This document summarizes research on two-dimensional solid-state nutation NMR experiments for determining quadrupole parameters of half-integer quadrupolar nuclei. It presents:
1) A complete series of simulated nutation spectra for spins I = 3/2 to I = 5 calculated using density matrix formalism to serve as fingerprints for parameter determination.
2) Applications of the method to 27Al in spodumene and 45Sc in Sc2(SO4)3 to determine their quadrupole parameters by comparing experimental spectra to simulations.
3) Discussion of experimental aspects like resonance offset and magic angle spinning and how they affect the nutation spectra.
This document summarizes a research article that develops a theoretical model to describe how decoherence effects rubidium vapor in an electromagnetically induced transparency (EIT) experiment. The model accounts for decoherence from both dephasing and population relaxation. It quantifies the impact of decoherence on various experimental measurements, including Faraday rotation, susceptibility, transmission, and coherence relationships. The model is in good agreement with previous experimental results. It also discusses how the model could be applied to other EIT-based experiments and how Faraday rotation could be used to detect single atoms.
The document analyzes five numerical integration methods (Euler, improved Euler, Verlet, Euler-Cromer, and RK4) for approximating the solution to a damped harmonic oscillator equation. The RK4 method was found to best match the analytic solution. RK4 was then used to study critical damping, steady state oscillations under a driving force, and the effects of sudden applied forces. Resonance curves were also generated. The document concludes that RK4 is the most accurate integration method for this problem.
The document summarizes key concepts from quantum mechanics and symmetries. It states that physical states are represented by rays in a Hilbert space, with observables represented by Hermitian operators. The probability of measuring a state is given by the inner product of the state vectors. Symmetries are represented by either unitary or antiunitary operators on the Hilbert space. Symmetries that can be continuously connected to the identity must be represented by unitary operators. Symmetries form a group, with transformations combining according to the group multiplication rule.
The phase plane of moving discrete breathersPaul Houle
1. The document studies discrete breathers in a periodic chain of 5 atoms coupled by quadratic-quartic springs. It uses discrete symmetries to simplify the analysis and visualize the phase space.
2. It finds that unlike even atom chains, odd atom chains have no energy threshold for breathers due to degenerate band edge modes. It observes long-lived "hopping breathers" that move chaotically in the separatrix region dividing pinned and moving states.
3. It applies linear stability analysis to measure quasi-phonon frequencies around breathers and map the stable and unstable manifolds of unstable breathers. This allows systematically launching moving or pinned breathers by perturbing stable or unstable breathers.
1. The document discusses nonadiabatic transitions in chemical systems using path integral representations.
2. It introduces a nonadiabatic path integral approach based on overlap integrals between nuclear configurations on different electronic surfaces, as an alternative to the traditional derivative coupling approach.
3. A nonadiabatic beads model is presented for calculating nonadiabatic partition functions using a quantum-classical mapping, where the Boltzmann operator is divided among discrete nuclear coordinates on different surfaces connected by overlap integrals.
A Route to Chaos for the Physical Double Pendulum by Daniel Berkowitz
This document summarizes research examining the route to chaos for the physical double pendulum. The Lagrangian and Hamiltonian are derived for the physical double pendulum system. Poincare sections show that as the parameter epsilon increases, the quasi-periodic tori in phase space collapse into points, indicating periodic motion of the bottom pendulum every n oscillations. Bifurcation diagrams also show splitting corresponding to period n orbits just before the onset of global chaos. The research demonstrates that the physical double pendulum exhibits ordered periodic motion just prior to becoming fully chaotic.
Phonon frequency spectrum through lattice dynamics and normal coordinate anal...Alexander Decker
The document discusses the lattice dynamics and normal coordinate analysis of the high-temperature superconductor Tl2Ca3Ba2Cu4O12. It presents the following key points:
1. Lattice dynamics calculations using the three-body force shell model reproduce observed Raman and infrared phonon frequencies reasonably well.
2. Normal coordinate analysis using Wilson's F-G matrix method yields vibrational frequencies in good agreement with experimental values and lattice dynamics calculations.
3. Potential energy distribution calculations confirm that the chosen vibrational frequencies make the maximum contribution to the potential energy of the material's normal coordinate frequencies.
Quantum Theory. Wave Particle Duality. Particle in a Box. Schrodinger wave equation. Quantum Numbers and Electron Orbitals. Principal Shells and Subshells. A Fourth Quantum Number. Effective nuclear charge
This document summarizes a research article that develops a theoretical model to describe how decoherence effects rubidium vapor in an electromagnetically induced transparency (EIT) experiment. The model accounts for decoherence from both dephasing and population relaxation. It quantifies the impact of decoherence on various experimental measurements, including Faraday rotation, susceptibility, transmission, and coherence relationships. The model is in good agreement with previous experimental results. It also discusses how the model could be applied to other EIT-based experiments and how Faraday rotation could be used to detect single atoms.
The document analyzes five numerical integration methods (Euler, improved Euler, Verlet, Euler-Cromer, and RK4) for approximating the solution to a damped harmonic oscillator equation. The RK4 method was found to best match the analytic solution. RK4 was then used to study critical damping, steady state oscillations under a driving force, and the effects of sudden applied forces. Resonance curves were also generated. The document concludes that RK4 is the most accurate integration method for this problem.
The document summarizes key concepts from quantum mechanics and symmetries. It states that physical states are represented by rays in a Hilbert space, with observables represented by Hermitian operators. The probability of measuring a state is given by the inner product of the state vectors. Symmetries are represented by either unitary or antiunitary operators on the Hilbert space. Symmetries that can be continuously connected to the identity must be represented by unitary operators. Symmetries form a group, with transformations combining according to the group multiplication rule.
The phase plane of moving discrete breathersPaul Houle
1. The document studies discrete breathers in a periodic chain of 5 atoms coupled by quadratic-quartic springs. It uses discrete symmetries to simplify the analysis and visualize the phase space.
2. It finds that unlike even atom chains, odd atom chains have no energy threshold for breathers due to degenerate band edge modes. It observes long-lived "hopping breathers" that move chaotically in the separatrix region dividing pinned and moving states.
3. It applies linear stability analysis to measure quasi-phonon frequencies around breathers and map the stable and unstable manifolds of unstable breathers. This allows systematically launching moving or pinned breathers by perturbing stable or unstable breathers.
