A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

High-Precision Determination of NMR Interaction Parameters by Measurement of Single Crystals: A Review of Classical and Advanced Methods

In this review, the process of extracting precise values for NMR interaction tensors from single crystal samples is systematically explored. Starting with a description of the orientation dependence of the considered interactions, i.e., chemical shift, dipolar, and quadrupole interaction, the techniques for acquiring and analysing single-crystal spectra are outlined. This includes the 'classical' approach, which requires the acquisition of three rotation patterns around three rotation axes that are orthogonal to each other, as well as more recent strategies aimed at reducing the number of required NMR spectra. One such strategy is the 'single-rotation method', which exploits the symmetry relations between tensors in the crystal structure to reduce the necessary amount of orientation-dependent data. This concept may be extended to additionally include the orientation of the goniometer axis itself in the data fit, which may be termed the 'minimal-rotation method'. Other, more exotic schemes, such as the use of specialised probe designs or the investigation of single crystals under magic-angle-spinning, are also briefly discussed. Actual values of NMR interaction tensors as determined from the various single-crystal methods have been collected and are provided in tables for spin I=1/2, I=1, and half-integer spins with I>1/2.

Quantifying the quadrupolar interaction by 45Sc-NMR spectroscopy of single crystals

Single crystals of the compound [ [Formula: see text] ]Cl₄ ⋅ 2H₂O were studied by ⁴⁵Sc-NMR, with the effect of the quadrupolar coupling interaction on the spectra of the spin-7/2 nucleus analysed in the hierarchical framework of perturbation theory. Orientation-dependent spectra acquired at B₀ = 17.6 T showed strong second-order effects due to the comparatively large coupling constant of χ = |14.613 ± 0.006| MHz, with an associated asymmetry parameter of η_Q = 0.540 9 ± 0.000 4. By analysing the splittings of the ±3/2 satellites, which in good approximation are subjected to first-order effects only, the full quadrupolar coupling tensor could be determined. The second-order effects caused by this tensor were calculated according to theoretical predictions for all orientations, and subtracted from both the centres of gravity of the satellites, and the central transitions. This allowed extraction of the full chemical shift tensor, with the eigenvalues being δ₁₁ = (5.6 ± 0.9) ppm, δ₂₂ = (12.4 ± 0.9) ppm, and δ₃₃ = (38.5 ± 0.9) ppm. In spectra acquired at a lower magnetic field of B₀ = 9.4 T, third-order effects could be detected, and similarly quantified using analytical expressions.

Quadrupole interactions: NMR, NQR, and in between from a single viewpoint

Nuclear spins with quantum numbers >1/2 can interact with a static magnetic field, or a local electric field gradient, to produce quantized energy levels. If the magnetic field interaction dominates, we are doing nuclear magnetic resonance (NMR). If the interaction of the nuclear electric quadrupole with electric field gradients is much stronger, this is nuclear quadrupole resonance (NQR). The two are extremes of a continuum, as the ratio of one interaction to the other changes. In this work, we look at this continuum from a single, unified viewpoint based on a Liouville-space approach: the direct method. This method does not require explicit operators and their commutators, unlike Hamiltonian methods. We derive both the quadrupole-perturbed NMR solution and also the Zeeman-perturbed NQR results. Furthermore, we examine the polarization of these signals, because this is different between pure NMR and pure NQR spectroscopy. Spin 3/2 is the focus here, but the approach is perfectly general and can be applied to any spin. Copyright © 2016 John Wiley & Sons, Ltd.

Spin-Spin Coupling between Quadrupolar Nuclei in Solids: (11)B-(75)As Spin Pairs in Lewis Acid-Base Adducts

