On the application of magic echo cycles for quadrupolar echo spectroscopy of spin-1 nuclei

Magic echo cycles are introduced for performing quadrupolar echo spectroscopy of spin-1 nuclei. An analysis is performed via average Hamiltonian theory showing that the evolution under chemical shift or static field inhomogeneity can be refocused simultaneously with the quadrupolar interaction using these cycles. Due to the higher convergence in the Magnus expansion, with sufficient RF power, magic echo based quadrupolar echo spectroscopy outperforms the conventional two pulse quadrupolar echo in signal to noise. Experiments highlighting a signal to noise enhancement over the entire bandwidth of the quadrupolar pattern of a powdered sample of deuterated polyethelene are shown.     

Resonant X-ray scattering study of quadrupole-strain coupling in DyB4

X-ray and neutron scattering studies were performed on DyB4 which exhibits both a quadrupolar ordering and a macroscopic lattice distortion. A forbidden reflection at 7.792 keV near the Dy L3 absorption edge is identified as a quadrupolar ordering peak, and the quadrupolar order and a monoclinic structural distortion develop concomitantly below 12.3 K as second-order-type phase transitions. Coupling between the quadrupolar order and the strain in DyB4 is directly demonstrated by observing that both order parameters are proportional to each other.     

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.     

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.     

High-field QCPMG NMR of large quadrupolar patterns using resistive magnets

Spectroscopy in a high magnetic field reduces second-order quadrupolar shift while increasing chemical shift. It changes the scale between quadrupolar and chemical shift of half-integer quadrupolar spins. The application of QCPMG multiple echo for acquiring large quadrupolar pattern under the high magnetic field of a 25 T resistive magnet is presented for acquiring large quadrupolar patterns. It shows that temporal field fluctuations and spatial homogeneity of the Keck magnet at the NHMFL contribute about ±20 ppm in line broadening. NMR patterns which have breadths of hundreds to thousands of kilohertz can be efficiently recorded using a combination of QCPMG and magnetic field stepping with negligible hindrance from the inhomogeneity and field fluctuations of powered magnets.     

Low-temperature spectrum and damping of elementary excitations in biquadratic exchange systems (S = 1)

The diagrammatic technique for the standard basis operators and the expansion in powers of 1/z (where z is the number of spins interacting with any given spin) are used to calculate the energies of magnons (in the ferromagnetic and quadrupolar phases) and quadrons (in the ferromagnetic phase) and the bipolar and quadrupolar order parameters (in both phases) in the (1/z)¹ approximation and at low temperatures. it is shown that magnons are damped with the exception of the (ferromagnetic and quadrupolar) case of the Schrödinger exchange model and quadrupolar phase; the quadrons are undamped only in a special case of the ferromagnetic phase, i.e. in the ferrogmatic case of the Schrödinger exchange model. In the limit K → 0, i.e. the biquadratic exchange interaction tends to zero, the spectrum of magnons and their damping are identical with the results by Balakhonov et al. The spin waves and quadrupolar order parameter are discussed in more detail in the following cases of the quadrupolar phase: ferromagnetic-quadrupolar limit, isotropic purely quadrupolar interactions, and quadrupolar-antiferromagnetic limit.     

Separated quadrupolar field experiment

We describe an NMR experiment that produces spectra correlating the first-order quadrupolar spectrum and the central transition spectrum of half-integer quadrupolar spins, allowing one to separate the quadrupolar parameters in overlapping spectra under both static and magic-angle-spinning conditions. Promising fields of applications include situations where the sample cannot easily be rotated, or where it cannot be rotated at the magic angle.     

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.     

Site-selective QPASS for the isolation of large quadrupolar coupling environments

Spectral editing of high spinning rate quadrupolar powder patterns observed using the QPASS experiment was achieved through the coupling of QPASS with the selective pi/2-RAPT enhancement sequence. The resulting pi/2-RAPT-QPASS sequence yields spectra that are dominated by the powder patterns form sites with large quadrupolar couplings thus reducing the overlap of patterns from multiple sites of different symmetry in a material. The 93Nb isotropic chemical shifts and quadrupolar coupling parameters were determined for the two niobium crystallographic sites in the layered KCa2Nb3O10. The asymmetric surface site in the structure was selectively enhanced and easily fit to second-order quadrupolar powder pattern with this method.     

