Simple Harmonic Oscillator Model Research Articles

Supramolecular self-sorting predicted by a simple harmonic oscillator model

Macrocycles that encapsulate two guests can self-sort those into homo- and heterodimers. We report here a family of self-sorting homobimetallic Pt(II) terpyridyl acetylide dimers secured together with a pair of...

Bekenstein-Smarr formula and quantization of Kerr black holes

By solving the zero surface equation and studying the black holes law, we derive the equivalent Bekenstein-Smarr formula for Kerr black holes. With the reduced phase space method and canonical transformation, we establish the quantum mechanical model for Kerr black hole which is a standard two-mode simple harmonic oscillator model. With the help of this model, we further derive the spectra of the event horizon area and corresponding eigen wave functions for the Kerr black holes, which are proved that the area of a black hole event horizon is quantized.

Determination of Stretching Frequencies by Isotopic Substitution Using Infrared Spectroscopy: An Upper-Level Undergraduate Experiment for an In-Person or Online Laboratory

A Fourier transform infrared (FTIR) experiment appropriate for an upper-level undergraduate laboratory such as chemical instrumentation is described. Students collect FTIR spectra of four protio-solvents and their deuterated analogues. In addition to qualitatively observing C–H and O–H peaks shift to lower energy upon deuteration, students apply a simple harmonic oscillator model to predict the magnitude of the shifts and gain an understanding of the relationship between reduced mass and vibrational frequencies. Vibrational calculations are then performed to corroborate the experimental results and allow students to visualize the normal modes involved via animations of the vibrations. This experiment is suitable to be used as either an in-person or online laboratory.

Thermally activated motion of cold excitons in conjugated polymers

AbstractThermally activated migration of localized cold excitons were investigated in archetypal conjugated polymers; trans‐polyacetylene, polythiophene, and poly(p‐phenylene vinylene). Room temperature thermal energy has been found to provide sufficient energy to avail substantial mobility for excitons. A simple undistorted but displaced harmonic oscillator model has been assumed to predict the activation energy for exciton hopping between identical sites in perfectly planar conjugated chains, promoted by vibrational modes with Bu symmetry. The activation energies of exciton hopping has been found to increase in the order of trans‐polyacetylene (6.4 meV), polythiophene (29.5 meV), and poly(p‐phenylene vinylene) (147.4 meV). A rough Monte‐Carlo simulation based on incoherent hopping mechanism reveals excitons can displace in an average of 2.17 nm in polythiophene whereas those distances are much smaller in poly(p‐phenylene vinylene) (0.24 nm) and trans‐polyacetylene (0.37 nm). The simulation results partially explain why polythiophene based polymers are better suited for organic photovoltaic applications whereas poly(p‐phenylene vinylene) are more appropriate for organic light emitting diode applications.

Characterizing the γ-Ray Variability of Active Galactic Nuclei with the Stochastic Process Method

Gamma-ray astronomy in the time domain has been by now progressed further as the variabilities of active galactic nuclei (AGNs) on different timescales have been reported a lot. We study the γ-ray variabilities of 23 jetted AGNs by applying a stochastic process method to the ∼12.7 yr long-term light curve (LC) obtained by the Fermi-Large Area Telescope (Fermi-LAT). In this method, the stochastically driven damped simple harmonic oscillator (SHO) and the damped random-walk (DRW) models are used to model the long-term LCs. Our results show that the long-term variabilities of 23 AGNs can be characterized well by both SHO and DRW models. However, the SHO model is restricted in the overdamped mode, and the parameters are poorly constrained. The SHO power spectral densities (PSDs) are the same as those of the typical DRW PSD. In the plot of the rest-frame timescale that corresponds to the broken frequency in the PSD versus black hole mass, the intrinsic, characteristic γ-ray timescales of 23 AGNs occupy almost the same space with the optical variability timescales obtained from the accretion disk emission. This suggests a connection between the jet and the accretion disk. As with the optical variability of the AGN accretion disk, the γ-ray timescale is also consistent with the thermal timescale caused by the thermal instability in the standard accretion disk of AGNs.

The Resonant Excitation Process in Commercial Quadrupole Ion Traps Revisited.

