Antisymmetric Vibration Research Articles

Sensitivity of the C–H Stretching Band in Raman Spectra to Phospholipid Order

ABSTRACTPhospholipid bilayers, which are a major component of cellular membranes and some drug delivery vehicles, can be in different states of order depending on the conformations of the hydrocarbon tails and their mutual arrangement. The important task of experimental characterization of phospholipid order is often addressed using Raman spectra of the C–H stretching bands. Such characterization uses some empirical relationships for apparent maxima in the spectra, although the origin of the sensitivity of the C–H band to phospholipid order and its model description remain unclear. Surely, a correct description of the sensitivity of the C–H band to phospholipid order is critical for its proper application. Here, we provide a description of the ordering sensitivity of the C–H stretching band using a polarized Raman experiment with hydrated aligned multibilayers of a saturated phospholipid. By this way, Raman contributions from symmetric and antisymmetric vibrations of CH2 were obtained in a model‐free manner. Experiments in a wide temperature range and the use of isotopic isolation helped us to consider separately the effects of conformational and lateral order of chains. The conformational sensitivity of the spectrum of antisymmetric vibrations was confirmed by DFT modeling. The outcomes of the study allowed us to provide recommendations for the use of the Raman spectrum of the C–H stretching band to characterize the conformational and lateral order of phospholipid‐containing materials.

Supercapacitor electrode properties of NiWO4 nanocrystals processed by thermal plasma and deposited on the FTO substrate

In this communication, nickel tungstate (NiWO4) nanocrystals were prepared using the coprecipitation method at 90 °C for 6 h and processed by thermal plasma (TP) at 600 °C for 1 h. X-ray diffraction (XRD), Rietveld refinement data, micro-Raman, and Fourier transform infrared (FT-IR) spectroscopies were employed to analyze their structure and optical data. The morphology was observed through field emission scanning electron microscopy (FE-SEM) images. Moreover, their supercapacitor electrode properties were investigated in detail. XRD patterns, Rietveld refinement, micro-Raman/FTIR spectra, and FE-SEM images confirm pristine NiWO4 nanocrystals with a monoclinic structure, symmetric and antisymmetric stretch vibrations of ONiO and OWO bonds, which are related to IR- and Raman-active vibrational modes, spherical shape with average crystal sized ~29.2 nm. These NiWO4 nanocrystals present a high specific capacitance of 351.26 F g−1 at 40 mV s−1, in comparison with NiWO4 nanocrystals processed in heat treatment conventional (120 F g−1 at 40 mV s−1), which can be applied as a stable cathodic electrode. These results suggest that NiWO4 nanocrystals processed by TP and deposited on FTO substrate are good candidates for use as electrodes for energy storage devices in the next generation of batteries.

Hydrogen bonding switched symmetric and anti-symmetric vibrations SRS of –CH2 in aqueous DEG solution

The symmetric and anti-symmetric stretching vibrations of C–H extensively exist in organic molecules, which constantly arise together in Raman spectroscopy. In this study, a method was presented to switch stimulated Raman scattering peak from symmetric vibrations (2870 cm−1) to anti-symmetric vibrations (2924 cm−1) of –CH2 by regulating the hydrogen bonding (HB) associated through intermolecular and intramolecular in diethylene glycol (DEG) molecules. Such dominant association structure can be converted from DEG-DEG to DEG-H2O in the aqueous DEG solutions. The interaction between H2O and DEG was investigated through the density functional theory calculation. The changes in intermolecular HB have effects on the intramolecular vibrational modes by the intermolecular forces and symmetry of the DEG molecule. This study has exciting potential prospect and can be extended to other mixing solutions.

Correlation between absorbed acetylene spectral characteristic and nature of ionic liquids studied by in situ ATR-FTIR spectroscopy

Ionic liquids (ILs) are believed to be a prospective replacement for molecular solvents and are suggested to be used for storing and/or purifying acetylene. Systematic studies of the nature of the interaction between acetylene and different ILs may be necessary to predict new types of ionic liquids. Therefore, in order to investigate the absorption of acetylene by ionic liquids, in situ ATR-FTIR spectroscopy has been applied for the first time. Two series of ILs, such as [C2‍–‍10MIM][BF4] and ILs based on imidazolium with various anions, have been considered. A detailed analysis of the spectra of absorbed acetylene is given, with special attention being paid to the basicity of IL anions as well as the influence of the cation alkyl chain length. It has been found that the magnitude of the band shift associated with the CH stretching vibration and the antisymmetric CH bending vibration of absorbed acetylene correlates well with the change in the Kamlet-Taft β parameter of the ILs. Moreover, neither the free volume nor the IL polarity plays a key role in acetylene solubility. In addition, the enthalpy and entropy of absorption have been calculated for several ILs.