Determination of the antiferroquadrupolar order parameters in UPd(3)

By combining accurate heat capacity and x-ray resonant scattering results we have resolved the long standing question regarding the nature of the quadrupolar ordered phases in UPd(3). The order parameter of the highest temperature quadrupolar phase has been uniquely determined to be antiphase Q{zx} in contrast to the previous conjecture of Q{x{2}-y{2}}. The azimuthal dependence of the x-ray scattering intensity from the quadrupolar superlattice reflections indicates that the lower temperature phases are described by a superposition of order parameters. The heat capacity features associated with each of the phase transitions characterize their order, which imposes restrictions on the matrix elements of the quadrupolar operators.     

Multiple-Rotor-Cycle QPASS Pulse Sequences: Separation of Quadrupolar Spinning Sidebands with an Application to La NMR

The quadrupolar phase-adjusted spinning sidebands (QPASS) pulse sequence has been recently demonstrated as a useful method for obtaining quadrupolar parameters with magic-angle spinning NMR. The sequence separates spinning sidebands by order in a two-dimensional experiment. A sheared projection of the 2D spectrum effectively yields the infinite spinning rate second-order quadrupolar powder pattern, which can be analyzed to determine quadrupolar coupling constants and asymmetry parameters. The RF power and spinning speed requirements of the original QPASS sequence make it an experimentally demanding technique. A new version of the sequence is demonstrated here and is shown to alleviate many problems associated with the original sequence. New solutions to the determining equations, based on the use of multiple rotor cycles in the QPASS sequence, lead to longer delays between the nine π pulses, provide less chance of pulse overlap, and allow for use of weaker RF field strengths that excite only the central quadrupolar transition. A three-rotor-cycle version of the new experiment is demonstrated on the ¹³⁹La nucleus.     

Curie temperature of a spin-one bethe lattice

The exact expression for the Curie temperature of a spin-1 Bethe lattice with dipolar and quadrupolar interactions is derived and the nature of variation of the Curie temperature with quadrupolar interaction constant is studied.     

Quadrupolar interactions in TmCu

A thorough study of the cubic rare-earth intermetallic compound TmCu has been performed in the paramagnetic phase in order to obtain reliable quadrupolar parameters. The two (tetragonal and trigonal) symmetry-lowering modes have been systematically investigated by means of all the available experiments: parastriction, elastic constants, third-order magnetic susceptability and magnetization curves. Complete consistency is found between the various determinations of the magnetoelastic and quadrupolar exchange parameters. As in other isomorphous compounds, these quadrupolar interactions appear to be strong enough to compete with the bilinear interactions.     

Selective detection of ordered sodium signals by a jump-and-return pulse sequence

A simple pulse sequence, derived from the shaped pulse optimally exciting the central transition of a spin 3/2, can be used to selectively detect ordered sodium with a given quadrupolar coupling. The pulse sequence consists of two pulses with opposite phases and separated by a delay, called a quadrupolar jump-and-return (QJR) sequence. This QJR sequence is tested with a phantom made of sodium ions in bacteriophage and in aqueous solution and its feasibility for contrast modification based on the quadrupolar coupling is demonstrated.     

Electric quadrupolar contribution to the nuclear spin-lattice relaxation of Ir in Fe

We report on the first quantitative determination of the electric quadrupolar contribution to the nuclear spin-lattice relaxation in a transition metal. For 186Ir and 189Ir in Fe we have determined the magnetic and the electric quadrupolar part of the relaxation for magnetic fields between 0.01 and 2 T. The quadrupolar part gives information on the role of the orbital motion of the electrons for the relaxation process. Our results prove that the unexpected high relaxation rates in Fe and their magnetic field dependence are due to a nonorbital relaxation mechanism.     

