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.     

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.     

Theoretical and experimental assessment of single- and multiple-quantum cross-polarization in solid state NMR

Continuous wave cross-polarization (CP) NMR experiments with magic angle spinning (MAS) are reviewed for the case of isolated spin pairs I-S, with spin quantum numbers I = ½ and S ½ (1/2, 3/2, …). For two spin-1/2 nuclei, the Hartmann-Hahn matching conditions are examined at various sample rotation rates ν_R, especially with regard to off-resonance behaviour. In addition to signal enhancement, the CPMAS experiment can be used for the selective determination of inter-nuclear distances between spin-1/2 nuclei. Theoretical examination of the CP transfers to single-quantum (1Q-CPMAS) and multiple-quantum (MQ-CPMAS) levels of quadrupolar nuclei is presented. Simple analytical formulae describing the Hartmann-Hahn matching under various experimental conditions are verified using computer simulations of the spin density matrix under MAS, and the experimental data. The strategies for the most efficient acquisition of 1Q-CPMAS and MQ-CPMAS spectra are extensively discussed.     

Separation of the ortho and para NMR signals in solid deuterium via DQ filtering

Double quantum (DQ) filtering is shown to lead to an effective separation of the NMR signals from the para (I = 1) and ortho (I = 2) molecules in solid deuterium. The separation is achieved by the pulse sequence 90_φ^°–t_pr–90_φ^°–t_ev–90_x^°–t, where the phase-cycled first two pulses create the DQ coherence. Two components are observed after the third pulse; the para signal shows the maximum at a short time t while the ortho signal reaches the maximum at a longer t. The observed signal can be expressed as ∑_I [F_I(t_pr − t) − F_I(t_pr + t)], where F_I(t) is a proper fitting function for the free induction signal of the para and ortho molecules (with I = 1 or 2, respectively). Numerical fits to experimental data at 4.2 and 2 K show that this method can be used to determine the ratio F₁(0)/F₂(0) and thus, because the initial value F_I(0) is proportional to the respective magnetization before the pulse sequence, the ortho and para concentrations in solid deuterium.     

Multiple-Quantum J-Resolved NMR Spectroscopy (MQ-JRES): Measurement of Multiple-Quantum Relaxation Rates and Relative Signs of Spin Coupling Constants

The technique of multiple-quantum J-resolved NMR spectroscopy (MQ-JRES) is introduced and applied to the spin system SI₃–M (such as in the example given here, the ¹³CH₃–¹²CH in alanine). The SI₃ spin system was excited to its highest quantum state (8S_yI_xI_yI_y), which consists of four coherences: quadruple quantum of (3I + S), double quantum of (3I − S), double quantum of (I + S), and zero quantum of (I − S). In the MQ spectrum generated from the projection onto the F₁ dimension, the resonances of the different multiple-quantum coherences are resolved by their coupling constants to the remote spin (M). The absorptive lineshapes in both F₁ and F₂ dimensions enable accurate measurements of transverse relaxation rates, and both amplitude and relative signs of the long-range coupling constants are to be derived from either frequency or time domain data. The selective detection of MQ-JRES spectra of the individual MQ coherences using either phase cycling or pulsed field gradients is presented.     

Evolution of a Model Quantum System¶ Under Time Periodic Forcing:¶Conditions for Complete Ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t→∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very non-generic η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator. Received: 1 November 2000 / Accepted: 5 February 2001     

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.     

N NMR Echo in NH4ClO4after the Pulse Sequence (θ1)x–τ–(θ2)y–t

We have derived a closed-form expression for the solid echo signal of quadrupolarI= 1 nuclei after the pulse sequence (θ₁)_x–τ–(θ₂)_y–tfor arbitrary values of the RF nutation frequency ω₁= γB₁and the quadrupolar frequency ω_Q. In the case of single crystals both the true echo term of this expression and its induction-signal-like terms are important as shown by experiments on¹⁴N nuclei in NH₄ClO₄crystal. Conditions for obtaining the maximal echo in powder samples are presented. A very lowB₁field together with long RF pulses may distort even the central part of the spectrum, resulting in strange looking apparent spectra.     