Frequency-domain hydroelastic stress analysis of floating structures under spatially inhomogeneous wave field

This paper presents a frequency-domain method for the stress analysis of flexible floating structures exposed to a spatially inhomogeneous wave field. A hydroelastic model is developed for the continuous structure, wherein discrete rigid modules are connected by elastic beams with equivalent stiffness. The inhomogeneous wave field is divided into wave regions characterized by constant wave parameters. By assigning distinct wave parameters to modules based on their respective positions, the global hydroelastic equation is established to capture the wave inhomogeneity. To compute the stress, external loads in the hydroelastic equation, comprising hydrodynamic and hydrostatic restoring pressures, as well as inertia forces, are mapped onto the finite element model of the continuous structure, enabling stress calculation by the quasi-static method. Variations in wave directions and heights along the longitudinal direction of the structure are introduced with the aim of exploring the effect of the wave inhomogeneity on the stress. A deliberately flexible barge is constructed to evaluate the stress resulting from global structural vibrations. The results demonstrate a significant contribution of wave inhomogeneity to the stress, particularly during structural resonance frequencies. Moreover, an exclusive phenomenon is observed whereby the inhomogeneous wave field induces stress due to anti-symmetric vibrations under beam sea conditions.

Structure, optical, colorimetric, and supercapacitor properties of anode α-CuWO4 crystals

In this work, alpha-copper tungstate (α-CuWO4) crystals were prepared by polymeric precursor method. The structural and optical data were performed by X-ray diffraction (XRD), Rietveld refinement, Fourier-transform infrared (FTIR), and ultraviolet–visible (UV–Vis) diffuse reflectance spectroscopies. Moreover, their colorimetric and supercapacitor properties were investigated. Pristine α-CuWO4 crystals present a triclinic structure, symmetric and antisymmetric stretch vibrations in the O–Cu–O and O–W–O bonds corresponding to IR-active modes and an optical bandgap value of 2.24 eV. Colorimetric results showed that the α-CuWO4 crystals exhibit a brown color to be employed in pigments coating floors. The α-CuWO4 crystals present a specific capacitance of 275.68F/g at 5 mV/s to be applied as a stable anodic electrode. These results suggest that α-CuWO4 is an excellent candidate as an electrode for energy storage devices in the next generation of batteries.

Enhanced linearity through high-order antisymmetric vibration for MEMS DC power sensor

We present an electric power meter that capitalizes on the interaction of electrothermal strain and mechanical vibration in a micro-electro-mechanical systems (MEMS) beam undergoing the antisymmetric mode of vibration. This is achieved by using a resonant bridge driven with an electrothermal modulation technique. The change in electrical power is monitored through the alteration in the mechanical stiffness of the structure, which is tracked electrostatically. The observed deflection profile of the beam under the influence of electrothermal effects shows that the deflection geometry due to buckling exhibits similar trends as the first symmetric vibrational mode, in contrast to the antisymmetric mode. Therefore, we compare two distinct vibrational modes, converting the compressive thermal stress generated by the input electrical power via Joule heating into a shift in the resonance frequency. By employing antisymmetric vibrational mode, the output of our device is consistently monotonic to the input electrical power, even when the microbeam is experiencing buckling deflections. In addition, the sensing operation based on antisymmetric modes yields only a 1.5% nonlinear error in the response curve, which is ten times lower than that of symmetric modes. The observed deformation shape of the resonator agrees with the results obtained from multi-physics finite simulations. Finally, this approach has the potential to be extended to other frequency-shift-based sensors, allowing for higher linearity.

State-resolved studies of CO2 sticking to CO2 ice.