1. The document discusses nonadiabatic transitions in chemical systems using path integral representations.
2. It introduces a nonadiabatic path integral approach based on overlap integrals between nuclear configurations on different electronic surfaces, as an alternative to the traditional derivative coupling approach.
3. A nonadiabatic beads model is presented for calculating nonadiabatic partition functions using a quantum-classical mapping, where the Boltzmann operator is divided among discrete nuclear coordinates on different surfaces connected by overlap integrals.
A Route to Chaos for the Physical Double Pendulum by Daniel Berkowitz
This document summarizes research examining the route to chaos for the physical double pendulum. The Lagrangian and Hamiltonian are derived for the physical double pendulum system. Poincare sections show that as the parameter epsilon increases, the quasi-periodic tori in phase space collapse into points, indicating periodic motion of the bottom pendulum every n oscillations. Bifurcation diagrams also show splitting corresponding to period n orbits just before the onset of global chaos. The research demonstrates that the physical double pendulum exhibits ordered periodic motion just prior to becoming fully chaotic.
Phonon frequency spectrum through lattice dynamics and normal coordinate anal...Alexander Decker
The document discusses the lattice dynamics and normal coordinate analysis of the high-temperature superconductor Tl2Ca3Ba2Cu4O12. It presents the following key points:
1. Lattice dynamics calculations using the three-body force shell model reproduce observed Raman and infrared phonon frequencies reasonably well.
2. Normal coordinate analysis using Wilson's F-G matrix method yields vibrational frequencies in good agreement with experimental values and lattice dynamics calculations.
3. Potential energy distribution calculations confirm that the chosen vibrational frequencies make the maximum contribution to the potential energy of the material's normal coordinate frequencies.
Quantum Theory. Wave Particle Duality. Particle in a Box. Schrodinger wave equation. Quantum Numbers and Electron Orbitals. Principal Shells and Subshells. A Fourth Quantum Number. Effective nuclear charge
Pseudoperiodic waveguides with selection of spatial harmonics and modesVictor Solntsev
A principle of selection of modes and their spatial harmonics in periodic waveguides and, in particular, in spatially developed slowing systems for multibeam traveling-wave tubes (TWTs) is elaborated. The essence of the principle is in the following: varying along the length of the system its period and at least one more parameter that determines the phase shift per period, one can provide constant phase velocity of one spatial harmonic and destroy other spatial harmonics, i.e., reduce their amplitudes substantially. In this case, variations of the period may be significant, and the slowing system becomes nonuniform, or pseudoperiodic; namely, one of the spatial harmonics remains the same as in the initial periodic structure. Relationships are derived for the amplitudes of the spatial-wave harmonics, interaction coefficient, and coupling impedance of the pseudoperiodic system. The possibility of the mode selection in pseudoperiodic slowing systems when the synchronism condition is satisfied for the spatial harmonic of one mode is investigated. The efficiency of suppressing spurious spatial harmonics and modes for linear and abrupt variation of spacing is estimated. The elaborated principle of selection of spatial harmonics and modes is illustrated by an example of a two-section helical-waveguide slowing system.
This document presents a statistical theory of the equilibrium elastic and thermal properties of crystals that have rotational degrees of freedom. Microscopic expressions are derived for strain and stress tensors using a local equilibrium ensemble. Statistical expressions are obtained for properties like specific heat of deformed medium at fixed strain, elastic moduli, temperature stress coefficients, and coefficients of thermal expansion. These properties are described by static correlation functions of dynamical quantities related to strain tensors. The local equilibrium distribution approach allows consistent accounting of spatial dispersion of elastic and thermal properties in inhomogeneous asymmetric media.
This document provides an overview of Lagrangian mechanics and constraints in classical mechanics. It defines different types of constraints including holonomic, non-holonomic, rheonomic, and scleronomic constraints. Generalized coordinates are introduced as a set of independent parameters that can describe the motion of a mechanical system with constraints. The configuration space is defined as a 3N-dimensional space where a point represents the configuration of a system of N particles. Constraints reduce the number of degrees of freedom from 3N coordinates to n generalized coordinates.
This document discusses Lagrangian dynamics and Hamilton's principle. It begins by introducing important notation conventions used in the chapter. It then provides an overview of Hamilton's principle and how it can be used to derive Lagrange's equations of motion. This allows problems to be solved in a general manner even when forces are difficult to express or some constraints exist. Examples are provided, including deriving the equation of motion for a simple pendulum using both Cartesian and cylindrical coordinates. The concept of generalized coordinates is also introduced to represent the degrees of freedom of a system.
1. The document discusses using the method of double-time Green's functions to study vacancy migration in a one-dimensional lattice. It derives expressions for the diffusion coefficient that account for phonon scattering and lattice rearrangement during vacancy motion.
2. The diffusion coefficient D is expressed using linear response theory as a correlation function involving the velocity operator. Approximations are made to simplify the calculation, yielding expressions for D involving spectral intensities and the mass operator and damping of the double-time Green's function.
3. The spectral intensity is expressed in terms of the mass operator and damping, and the integral of the diffusion coefficient expression is evaluated in the nearest-neighbor approximation, resulting in final expressions for the diagonal and
Central problem in mechanics is describing a system's mechanical state and how it evolves over time. Three formulations include Galileo/Newton using coordinates and velocities, Lagrange using generalized coordinates and velocities, and Hamilton using positions and momenta in phase space.
Time homogeneity leads to conservation of energy from the Lagrangian not explicitly depending on time. Space homogeneity leads to conservation of momentum from the Lagrangian being independent of position coordinates. Noether's theorem links symmetries like time and space homogeneity to conserved quantities like energy and momentum.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
Classical mechanics analysis of the atomic wave and particulate formstheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The document discusses Lagrange's equations for describing the motion of particles and systems with constraints. It provides an example of using generalized coordinates to derive the equation of motion for a simple pendulum in terms of the angular coordinate φ. The Lagrangian approach eliminates constraint forces and allows problems to be solved in any coordinate system using Lagrange's equations.
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
Detection of an_unindentified_emission_line_in_the_stacked_x_ray_spectrum_of_...Sérgio Sacani
1. Researchers detected a previously unknown emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. 2. The line was detected at an energy of 3.55-3.57 keV and was seen independently in subsamples of clusters. 3. The line was also detected in Chandra observations of the Perseus cluster but not in observations of the Virgo cluster. 4. The nature of this line is unclear - it could be a thermal line from an undetected element, or potentially the decay line of a hypothesized dark matter particle called a sterile neutrino. Further observations are needed to determine the origin of the line.
Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.
This document discusses uncertainty principles and their application to the double slit experiment. It summarizes Heisenberg's uncertainty principle and its limitations in describing position and momentum spreads. It then applies various uncertainty inequalities to analyze Bohr's argument that an interference pattern requires not knowing which slit particles pass through. Local uncertainty principles assert that low momentum uncertainty implies not only large position uncertainty, but low probability of localization. The document analyzes applying these principles to justify Bohr's response to Einstein's proposed resolution of the double slit ambiguity.
A Calabi-Yau manifold is a smooth space that represents a deformation which smooths out an orbifold singularity. This document discusses superstring theory and fermions in string theories. It introduces the spinning string action and shows that the Neveu-Schwarz model contains a tachyon ground state while the Ramond model contains massless fermions. Combining the two sectors using the Gliozzi-Scherk-Olive projection results in a model with N=1 supersymmetry in ten dimensions.
Dielectrics in a time-dependent electric field: density-polarization functi...Claudio Attaccalite
In presence of a time-dependent macroscopic electric field the electron dynamics of dielectrics cannot be described by the time-dependent density only. We present a real-time formalism that has the density and the macroscopic polarization P as key quantities. We show that a simple local function of P already captures long-range correlation in linear and non-linear optical response functions.
This document summarizes research on the effect of a transverse magnetic field (TMF) on the absorption spectra of a degenerate two-level system. It investigates this effect both analytically and numerically using a rubidium vapor system. It finds that in both the Hanle and pump-probe configurations, the absorption spectra are split in the presence of a TMF, with the splitting proportional to the magnitude of the TMF. Collisional effects reinforce the population redistribution due to the TMF, leading to increased splitting. The splitting can be explained by setting the quantization axis along the total magnetic field, which changes the effective light polarization and creates new two-photon detuned subsystems. Additionally, in the pump-
Detection of an_unidentified_emission_line_in_the_stacked_xray_spectrum_of_ga...Sérgio Sacani
This document describes the detection of an unidentified emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. The line was detected at an energy of 3.55-3.57 keV in independent analyses of the MOS and PN instruments. The line was also seen in Chandra observations of the Perseus cluster. Possible explanations discussed include an atomic transition in thermal plasma, or the decay of sterile neutrino dark matter particles. However, the origin is unclear and requires further observation.
A STATISTICAL NOTE ON SYSTEM TRANSITION INTO EQUILIBRIUM [03-21-2014]Junfeng Liu
1) The document presents a statistical perspective on how an enclosed system of irregularly moving particles evolves over time to reach an equilibrium state of uniform particle distribution.
2) It introduces the concept of "nature's law of chaotic mixture" to describe this process, where the system's temporal evolution can be modeled as a hierarchy of transition probabilities between spatial regions.
3) Numerical simulations demonstrate that after sufficient time steps allowing for chaotic particle motions, the system does achieve an overall equilibrium state where particle proportions are uniform across all regions, consistent with the proposed statistical framework.
This document analyzes the normal vibration modes of s-triazine through infrared and Raman spectral analysis and ab initio force field calculations. It finds that the 6-31G basis set provides the best results for frequency and intensity calculations for s-triazine. It assigns the observed IR and Raman active vibrations and identifies the inactive w4 and w5 modes based on gas phase IR and condensed phase Raman combination bands. It describes the vibration modes of s-triazine using a standing wave model to aid in visualizing the complex ring vibrations, finding coupling between some ring modes and motions of the triazine hydrogen atoms.
Este documento proporciona información sobre los sistemas de seguimiento y control de accidentes en una empresa. Explica cómo clasificar accidentes según su gravedad, forma, agente causante y ubicación de la lesión. También describe índices estadísticos como el índice de frecuencia, gravedad e incidencia que miden la accidentalidad. Recomienda el método de líneas límite para controlar el índice de frecuencia y detectar cambios en las condiciones de seguridad.
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https://www.facebook.com/thuvienluanvan01
tai lieu tong hop, thu vien luan van, luan van tong hop, do an chuyen nganh
Pseudoperiodic waveguides with selection of spatial harmonics and modesVictor Solntsev
A principle of selection of modes and their spatial harmonics in periodic waveguides and, in particular, in spatially developed slowing systems for multibeam traveling-wave tubes (TWTs) is elaborated. The essence of the principle is in the following: varying along the length of the system its period and at least one more parameter that determines the phase shift per period, one can provide constant phase velocity of one spatial harmonic and destroy other spatial harmonics, i.e., reduce their amplitudes substantially. In this case, variations of the period may be significant, and the slowing system becomes nonuniform, or pseudoperiodic; namely, one of the spatial harmonics remains the same as in the initial periodic structure. Relationships are derived for the amplitudes of the spatial-wave harmonics, interaction coefficient, and coupling impedance of the pseudoperiodic system. The possibility of the mode selection in pseudoperiodic slowing systems when the synchronism condition is satisfied for the spatial harmonic of one mode is investigated. The efficiency of suppressing spurious spatial harmonics and modes for linear and abrupt variation of spacing is estimated. The elaborated principle of selection of spatial harmonics and modes is illustrated by an example of a two-section helical-waveguide slowing system.
This document presents a statistical theory of the equilibrium elastic and thermal properties of crystals that have rotational degrees of freedom. Microscopic expressions are derived for strain and stress tensors using a local equilibrium ensemble. Statistical expressions are obtained for properties like specific heat of deformed medium at fixed strain, elastic moduli, temperature stress coefficients, and coefficients of thermal expansion. These properties are described by static correlation functions of dynamical quantities related to strain tensors. The local equilibrium distribution approach allows consistent accounting of spatial dispersion of elastic and thermal properties in inhomogeneous asymmetric media.
This document provides an overview of Lagrangian mechanics and constraints in classical mechanics. It defines different types of constraints including holonomic, non-holonomic, rheonomic, and scleronomic constraints. Generalized coordinates are introduced as a set of independent parameters that can describe the motion of a mechanical system with constraints. The configuration space is defined as a 3N-dimensional space where a point represents the configuration of a system of N particles. Constraints reduce the number of degrees of freedom from 3N coordinates to n generalized coordinates.
