Field-dependent nuclear relaxation of spins 1/2 induced by dipole-dipole couplings to quadrupole spins: LaF3 crystals as an example

A general theory of spin-lattice nuclear relaxation of spins I=1/2 caused by dipole-dipole couplings to quadrupole spins S1, characterized by a non-zero averaged (static) quadrupole coupling, is presented. In multispin systems containing quadrupolar and dipolar nuclei, transitions of spins 1/2 leading to their relaxation are associated through dipole-dipole couplings with certain transitions of quadrupole spins. The averaged quadrupole coupling attributes to the energy level structure of the quadrupole spin and influences in this manner relaxation processes of the spin 1/2. Typically, quadrupole spins exhibit also a complex multiexponential relaxation sensed by the dipolar spin as an additional modulation of the mutual dipole-dipole coupling. The proposed model includes both effects and is valid for an arbitrary magnetic field and an arbitrary quadrupole spin quantum number. The theory is applied to interpret fluorine relaxation profiles in LaF3 ionic crystals. The obtained results are compared with predictions of the 'classical' Solomon relaxation theory.     

Electron spin-echo envelope modulation at S-band. 3: Analysis of nuclear quadrupole interaction effects

In the present paper the nuclear modulation of electron spin echo signals at S-band is investigated in the case of interacting nuclei with a quadrupole moment high enough to cause nuclear quadrupole couplings not negligible with respect to the nuclear Zeeman and dipolar hyperfine couplings. Both the two-pulse and three-pulse electron spin echo envelope modulation (ESEEM) due to²⁷Al and¹⁴N are simulated at different values of the nuclear quadrupole coupling by numerical diagonalization of the nuclear Hamiltonians. The behavior of their amplitude and periods is discussed on the basis of the ratios between the strengths of the nuclear quadrupole interaction and the nuclear Zeeman and the dipolar hyperfine interactions. The interpretation of their trends in terms of the eigenfunctions and eigenvectors of the nuclear Hamiltonians is carried out by using analytical equations obtained by perturbation approaches. First order perturbation treatments for integer and half-integer nuclear spin quantum numbers are developed when the nuclear quadrupole coupling is the main interaction. A discussion on the limits of the interpretation based on the perturbation approach is also given by comparing the magnitude Fourier transform of the patterns calculated by exact diagonalization and analytical equations.     

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.     

A simple theoretical treatment of quadrupolar effects on magic-angle-spinning solid-state NMR spectra of spin-1/2 nuclei in the limit of large quadrupole coupling constants

A simple approach is discussed for studying the effect of quadrupolar nuclei on the magic-angle-spinning solid-state NMR lines of spin-1/2 nuclei in the limit of large quadrupole coupling constants. Equations are derived both for the isotropic shifts and the Pake-like powder patterns for any quadrupolar spin and for arbitrary orientations of the internuclear vector with respect to the unique axis of an axially symmetric quadrupole tensor. First-order effects due to a small Zeeman perturbation on these lines are explored, as well as deviations from axial symmetry in the electric field gradient when S = 3/2 quadrupolar nuclei are involved. Spectral parameters likely to be observed in the case of coupling between ³¹P and ²⁰¹Hg are also discussed.     

Heteronuclear dipolar recoupling of half-integer quadrupole nuclei under fast magic angle spinning

An experimental method for the heteronuclear dipolar recoupling of half-integer quadrupole nuclei is proposed. The idea is to manipulate the central transition based on the recoupling technique of spin-polarization-inversion rotary resonance. This method allows the extraction of structural parameters under fast magic-angle spinning. Its validity has been examined by the average Hamiltonian theory and numerical simulations. The initial rotational-echo dephasing arising from the dipolar evolution can be approximated by a parabolic function, from which the heteronuclear van Vleck second moment can be estimated. A factor, estimated from two-spin simulations, is required to account for the effects of the quadrupolar coupling and is rather independent of the geometry and the orders of the spin systems. Our method can facilitate the structural characterization of materials containing half-integer quadrupole nuclei under high-resolution condition. Experimental verification has been carried out on two aluminophosphate systems, namely, AlPO₄-5 and AlPO₄-11.     

