CS Approximation Research Articles

Strain rate dependence of SM490B at room temperature and application to Cowper-Symonds constitutive models

Steel is widely used as a constituent material for various structures such as automobiles and ships. To perform high precision analysis including high strain rate behavior, an understanding for the strain rate dependence of material strength becomes very important. The purpose of this study is to evaluate the strain rate dependence of material strength with rolled steels for welded structure, JIS SM490B (ASTM E). We investigated the deformation characteristics at room temperature by performing compression tests at a wide range of strain rates and applied the obtained experimental results to the material constitutive model using an optimization method of Nelder-Mead method. The quasi-static tests were conducted using a universal testing machine at the strain rate of 10-3, 10-2 and 10-1 s-1. The impact test was conducted using a split Hopkinson pressure bar apparatus at the strain rate of approximately 103 s-1. As the results of the compression tests, it was confirmed that SM490B has a positive strain rate dependence of material strength. The Cowper-Symonds constitutive model showed good agreement with the experimental results up to the strain of 20%. However, the error became larger between experimental results and CS approximation as the strain increases to 20% or more.

State-to-state quantum dynamics studies of the H + LiH+ → Li+ + H2 reaction

ABSTRACTQuantum dynamical calculations of the H + LiH+ → Li+ + H2 reaction were performed based on the potential energy surface (PES) reported by Dong et al. (RSC Adv. 7, 7008 (2017)) using the time-dependent quantum wave packet method in collision energy range from 0.01 to 1.0 eV. Dynamics properties such as reaction probability, integral cross section, differential cross section (DCS), and thermal rate constant of the H + LiH+ → Li+ + H2 reaction were reported at the state-to-state level of theory and compared with available theoretical calculations. The results indicated that present values are in good agreement with results obtained from the quasi-classical trajectory method. However, large differences can be found between present values and previous quantum results. This can be attributed to the different PESs used in the calculation and the CS approximation was adopted in previous theoretical studies. In addition, the ‘rebound’ reaction mechanism was proposed in previous theoretical studies in a high collision energy range. However, the DCS scattering signals calculated in the present work indicated that complex-forming and direct abstract reaction mechanisms are dominant in low and high collision energies, respectively.

Accurate integral cross sections for the H + CO2 → OH + CO reaction

A full-dimensional quantum dynamics calculation has been carried out to yield the coupled-channel reaction probabilities with total angular momentum J up to 60 for the H + CO2 → OH + CO reaction on the PIP-NN PES potential energy surface. Well converged integral cross section (ICSs) for collision energy up to 2.4 eV were obtained for the ground initial state. As compared with previous study, it is found that the CS approximation substantially underestimated ICSs, which is inadequate for the title reaction. In addition, the agreement between QM and QCT ICSs is quite satisfactory, indicating that the QCT method is rather accurate on describing dynamics of this complex-forming reaction.

Some Observations about the MOLSCAT

For calculation of cross sections for collisional transitions between rotational levels in a molecule, a computer code, MOLSCAT is available. For the transitions between rotational levels in H2CS due to collisions with He atom, we have calculated cross sections under the CS approximation, for example, for total energy 11 cm−1. The calculations have been done for the single energy 11 cm−1 and for ten combinations. We have found that the cross sections for the single energy 11 cm−1, differ from those in the first seven combinations, but are in agreement with those in the last three combinations. The reason for the difference in the results appears that the MOLSCAT uses the intermediate data of calculations for one energy, in the calculations for other energies. The agreement with the last three combinations may be understood that when the energy of combination is in the decreasing trend, the cross sections for the first (common) energy are equal. It may be suggested to run the MOLSCAT for a single energy at a time.

Dynamical properties of S(3P) + HD reaction on 13A″state and their quantum wavepacket calculation

The time‐dependent wavepacket method is used to study the reaction dynamics of S(3P) + HD (v = 0, 1, 2) on the adiabatic 13A″ potential energy surface constructed by Han and coworkers [J. Chem. Phys. 2012, 136, 094308]. The reaction probabilities and integral cross sections as a function of collision energy are obtained and discussed. The results calculated by using the CC and the CS approximation have been compared, which suggests that for this direct abstraction reaction, the cheaper CS approximation calculation is valid enough in the quantum calculation. The investigation also shows that the reaction probabilities and integral cross sections tend to increase with collision energy. By analyzing the v‐dependent behavior of the integral cross sections, the significant effect of the vibrational excitation of HD is found. Also found in the calculation is a significant resonance feature in the reaction probabilities versus collision energy. © 2014 Wiley Periodicals, Inc.