Internal vibrations may affect the adsorption, scattering, and reactions of molecules impinging onto a surface. The energy of the ν 3 antisymmetric stretch vibration of CO2 slightly exceeds the desorption energy of CO2 bound to CO2 ice. We use supersonic molecular beam techniques and rovibrationally state-resolved excitation to determine whether this vibration affects condensation of gas phase CO2 to its ice. We detect sticking and CO2 ice formation using RAIRS and quantify the sticking probability using the King and Wells method with modulation of the vibrational excitation and Fourier transform based detection. We find that the influence of this vibration on the structure of the formed ice and on the sticking probability is negligible under our conditions. Based on our detection limit, we quantify the weighted average sticking probability at approximately 0.9 and the difference between the state-resolved and weighted average sticking probability as below 0.03%.

Design and compression-induced bandgap evolution of novel polygonal negative stiffness metamaterials

The paper proposes a novel polygonal negative stiffness (NS) metamaterial composed of folded beams and stiffening walls, which exhibits excellent performance in attenuating waves over a broad frequency range and shows compression-induced bandgap evolution characteristics. Dispersion relation and bandgaps for the propagation of elastic waves in NS metamaterials are calculated utilizing a semi-analytical periodic spectral method (PS-SAFE). The results demonstrate that broadband bandgaps can be achieved in the proposed polygonal NS metamaterials. Mode analysis and transmission spectrum are utilized to explain the underlying mechanisms of wave attenuation. Furthermore, appropriate deformation conditions are applied to the NS metamaterials to realize compression-induced bandgap evolution, which shows that compression-induced configuration variations significantly impact the frequency range and bandwidth of the bandgaps. As the deformation increases, folded beams enter the elastic instability stage, and the internal stress change strongly influence the vibration displacement modes, leading to various symmetric and antisymmetric displacement vibration modes within a similar frequency range, forming low-frequency broadband bandgaps. This work provides valuable insights into the design of advanced materials for low-frequency broadband wave attenuation.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

Antenna-coupled infrared nanospectroscopy of intramolecular vibrational interaction

Many photonic and electronic molecular properties, as well as chemical and biochemical reactivities are controlled by fast intramolecular vibrational energy redistribution (IVR). This fundamental ultrafast process limits coherence time in applications from photochemistry to single quantum level control. While time-resolved multidimensional IR-spectroscopy can resolve the underlying vibrational interaction dynamics, as a nonlinear optical technique it has been challenging to extend its sensitivity to probe small molecular ensembles, achieve nanoscale spatial resolution, and control intramolecular dynamics. Here, we demonstrate a concept how mode-selective coupling of vibrational resonances to IR nanoantennas can reveal intramolecular vibrational energy transfer. In time-resolved infrared vibrational nanospectroscopy, we measure the Purcell-enhanced decrease of vibrational lifetimes of molecular vibrations while tuning the IR nanoantenna across coupled vibrations. At the example of a Re-carbonyl complex monolayer, we derive an IVR rate of (25±8) cm-1 corresponding to (450±150) fs, as is typical for the fast initial equilibration between symmetric and antisymmetric carbonyl vibrations. We model the enhancement of the cross-vibrational relaxation based on intrinsic intramolecular coupling and extrinsic antenna-enhanced vibrational energy relaxation. The model further suggests an anti-Purcell effect based on antenna and laser-field-driven vibrational mode interference which can counteract IVR-induced relaxation. Nanooptical spectroscopy of antenna-coupled vibrational dynamics thus provides for an approach to probe intramolecular vibrational dynamics with a perspective for vibrational coherent control of small molecular ensembles.

MLRNet: Combining the Physics-Motivated Potential Models with Neural Networks for Intermolecular Potential Energy Surface Construction.

