6), which is shown to be rather inefficient. As far as we know, it has never been reported for quadrupolar nuclei withI≥ 3/2 in spin locking experiments. The method can also be easily extended to other areas of spectroscopy.     

Rotor-synchronized acquisition of quadrupolar satellite-transition NMR spectra: practical aspects and double-quantum filtration

The very broad resonances of quadrupolar (spin I > 1/2) nuclei are resolved by magic angle spinning (MAS) into a large number of spinning sidebands, each of which often remains anisotropically broadened. The quadrupolar interaction can be removed to a first-order approximation if the MAS NMR spectrum is acquired in a rotor-synchronized fashion, aliasing the spinning sidebands onto a centreband and thereby increasing the signal-to-noise ratio in the resulting, possibly second-order broadened, spectrum. We discuss the practical aspects of this rotor-synchronization in the direct (t(2)) time domain, demonstrating that the audiofrequency filters in the receiver section of the spectrometer have a significant impact on the precise timings needed in the experiment. We also introduce a novel double-quantum filtered rotor-synchronized experiment for half-integer spin quadrupolar (spin I = 3/2, 5/2, etc.) nuclei that makes use of central-transition-selective inversion pulses to both excite and reconvert double-quantum coherences and yields a simplified spectrum containing only the ST(1) (m(I) = +/-1/2 <--> +/-3/2) satellite-transition lineshapes. For spin I = 5/2 nuclei, such as (17)O and (27)Al, this spectrum may exhibit a significant resolution increase over the conventional central-transition spectrum.