Three-sublattice model that takes into account the spin density anisotropy, short-range order, and dimensioned factor for the binary Fe-Cr(V, Mo) systems

The diffusion properties of the elements entering into the composition of austenitic and ferritic steels are analyzed as functions of the temperature and the time evolution of induced radioactivity. As compared to austenitic steels, ferritic steels are shown to have better diffusion properties and swelling resistance during operation at temperatures up to 650°C under irradiation conditions and can be used as structural materials in the core of a fast neutron reactor. Experimental and calculated (Calphad method) data on the Fe-Cr phase diagram and the thermodynamic properties and the experimental data on the short-range order in Fe-Cr alloys are analyzed, and it is concluded that they are conflicting and that models taking into account the short-range order should be developed. The results of quantum-mechanical calculations of the average magnetic moment for ferromagnetic (FM) iron as a function of the volume are analyzed and used to introduce a concept of partial magnetic moments of the iron atoms located in the first four coordination spheres (1–4 CS). The values of these moments are calculated. The concept of the partial magnetic moments of iron atoms agrees qualitatively with the experimental data on the spin-density anisotropy of the bcc lattice of pure iron. This concept is used to formulate a three-sublattice model for binary FM alloys of Fe-M systems (M is an alloying paramagnetic element). An extended cell whose sites contain 8 bcc cells and 16 atoms per cell and that is isomorphic to a DO3-type crystal lattice is considered. A functional is constructed for the internal energy on the extended 16-atom cell. The dimensioned factor is taken into account in a self-consistent manner, by the expansion of the interaction energies of atoms of both components located in different CSs in the atom displacement with respect to ideal lattice sites. The dimensioned factor is taken into account in the three-sublattice model to obtain a set of equations of state for bcc FM binary iron-rich alloys in a 1–3 CS approximation. The obtained estimates demonstrate that the anisotropy of the distribution of magnetic moments in the bcc iron lattice is responsible for the appearance of a short-range order in bcc FM iron-rich Fe-Cr (V, Mo) binary alloys.

Stringent test of the statistical quasiclassical trajectory model for the H3+ exchange reaction: A comparison with rigorous statistical quantum mechanical results

A complete formulation of a statistical quasiclassical trajectory (SQCT) model is presented in this work along with a detailed comparison with results obtained with the statistical quantum mechanical (SQM) model for the H+ +D2 and H+ +H2 reactions. The basic difference between the SQCT and the SQM models lies in the fact that trajectories instead of wave functions are propagated in the entrance and exit channels. Other than this the two formulations are entirely similar and both comply with the principle of detailed balance and conservation of parity. Reaction probabilities, and integral and differential cross sections (DCS's) for these reactions at different levels of product's state resolution and from various initial states are shown and discussed. The agreement is in most cases excellent and indicates that the effect of tunneling through the centrifugal barrier is negligible. Some differences are found, however, between state resolved observables calculated by the SQCT and the SQM methods which makes use of the centrifugal sudden (coupled states) approximation (SQM-CS). When this approximation is removed and the full close coupling treatment is used in the SQM model (SQM-CC), an almost perfect agreement is achieved. This shows that the SQCT is sensitive enough to show the relatively small inaccuracies resulting from the decoupling inherent to the CS approximation. In addition, the effect of ignoring the parity conservation is thoroughly examined. This effect is in general minor except in particular cases such as the DCS from initial rotational state j=0. It is shown, however, that in order to reproduce the sharp forward and backward peaks the conservation of parity has to be taken into account.

Rotationally inelastic collisions of LiH (X 1Σ +) with H: State-to-state inelastic rotational cross-section

The potential energy surface describing the interaction between the hydrogen atom and the rotating LiH molecule has been calculated very accurately and presented in our previous work (J. Mol. Struct. (THEOCHEM) 678 (2004) 11). The rigid rotor potential already fitted analytically and expanded in terms of Legendre polynomials, is employed here to evaluate the pure state-to-state rotational cross-sections over a range of energies of astrophysical interest varying from 10 to 3000 cm −1. An exact close coupling and CS calculation of the state to state rotational cross-section was done first in the range of small energy, from 10 to 100 cm −1. The comparison of both results shows the good agreement between them and leads to a generalisation of such calculation for higher energy using the CS approximation. We determine the state to state rotational cross-section for transitions from j=0, 1, 2, 3, 4 to all accessible final states j′, at total energy varying between 10 and 3000 cm −1. The presented cross-sections were found to be very large even for low energy and large Δ jj′ . Thus indicates the important probability to produce the heated LiH molecule in excited states by collisions with H and explain the strong anisotropy discussed in our previous paper.