On the environmental decoherence and spin interference in mesoscopic loop structures

Mechanisms of ‘environmental decoherence’ such as surface scattering, Elliot–Yafet process and precession mechanisms, as well as their influence on the spin phase relaxation are considered and compared. It is shown that the ‘spin ballistic’ regime is possible, when the phase relaxation length for the spin part of the wave function (L^(s)) is much greater than the phase relaxation length for the ‘orbital part’ (L^(e)). In the presence of an additional magnetic field, the spin part of the electron's wave function (WF) acquires a phase shift due to additional spin precession about that field. If the structure length L is chosen to be L^(s)>L>L^(e), it is possible to ‘wash out’ the quantum interference related to the phase coherence of the ‘orbital part’ of the WF, retaining at the same time that related to the phase coherence of the spin part and, hence, to reveal corresponding conductance oscillations.     

Simulation of CPMAS Signals at High Spinning Speeds

The spin dynamics of anS( )I_Nsystem during the CP mixing time of continuous wave and variable amplitude cross-polarization magic angle spinning (CWCPMAS and VACPMAS) experiments is discussed. The signal enhancement of a low abundantSspin, coupled to a set ofN= 6 coupled spins withI= , is evaluated as a function of the length of the mixing time. For CWCPMAS this signal is first evaluated in the frequency domain and then transformed to the time domain. These calculations provide some additional insight into the CP spin dynamics and enable a practical approach toward the evaluation of CP signals of large spin systems. In addition the adiabatic character of the ramped VACPMAS experiments is discussed andS-spin signals of a spin system withN= 6 are simulated. Estimates of the upper bounds of the CP signals as a function of the number ofIspins in anS( )I_Nsystem are given and compared with the calculated values.     

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.     

Relaxation-induced dipolar exchange with recoupling-An MAS NMR method for determining heteronuclear distances without irradiating the second spin

A new magic-angle spinning NMR method for distance determination between unlike spins, where one of the two spins in question is not irradiated at all, is introduced. Relaxation-induced dipolar exchange with recoupling (RIDER) experiments can be performed with conventional double-resonance equipment and utilize the familiar π-pulse trains to recouple the heteronuclear dipolar interaction under magic-angle spinning conditions. Longitudinal relaxation of the passive spin during a delay between two recoupling periods results in a dephasing of the heteronuclear coherence and consequently a dephasing of the magnetization detected after the second recoupling period. The information about the dipolar coupling is obtained by recording normalized dephasing curves in a fashion similar to the REDOR experiment. At intermediate mixing times, the dephasing curves also depend on the relaxation properties of the passive spin, i.e., on single- and double-quantum longitudinal relaxation times for the case of I = 1 nuclei, and these relaxation times can be estimated with this new method. To a good approximation, the experiment does not depend on possible quadrupolar interactions of the passive spin, which makes RIDER an attractive method when distances to quadrupolar nuclei are to be determined. The new method is demonstrated experimentally with ¹⁴N and ²H as heteronuclei and observation of ¹³C in natural abundance.     

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.     

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.     

Direct Evidence of Nucleation During the Induction Period of Polyethylene Crystallization by SAXS

Direct evidence that nuclei are formed during the induction period of crystallization is obtained for the first time by means of small-angle X-ray scattering (SAXS). Polyethylene (PE) was used as a model crystalline polymer. The nucleating agent was mixed with PE in order to increase the scattering intensity I _x from nuclei as large as 10⁴ times bigger than usual. I _x increased soon after quenching to the crystallization temperature from the melt and saturated after some time. A new theory is proposed to estimate the size of the nuclei N, the number density distribution of nuclei with N at time t, f(t,N), and the induction time τ _i, by analyzing the SAXS scattering intensity. The volume-averaged size of the nuclei was nearly the same as that of critical nuclei and does not change so much with time during the induction period. Lamellae start stacking much later than nuclei start forming.     

Influencing the satellite transitions of half-integer quadrupolar nuclei for the enhancement of magic angle spinning spectra