MAS NMR with and without double-quantum filtration at and near the n=0 rotational resonance condition

Spectral lineshapes of MAS NMR spectra of dipolar (re)coupled spin pairs exhibiting considerable chemical shielding anisotropies at and near the so-called n=0 rotational resonance (R2) condition are considered. The n=0 R2 condition is found to be not extremely sharp. Anisotropic interaction parameters such as chemical shielding tensor orientations and the magnitude of the dipolar coupling constant remain sensitively encoded in such lineshapes even when differences in isotropic chemical shielding values of up to 400 Hz (corresponding to ca. half the size of the dipolar coupling constant) are present. Additional double-quantum filtration (DQF) may enhance the sensitivity of spectral lineshapes to anisotropic interaction parameters for even larger differences in isotropic chemical shielding values. The dependence of the DQF efficiency on spin-system parameters as well as on external parameters (Larmor and MAS frequencies) is investigated. Away from R2 conditions a trend to lower DQF efficiencies is found whereas some spin-system parameters are more sensitively encoded in the corresponding spectral lineshapes. Our study is based on numerical simulations, with the known parameters of the 31P spin pair in Na4P2O7.10H2O representing our model case.     

Effective dipolar couplings determined by dipolar dephasing of double-quantum coherences

It is shown how homonuclear distances and homonuclear dipolar lattice sums between spin-1/2 nuclei can be measured by a pulsed solid-state NMR experiment under magic-angle spinning conditions. The presented technique is based on double-quantum coherence filtering. Instead of measuring a build-up of double-quantum coherence the pulse sequence is designed to dephase double-quantum coherence. This is achieved by exciting double-quantum coherence either with the help of the through-space dipolar coupling or the through-bond dipolar coupling while the dephasing relies on the through-space dipolar coupling as selected by a gamma-encoded pulse sequence from the C/R symmetry class. Since dephasing curves can be normalized on zero dephasing, it is possible to analyze the initial dephasing regime and hence determine dipolar lattice sums (effective dipolar couplings) in multiple-spin systems. A formula for the effective dipolar coupling is derived theoretically and validated by numerical calculations and experiments on crystalline model compounds for (13)C and (31)P spin systems. The double-quantum dephasing experiment can be combined with constant-time data sampling to compensate for relaxation effects, consequently only two experimental data points are necessary for a single distance measurement. The phase cycling overhead for the constant-time experiment is minimal because a short cogwheel phase cycle exists. A 2D implementation is demonstrated on [(13)C(3)]alanine.     

Rotor-encoded heteronuclear MQ MAS NMR spectroscopy of half-integer quadrupolar and spin I=1/2 nuclei

A new two-dimensional heteronuclear multiple-quantum magic-angle spinning (MQ MAS) experiment is presented which combines high resolution for the half-integer quadrupolar nucleus with information about the dipolar coupling between the quadrupolar nucleus and a spin I=1/2 nucleus. Homonuclear MQ coherence is initially created for the half-integer quadrupolar nucleus by a single pulse as in a standard MQ MAS experiment. REDOR recoupling of the heteronuclear dipolar coupling then allows the creation of a heteronuclear multiple-quantum coherence comprising multiple- and single-quantum coherence of the quadrupolar and spin I=1/2 nucleus, respectively, which evolves during t1. Provided that the t1 increment is not rotor synchronized, rotor-encoded spinning-sideband patterns are observed in the indirect dimension. Simulated spectra for an isolated IS spin pair show that these patterns depend on the recoupling time, the magnitude of the dipolar coupling, the quadrupolar parameters, as well as the relative orientation of the quadrupolar and dipolar principal axes systems. Spectra are presented for Na2HPO4, with the heteronuclear 23Na-1HMQ MAS experiments beginning with the excitation of 23Na (spin I=3/2) three-quantum coherence. Coherence counting experiments demonstrate that four- and two-quantum coherences evolve during t1. The heteronuclear spinning-sideband patterns observed for the three-spin H-Na-H system associated with the Na(2) site are analyzed. For an IS2 system, simulated spectra show that, considering the free parameters, the spinning-sideband patterns are particularly sensitive to only, first, the angle between the two IS internuclear vectors and, second, the two heteronuclear dipolar couplings. It is demonstrated that the proton localization around the Na(2) site according to the literature crystal structure of Na2HPO4 is erroneous. Instead, the experimental data is consistent with two alternative different structural arrangements, whereby either there is a deviation of 10 degrees from linearity for the case of two identical Na-H distances, or there is a linear arrangement, but the two Na-H distances are different. Furthermore, the question of the origin of spinning-sidebands in the (homonuclear) MQ MAS experiment is revisited. It is shown that the asymmetric experimental MQ sideband pattern observed for the low-C(Q) Na(2) site in Na(2)HPO4 can only be explained by considering the 23Na chemical shift anisotropy.     