The collision-induced resonant excitation process in real quadrupole ion traps is revisited theoretically and experimentally by explicitly including in the discussion the influence of higher order potential impurities. This includes mainly the dependence of the secular oscillation frequency fion on the ion's oscillation amplitude zmax. Due to frequency calibration, commercial ion traps use excitation frequencies fexc that are higher than the theoretical secular oscillation frequency fion. This may lead to switching in frequency order between fexc and fion that can allow ions to stay longer in on-resonance. It is also found that there is a most efficient but also a harshest excitation frequency, which are not identical. These phenomena are explained and described with a simple harmonic oscillator model and precise numerical calculations, using the trajectory simulation program ITSIM 5.0. Experimental MS2 have been performed with the thermometer ion leucine-enkephalin, which are then in line with expectations from the trajectory calculations. The important difference to the existing literature is that, here, overexcitation is characterized by the observed a4/b4 fragment-ion ratio, while the fragmentation efficiency was kept constant. By slightly increasing the excitation frequency one can obtain drastically different effective collisional temperatures. This knowledge gives even commercial ion traps, without instrument adjustments, the possibility of producing energetically versatile fragment ion spectra. It is also shown that the damped driven harmonic oscillator cannot be used as a simplified model of the motion during the resonant excitation process in real ion traps.

Inverted harmonic oscillator dynamics of the nonequilibrium phase transition in the Dicke model.

We show how the dynamics of the Dicke model after a quench from the ground-state configuration of the normal phase into the superradiant phase can be described for a limited time by a simple inverted harmonic oscillator model and that this limited time approaches infinity in the thermodynamic limit. Although we specifically discuss the Dicke model, the presented mechanism can also be used to describe dynamical quantum phase transitions in other systems and presents an opportunity for simulations of physical phenomena associated with an inverted harmonic oscillator.

MEMS displacement generator for atomic force microscopy metrology

Atomic force microscopy enables three-dimensional high-resolution imaging of surfaces with nanoscale features. In order to obtain the quantitative information about surface geometry, the atomic force microscope’s scanning system must be calibrated. This is usually done by using calibration samples of known and/or defined shape based on either lithographic or crystal structures. In this work we report on a microelectromechanical device, referred to as a displacement generator, whose vertical deflection is controlled electronically. The designed, fabricated and applied device is formed out of a silicon nitride doubly clamped lever, referred to as a microbridge, with a deposited pair of platinum strips. When the MEMS displacement generator is immersed in a magnetic field and when it is electrically biased, the associated Lorentz force induces a structural displacement. In the presented design, the silicon nitride microbridges were fabricated on a (110) silicon wafer in a Wheatstone bridge configuration. A second reference cantilever was mechanically supported by the silicon substrate. In this way, a highly symmetrical structure was fabricated, making it possible to control precisely deflection in Z direction with sub-nanometre precision. The cantilever’s high resonance frequency, of ca. 500 kHz, makes the constructed device insensitive to external vibration sources which are typically at much lower frequencies. As the stage function can be described using the simple harmonic oscillator model, it is clear that the system can operate with sub-nanometre resolution, which, for the purpose of microscope calibration, is extremely beneficial. By placing of the atomic force microscope tip on the actuated reference device it is possible to determine the response of the system over a wide frequency bandwidth. In this work we will describe the fabrication process of the MEMS displacement generator, interferometric and traceable investigations of thermomechanical and electromagnetic actuation schemes. Moreover, we will present the results of the calibration of an atomic force microscope operating in contact and intermittent contact modes.