A physics-based machine learning model called MLRNet has been developed to construct the high-accuracy two-body intermolecular potential energy surface (IPES). The outputs of the neural network are integrated into the physically realistic Morse/long-range (MLR) function, which ensures that the MLRNet has meaningful extrapolation at both short and long ranges and solves the asymptotic problem in common neural network potential (NNP) models. The neural network representation of the MLR parameters is more flexible and more efficient than the polynomial expansion in the conventional mdMLR model, especially for systems containing nonrigid monomer(s). The present work illustrates the basic framework of the current MLRNet model, including (i) how to combine the physically meaningful MLR function with different possible NN structures, (ii) the preservation of permutation symmetry, and (iii) the predetermination of the long-range function uLR. We choose two realistic systems to demonstrate the performance of MLRNet: the three-dimensional IPES of CO2-He including the CO2 antisymmetric vibration Q3 and the six-dimensional IPES of the H2O-Ar system. In both cases, the fitting errors of the MLRNet are several times smaller than those of the conventional mdMLR model. Both short-range and long-range extrapolation tests were performed to illustrate the extrapolation ability of the MLRNet and its damping function version. Moreover, for the 6-D H2O-Ar system, the MLRNet only needs 1596 trainable parameters, which is almost equal to the number needed for the 5-D mdMLR model (1509) and half that needed for the PIP-NN model (3501) within similar accuracy, which illustrates the model efficiency in high-dimensional IPES fitting.

Structural, morphological, and magneto-optical investigations of pure and (Sn, Zn) co-doped CuO nanoparticles: A novel corrosion inhibitor in acidic media

The present study reports work to synthesize metal oxide semiconductors of pure and (Sn, Zn) co-doped copper oxide (CuO) nanoparticles, with the chemical formula of Cu1–2xSnxZnxO (x = 0.000, 0.005, 0.010, and 0.020), via the co-precipitation method. The structural, morphological, optical, and magnetic properties were characterized by X-ray diffraction (XRD), transmission electron microscopy (TEM), Fourier transform infrared spectroscopy (FTIR), ultraviolet-visible spectroscopy (UV–vis), photoluminescence spectroscopy (PL), and vibrating-sample magnetometer (VSM), respectively. XRD results revealed the successful preparation of the monoclinic CuO phase, with small traces of secondary phase SnO appearing at samples of x = 0.010 and x = 0.020. The size of the prepared nanoparticles, obtained from XRD and TEM, showed an overall decrease when increasing the (Sn, Zn) co-doping concentrations, leading to an increase in the surface area appropriate for magnetic applications. At samples of x = 0.005 and x = 0.010, the co-dopants changed morphology from large spheres into nanorods and nanokernels. The diameters of the nanorods increased from 8.5 nm to 24.15 nm, while their lengths decreased from 51 nm to 48 nm. However, a flower-shaped nanostructure was observed in the x = 0.020 sample. FTIR confirmed the fingerprint vibration modes represented by antisymmetric vibration νas(Cu─O) in the crystal lattice. The first decrease in size was accompanied by a decrease in the energy gap caused by the lattice shrinkage, which in turn is caused by the vacancies and energy states in the bandgap stimulated by Sn and Zn ions. Following this decrease, there was an increase in the energy gap because of the shift in Fermi level to a higher energy state that is suitable for solar cell applications. PL results confirmed the excitation dependence nature of these nanoparticles. Correspondingly, the prepared nanoparticles can be utilized as potential applicants for blue chip and near-UV white light-emitting devices. VSM data revealed that paramagnetic and weak ferromagnetic behavior coexist in the co-doped nanoparticle samples. Through the electrochemical impedance spectroscopy and polarization measurements, the anticorrosion performance of these nanoparticles on mild steel has been studied in 0.5 M HCl solutions at 30 °C. The impedance results revealed that the corrosion process takes place under activation control. The polarization measurements showed that the prepared nanoparticles behave as a mixed-type inhibitor with 83 % efficiency in the x = 0.020 sample.

FTIR spectroscopy as a convenient tool for detection and identification of airborne Cr(VI) compounds arising from arc welding fumes

Given the significant presence of the carcinogenic Cr(VI) in arc welding fumes from stainless steels, it is also important, in addition to estimating the Cr(VI) levels, to identify Cr(VI) compounds, as it throws light on the mechanistic pathways towards fume formation. FTIR data is presented in this paper for arc welding fumes collected from Manual Metal Arc Welding (MMA), Flux Cored Arc Welding (FCAW) and Solid Wire Welding (Metal Inert/ Active Gas Welding [MIG/ MAG]). For MMA and FCAW samples, clear spectra corresponding to Na, K, dichromates was observed at wave number of around 725-740 cm-1 and at 890-900 cm-1. Chromate species were also observed at around 850-855 cm-1, as was evidence of CrO3 (chromium trioxide) too (950-970 cm-1). The identification of these compounds was done by carefully identifying the Cr-O-Cr anti-symmetric vibrations, the symmetric stretching of the CrO4 tetrahedra, and the stretching vibrations of the planar CrO3 structure for the chromium trioxide. All the above compounds were volatile, and present as nanoparticles in welding fumes, thereby potentially causing significant harm to the welders. Additionally, crystalline phases (Fe-Mn spinels) were also observed through powder XRD, and the data was compared with ion chromatography estimates for Cr(VI) and found to be consistent.