Electric quadrupole interactions and the γ–α phase transition in Ce: the role of conduction electrons

We present a theoretical model of the “isostructural" - phase transition in Ce which is based on quadrupolar interactions due to coupled charge density fluctuations of 4f electrons and of conduction electrons. The latter are treated in tight-binding approximation. The - transition is described as an orientational ordering of quadrupolar electronic densities in a structure. The quadrupolar order of the conduction electron densities is complementary to the quadrupolar order of 4f electron densities. The inclusion of conduction electrons leads to an increase of the lattice contraction at the - transition in comparison to the sole effect of 4f electrons. We calculate the Bragg scattering law and suggest synchrotron radiation experiments in order to check the structure. Received 21 September 1999 and Received in final form 2 May 2000     

Li NMR in lithium borate glasses

In anticipation of using fluctuations in the nuclear dipolar and quadrupolar interaction as a probe of lithium ion motion in lithium borate glasses, the static values of these interactions were measured using a variety of echo techniques. The static quadrupolar echo spectrum of ⁷Li and a calculation of the dipolar interaction in crystalline Li₂B₄O₇ (same chemical composition as the glass under study) were used to estimate the strength of the two interactions. These indicate that the dipolar and quadrupolar interactions for ⁶Li will be of similar size and the dipolar interaction will be dominated by the unlike spin interaction between the ⁶Li and the ¹⁰B, ¹¹B spins. An appropriate theoretical model is proposed and explicit expressions for the echo amplitude are calculated in terms of the dipolar and quadrupolar second moments. This single spin model takes into account the quadrupolar interaction but treats the dipolar interaction as an effective magnetic field. Experimental results are presented which show the essential validity of the model and measurements lead to reasonable values for the dipolar and quadrupolar second moments. The relative merits of the various echo techniques are discussed.     

Phase diagrams of spin-1 Ising model with bilinear and quadrupolar interactions under magnetic field. Two-particle cluster approximation

The spin-1 Ising model with bilinear and quadrupolar short-range interactions under magnetic field is investigated within the two-particle cluster approximation. It is shown that for those values of the quadrupolar interaction when at zero magnetic field the system undergoes a temperature phase transition between quadrupolar and paramagnetic phases, a triple point may exist in the temperature vs. magnetic field phase diagrams, necessarily along with a critical point. It is also shown that the critical points in the temperature vs. magnetic field phase diagrams of the investigated model can be of three types.     

High resolution heteronuclear correlation NMR spectroscopy between quadrupolar nuclei and protons in the solid state

A high resolution two-dimensional solid state NMR experiment is presented that correlates half-integer quadrupolar spins with protons. In this experiment the quadrupolar nuclei evolve during t1 under a split-t1, FAM-enhanced MQMAS pulse scheme. After each t1 period ending at the MQMAS echo position, single quantum magnetization is transferred, via a cross polarization process in the mixing time, from the quadrupolar nuclei to the protons. High-resolution proton signals are then detected in the t2 time domain during wPMLG5* homonuclear decoupling. The experiment has been demonstrated on a powder sample of sodium citrate and 23Na-1H 2D correlation spectra have been obtained. From the HETCOR spectra and the regular MQMAS spectrum, the three crystallographically inequivalent Na+ sites in the asymmetric unit were assigned. This MQMAS-wPMLG HETCOR pulse sequence can be used for spectral editing of half-integer quadrupolar nuclei coupled to protons.     

6Li NMR in lithium borate glasses

In anticipation of using fluctuations in the nuclear dipolar and quadrupolar interaction as a probe of lithium ion motion in lithium borate glasses, the static values of these interactions were measured using a variety of echo techniques. The static quadrupolar echo spectrum of ⁷Li and a calculation of the dipolar interaction in crystalline Li₂B₄O₇ (same chemical composition as the glass under study) were used to estimate the strength of the two interactions. These indicate that the dipolar and quadrupolar interactions for ⁶Li will be of similar size and the dipolar interaction will be dominated by the unlike spin interaction between the ⁶Li and the ¹⁰B, ¹¹B spins. An appropriate theoretical model is proposed and explicit expressions for the echo amplitude are calculated in terms of the dipolar and quadrupolar second moments. This single spin model takes into account the quadrupolar interaction but treats the dipolar interaction as an effective magnetic field. Experimental results are presented which show the essential validity of the model and measurements lead to reasonable values for the dipolar and quadrupolar second moments. The relative merits of the various echo techniques are discussed.