This document discusses Lagrangian dynamics and Hamilton's principle. It begins by introducing important notation conventions used in the chapter. It then provides an overview of Hamilton's principle and how it can be used to derive Lagrange's equations of motion. This allows problems to be solved in a general manner even when forces are difficult to express or some constraints exist. Examples are provided, including deriving the equation of motion for a simple pendulum using both Cartesian and cylindrical coordinates. The concept of generalized coordinates is also introduced to represent the degrees of freedom of a system.
1. The document discusses using the method of double-time Green's functions to study vacancy migration in a one-dimensional lattice. It derives expressions for the diffusion coefficient that account for phonon scattering and lattice rearrangement during vacancy motion.
2. The diffusion coefficient D is expressed using linear response theory as a correlation function involving the velocity operator. Approximations are made to simplify the calculation, yielding expressions for D involving spectral intensities and the mass operator and damping of the double-time Green's function.
3. The spectral intensity is expressed in terms of the mass operator and damping, and the integral of the diffusion coefficient expression is evaluated in the nearest-neighbor approximation, resulting in final expressions for the diagonal and
Central problem in mechanics is describing a system's mechanical state and how it evolves over time. Three formulations include Galileo/Newton using coordinates and velocities, Lagrange using generalized coordinates and velocities, and Hamilton using positions and momenta in phase space.
Time homogeneity leads to conservation of energy from the Lagrangian not explicitly depending on time. Space homogeneity leads to conservation of momentum from the Lagrangian being independent of position coordinates. Noether's theorem links symmetries like time and space homogeneity to conserved quantities like energy and momentum.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
Classical mechanics analysis of the atomic wave and particulate formstheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The document discusses Lagrange's equations for describing the motion of particles and systems with constraints. It provides an example of using generalized coordinates to derive the equation of motion for a simple pendulum in terms of the angular coordinate φ. The Lagrangian approach eliminates constraint forces and allows problems to be solved in any coordinate system using Lagrange's equations.
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
Detection of an_unindentified_emission_line_in_the_stacked_x_ray_spectrum_of_...Sérgio Sacani
1. Researchers detected a previously unknown emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. 2. The line was detected at an energy of 3.55-3.57 keV and was seen independently in subsamples of clusters. 3. The line was also detected in Chandra observations of the Perseus cluster but not in observations of the Virgo cluster. 4. The nature of this line is unclear - it could be a thermal line from an undetected element, or potentially the decay line of a hypothesized dark matter particle called a sterile neutrino. Further observations are needed to determine the origin of the line.
Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.
This document discusses uncertainty principles and their application to the double slit experiment. It summarizes Heisenberg's uncertainty principle and its limitations in describing position and momentum spreads. It then applies various uncertainty inequalities to analyze Bohr's argument that an interference pattern requires not knowing which slit particles pass through. Local uncertainty principles assert that low momentum uncertainty implies not only large position uncertainty, but low probability of localization. The document analyzes applying these principles to justify Bohr's response to Einstein's proposed resolution of the double slit ambiguity.
A Calabi-Yau manifold is a smooth space that represents a deformation which smooths out an orbifold singularity. This document discusses superstring theory and fermions in string theories. It introduces the spinning string action and shows that the Neveu-Schwarz model contains a tachyon ground state while the Ramond model contains massless fermions. Combining the two sectors using the Gliozzi-Scherk-Olive projection results in a model with N=1 supersymmetry in ten dimensions.
Dielectrics in a time-dependent electric field: density-polarization functi...Claudio Attaccalite
In presence of a time-dependent macroscopic electric field the electron dynamics of dielectrics cannot be described by the time-dependent density only. We present a real-time formalism that has the density and the macroscopic polarization P as key quantities. We show that a simple local function of P already captures long-range correlation in linear and non-linear optical response functions.
This document summarizes research on the effect of a transverse magnetic field (TMF) on the absorption spectra of a degenerate two-level system. It investigates this effect both analytically and numerically using a rubidium vapor system. It finds that in both the Hanle and pump-probe configurations, the absorption spectra are split in the presence of a TMF, with the splitting proportional to the magnitude of the TMF. Collisional effects reinforce the population redistribution due to the TMF, leading to increased splitting. The splitting can be explained by setting the quantization axis along the total magnetic field, which changes the effective light polarization and creates new two-photon detuned subsystems. Additionally, in the pump-
Detection of an_unidentified_emission_line_in_the_stacked_xray_spectrum_of_ga...Sérgio Sacani
This document describes the detection of an unidentified emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. The line was detected at an energy of 3.55-3.57 keV in independent analyses of the MOS and PN instruments. The line was also seen in Chandra observations of the Perseus cluster. Possible explanations discussed include an atomic transition in thermal plasma, or the decay of sterile neutrino dark matter particles. However, the origin is unclear and requires further observation.
A STATISTICAL NOTE ON SYSTEM TRANSITION INTO EQUILIBRIUM [03-21-2014]Junfeng Liu
1) The document presents a statistical perspective on how an enclosed system of irregularly moving particles evolves over time to reach an equilibrium state of uniform particle distribution.
2) It introduces the concept of "nature's law of chaotic mixture" to describe this process, where the system's temporal evolution can be modeled as a hierarchy of transition probabilities between spatial regions.
3) Numerical simulations demonstrate that after sufficient time steps allowing for chaotic particle motions, the system does achieve an overall equilibrium state where particle proportions are uniform across all regions, consistent with the proposed statistical framework.
This document analyzes the normal vibration modes of s-triazine through infrared and Raman spectral analysis and ab initio force field calculations. It finds that the 6-31G basis set provides the best results for frequency and intensity calculations for s-triazine. It assigns the observed IR and Raman active vibrations and identifies the inactive w4 and w5 modes based on gas phase IR and condensed phase Raman combination bands. It describes the vibration modes of s-triazine using a standing wave model to aid in visualizing the complex ring vibrations, finding coupling between some ring modes and motions of the triazine hydrogen atoms.
Este documento proporciona información sobre los sistemas de seguimiento y control de accidentes en una empresa. Explica cómo clasificar accidentes según su gravedad, forma, agente causante y ubicación de la lesión. También describe índices estadísticos como el índice de frecuencia, gravedad e incidencia que miden la accidentalidad. Recomienda el método de líneas límite para controlar el índice de frecuencia y detectar cambios en las condiciones de seguridad.
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Antiguamente, los monjes copiaban y decoraban los textos religiosos a mano. En 1450, Johannes Gutenberg inventó la imprenta en Alemania usando tipos móviles de metal, prensa de vino e imprimiendo con tinta más clara. Gutenberg publicó la Biblia de las 42 líneas, la primera obra impresa. La imprenta se expandió por Europa, llegando a España en 1472 con la publicación de la obra Sigonal de Águilafuente.