Optical, impedance, and relaxation response of Fe3+ doped double perovskite solid solution

In this report, the effect of Fe3+-incorporation on the crystal structure, vibrational, optical, impedance, modulus, and ferroelectric responses of BaBi1.8Fe0.2TiO6 was discussed. According to XRD and SEM studies, the crystal structure is monoclinic single-phase polycrystalline. The vibrational modes identification from the FTIR spectrum concedes different modes of symmetric and/or antisymmetric vibrations which are usual characteristics of perovskite compounds. The UV-visible study outlines strong absorption and weak reflection near UV and the visible spectrum has a cutoff wavelength of 560 nm. The Tauc and Wood relation estimated the direct band gap energy in the absorbance spectra is 2.70 eV, while the diffused reflectance spectroscopy (DRS) method revealed its value in the reflectance spectra as Eg = 2.84 eV. The resulting bandgap energies suggest that they could be used in photocatalysis. The impedance and modulus analysis supports the semiconducting nature and non-Debye kind relaxation phenomenon in the compound. The PE hysteresis loop with non-zero remanent polarization indicates the material's ferroelectric property, which is suitable for memory storage applications.

On the Cooperative Origin of Solvent-Enhanced Symmetry-Breaking Charge Transfer in a Covalently Bound Tetracene Dimer Leading to Singlet Fission.

Singlet fission in covalently bound acene dimers in solution is driven by the interplay of excitonic and singlet correlated triplet 1(TT) states with intermediate charge-transfer states, a process which depends sensitively on the solvent environment. We use high-level electronic structure methods to explore this singlet fission process in a linked tetracene dimer, with emphasis on the symmetry-breaking mechanism for the charge-transfer (CT) states induced by low-frequency antisymmetric vibrations and polar/polarizable solvents. A combination of the second-order algebraic diagrammatic construction (ADC(2)) and density functional theory/multireference configuration interaction (DFT/MRCI) methods are employed, along with a state-specific conductor-like screening model (COSMO) solvation model in the former case. This work quantifies, for the first time, an earlier mechanistic proposal [Alvertis et al., J. Am. Chem. Soc. 2019, 141, 17558] according to which solvent-induced symmetry breaking leads to a high-energy CT state which interacts with the correlated triplet state, resulting in singlet fission. An approximate assessment of the nonadiabatic interactions between the different electronic states underscores that the CT states are essential in facilitating the transition from the bright excitonic state to the 1(TT) state leading to singlet fission. We show that several types of symmetry-breaking inter- and intra-fragment vibrations play a crucial role in a concerted mechanism with the solvent environment and with the symmetric inter-fragment torsion, which tunes the admixture of excitonic and CT states. This offers a new perspective on how solvent-induced symmetry-breaking CT can be understood and how it cooperates with intramolecular mechanisms in singlet fission.

Direct Probing of Vibrational Interactions in UiO-66 Polycrystalline Membranes with Femtosecond Two-Dimensional Infrared Spectroscopy

UiO-66 is a benchmark metal-organic framework that holds great promise for the design of new functional materials. In this work, we perform two-dimensional infrared measurements on polycrystalline membranes of UiO-66 grown on c-sapphire substrates. We study the symmetric and antisymmetric stretch vibrations of the carboxylate groups of the terephthalate linker ions and find that these vibrations show a rapid energy exchange and a collective vibrational relaxation with a time constant of 1.3 ps. We also find that the symmetric vibration of the carboxylate group is strongly coupled to a vibration of the aromatic ring of the terephthalate ion. We observe that the antisymmetric carboxylate vibrations of different terephthalate linkers show rapid resonant (Förster) energy transfer with a time constant of ∼1 ps.

