Quadrupolar Spin Research Articles

Theoretical study of homonuclear J coupling between quadrupolar spins: Single-crystal, DOR, and J-resolved NMR

The theory describing homonuclear indirect nuclear spin–spin coupling (J) interactions between pairs of quadrupolar nuclei is outlined and supported by numerical calculations. The expected first-order multiplets for pairs of magnetically equivalent (A2), chemically equivalent (AA′), and non-equivalent (AX) quadrupolar nuclei are given. The various spectral changeovers from one first-order multiplet to another are investigated with numerical simulations using the SIMPSON program and the various thresholds defining each situation are given. The effects of chemical equivalence, as well as quadrupolar coupling, chemical shift differences, and dipolar coupling on double-rotation (DOR) and J-resolved NMR experiments for measuring homonuclear J coupling constants are investigated. The simulated J coupling multiplets under DOR conditions largely resemble the ideal multiplets predicted for single crystals, and a characteristic multiplet is expected for each of the A2, AA′, and AX cases. The simulations demonstrate that it should be straightforward to distinguish between magnetic inequivalence and equivalence using J-resolved NMR, as was speculated previously. Additionally, it is shown that the second-order quadrupolar-dipolar cross-term does not affect the splittings in J-resolved experiments. Overall, the homonuclear J-resolved experiment for half-integer quadrupolar nuclei is demonstrated to be robust with respect to the effects of first- and second-order quadrupolar coupling, dipolar coupling, and chemical shift differences.

Distance Measurements to Metal Ions and Other Quadrupolar Spins by Magic Angle Spinning Solid State NMR

AbstractThe characterization of metal sites in many types of materials is essential for understanding many aspects of chemistry and biochemistry, such as catalysis, enzymatic activity, and coordination. Of the many approaches to study such sites, this review deals with the utilization of dipolar recoupling in the context of magic angle spinning solid state NMR for qualitative and, in particular, quantitative distance measurements between metals and other quadrupolar nuclei to their surrounding environment. It will focus on the development of techniques, on their applicability towards experimental realizations, on key applications to materials and biological systems, and on current challenges and limitations.

Central-transition double-quantum sideband NMR spectroscopy of half-integer quadrupolar nuclei: estimating internuclear distances and probing clusters within multi-spin networks

We introduce a strategy to estimate the size of clusters of recoupled homonuclear half-integer quadrupolar nuclei under magic-angle spinning (MAS) conditions, by combining double-quantum (2Q) sideband NMR experiments with an approximate numerical analysis based on the summation of all spin-pairs present over a given radius of the structure. The experiment relies solely on the evolution of homonuclear 2Q coherences (2QC) among the central-transitions (CT) of half-integer spins and is suitable for probing clusters in network structures, such as those encountered in large groups of oxide-based materials. Experimental (11)B, (23)Na and (27)Al NMR results are presented on bis(catecholato)diboron, Na2SO4 and Al2O3, respectively; in each case, the growth of the spin-cluster size was monitored from a series of experiments that employed progressively lengthened 2QC excitation intervals. Our new approach is the first option for probing larger constellations of half-integer spins; it provides similar information as the "multiple-quantum spin counting" experiment, which is well-established for spin-1/2 applications but has hitherto not been demonstrated for half-integer spins undergoing MAS. We also discuss various options for determining the internuclear distance within a (nearly) isolated pair of half-integer spins by comparing the experimental 2Q sideband NMR spectra with results from numerical simulations involving various degrees of approximation.

Longitudinal magnetization reversal in ferromagnets with Heisenberg exchange and strong single-ion anisotropy

We analyze theoretically the novel pathway of ultrafast spin dynamics for ferromagnets with high enough single-ion anisotropy. This longitudinal spin dynamics includes the coupled oscillations of the modulus of the magnetization together with the quadrupolar spin variables, which are expressed through quantum expectation values of operators bilinear on the spin components. Even for a simple single-element ferromagnet, such dynamics can lead to a magnetization reversal under the action of an ultrashort laser pulse.