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El documento describe el Filesystem Hierarchy Standard (FHS), un estándar que define la estructura básica de directorios en sistemas Linux y Unix. El FHS fue desarrollado originalmente para estandarizar la organización de directorios entre distribuciones Linux y desde entonces ha sido adoptado por muchos sistemas Unix. El documento explica los principales directorios definidos por el FHS como /bin, /dev, /etc, /home, /lib, /opt, /usr y otros.
Los ciclos económicos son fenómenos propios de las economías basadas en el dinero y la actividad comercial. Se componen de cuatro fases: auge, recesión, depresión y recuperación. La recesión es la fase descendente del ciclo donde cae la inversión, producción y empleo. La depresión es el punto más bajo con alto desempleo y baja demanda. La recuperación es la fase ascendente donde se renueva el capital impulsando el crecimiento económico.
Arzúa es una villa gallega situada en la provincia de A Coruña. Es conocida como el "Pórtico de Santiago" ya que es la última parada para los peregrinos antes de llegar a Santiago de Compostela. La economía se basa principalmente en la ganadería, agricultura y turismo relacionado con el Camino de Santiago. Arzúa también es famosa por sus quesos con denominación de origen.
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Classical and Quasi-Classical Consideration of Charged Particles in Coulomb F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb centres is carried out. It is shown that the proton and electron can to create a stable connection with the dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom coincide with the Born orbits.
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
Similar to Journ of Magn Res 72, 62-74 (1987) (20)
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB F...
Journ of Magn Res 72, 62-74 (1987)
1. JOURNAL OF MAGNETIC RESONANCE 71,62-14 (1987)
Two-DimensionalSolid-StateNutation NMR of Half-Integer
Quadrupolar Nuclei
A. P. M. KENTGENS, J. J. M. LEMMENS, F. M. M. GEURTS, AND W. S. VEEMAN
Department of Molecular Spectroscopy, Faculty of Science, University of Nijmegen, Toernooiveld,
6525 ED Nijmegen, The Netherlands
Received May 13, 1986; revised July 11, 1986
The two-dimensional solid-state nut&ion NMR experiment for the determination of
quadrupole parameters, asintroduced by Samoson and Lippmaa is evaluated. A complete
series of spectra (for spin Z = f, i, f, and p) resulting from density-matrix calculations is
presented, and some experimental aspectsof the method am discussed.Finally applications
of the method to *‘Al (I = 3) in spodumene and to 45Sc(Z = i) in Sc2(SO& are shown.
Q 1987 Academic Pres, Inc.
INTRODUCTION
The majority of elements’ in the periodic table have nuclei with half-integer quad-
rupole spins. It is therefore not surprising that there is interest in high-resolution NMR
spectra of quadrupole nuclei in solids. Especially the recent studies of zeolites, clays,
and ceramics have focused attention on obtaining structural information from NMR
spectra of quadrupolar spins.
In comparison to nuclear spins with spin quantum number I = 1, the NMR spectra
of quadrupolar spins contain two new features: ( 1)several transitions are possible and
(2) the transition frequencies are determined not only by the interaction between the
magnetic moment of the nucleus and the static external field but also by the interaction
between the nuclear electric quadrupole moment and the electric field gradient of the
surroundings of the particular nucleus. The chemical information which an NMR
spectrum of a quadrupolar spin offers is not limited to the chemical-shift data, but
such a spectrum can also provide the parameters that describe the quadrupolar inter-
action. These parameters depend on the local symmetry around the nucleus in con-
sideration and thus give direct structural information.
Traditionally, quadrupole interaction parameters can be determined by NQR. There
is usually a small magnetic field or none, and the nuclear spin levels are split mainly
by the quadrupole interaction. The frequencies of the transitions between these levels
provide the quadrupole parameters. The disadvantages of NQR are its low sensitivity
when the quadrupole interaction is rather small (0- 10 MHz) and the wide frequency
range one has to search for possible resonances. Both of these disadvantages can be
overcome in principle, at least for small quadrupole interactions, in an NMR ex-
periment.
0022-2364187 $3.00
Copyrishl8 1987 by Academic F’ms, Inc.
AU rights of reprcduction in any form -ed.
62
2. NMR OF QUADRUF’OLAR NUCLEI 63
The description of the NMR spectra of quadrupolar spins depends strongly on the
spin quantum number Zand the relative magnitude of the Zeeman and the quadrupole
interactions. Here we limit our discussion to half-integer spins (I = 5,I, 5,:) and assume
the quadrupole interaction to be small with respect to the Zeeman interaction, The
(anisotropic) contribution of the quadrupole term to the NMR transition frequencies
can then be calculated by perturbation theory.
In high magnetic fields all Zeeman transitions WZ,m’ shift in first order because of
the quadrupole interaction except the f, - 1 transition which experiences a (much
smaller) second-order shift. As a result of this, typical spectra of polycrystalline samples
containing quadrupolar spins give characteristic powder patterns for the 1, - 1 tran-
sition whereas all other transitions are usually broadened beyond detection (I). Because
the magic angle has no “magic” properties for the second-order quadrupolar inter-
action, MAS will not average this interaction, but yields powder patterns for the 1,
- f transition which are approximately four times narrower than for static
samples (2).
Although, in principle, it is possible to extract information about the electric field
gradient from the spectra of static or spinning polycrystalline samples, in practice the
powder patterns are often blurred by a spread in chemical shift and/or by the presence
of more than one quadrupole interaction. To overcome these problems Samoson and
Lippmaa (3) introduced a simple two-dimensional experiment which allows one to
separate the quadrupole interaction from the chemical-shift interaction. This technique
is based on a nutation experiment (4) where the evolution of the spin system in the
presence of a radiofrequency field Br is studied in the rotating frame. This evolution
yields a low-field (B, - O.OOl&) NMR spectrum, the nutation spectrum, with the
sensitivity of the high-field spectrometer. In this respect it can directly be compared
to the zero-field NMR technique developed by Pines and co-workers (5) where the
sample is pneumatically shifted in and out of the magnet. The advantage of the nutation
experiment is that, in contrast to the zero-field technique, there is no limitation on
the T, spin-lattice relaxation time. A nutation spectrum, however, is more difficult
to analyse than a zero-field spectrum.