Double frequency sweeps can induce spin transitions in a set of satellites of a half-integer quadrupolar nucleus by simultaneously passing through resonance for a satellite pair. It is shown that by transferring population from the outer spin levels to the inner |1/2 and |−1/2 levels an increased intensity for central transition spectra is obtained. Although Magic Angle Spinning in principle interferes with this process, and the adiabaticity of the passages is different for every crystallite in a powder, enhanced spectra with undistorted line shapes are obtained for I=3/2 (²³Na) and 5/2 (²⁷Al) spins experiencing quadrupolar interactions with ω_Q in the range 0.1–3 MHz. Even at spinning speeds up to 30 kHz significant enhancements are obtained. An analysis of the combined effects of double frequency sweeps (DFS) and MAS indeed shows strongly different effects for different crystallites in powder ranging from no gain at all to the theoretical maximum gain of 2I. As the effects are randomly distributed over all orientations on a sphere this is averaged over the whole line shape. Therefore, undistorted powder patterns are obtained enhanced by the average gain over the individual crystallites. Saturation of the satellite transitions, which can only be achieved if spin–spin relaxation is sufficiently strong, leads to identical results. Optimization of the sweeps should be toward an optimal effect on the population transfer to the central levels and chosen short with respect to spin–lattice relaxation times.     

Electric quadrupole interactions in non-axially symmetric electric field gradients of spinI=2 and 3 nuclei in polycrystalline sources

Calculations and formulae are presented for the purpose of understanding and analyzing Perturbed Angular Correlation (PAC) experiments that use spin-two and spin-three probe nuclei. For electric quadrupole interactions of probe nuclei in a polycrystalline source with non-axially symmetric electric field gradients, the interaction frequencies and perturbation functions for spin-two and spin-three nuclei differ qualitatively and quantitatively from those for spin-five-halves nuclei. These differences result primarily because them-states of integral-spin nuclei are nondegenerate in an asymmetric electric field gradient. To help the experimenter deal with the added complexity of the integral-spin casesI=2 and 3, closed-form expressions are provided for the energy eigenvalues and the eigenvectors as a function of the asymmetry parameter η. To deal with the problem of calculating the perturbation functionsG _kk (t) for η>0, the summation overm-state quantum numbers is formulated in terms ofa _n′,n ^(k) -coefficients. These coefficients are analogous to theS _kn -coefficients used in the case for η=0. To illustrate the differences between the half-integral-spin caseI=5/2 and the integral-spin casesI=2 and 3, energy difference diagrams anda _n′,n′ ⁽²⁾ diagrams are presented and discussed.     

Tris(dimethylamino)-cyclopropenium radical dication

Tris(dimethylamino)-cyclopropenium cation (I^⊕) undergoes a facile oxidation to the radical dication (I^2⊕). The E.S.R. spectrum of I^2⊕, which covers a range of almost 200 gauss, has been analysed in terms of the coupling constants a _N =7·51 (three ¹⁴N nuclei) and a _H ^CH₃ =8·16 gauss (eighteen protons). The line-width can be expressed as

where M _I(N) is the magnetic quantum number for the set of three equivalent ¹⁴N nuclei. A simple molecular orbital model of the trisamino-cyclopropenium system satisfactorily accounts for some physico-chemical properties of I^⊕. The E.S.R. results for I^2⊕ are compared with those for the iso-π-electronic hexamethyl[3]radialene radical anion (II^?).     

Muon addition to nitrogen,oxygen and other atoms: Considerations for level crossing resonance studies

The purpose of this note is to examine the conditions under which muon level crossing resonance with quadrupolar nuclei may be used to characterise the elusive diamagnetic fraction which is formed when positive muons are stopped in various media and associated chemically with the host molecules. A potential difficulty is identified for nuclei having integral spin, which may explain why cross polarisation to¹⁴N has not yet been detected. The general suitability of nuclei with half-integral spin (I≥3/2) is illustrated with the case of¹⁷O, and suggestions are made for future studies with other nuclei, including species such as molecular ions and defect complexes (muon-impurity pairs).     

Effect of localized spins in coherent transport through quantum dots

The effect of localized spins on the quantum coherence in solids is discussed. A quantum dot with an odd number of electrons can be a model system for a localized spin. It is experimentally shown that a spin flip scattering by a quantum dot pulls the trigger of quantum decoherence. On the other hand, spin flip scattering is the basic process to construct the Kondo singlet state around a magnetic impurity. Through an interference effect of the Kondo state (the Fano–Kondo effect) in a side-coupled dot system, we show experimentally that the Kondo singlet state is quantum mechanically coherent. The analysis of the Fano–Kondo lineshape indicates the locking of the phase shift to π/2, which is in agreement with theoretical predictions. The Fano–Kondo effect is also observed in an Aharonov–Bohm ring, in which a quantum dot is embedded, and also indicates the phase shift locking to π/2.