Nuclear magnetic relaxation induced by exchange-mediated orientational randomization: Longitudinal relaxation dispersion for a dipole-coupled spin-1/2 pair

In complex biological or colloidal samples, magnetic relaxation dispersion (MRD) experiments using the field-cycling technique can characterize molecular motions on time scales ranging from nanoseconds to microseconds, provided that a rigorous theory of nuclear spin relaxation is available. In gels, cross-linked proteins, and biological tissues, where an immobilized macromolecular component coexists with a mobile solvent phase, nuclear spins residing in solvent (or cosolvent) species relax predominantly via exchange-mediated orientational randomization (EMOR) of anisotropic nuclear (electric quadrupole or magnetic dipole) couplings. The physical or chemical exchange processes that dominate the MRD typically occur on a time scale of microseconds or longer, where the conventional perturbation theory of spin relaxation breaks down. There is thus a need for a more general relaxation theory. Such a theory, based on the stochastic Liouville equation (SLE) for the EMOR mechanism, is available for a single quadrupolar spin I = 1. Here, we present the corresponding theory for a dipole-coupled spin-1/2 pair. To our knowledge, this is the first treatment of dipolar MRD outside the motional-narrowing regime. Based on an analytical solution of the spatial part of the SLE, we show how the integral longitudinal relaxation rate can be computed efficiently. Both like and unlike spins, with selective or non-selective excitation, are treated. For the experimentally important dilute regime, where only a small fraction of the spin pairs are immobilized, we obtain simple analytical expressions for the auto-relaxation and cross-relaxation rates which generalize the well-known Solomon equations. These generalized results will be useful in biophysical studies, e.g., of intermittent protein dynamics. In addition, they represent a first step towards a rigorous theory of water (1)H relaxation in biological tissues, which is a prerequisite for unravelling the molecular basis of soft-tissue contrast in clinical magnetic resonance imaging.

Ultra-Wideline Solid-State NMR Spectroscopy

Although solid-state NMR (SSNMR) provides rich information about molecular structure and dynamics, the small spin population differences between pairs of spin states that give rise to NMR transitions make it an inherently insensitive spectroscopic technique in terms of signal acquisition. Scientists have continuously addressed this issue via improvements in NMR hardware and probes, increases in the strength of the magnetic field, and the development of innovative pulse sequences and acquisition methodologies. As a result, researchers can now study NMR-active nuclides previously thought to be unobservable or too unreceptive for routine examination via SSNMR. Several factors can make it extremely challenging to detect signal or acquire spectra using SSNMR: (i) low gyromagnetic ratios (i.e., low Larmor frequencies), (ii) low natural abundances or dilution of the nuclide of interest (e.g., metal nuclides in proteins or in organometallic catalysts supported on silica), (iii) inconvenient relaxation characteristics (e.g., very long longitudinal or very short transverse relaxation times), and/or (iv) extremely broad powder patterns arising from large anisotropic NMR interactions. Our research group has been particularly interested in efficient acquisition of broad NMR powder patterns for a variety of spin-1/2 and quadrupolar (spin > 1/2) nuclides. Traditionally, researchers have used the term "wideline" NMR to refer to experiments yielding broad (1)H and (2)H SSNMR spectra ranging from tens of kHz to ∼250 kHz in breadth. With modern FT NMR hardware, uniform excitation in these spectral ranges is relatively easy, allowing for the acquisition of high quality spectra. However, spectra that range in breadth from ca. 250 kHz to tens of MHz cannot be uniformly excited with conventional, high-power rectangular pulses. Rather, researchers must apply special methodologies to acquire such spectra, which have inherently low S/N because the signal intensity is spread across such large spectral breadths. We have suggested the term ultra-wideline NMR (UWNMR) spectroscopy to describe this set of methodologies. This Account describes recent developments in pulse sequences and strategies for the efficient acquisition of UWNMR spectra. After an introduction to anisotropically broadened NMR patterns, we give a brief history of methods used to acquire UWNMR spectra. We then discuss new acquisition methodologies, including the acquisition of CPMG echo trains and the application of pulses capable of broadband excitation and refocusing. Finally, we present several applications of UWNMR methods that use these broadband pulses.

Spin coherent states in NMR quadrupolar system: experimental and theoretical applications

Working with nuclear magnetic resonance (NMR) in quadrupolar spin systems, in this paper we transfer the concept of atomic coherent state to the nuclear spin context, where it is referred to as pseudo-nuclear spin coherent state (pseudo-NSCS). Experimentally, we discuss the initialization of the pseudo-NSCSs and also their quantum control, implemented by polar and azimuthal rotations. Theoretically, we compute the geometric phases acquired by an initial pseudo-NSCS on undergoing three distinct cyclic evolutions: (i) the free evolution of the NMR quadrupolar system and, by analogy with the evolution of the NMR quadrupolar system, that of (ii) single-mode and (iii) two-mode Bose-Einstein Condensate like system. By means of these analogies, we derive, through spin angular momentum operators, results equivalent to those presented in the literature for orbital angular momentum operators. The pseudo-NSCS description is a starting point to introduce the spin squeezed state and quantum metrology into nuclear spin systems of liquid crystal or solid matter.