In this paper we discuss the calculation of these nutation spectra using the density
matrix formalism, and present a number of simulated spectra for all half-integer quad-
rupole spins (I = $to Z = 5) which can be used as fingerprints. In a previous paper on
27A1nutation NMR of zeolites (6) it was shown how nutation NMR can be used to
separate the signals from 27A1with a large quadrupole interaction from the Al signals
with a small quadrupole interaction. Here we want to show that by comparing an
experimental spectrum to a set of simulated spectra one can determine the quadrupole
parameters. This will be demonstrated for 27Al in spodumene and for 45Scin S@SO4)3.
Further, some experimental aspects of the method will be discussed.
THEORY
The pulse scheme of the experiment is outlined in Fig. 1. During the evolution
period tl , an rf field is present and the system evolves under the secular Hamiltonian
&“, in the rotating frame, assuming that the sample is static (i.e., no MAS):
9, =~o~+&“~+‘zQ=(n,-a)zz-n,zx+n~(3z~-z(z+ 1)) 111
3. 64 KENTGENS ET AL.
b F, -projection
FIG. 1. (a) Pulse scheme of the experiment; a free induction decay is acquired during tzas a function of
the pulse length tI . Subsequent Fourier transformation of the signals obtained yields a typical 2D powder
pattern as shown in (d). This spectrum was calculated for Z = f with a ratio Q&l,+ = 0.45 and 9 = 0. The
F2 projection of this pattern (c) gives the second-order quadrupole powder pattern, and the F, projection
(b) is a characteristic nutation spectrum for the ratio Q,$&.
with
% = ;;;!;)(3 cos28- 1+ 7jsin28 cos 2@) = Qo(3 cos2B- 1+ 7)sin28cos 2@)
where Q,+= y& and B and @are the polar angles orienting the magnetic field &, in
the principal axis system of the field gradient, and 1 represents the asymmetry param-
eter. Here dipolar interactions and the nonsecular part of the quadrupole Hamiltonian
have been neglected. During t2 there is no rf field present and the FID of the f, - f
transition is acquired (assuming that all other transitions are too broad to be detected).
The system is now governed by the Hamiltonian S2:
z2=&“z+2Yf3+‘zQ PI
where SF’=and Sacs are the Zeeman and the chemical-shift interaction and So is the
quadrupolar interaction which only contributes in second order to the lineshape of
4. NMR OF QUADRUPOLAR NUCLEI 65
the central 4, - 4 transition (7). With the knowledge of these Hamiltonians we can
calculate the signal s(t,, fz) using the density-matrix formalism. At time tl = 0 we
start with the equilibrium density matrix u(O)in the high-temperature approximation.
As one can seethe dominating Zeeman interaction X’z is not present in 3?‘, . So if we
irradiate close enough to resonance (zOf small) then %‘o and 9’ti are the most im-
portant terms of 2,. This means that the eigenfunctions of ~3?rdepend on the ratio
of Qo and Q,. From the Liouville-von Neumann equation we get at time tl
u(tl) = exp(-i~ltl/~)u(0)ex~i~,~,/~). [31
Starting in a basis of eigenfunctions of Z,, we get after (numerical) diagonalization of
z,, using the orthogonal transformations T
u(tr) = Texp(-iE~l/ti)~+~(0)Texp(iE~I/~)~+. [41
Here Z?represents the diagonalized matrix of X1 giving the eigenvalues
Ej= C Z TpjzlpqTqj 151
P 4
and eigenfunctions
lj) = C Tij14 m). [61
We assume that we only detect the coherence between the f and the - f states
during t2, so only the element ~r(t~)~~~,-~,~has to be evaluated. The signal of the central
transition then can be calculated as a function of tl and t2:
- a(t1)1/2,-1,2exp(-i~2t2)
= C(R-*,2,,,2)i,jeXp(i~2iitl)eXp(-i~2t~)
i,j
171
where Q2 represents the Larmor precession of the spins during t2 and Q2,
= (Ei - Ej)/h is the transition frequency in the rotating frame between the states Ii)
and /j). The coefficients (R-1/2,1/&
(R-1/2,1/& = Tl/2,iT-l/2,j C Tk,iTk,jdO)k,k 181
k
represent the contributions of the coherence between Ii) and lj) in the rotating frame
during tl to the 4, - 1 coherence detected during t2.
To summarize, we have during tl a system with eigenfunctions and eigenvalues
which depend on the ratio of no and C&r(Fig. 2). The coherences between all these
levels, with frequency a,,, develop during tl and give a certain contribution to the 1,
-1 coherence detected during tz. Therefore, a two-dimensional Fourier transformation
of the acquired signal will give a characteristic powder pattern (Fig. Id), whose pro-
jection on the F2 axis (Fig. lc) yields the normal powder lineshape due to the combined
effect of chemical-shift anisotropy and quadrupole interaction. Projection onto the F,
axis (Fig. 1b) gives the nutation spectrum which depends on the quadrupole parameters
e2qQ and 1, the spin quantum number Zand the rf field strength B, , and is independent
5. 66 KENTGENS ET AL.
energy (MHz)
.04
.03
-.os
0 25 5 1.5
I
Ok-----
:;‘::I:-
-%?ii-xiy6 (kHz) -
FIG. 2. Energy diagram of an isolated spin I = $ in the presence of a magnetic field B with quadrupole
parameters e2qQ/h = 1 MHz and t) = 0. On the left is the low-field situation Zz < &“o and on the right is
the situation &“= $ Zo, where X’o appears to be negligible.
of the chemical shift. As we shall seelater, and aswe reported before (6), the nutation
spectra show more detail than MAS NMR spectra and therefore we concentrate on
the nutation spectra instead of on the whole 2D powder pattern.
In the extreme cases l.%‘o[6 (%‘A and l%‘o[ + I%‘,.rjthe nutation spectrum consists
of a single line. In the first situation X’o may be neglected with respect to Xti and the
nutation frequency is simply f&r. In the second situation straightforward perturbation
theory shows that the nutation frequency becomes (I + $)fiti (6-84b).
In intermediate cases, I~??‘ol- IsA, the spectra are complicated and several peaks
can occur because many transition frequencies t&in the rotating frame exist. In addition
the nutation spectra will be powder patterns because of the anisotropic nature of the
quadrupole interaction. Therefore for intermediate casesthe experimentally obtained
nutation spectra have to be compared to simulated spectra, calculated as de-
scribed above.