Theoretical study on thermodynamic properties of NO gas

<sec>Nitric oxide (NO) is one of atmospheric molecules of interest and has attracted considerable attention due to its important role in the chemical process taking place in a flow field of hypersonic vehicle, in which the thermodynamic properties are required in the calculation of the aerothermodynamic flow field. Moreover, the total internal partition function is the key to calculating the thermodynamic properties of high-temperature gases. For diatomic molecules, according to the product approximation, the total internal partition function is split into three parts: electronic, vibration and rotation partition function. In this paper, by using the quantum statistical ensemble theory based on some classical thermodynamic and statistical formulae, the thermodynamic properties of NO are analyzed and discussed.</sec><sec>Firstly, in order to obtain an accurate energy of molecule, the variational algebraic method (VAM) is employed to calculate the full vibrational energy, the resultis in good agreement with the experimental result and thus yielding the realistic predictions of the unobserved higher vibrational energy that converges to the dissociation limit. Secondly, an attempt is to use the full VAM vibrational energy, the Rydberg-Klein-Rees (RKR) vibrational energy, the simple Harmonic oscillator (SHO) model and the quantum-mechanical vibrational energy obtained by the multiconfiguration self-consistent-field (MCSCF) to calculate the vibrational partition function. Then, with the rotational contributions from the Müller-McDowell formula, the internal partition function can be determined by combining the product of electronic, vibration and rotation partition functions. Thirdly, according to the thermodynamic and statistical formulae, it is easy to calculate the internal energy, entropy and heat capacity for the NO molecule in a range of 1000－5000 K. Comparison of different calculated heat capacities with the experimental ones reveals the heat capacity, of which vibrational contributions determined by the full VAM vibrational energy accord better with the experimental ones, with the maximum relative error being no more than 2.4%, whereas it can be seen that those thermodynamic results evaluated from the SHO model attest to a failure for the summation of infinite vibrational energy. The thermodynamic results of NO may have proper applications in areas that can be of great importance in theoretical and (or) experimental aspects.</sec>

Vibrational spectroscopy of protonated amine-water clusters: tuning Fermi resonance and lighting up dark states.

Strong coupling between stretching fundamentals and bending overtones of vibrational modes, known as Fermi resonance (FR), has been observed for proton motions in the protonated trimethylamine-water cluster. To investigate the role of FR, we examined the vibrational spectra of other three protonated ammonia/amine-water clusters, including the NH4+ ion and its mono- and di-methylated analogues, respectively, with and without argon tagging. In these systems, a simple frequency-scaled harmonic oscillator model will predict only one strong band between 2600 and 3200 cm-1 uniquely due to the hydrogen-bonded NH stretching fundamental for a given conformer. In the experimental vibrational spectra, however, multiple sharp bands were observed. Such a discrepancy often leads to the notions of the coexistence of multiple conformers and/or the appearance of an overtone state as a result of FR. In this work, we applied a discrete variable representation (DVR) implementation of ab initio anharmonic algorithms and demonstrated how one N-H+ stretching fundamental can lead to multiple bands as a result of intrinsic anharmonic couplings. A prominent effect of tuning these FR bands and lighting up dark overtone states in this wide frequency range was investigated by changing the number of methyl groups in the protonated amine moiety. The effect of Ar-tagging was also analyzed and decent agreement between the experimental and simulated spectra certified the above-mentioned simple pictures. We also found that the coupling constant for trimethylamine is the largest among these protonated amine-water clusters, and the overall coupling strength decreases as the hydrogen-bonded NH stretching frequency redshifts in the order of dimethylamine, methylamine, and ammonia.

Velocity echoes in water

A three-point velocity correlation function ⟨v(t1 + t2)v2(t1)v(0)⟩ is introduced for a better understanding of the recent 2D-Raman-THz spectroscopy of the intermolecular degrees of freedoms of water and aqueous salt solutions. This correlation function reveals echoes in the presence of inhomogeneous broadening, which are coined “velocity echoes.” In analogy to the well-known two-point velocity correlation function ⟨v(t)v(0)⟩, it reflects the density of states (DOS) of the system under study without having to amend them with transition dipoles and transition polarizabilities. The correlation function can be calculated from equilibrium trajectories and converges extremely quickly. After deriving the theory, the information content of the three-point velocity correlation function is first tested based on a simple harmonic oscillator model with Langevin dynamics. Subsequently, velocity echoes of TIP4P/2005 water are calculated as a function of temperature, covering ambient conditions, the supercooled regime and amorphous ice, as well as upon addition of various salts. The experimentally observed trends can be reproduced qualitatively with the help of computationally very inexpensive molecular dynamics simulations.

A Semi-Classical View on Epsilon-Near-Zero Resonant Tunneling Modes in Metal/Insulator/Metal Nanocavities.