Measuring dipolar and J coupling between quadrupolar nuclei using double-rotation NMR

Among the interactions which govern NMR spectra, spin-spin coupling interactions provide the most direct form of structural information which is of interest to chemists. Dipolar coupling may be used to measure internuclear distances directly and J coupling may be used to identify bonding interactions and provide insights into the nature of the bonds. It is well known that the presence of a quadrupolar interaction reintroduces the dipolar interaction in spinning samples; however, similarly to the J coupling, this information is often lost if the observed nucleus is quadrupolar due to quadrupolar spectral broadening. Here we show for multiple spin pairs that double-rotation (DOR) NMR fully removes the effects of the quadrupolar interaction on the NMR spectrum leaving only the effects of dipolar and J couplings. We also demonstrate that the J coupling multiplets do not disappear for quadrupolar A2 spin pairs as they do for spin-1/2 nuclei. With DOR NMR, it is then straightforward to measure homonuclear J coupling constants between magnetically equivalent quadrupolar nuclei. A deeper understanding of the origins of the magnitudes and dominant mechanisms of J coupling for quadrupolar spin pairs in a series of related compounds is obtained by decomposing computed J coupling constants into their major molecular orbital contributions.

Preparing Pseudo-Pure States in a Quadrupolar Spin System Using Optimal Control

Pseudo-pure state (PPS) preparation is crucial in nuclear magnetic resonance quantum computation. There have been some methods in spin-1/2 systems and a few attempts in quadrupolar spin systems. As optimal control via gradient ascent pulses engineering (GRAPE) has been widely used in quantum information science, we apply this technique to PPS preparation in quadrupolar spin systems. This approach shows an effective and fast quantum control method for both the state preparation and the realization of quantum gates in quadrupolar systems.

Concept of effective Hamiltonians for transitions in multi-level systems

Employing the concept of effective Hamiltonians, an analytical theory is introduced to describe transitions in a multi-level system in nuclear magnetic resonance (NMR) spectroscopy. Specifically, the discussion is centered towards the treatment of selective and non-selective excitations in static quadrupolar spin (I > 1/2) systems. To this end, effective radiofrequency (RF) Hamiltonians based on the spherical tensor formalism are proposed for describing transitions in both integral (I = 1, 2 and 3) and half-integral (I = 3/2, 5/2 and 7/2) quadrupolar spins. The optimum conditions desired for selective excitation in a multi-level system are derived pedagogically from first principles and presented through analytical expressions. Employing suitable model systems, the derived optimum conditions are substantiated through rigorous numerical simulations based on the spherical tensor formalism. The theory presented provides a framework for describing selective and non-selective RF pulses and could improve our understanding of multiple-pulse experiments involving quadrupolar nuclei.

Direct Investigation of Covalently Bound Chlorine in Organic Compounds by Solid‐State 35Cl NMR Spectroscopy and Exact Spectral Line‐Shape Simulations