Relaxation of a quadrupolar nucleus in a strong magnetic field

Longitudinal relaxation of spin systems (S = 1, 3/2, 3) containing isolated quadrupolar nuclei are studied. The characteristic features of the relaxation behavior are identified in strong magnetic fields in the presence of chemical shift anisotropy. Two mechanisms are established that favor involvement of high rank multipoles in the relaxation process: an autocorrelation mechanism and a cross correlation mechanism. Multipoles of odd rank are involved in the relaxation of nuclei with spin S > 1 as a result of the autocorrelation mechanism (in this case, due to quadrupole interactions). The cross correlation mechanism, due to correlations between the chemical shift anisotropy and the quadrupole interactions, favors the appearance of multipoles of even rank. Expressions are presented for the multipole cross relaxation and longitudinal relaxation rates for spin systems with S = 1, 3/2, 3. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 1, pp. 18–25, January–February, 2006.     

Through-space R3-HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR

We present several new methods that allow to obtain through-space 2D HETCOR spectra between spin-1/2 and half-integer quadrupolar nuclei in the solid state. These methods use the rotary-resonance concept to create hetero-nuclear coherences through the dipolar interaction instead of scalar coupling into the HMQC and refocused INEPT experiments for spin n/2 (n>1). In opposite to those based on the cross-polarization transfer to quadrupolar nuclei, the methods are very robust and easy to set-up.     

Analysis of correlation patterns in hyperfine sublevel correlation spectroscopy of<Emphasis Type="Italic">S</Emphasis>=1/2,<Emphasis Type="Italic">I</Emphasis>=3/2 systems

The properties of two-dimensional four-pulse electron spin echo envelope modulation spectra of nuclei with nuclear spinI=3/2 in disordered systems are discussed by means of calculated lithium hyperfine sublevel correlation spectra. Since a case of small coupling is treated, the multifrequency transition components appear only in the first quadrant of the two-dimensional spectra. The influence of the hyperfine and nuclear quadrupole interaction parameters on the cross peak ridges in the two-dimensional powder pattern is analyzed. Furthermore the possibility to determine the magnitude of the dipolar hyperfine and the nuclear quadrupole coupling from the cross peak line shapes is investigated. In this context the potential of those spectral features representing the so-called multiquantum transitions for the interpretation of the spectra is demonstrated.     

Spin diffusion of dipolar energy in NQR

The theory of spin diffusion was extended to the case of nuclear dipolar order in solids containing paramagnetic impurities and nuclei with spin I > 1/2 having nuclear quadrupole moment. We show that spin diffusion process of dipolar order takes place in solids containing paramagnetic impurities. At the start of relaxation process, the direct relaxation regime is realized with non-exponential time dependence. Then the relaxation regime will be changed to diffusion-limited one. Using obtained expressions for the spin lattice relaxation times for these two relaxation regimes, the diffusion coefficient of the dipolar order in nuclear quadrupole resonance can be estimated from experimental data.     