Metal/Insulator/Metal nanocavities (MIMs) are highly versatile systems for nanometric light confinement and waveguiding, and their optical properties are mostly interpreted in terms of surface plasmon polaritons. Although classic electromagnetic theory accurately describes their behavior, it often lacks physical insight, leaving some fundamental aspects of light interaction with these structures unexplored. In this work, we elaborate a quantum mechanical description of the MIM cavity as a double barrier quantum well. We identify the square of the imaginary part κ of the refractive index ñ of the metal as the optical potential and find that MIM cavity resonances are suppressed if the ratio n/κ exceeds a certain limit, which shows that low n and high κ values are desired for strong and sharp cavity resonances. Interestingly, the spectral regions of cavity mode suppression correspond to the interband transitions of the metals, where the optical processes are intrinsically non-Hermitian. The quantum treatment allows to describe the tunnel effect for photons and reveals that the MIM cavity resonances can be excited by resonant tunneling via illumination through the metal, without the need of momentum matching techniques such as prisms or grating couplers. By combining this analysis with spectroscopic ellipsometry on experimental MIM structures and by developing a simple harmonic oscillator model of the MIM for the calculation of its effective permittivity, we show that the cavity eigenmodes coincide with low-loss zeros of the effective permittivity. Therefore, the MIM resonances correspond to epsilon-near-zero (ENZ) eigenmodes that can be excited via resonant tunneling. Our approach provides a toolbox for the engineering of ENZ resonances throughout the entire visible range, which we demonstrate experimentally and theoretically. In particular, we apply our quantum mechanical approach to asymmetric MIM superabsorbers and use it for configuring broadly tunable refractive index sensors. Our work elucidates the role of MIM cavities as photonic analogues to tunnel diodes and opens new perspectives for metamaterials with designed ENZ response.

Resolving fine electromechanical structure of collagen fibrils via sequential excitation piezoresponse force microscopy

Collagen is the main protein in extracellular matrix that is found in many connective tissues, and it exhibits piezoelectricity that is expected to correlate with its hierarchical microstructure. Resolving fine electromechanical structure of collagen, however, is challenging, due to its weak piezoresponse, rough topography, and microstructural hierarchy. Here we adopt the newly developed sequential excitation strategy in combination with piezoresponse force microscopy to overcome these difficulties. It excites the local electromechanical response of collagen via a sequence of distinct frequencies, minimizing crosstalk with topography, followed by principal component analysis to remove the background noise and simple harmonic oscillator model for physical analysis and data reconstruction. These enable us to acquire high fidelity mappings of fine electromechanical response at the nanoscale that correlate with the gap and overlap domains of collagen fibrils, which show substantial improvement over conventional piezoresponse force microscopy techniques. It also embodies the spirit of big data atomic force microscopy that can be readily extended into other applications with targeted data acquisition.

Mapping intrinsic electromechanical responses at the nanoscale via sequential excitation scanning probe microscopy empowered by deep data.

Ever-increasing hardware capabilities and computation powers have enabled acquisition and analysis of big scientific data at the nanoscale routine, though much of the data acquired often turn out to be redundant, noisy and/or irrelevant to the problems of interest, and it remains nontrivial to draw clear mechanistic insights from pure data analytics. In this work, we use scanning probe microscopy (SPM) as an example to demonstrate deep data methodology for nanosciences, transitioning from brute-force analytics such as data mining, correlation analysis and unsupervised classification to informed and/or targeted causative data analytics built on sound physical understanding. Three key ingredients of such deep data analytics are presented. A sequential excitation scanning probe microscopy (SE-SPM) technique is first developed to acquire high-quality, efficient and physically relevant data, which can be easily implemented on any standard atomic force microscope (AFM). Brute-force physical analysis is then carried out using a simple harmonic oscillator (SHO) model, enabling us to derive intrinsic electromechanical coupling of interest. Finally, principal component analysis (PCA) is carried out, which not only speeds up the analysis by four orders of magnitude, but also allows a clear physical interpretation of its modes in combination with SHO analysis. A rough piezoelectric material has been probed using such a strategy, enabling us to map its intrinsic electromechanical properties at the nanoscale with high fidelity, where conventional methods fail. The SE in combination with deep data methodology can be easily adapted for other SPM techniques to probe a wide range of functional phenomena at the nanoscale.