35/37Cl NMR spectroscopy studies of organic systems are very rare, with only a few neat liquids having been studied.1 The lack of chlorine NMR spectroscopy data may be explained by the fact that 35Cl and 37Cl are quadrupolar (spin I=3/2) and low-frequency isotopes. The quadrupole moments of the chlorine nuclei couple with the electric field gradient (EFG) tensor at the nuclei; this phenomenon is known as the quadrupolar interaction (QI). The quadrupolar coupling constant, CQ, and the quadrupolar asymmetry parameter, ηQ, describe the magnitude and asymmetry of the QI. In solution, one of the consequences of the QI is fast relaxation, which means that the 35/37Cl NMR signals for covalently bound chlorines are very broad and are of low intensity.1 For these reasons, chemically distinct chlorine sites are very difficult to distinguish with solution NMR spectroscopy. However, in the solid state, nuclear spin relaxation is typically slower, thus enabling higher quality 35Cl NMR spectra to be collected, at least in principle. Unfortunately, the magnitude of the QI for covalently bound chlorines is very large because of the substantial, anisotropic EFG at the Cl atom, owing mainly to its electronic configuration when it forms a chlorine–carbon bond. Conventional wisdom is that such chlorine sites cannot be studied in powders by solid-state NMR spectroscopy as the central transition (CT; mI=1/2↔−1/2) can span tens of megahertz in typical commercially available magnetic fields. For this reason, only ionic chlorides2 and inorganic chlorides3 have been studied, as the EFG at these chlorides is often an order of magnitude smaller than at covalently bound chlorine atoms in organic molecules. The bonding environments for these types of chlorine atoms are substantially different from the environments in those chloride-containing molecules that have been studied previously.2, 3 A partial 35Cl NMR spectrum for hexachlorophene has been briefly mentioned in the literature.4 On the other hand, most of the interesting chlorine chemistry occurs when Cl is covalently bound to a carbon atom, where the chlorine atom often acts as a leaving group. Chlorine atoms are also important in many organic pharmaceuticals as well as in crystal design applications where they can form halogen bonds.5 Recent studies show that covalently bound chlorine is also important in biological chemistry where, for example, the tryptophan 7-halogenase was found to selectively chlorinate tryptophan moieties.6

Measurement of statistical nuclear spin polarization in a nanoscale GaAs sample

We measure the statistical polarization of quadrupolar nuclear spins in a sub-micrometer (0.6 um^3) particle of GaAs using magnetic resonance force microscopy. The crystalline sample is cut out of a GaAs wafer and attached to a micro-mechanical cantilever force sensor using a focused ion beam technique. Nuclear magnetic resonance is demonstrated on ensembles containing less than 5 x 10^8 nuclear spins and occupying a volume of around (300 nm)^3 in GaAs with reduced volumes possible in future experiments. We discuss how the further reduction of this detection volume will bring the spin ensemble into a regime where random spin fluctuations, rather than Boltzmann polarization, dominate its dynamics. The detection of statistical polarization in GaAs therefore represents an important first step toward 3D magnetic resonance imaging of III-V materials on the nanometer-scale.

Transport and Accumulation of Spin Anisotropy

We show that spin anisotropy can be transferred to an isotropic system by transport of a spin-quadrupole moment. We derive the quadrupole moment current and continuity equation and study a spin-valve structure consisting of two ferromagnets coupled to a quantum dot probing an impurity spin. The quadrupole backaction on their coupled spin results in spin torques and anisotropic spin relaxation which do not follow from standard spin-current considerations. We demonstrate the detection of the impurity spin by charge transport and its manipulation by electric fields.

Transmission of excitations in a spin-1 Bose-Einstein condensate through a barrier

We investigate tunneling of excitations across a potential barrier separating two spin-1 Bose-Einstein condensates. Using the mean-field theory at the absolute zero temperature, we determine transmission coefficients of excitations in the saturated magnetization state and unsaturated magnetization states. All excitations except the quadrupolar spin mode in the saturated magnetization state show the anomalous tunneling phenomenon characterized as perfect tunneling in the low momentum limit through a potential barrier. The quadrupolar spin mode in the saturated magnetization state, whose spectrum is massive, shows total reflection. We discuss properties common between excitations showing the anomalous tunneling phenomenon. Excitations showing perfect tunneling have gapless spectrum in the absence of the magnetic field, and their wave functions in the low energy limit are the same as the condensate wave function.

Observation of electric quadrupole spin resonance of Ho3+ impurity ions in synthetic forsterite

The anomalous dependence of the electron paramagnetic resonance (EPR) line shape on the microwave power in the resonator has been found when studying the continuous-wave EPR spectra of impurity holmium ions in synthetic forsterite on an ELEXSYS E 580 EPR spectrometer. The power-threshold transition from the conventional lines being the derivatives of the spectral line contours to the spectral line contours themselves has been observed as the power increased. The properties of the anomalous EPR lines are qualitatively explained assuming that the resonance electric quadrupole transitions take place between the electron spin levels.