Solid-state (11)B NMR measurements of Lewis acid-base adducts of the form R3AsBR'3 (R = Me, Et, Ph; R' = H, Ph, C6F5) were carried out at several magnetic field strengths (e.g., B0 = 21.14, 11.75, and 7.05 T). The (11)B NMR spectra of these adducts exhibit residual dipolar coupling under MAS conditions, allowing for the determination of effective dipolar coupling constants, Reff((75)As,(11)B), as well as the sign of the (75)As nuclear quadrupolar coupling constants. Values of Reff((75)As,(11)B) range from 500 to 700 Hz. Small isotropic J-couplings are resolved in some cases, and the sign of (1)J((75)As,(11)B) is determined. Values of CQ((75)As) measured at B0 = 21.14 T for these triarylborane Lewis acid-base adducts range from -82 ± 2 MHz for Et3AsB(C6F5)3 to -146 ± 1 MHz for Ph3AsBPh3. For Ph3AsBH3, two crystallographically nonequivalent sites are identified with CQ((75)As) values of -153 and -151 ± 1 MHz. For the uncoordinated Lewis base, Ph3As, four (75)As sites with CQ((75)As) values ranging from 193.5 to 194.4 ± 2 MHz are identified. At these applied magnetic field strengths, the (75)As quadrupolar interaction does not satisfy high-field approximation criteria, and thus, an exact treatment was used to describe this interaction in (11)B and (75)As NMR spectral simulations. NMR parameters calculated using the ADF and CASTEP program packages support the experimentally derived parameters in both magnitude and sign. These experiments add to the limited body of literature on solid-state (75)As NMR spectroscopy and serve as examples of spin-spin-coupled quadrupolar spin pairs, which are also rarely treated in the literature.

Second-order quadrupole interaction based detection of ultra-slow motions: Tensor operator framework for central-transition spectroscopy and the dynamics in hexagonal ice as an experimental example

The second-order quadrupolar broadening of the central transition of nuclear probes with half-integer spins I is demonstrated to be useful to detect ultraslow molecular motions. On the basis of density matrix calculations explicit expressions are derived for quadrupolarly modulated sin-sin and cos-cos signals of selectively excited nuclei with I=3/2, 5/2, 7/2, and 9/2. These correlation functions are suitable for implementation in two-dimensional exchange spectroscopy as well as for stimulated-echo experiments. As an application, ¹⁷O measurements of the reorientational correlation function of water molecules in hexagonal ice are presented.

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 (139)La spectra of La2O3.

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 (23)Na 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.

QUEST-QUadrupolar Exact SofTware: a fast graphical program for the exact simulation of NMR and NQR spectra for quadrupolar nuclei

We present a new program for the exact simulation of solid-state NMR spectra of quadrupolar nuclei in stationary powdered samples which employs diagonalization of the combined Zeeman-quadrupolar Hamiltonian. The program, which we call QUEST (QUadrupolar Exact SofTware), can simulate NMR spectra over the full regime of Larmor and quadrupolar frequency ratios, which encompasses scenarios ranging from high-field NMR to nuclear quadrupole resonance (NQR, where the Larmor frequency is zero) and does not make use of approximations when treating the quadrupolar interaction. With the use of the fast powder averaging scheme of Alderman, Solum, and Grant, exact NMR spectral simulations are only marginally slower than the second-order perturbation theory counterpart. The program, which uses a graphical user interface, also incorporates chemical shift anisotropy and non-coincident chemical shift and quadrupolar tensor frames. The program is validated against newly-acquired experimental data through several examples including: the low-field (79/81)Br NMR spectra of CaBr(2), the (14)N overtone NMR spectrum of glycine, the (187)Re NQR spectra of Re(2)(CO)(10), and lastly the (127)I overtone NQR spectrum of SrI(2), which, to the best of our knowledge, represents the first direct acquisition of an overtone NQR spectrum for a powdered sample.

Definitive solid-state 185/187Re NMR spectral evidence for and analysis of the origin of high-order quadrupole-induced effects for I=5/2

Rhenium-185/187 solid-state nuclear magnetic resonance (SSNMR) experiments using NaReO(4) and NH(4)ReO(4) powders provide unambiguous evidence for the existence of high-order quadrupole-induced effects (HOQIE) in SSNMR spectra. Fine structure, not predicted by second-order perturbation theory, has been observed in the (185/187)Re SSNMR spectrum of NaReO(4) at 11.75 T, where the ratio of the Larmor frequency (ν(0)) to the quadrupole frequency (ν(Q)) is ∼2.6. This is the first experimental observation that under static conditions, HOQIE can directly manifest in SSNMR powder patterns as additional fine structure. Using NMR simulation software which includes the quadrupole interaction (QI) exactly, extremely large (185/187)Re nuclear quadrupole coupling constants (C(Q)) are accurately determined. QI parameters are confirmed independently using solid-state (185/187)Re nuclear quadrupole resonance (NQR). We explain the spectral origin of the HOQIE and provide general guidelines that may be used to assess when HOQIE may impact the interpretation of the SSNMR powder pattern of any spin-5/2 nucleus in a large, axially symmetric electric field gradient (EFG). We also quantify the errors incurred when modeling SSNMR spectra for any spin-5/2 nucleus within an axial EFG using second-order perturbation theory. Lastly, we measure rhenium chemical shifts in the solid state for the first time.