<Superscript>23</Superscript>Na and <Superscript>27</Superscript>Al NMR Study of Structure and Dynamics in Mordenite

Temperature dependencies of ²⁷Al and ²³Na nuclear magnetic resonance spectra and spin–lattice relaxations in mordenite have been studied in static and magic angle spinning regimes. Our data show that the spin–lattice relaxations of the ²³Na and ²⁷Al nuclei are mainly governed by interaction of nuclear quadrupole moments with electric field gradients of the crystal, modulated by translational motion of water molecules in the mordenite channels. At temperatures below 200 K, the dipolar interaction of nuclear spins with paramagnetic impurities becomes an important relaxation mechanism of the ²³Na and ²⁷Al nuclei.     

Sign of nuclear electric quadrupole coupling constants in solids fromγ-ray anisotropy measurements after capture of polarized neutrons

An experimental method is presented which allows the determination of the sign of nuclear electric quadrupole interactions in solids. Activated target nuclei with a purely dipolar spin polarization are produced by capture of polarized thermal neutrons. The quadrupole coupling of the nuclei to the electric field gradient tensor in the target crystal converts this (dipolar) polarization partly into a (quadrupolar) alignment, which can be measured by the anisotropy of the-ray emission in a succeeding nuclear transition. The sign of the alignment created in this reorientation process depends on the sign of the electric quadrupole interaction. The reorientation effect can be enhanced by selective induction of nuclear magnetic resonance transitions. The method has been applied to measure the sign of the crystal electric field gradient (efg) in tetragonal MgF₂. Further, the sign of an efg in cubic CaF₂ originating from a¹⁹F interstitial adjacent to the activated²⁰F probe nucleus has been determined. The method is in principle applicable to a considerable number of nuclides.     

Residual 31P, 35,37Cl dipolar coupling in 31P MAS spectra of chlorocyclophosphazenes

In ³¹P MAS NMR spectra of chlorocyclophosphazenes, characteristic splittings have been observed for PCl or PCl₂ groups. At different applied magnetic fields, the fine structure and total width of the patterns change in a characteristic way, demonstrating that the splittings are due to indirect spin–spin and residual dipolar interactions with the chlorine nuclei directly bonded to phosphorus. For trans-nongeminal N₃P₃Cl₃(NMe₂)₃ and N₃P₃Cl₆ as examples, the spectra have been analyzed to obtain information on chlorine nuclear quadrupole coupling constants and ^35,37Cl, ³¹P indirect spin–spin coupling constants. Neglect of these interactions may result in misinterpretations of the multiplicity in ³¹P MAS spectra of chlorophosphazenes.     

Improved narrowband dipolar recoupling for homonuclear distance measurements in rotating solids

Recovery of the magnetic dipolar interaction between nuclei bearing the same gyromagnetic ratio in rotating solids can be promoted by synchronous rf irradiation. Determination of the dipolar interaction strength can serve as a tool for structural elucidation in polycrystalline powders. Spinning frequency dependent narrow-band (nb) RFDR and SEDRA experiments are utilized as simple techniques for the determination of dipolar interactions between the nuclei in coupled homonuclear spin pairs. The magnetization exchange and coherence dephasing due to a fixed number of rotor-synchronously applied pi-pulses is monitored at spinning frequencies in the vicinity of the rotational resonance (R(2)) conditions. The powder nbRFDR and nbSEDRA decay curves of spin magnetizations and coherences, respectively, as a function of the spinning frequency can be measured and analyzed using simple rate equations providing a quantitative measure of the dipolar coupling. The effects of the phenomenological relaxation parameters in these rate equations are discussed and an improved methodology is suggested for analyzing nbRFDR data for small dipolar couplings. The distance between the labeled nuclei in the 1,3-(13)C(2)-hydroxybutyric acid molecule is rederived using existing nbRFDR results and the new simulation procedure. A nbSEDRA experiment has been performed successfully on a powder sample of singly labeled 1-(13)C-L-leucine measuring the dipolar interaction between the labeled carboxyl carbon and the natural abundant beta-carbon. Both narrowband techniques are employed for the determination of the nuclear distances between the side-chain carbons of leucine and its carbonyl carbon in a tripeptide Leu-Gly-Phe that is singly (13)C-labeled at the leucine carbonyl carbon position.     