Interpolation by fast Wigner transform for rapid calculations of magnetic resonance spectra from powders

We introduce a novel interpolation strategy, based on nonequispaced fast transforms involving spherical harmonics or Wigner functions, for efficient calculations of powder spectra in (nuclear) magnetic resonance spectroscopy. The fast Wigner transform (FWT) interpolation operates by minimizing the time-consuming calculation stages, by sampling over a small number of Gaussian spherical quadrature (GSQ) orientations that are exploited to determine the spectral frequencies and amplitudes from a 10-70 times larger GSQ set. This results in almost the same orientational averaging accuracy as if the expanded grid was utilized explicitly in an order of magnitude slower computation. FWT interpolation is applicable to spectral simulations involving any time-independent or time-dependent and noncommuting spin Hamiltonian. We further show that the merging of FWT interpolation with the well-established ASG procedure of Alderman, Solum and Grant [J. Chem. Phys. 134, 3717 (1986)] speeds up simulations by 2-7 times relative to using ASG alone (besides greatly extending its scope of application), and between 1-2 orders of magnitude compared to direct orientational averaging in the absence of interpolation. Demonstrations of efficient spectral simulations are given for several magic-angle spinning scenarios in NMR, encompassing half-integer quadrupolar spins and homonuclear dipolar-coupled (13)C systems.

Diagonalization-free implementation of spin relaxation theory for large spin systems

The Liouville space spin relaxation theory equations are reformulated in such a way as to avoid the computationally expensive Hamiltonian diagonalization step, replacing it by numerical evaluation of the integrals in the generalized cumulant expansion. The resulting algorithm is particularly useful in the cases where the static part of the Hamiltonian is dominated by interactions other than Zeeman (e.g. in quadrupolar resonance, low-field EPR and Spin Chemistry). When used together with state space restriction tools, the algorithm reported is capable of computing full relaxation superoperators for NMR systems with more than 15 spins.

Electrodynamics of finite-size particles with arbitrary spin

We construct the most general Hamiltonian for the electromagnetic interaction of the finite-size particle-like nucleus with arbitrary spin, magnetic dipole, and electric quadrupole moments. It includes all the terms, which are important for obtaining atomic energy levels up to the order ${\ensuremath{\alpha}}^{6}$. The result is verified against spin $s=0$, $1/2$, and $1$ cases, where the Foldy-Wouthuysen transformation is performed of the corresponding relativistic equation.

Effect of the Tensor Force on Charge-Exchange Spin-Dependent Multipole Excitations

The charge-exchange spin-dipole (SD) and spin-quadrupole (SQ) strength functions of 90Zr are calculated with and without the tensor terms of the Skyrme interaction in self-consistent HF+RPA approach. It is found that, in SD and SQ transitions, the RPA correlations associated with the tensor terms shin dramatically the strengths of (Ylσ)λ = l−1 and (Ylσ)λ = 1 modes upward and downward, respectively, and also shift the strengths of (Ylσ)λ = l + 1 modes upward. The coupling between (Yl = λ − 1σ)λ and (Yl = λ + 1σ)λ modes arising from the tensor correlation is noticeable. The RPA tensor correlations produce strengths of SD and SQ modes, which are distributed in a much wider energy range, and the (Ylσ)λ = l − 1 modes dominate the high energy part of the strength functions. These energy shifts and coupling effects of different modes can be understood qualitatively by expressing a finite range tensor force in a separable form.

ESTIMATION OF DOMAIN SIZE IN NANO-FERROELECTRICS FROM NMR T1 MEASUREMENTS

ABSTRACT The spin lattice relaxation of I = 3/2 quadrupolar spin system due to domain walls in order-disorder ferroelectrics has been studied and a general method is proposed for the measurement of domain width in nano ferroelectrics. Based on the fact that electric polarization undergoes spiral orientation as one moves from one domain to the other, it is assumed that at low temperatures the spins at and near domain walls undergo relaxation due to possible easy reorientation of electric polarization in domain walls even though such a relaxation in the main body of the domain has almost ceased. The spins present inside the domain undergo relaxation through transfer of magnetization to the domain walls through a spin diffusion process by nearest neighbour interaction. Rate equations for spin populations are formed by representing the ferroelectric domain by a one-dimensional chain of equidistant spins having dipolar coupling. Spin populations are calculated as a function of time for different ratios of quadrupolar to dipolar transition probabilities for a sample subjected to selective rf pulse. Expression for spin- lattice relaxation time T 1 is derived in terms of domain width and ratio of quadrupolar to dipolar transition probabilities. It is found that the domain width can be estimated provided the value of spin lattice relaxation time T 1 is known for the corresponding crystal with normal sized grains. The results are quite general and can be applied to any order disorder ferroelectric with nano sized domains and having spin I = 3/2 nuclei.