Exact calculation of the response of a quadrupolar nucleus to radio frequency irradiation

The effect of a pulse in NMR is usually considered as a rotation of the frame of reference of the spin system. For spins-1/2, this concept is an important and very useful tool. The assumption behind this concept is that while the radio frequency irradiation is on, this term dominates all other interactions. Although this is usually true for spins-1/2, typical interactions for a quadrupolar nucleus can be very large and the assumption is no longer valid. The full solution is complex, but two extreme cases are already solved. If the quadrupole interaction is very small, then the assumption is valid and the pulse does act like a rotation of the frame of reference. At the other extreme, if the interaction is large and the spin, I, is half-integral, then the central transition remains relatively narrow and can be treated as a fictitious spin-1/2. The pulse then acts as a rotation, but with a scaling factor of I + 1/2. This paper treats the general case, where no approximations are made. The effects can be observed in a nutation experiment, in which the observed signal is plotted as a function of pulse width, in a simple one-pulse experiment. If the pulse acts as a rotation, then the nutation plot will be a sine wave, but otherwise it will be a sum of sinusoids. This is true even for a single-crystal sample with a single quadrupolar coupling. If the sample is a powder, then the nutation plot will be a sum of many sinusoids, since the quadrupole coupling will vary with the powder average. This paper sketches out the theory of these effects based on a full and exact description of a quadrupolar system and illustrates it with some nutation spectra of 23Na in a powdered sample of sodium nitrate.

Probing quadrupolar nuclei by solid-state NMR spectroscopy: recent advances

Solid-state nuclear magnetic resonance (NMR) of quadrupolar nuclei has recently undergone remarkable development of capabilities for obtaining structural and dynamic information at the molecular level. This review summarizes the key achievements attained during the last couple of decades in solid-state NMR of both integer spin and half-integer spin quadrupolar nuclei. We provide a concise description of the first- and second-order quadrupolar interactions, and their effect on the static and magic angle spinning (MAS) spectra. Methods are explained for efficient excitation of single- and multiple-quantum coherences, and acquisition of spectra under low- and high-resolution conditions. Most of all, we present a coherent, comparative description of the high-resolution methods for half-integer quadrupolar nuclei, including double rotation (DOR), dynamic angle spinning (DAS), multiple-quantum magic angle spinning (MQMAS), and satellite transition magic angle spinning (STMAS). Also highlighted are methods for processing and analysis of the spectra. Finally, we review methods for probing the heteronuclear and homonuclear correlations between the quadrupolar nuclei and their quadrupolar or spin-1/2 neighbors.

Practical comparison of sensitivity and resolution between STMAS and MQMAS for 27Al

An experimental comparison of sensitivity and resolution of satellite transition (ST) MAS and multiple quantum (MQ) MAS was performed for (27)Al (I=5/2) using several pulse sequences with a z-filter and SPAM, and two inorganic samples of kaolin (Al(2)Si(2)O(5)(OH)(4)) and glass (43.1CaO-12.5Al(2)O(3)-44.4SiO(2)). Six pulse sequences of STMAS (double-quantum filter-soft pulse added mixing=DQF-SPAM, double-quantum filter=DQF, double-quantum=DQ) and MQMAS (3QMAS-z-filter=3Qz, 3QMAS-SPAM=3Q-SPAM, 5QMAS-z-filter=5Qz) are employed. All experiments have been conducted utilizing a static field of 16.4T (700MHz for (1)H) and a rotor spinning frequency of 20kHz. Dependence of S/N ratios as a function of radio frequency (r.f.) field strengths indicates that strong r.f. fields are essential to obtain a better S/N ratio in all experiments. High sensitivity is obtained in the following order: DQF-SPAM, DQF, DQ, 3QSPAM, and 3Qz, although the degree of sensitivity enhancement given by STMAS for glass is slightly smaller than that for kaolin. This might be due to the different excitation and conversion efficiencies of ST and MQ coherences as a function C(q) values because quadrupolar interaction of the glass are widely distributed, or to motional broadening caused by framework flexibility in the structure of glass. With respect to resolution, the full widths at half maximum (FWHM) of F(1) projections of DQF-STMAS and 3QMAS spectra for kaolin are found to be comparable, which agrees with a simulated result reported in a literature. For glass, the STMAS possess slightly wider line widths than 3QMAS. However, because such a difference in line widths of STMAS and 3QMAS spectra is substantially small, we have concluded that STMAS and 3QMAS have comparable resolution for crystalline and non-crystalline materials.