Ordering in nematic liquid crystals from NMR cross-polarization studies

The measurement of dipolar couplings between nuclei is a convenient way of obtatining directly liquid crystalline ordering through NMR since the coupling is dependent on the average orientation of the dipolar vector in the magnetic field which also aligns the liquid crystal. However, measurement of the dipolar coupling between a pair of selected nuclei is beset with problems that require special solutions. In this article the use of cross polarization for measuring dipolar couplings in liquid crystals is illustrated. Transient oscillations observed during cross polarization provide the dipolar couplings between essentially isolated nearest neighbour spins which can be extracted for several sites simultaneously by employing two-dimensional NMR techniques. The use of the method for obtaining heteronuclear dipolar couplings and hence the order parameters of liquid crystals is presented. Several modifications to the basic experiment are considered and their utility illustrated. A method for obtaining proton-proton dipolar couplings, by utilizing cross polarization from the dipolar reservoir, is also presented.     

Measuring (13)C-(2)D dipolar couplings with a universal REDOR dephasing curve

A (13)C-observe REDOR experiment is described which allows (13)C-(2)D dipolar couplings to be obtained by a universal dipolar dephasing curve. Previous (13)C-observe REDOR experiments on (13)C-(2)D spin pairs generally relied on numerical simulations to obtain the dipolar coupling. The REDOR experiment described in this article is based on a deuterium composite pulse, and the data analysis eliminates the need for numerical simulations and is the same as the traditional REDOR analysis performed on pairs of spin-12 nuclei. Copyright 2000 Academic Press.     

Homonuclear Dipolar Coupling and CPMG Spin-Echoes in NQR

The nuclear quadrupole resonance signals from both a single crystal and a powder sample of spin-1 nuclei under a Carr?CPurcell?CMeiboom?CGill are modeled numerically. While the single crystal clearly shows the effects of dipolar coupling for on-resonant pulses, the powder does not. However, for certain off-resonant conditions, the powder sample exhibits the same response as the single crystal. Experimentally, this corresponds to the observation of a rapid decay at these conditions. Using a powder sample of NaNO₂, the functional form of the echo train, when the dipolar coupling is not refocused, is clearly different than the decay for an on-resonance sequence, and can be used to characterize the dipolar coupling.     

Quadrupole relaxation enhancement--application to molecular crystals

A general theory of field dependent spin-lattice relaxation for nuclei of the spin quantum number 1/2 (1H, 19F, 13C) caused by dipole-dipole interactions with neighboring quadrupolar nuclei (nuclei possessing a quadrupolar moment) is presented. The theory is valid for arbitrary motional conditions and should be treated as a quadrupolar counterpart of the paramagnetic relaxation enhancement theory. When the energy level splitting of the dipolar spin (I=1/2) matches one of the transition frequencies of the quadrupolar nuclei one can observe a local enhancement of the dipolar spin relaxation (referred to as "quadrupolar peaks"). To see such effects the dynamics modulating the spin interactions has to be relatively slow. This brings the system beyond the validity range of perturbation approaches and requires the stochastic Liouville equation to be applied. The presented theory describes the quadrupolar relaxation enhancement (QRE) for an arbitrary spin quantum number of the quadrupolar nuclei and includes the asymmetry of the quadrupolar coupling. It has been applied to interpret the shape of magnetization curves (amplitude of 1H magnetization versus magnetic field) for the molecular crystal [C3N2H5]6[Bi4Br18] ([C3N2H5]-imidazolium). The magnetization curves show several dips (local minima) attributed to 1H-14N quadrupolar relaxation enhancement effects. In addition, as a limiting case a perturbation approach to QRE has been presented and its validity conditions have been discussed.