Figures 3-6 each show a series of calculated nutation spectra for different ratio
t&&.r, for the different spins Z = 5 to Z = 3. In all spectra the asymmetry parameter
71and the resonance offset are assumed to be 0. These spectra were calculated for 7500
crystallite orientations, taking up to 5 minutes computer time on a mainframe (NAS
9060). The spectra appear to be very characteristic powder patterns so for these in-
termediate cases with !& known, one can determine no (no = e2gQ/h8Z(2Z - 1)).
Because no depends on the nuclear quadrupole moment Q and the spin quantum
number Z, and with Q,+limited for experimental reasons, what range of electric field
gradients eq can be determined without getting in the extreme situation with only one
nutation frequency left will depend on the nucleus in question.
Figure 7 shows the effect of the change of the asymmetry parameter 17for a spin
Z = $ with G-J& = 0.6. The calculations are for 100 X 100 crystallite orientations
taking up to 10 minutes computer time. The overall appearance of the powder pattern
6. NMR OF QUADRUPOLAR NUCLEI 67
I = 3/2
i, Orf 2 Orf
FIG. 3. Calculated nutation spectra for an isolated spin I = 3 as function of the ratio &/&, with 9 = 0.
No resonance offset is taken into account. The scale factors of the spectra are 1, 4.7, 15.2, 14.9, 10.2, 7.3,
5.7, and 3.0 with respect to the G@, = 0 spectrum. In all spectra a Lorentzian line broadening of 2.5 kHz
was applied.
remains the same but the intensity of some of the lines changes drastically. So it is
also possible to get a good estimate of T)from the nutation spectra.’
So far it has been assumed that the sample is static, i.e., no magic-angle spinning
applied. Magic-angle spinning would be desirable in view of the resolution along F2.
However, numerical computer calculations of the simulated nutation spectra under
MAS conditions becomes very time-consuming because the change of the orientation
of the sample during tl and thus of &“r cannot be neglected. Experimentally it has
been observed that the nutation spectra change with MAS. The discussion of these
spectra and their interpretation will be postponed to a later publication.
EXPERIMENTAL ASPECTS
Resonance ofiet. When the system is irradiated on resonance (Q = s2,) then the
matrix of &“r in Eq. [2] becomes symmetrical with respect to both diagonals of the
matrix. As a result of this
’ A complete set of simulated spectra with asymmetry parameter variation for all spins(I = $to I = $)
can be obtained on request to the authors.
7. I = 5/2
c,
r
Qf 34f
FIG 4. As in Fig. 3, but for spin I = $. Scale factors: 1, 7.4, 8.2, 25.0, 25.1, 19.4, 15.7, and 6.9.
0.6
0.05
0
0 Qf Wf
FIG. 5. As in Fig. 3, but for spin I = $. Scale factors: 1, 6.7, 16.0, 22.0, 39.6, 32.6, 26.2, and 12.7.
68
8. -0
0 Orf %f
FIG. 6. As in Fig. 3, but for Z = f. Scale factors: 1, 12.1, 25.2, 19.8, 72.0, 61.6, 52.3, and 25.7.
FIG. 7. Nutation spectra for a spin Z = f as lknction of the asymmetry parameter t (&/Q, = 0.6). The
overall appearance of the powder pattern remains the same, but the intensity of some lines decreaseswith
increasing 9.
69
9. 70 KENTGENS ET AL.
and
W1/2,1,2h,j = -v-1/2,1/2)j,i i#j
(R-1/2,I&j = 0 i=j.
Substitution in Eq. [7] shows that the signal then becomes
[91
WI, f2) = 2 U- I,z,l,z)i,jsin QijtlexP(-ifizt2).
iJ
[lOI
The amplitude of the signal detected during t2 is sine modulated which allows us to
obtain pure absorption spectra (9). The presence of an off-resonance term in z1 lowers
the symmetry of its matrix and the modulation now becomes a phase-modulation,
which cannot be changed to amplitude-modulation by phase cycling. Thus there will
be dispersive contributions to the lineshapes which can easily distort powder patterns.
Another effect of off-resonance irradiation is the appearance of a dispersive line at
fir = 0. This is due to the magnetization component along the BeEfield in the rotating
frame which does not evolve during tl .
Both effects of off-resonance irradiation can even be observed for spins with Z = 5
for which the signal in such a case can be written as
S(tl, t2) - [A( 1- cos QeEtl)+ iB sin !&&l]exp(-iQ2t2) [Ill
where A = Q,AQ/2(Q$ + AQ2) and B = Qd2w. When A0 = Q - Q0# 0 then
A # 0 and thus a constant and a cos fled1 term is introduced causing, respectively,
the Q, = 0 signal and the phase modulation. To avoid complications it is clearly
recommendable to irradiate on resonance, i.e., with the excitation frequency Q equal
to the Larmor frequency yB0. That “on resonance” in this case does not mean at the
frequency of the 1, - 4 transition is shown by the following magic-angle spinning
experiment. Here the Larmor frequency lies outside of the quadrupole lineshape (2)
and Fig. 8 shows the nutation spectra of NaN02 (e2qQ/h = 1.1 MHz, 9 = 0.1) as a
function of the excitation frequency. In this case we are in the extreme situation
no b Qti and MAS does not influence the nutation spectrum. It is clear that the line
at Qi = 0 increases relative to the line at 2Qfi. The line at Bi = 0 is minimal when we
irradiate outside the powder pattern near the Larmor frequency.
Line broadening. An important cause for line broadening in the F, dimension is
the inhomogeneity of the rf magnetic field. The effect of rf inhomogeneity cannot
simply be described with a Lorentzian broadening exp(-tr/rJ because a spread in !&
will also cause a spread in the ratio Qo/s2ti, and will thus change the whole nutation
spectrum. Furthermore the lineshapes due to rfinhomogeneity will not be Lorentzian
but asymmetric (see Ref. (ZO), Fig. 3). Only in the extreme case Q, $ no, do we get
a line at nutation frequency Qti with a linewidth directly determined by the rf inho-
mogeneity. In the other extreme case Qd 4 no there will be a line at (I + f)&, so the
linewidth will be (I + 4)~ times the rf inhomogeneity.