Lamb waves in elastic–plastic plate containing voids

Abstract Propagation of Lamb waves has been studied in an infinite plate of finite thickness and composed of elastic–plastic material containing void pores. The plastic deformation in the plate material is considered through a single-plane and direction. Assuming a linear relation between the elastic and plastic parts of the change in void volume fraction and using the stress-free boundary surfaces of the plate, the dispersion relations corresponding to the symmetric and antisymmetric vibrations with respect to the mid-plane of the plate are derived. The corresponding dispersion relations of Lamb modes in the elastic–plastic plate have been reduced by neglecting the presence of voids. Some earlier known results are recovered from the present formulation under particular cases. The behaviour of symmetric and antisymmetric family of Lamb modes are depicted pictorially against the frequency, slip-angle, thickness of the plate and void parameters in a specific plate material. It is found that the plasticity and the presence of the voids in the plate material affect the propagation of Lamb modes. In particular, at the slip-angle $45^o$, the $S_0$- and $A_0$- modes exhibit symmetry, but $S_0$ is bell shaped and $A_0$ is parabolic.

Train-Induced Dynamic Behavior and Fatigue Analysis of Cable Hangers for a Tied-Arch Bridge Based on Vector Form Intrinsic Finite Element

This paper presents train-induced dynamic response and fatigue damage analyses for the hangers of a tied-arch railway bridge. A train–bridge interaction analysis is carried out using the vector form intrinsic finite element method, through which the inertial force effect of the moving train can be effectively analyzed. The responses of the bridge deck and its hangers are investigated at resonant train speeds. Significantly larger stresses are observed on the shorter hangers near the arch anchorages, which is primarily caused by the first two anti-symmetrical vibration modes. The fatigue damage to the hangers is estimated using the Palmgren-Miner model (PMM) for linear fatigue damage accumulation and the continuum damage mechanics (CDM) method for nonlinear accumulation. The mean stress effect is considered in the S–N curve primarily by Smith–Watson–Topper equation in terms of the effective stress range with a zero-mean stress. Two probability distributions for train speed are considered for the current and future operating conditions with mean speeds of 220 and 300[Formula: see text]km/h, respectively. This study found that the fatigue lives estimated by the nonlinear CDM are significantly shorter than those estimated by the linear PMM. It also found that the shortest hanger reflects the shortest fatigue life at the current operating speed, whereas the longer hanger near the third point of the bridge deck may have the shortest fatigue life at an increased speed in the future.

A Simplified Method of Microscopic Polarizability Tensor Differential of Hyper-Raman Spectroscopy Based on the Bond Additivity Model

Coherent anti-Stokes Raman spectroscopy (CARS) and Ccherent anti-Stokes hyper-Raman spectroscopy (CAHRS), as other high-order nonlinear spectroscopy techniques, are widely exploited in many research fields, such as dynamic processes, gene expression spectrum screening, high-resolution spectroscopy, and nonlinear high-resolution imaging. However, it is difficult to make a quantitative analysis of the spectral signals that involve a large number of high-order micropolarizability tensors. It is reported that the CARS and CAHRS microscopic hyperpolarizability tensor elements can be decomposed into the product of the differentiation of Raman microscopic polarizability tensor α′i′j′ and hyper-Raman microscopic polarizability tensor β′i′j′k′ so that the high-order spectra can be simplified to the analysis of low-order spectra. In this paper, we use the bond additivity model (BAM) combined with experimental corrections to address the carbon dioxide (CO2) molecule and present the simplified scheme for differentiation of hyper-Raman microscopic polarizability tensor elements β′i′j′k′. Taking advantage of this approach, combined with the experimental correction, the differentiation of Hyper-Raman microscopic polarizability tensor elements β′i′j′k′ of the CO2 is obtained and the expressions of β′i′j′k′ for antisymmetric vibrations of CO2 are deduced. Finally, substituting the differentiation of Raman microscopic polarizability tensor elements α′i′j′ reported in the literature into the ratio above can obtain the proportional relationship between the microscopic polarizability tensor elements of CARS and CAHRS of the CO2. This method can provide the basis for the quantitative analysis of high-order nonlinear spectral profiles.