Repulsive Casimir force

The Casimir force between two parallel thick plates, one perfectly dielectric, the other purely magnetic, has been calculated long ago by Boyer [T. H. Boyer, Phys. Rev. A {\bf 9}, 2078 (1974)]. Its most characteristic property is that it is repulsive. The problem is actually delicate and counterintuitive, since it implies, for instance, that in the boundary layer of the electric plate the square $E^2$ of the electric field becomes a negative quantity. In the present paper we analyze the problem by first considering the simple harmonic oscillator model introduced by us earlier [J. S. H{\o}ye {\it et al.}, Phys. Rev. E {\bf 67}, 056116 (2003); Phys. Rev. A {\bf 94}, 032113 (2016)]. Extension of this model shows how the repulsive behavior can be understood on a microscopic basis, due to the duality between canonical and mechanical momenta in presence of the electromagnetic vector potential. This duality corresponds to the TM and TE modes in electrodynamics. We analyze the generalized Boyer case where the permittivities and permeabilities of the parallel plates are arbitrary. In this respect we first find the induced interaction between a pair of particles with given electric and magnetic polarizabilities and then find it for a pair of parallel plates. The method used for our evaluations is the statistical mechanical one that we have introduced and applied earlier. Whether the pair of particles or plates attract or repel each other depends on their polarizabilities or permittivities and permeabilities respectively. For equal particles or equal plates there is always attraction.

The ‘Miracle’ of Applicability? The Curious Case of the Simple Harmonic Oscillator

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.

Application of one-dimensional semiclassical transition state theory to the CH3OH + H ⇌ CH2OH/CH3O + H2 reactions.

The rate constants of the two branches of H-abstractions from CH3OH by the H-atom and the corresponding reactions in the reverse direction are calculated using the one-dimensional semiclassical transition state theory (1D SCTST). In this method, only the reaction mode vibration of the transition state (TS) is treated anharmonically, while the remaining internal degrees of freedom are treated as they would have been in a standard TS theory calculation. A total of eight ab initio single-point energy calculations are performed in addition to the computational cost of a standard TS theory calculation. This allows a second-order Richardson extrapolation method to be employed to improve the numerical estimation of the third- and fourth-order derivatives, which in turn are used in the calculation of the anharmonic constant. Hindered-rotor (HR) vibrations are identified in the equilibrium states of CH3OH and CH2OH, and the TSs of the reactions. The partition function of the HRs are calculated using both a simple harmonic oscillator model and a more sophisticated one-dimensional torsional eigenvalue summation (1D TES) method. The 1D TES method can be easily adapted in 1D SCTST computation. The resulting 1D SCTST with 1D TES rate constants show good agreement to previous theoretical and experimental works. The effects of the HR on rate constants for different reactions are also investigated.This article is part of the theme issue 'Modern theoretical chemistry'.

Phenomenology of inflationary models

There are many inflationary models compatible with observational data. One can investigate inflationary models by looking at their general features, which are common in most of the models. Here we have investigated some of the single-field models without considering their origin in order to find the phenomenology of them. We have shown how to adjust the simple harmonic oscillator model in order to be in good agreement with observational data.

Simple Harmonic Oscillator Model of O2 Molecule in Vacuum: A Classical Molecular Dynamics Study

The simple harmonic oscillator model allows a basic understanding of all processes and can be used to analyse optical vibrational modes and electronic transitions in atoms, molecules and crystals, in order to derive general properties of harmonic generation to all orders. In particular, we are done to investigate single, ten, and twenty harmonic oscillator of O2 molecule using GROMACS 4.55. Time is the important factor in simulation because the best result of simulation can be obtained by increasing the number of steps and by decreasing the timesteps. Since the properties only depend on time and not on the specific microscopic model, they can also be adopted for the quantum-mechanical description by using in the classical molecular dynamics.Â

Effect of 10B isotope and vacancy defects on the phonon modes of two-dimensional hexagonal boron nitride

We report the details of the effects of the 10B isotope and those of B and N vacancies combined with the isotope on the phonon modes of two-dimensional hexagonal boron nitride (h-BN). The phonon density of states and localization problems are solved using the forced vibrational method, which is suitable for an intricate and disordered system. We observe an upward shift of Raman-active E2g-mode optical phonons (32 cm−1) for a 100% 10B isotope, which matches well with the experiment and simple harmonic oscillator model. However, a downward shift of E2g-mode phonons is observed for B or N vacancies and the combination of the isotope and vacancy-type disordered BN. Strong localized eigenmodes are found for all types of defects, and a typical localization length is on the order of ∼7 nm for naturally occurring BN samples. These results are very important for understanding the heat dissipation and electron transport properties of BN-based nanoelectronics.