Recycle delay. Another important experimental aspect in the nutation experiment
is that one has to ensure that the recycle delay or relaxation delay is long enough for
the system to return to equilibrium before the next pulse arrives. If the recycle delay
is short with respect to T, then the build up of magnetization along the z axis is
incomplete. This will distort the pure sine amplitude modulation of the FID
10. NMR OF QUADRUPOLAR NUCLEI 71
NaN02
a b
1
F1 %f 0
C
,
5 *%f 0
FIG. 8. MA!3 spectra of NaN02 together with their nut&ion spectra as a function of the excitation frequency
fi. (a) Off-resonance irradiation; in the nutation spectrum we seea large line at R, = 0. (b) Irradiation close
to the Larmor frequency (outside the powder pattern) gives the best nutation spectrum. (c) With the carrier
frequency equal to the average 4, - f transition frequency (in the middle of the powder pattern) the line at
ti, = 0 in the nutation spectrum increases again.
(Eq. [ lo]). Fourier transformation of a distorted sin QIt wave will give aline at frequency
Q, plus a number of harmonics at 2%) 3!4, . . . . The number and amplitude of
these harmonics depends on the distortion of the sine wave, and they can easily be
mistaken for components with a large quadrupole frequency. Figure 9 shows this effect
- F2
FIG. 9. 2D nutation spectrum of ‘Li in Lick where the recycle delay (0.25 s) is short with respect to T, .
Li has a small no in Lick so only one line at t& is expected. Because of the short recycle delay, however,
there are several harmonics of this frequency present.
11. 72 KENTGENS ET AL,
for the ‘Li nutation spectrum of LiCl. Here Li has a very small quadrupole interaction,
but due to the short repetition rate we see not only a line at 52, = Qfi but also at
Q, = 2&, 3&, and 4Q+ In fact this experiment is proposed asa method to determine
Q,.rin solution, for nuclei with long T, and low natural abundance (II).
EXPERIMENTAL
NaN02, LiCl, and SCZ(SO& were all commercially available chemicals. NaN02
and spodumene spectra were recorded on a Bruker CXP-300 with, respectively, 64
and 128 tl increments of 2 ~.ls.A standard Bruker probe with an rf field of 36 kHz
was employed. LiCl and Sc#O.& spectra were recorded on a Bruker WM-500 (with
256 t, increments of 2 and 1.5 ps, respectively). Here a specially constructed probe
equipped with a 6 X 12 mm solenoid perpendicular to BO, operating with an rf field
strength up to 70 kHz, was used.
SPODUMENE
J‘_~-_--
/ L-...cd.--L.-,.,--r--r- L+------- .~_
, I I0 Yf 150
- KHz
FIG. 10. (a) 27AIspectrum of the central transition of spodumene recorded at 78.2 MHz. (b) F, projection
of the 2D nutation spectrum of spodumene recorded on a Bruker CXP-300 (128 t, increments of 2 ps and
Q, = 36 kHz). (c) Simulated spodumene spectrum using the quadrupole parameters e*qQlh = 2.95 MHz
and 7 = 0.94 (12) and a Lorentzian broadening of 2.5 kHz.
12. NMR OF QUADRUPOLAR NUCLEI 73
Sc*(SO&
SC 1=7/2
200 khz
-F 2
FIG. 11. 2D nutation spectrum of 45Scin SC&SO& recorded on a Bruker WM 500, together with its F,
projection (256 tl increments of I .5 ps with a 70 kHz rf field).
RESULTS
To demonstrate the effectiveness of the nutation experiment, Fig. 10a displays the
NMR spectrum of 27Al in powdered spodumene recorded on a Bruker CXP-300 at
78.2 MHz. The spectrum consists of one featureless line 5 kHz wide. It is clear that
no accurate quadrupole interaction parameters can be extracted from this spectrum.
In addition, MAS does not solve the problem; again we see a rather featureless line,
1.5 kHz wide. In Ref. (2~) it is shown that this MAS spectrum can be reproduced
theoretically (using a large line broadening) with the known quadrupole parameters
(e*qQ/h = 2.95 MHz, 1 = 0.94 (Z2)), but it will be 1c ear that it is almost impossible
to do so without any preknowledge of the quadrupole parameters. The result of the
2D nutation experiment, however, appears to be a well structured pattern. From the
F1 projection of this pattern (Fig. lob) we can estimate the magnitude of the quadrupole
parameters e*qQ and rl using our set of calculated spectra. It appears that flo/Q,.r - 1
with an rf field strength of 36 kHz; this means that e*qQ/h - 3 + 0.5 MHz, and that
I) must be between 0.8 and 1.
Another example is the 45Sc(I = $) nutation spectrum of Scz(SO& (Fig. 11). The
spectrum of the central transition measured at 121.5 MHz is 2.6 kHz wide and shows
no structure, MAS narrows the spectrum to 600 Hz. The MAS spectrum recorded at
43.8 MHz does show some structure from which it becomes clear that there must be
siteswith different quadrupole parameters but the same chemical shift. The 2D nutation
spectrum is well structured (Fig. 11). Comparing its projection to our set of calculated
spectra reveals that the major constituent has a e*qQ/h - 2 f. 0.5 MHz with a low
13. 74 KENTGENS ET AL.
asymmetry parameter (7 between 0.1 and 0.4). P. P. Man recently also showed an
intermediate case for Mn in KMn04 (8~).
CONCLUSIONS
The 2D nutation method appears to be very useful for the determination of quad-
rupole interaction parameters of half-integer quadrupolar nuclei in a field gradient,
especially when these parameters cannot be determined from MAS experiments. The
method combines the sensitivity of high-field measurements with the information one
gets from experiments at low or zero field. The spectra can easily be simulated with
a straightforward density matrix calculation. To get high sensitivity and maximum
resolution in the F2 dimension, which is advantageous if nuclei with different chemical
shifts are present, one preferably performs the experiment in the highest available
magnetic field. Because there is no obvious need for MAS the method will also be
very suited for high-temperature studies (e.g., of zeolites).
ACKNOWLEDGEMENTS
Mr. P. van Dael and Dr. C. Haasnoot, of the SON hf-NMR facility at the University of Nijmegen, are
thanked for their support with the experiments on the 500 MHz spectrometer. We thank Mr. J. W. M. van
OS and Mr. P. van Dijk for their technical assistance and Prof. dr. E. de Boer for critically reading the
manuscript. Dr. B. de Jong kindly supplied the spodumene sample. This work was carried out under the
auspices of the Netherlands Foundation of Chemical Research (SON) and with the aid of the Netherlands
Organization for the Advancement of Pure Research (ZWO).
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