Time-Dependent Wave Packet Study of the O + O2 (v = 0, j = 0) Exchange Reaction

Time-dependent wave packet calculations were carried out to study the O + O2 (v = 0, j = 0) exchange reaction on the Siebert−Schinke−Bitterova potential energy surface. Because of the presence of a deep well supporting quasistable ozone complexes, it is found that one needs to propagate wave packets up to 20 ps of time to fully converge the pronounced resonance structures in the total reaction probabilities. We calculated the total reaction probability for total angular momentum J = 0 for collision energies up to 0.6 eV, and the integral cross section for collision energies up to 0.4 eV under the centrifugal-sudden approximation. To assess the accuracy of the CS approximation for the reaction, we calculated fully converged cross sections up to a collision energy of 0.04 eV. It is found that (a) both fully converged and centrifugal-sudden cross sections are full of a resonance structure, although not as pronounced as for the J = 0 reaction probability, and (b) the centrifugal-sudden approximation can only be used to accurately calculate the thermal rate constant for the reaction, but not the integral cross section.

Quantum scattering studies of the Λ doublet resolved rotational energy transfer of OH(X 2Π) in collisions with He and Ar

Three dimensional potential energy surfaces for the collision systems OH(X 2Π)+He and OH(X 2Π)+Ar have been calculated using the coupled electron pair approximation (CEPA) and large basis sets. The asymptotically degenerate 2Πx and 2Πy states split into two states of 2A′ and 2A″ symmetry, respectively, when the C∞v symmetry is lifted by the approach of the noble gas atom. The average and half difference of the calculated points on the A″ and A′ potential energy surfaces were fitted to analytical functions, which were then vibrationally averaged. These potential energy surfaces have been used in quantum scattering calculations of cross sections for collision induced rotationally inelastic transitions. Test calculations showed that the cross sections obtained from exact close-coupling calculations (CC) and within the coupled states approximation (CS) are in close agreement for these systems, and therefore the CS approximation has been used in all further calculations. Rotational transitions with Λ doublet resolution show, within the same spin–orbit manifold and at low collision energies, a propensity to populate preferentially the e final levels in the F1(2Π3/2) state and an e/f conserving propensity in the F2(2Π1/2) state, while transitions between the two spin–orbit manifolds show a parity conserving propensity. For the v=2 vibrational level kinetic rate coefficients were calculated for a large range of temperatures. The calculated cross sections are in excellent agreement with recent measurements of Schreel, Schleipen, Epping, and ter Meulen.

OPACITY FUNCTION FOR ROTATIONALLY INELASTIC SCATTERING OF Ar+N2 SYSTEM IN IT'S LONG-RANGE POTENTIALS

A new method is proposed to deal with the rotationally inelastic scattering of A+BC collision system. The matrix element 0,l/?)t|R′> is obtained analytically, the centrifugal potential takes the CS approximation, the rotationally term in the total Hamiltonian is treated by the Fourier transform and the time-dependent Schrodinger equation is approximated by the SOD method. Using this method the opacity function for Ar+N2 scattering system is calculated. The results show that the method introduced in the present paper is more suitable than the CCWP method in calculating the transition for long-range Ar+N2 potentials. And the calculational efficiency of our method is higher than the latter.

Self-consistent-field theory for hard-sphere chains close to hard walls

A continuum theory for confined hard-sphere polymers is presented. Starting from fundamental relations and applying defined approximations, a constitutive relation for the conformation probability (analogous to the result in mean-field lattice theories) is developed. The main problem of hard-sphere correlations is attacked by two approximate methods: First, using the Carnahan–Starling equation of state and local volume fractions (CS). Second, by an extension of the lattice theory to spherical components with unequal volumes (LATT). The agreement with Monte Carlo simulations is good for both approximations at low densities, but becomes only qualitative at the higher concentrations. The CS approximation seems to be favored over the LATT approach at the higher concentrations when correlation becomes more important. Both free and grafted chains are treated. The influence of chain length, grafting density, solvent concentration, solvent chain length, and surface curvature on the segment distribution is investigated.

Variational calculation of integral cross sections for the reaction F+H 2→HF+H

Accurate reactive scattering calculations of integral cross sections for the reaction F+H 2 were carried out by utilizing the latest basis contraction techniques developed for the S matrix Kohn variational method. Convergence with respect to increasing the number of K states ( K max) for high J calculations were tested. Calculations using K max2 gave accurate integral cross sections. Computed cross sections using variational CS approximation ( K maxO) were also shown and are correct semi-quantitatively

Spherical wave close coupling wave packet formalism for gas phase nonreactive atom–diatom collisions

In this paper we discuss the use of the total angular momentum representation in the close coupling-wave packet (CCWP-J) method for solving the time dependent Schrödinger equation for inelastic, nonreactive gas phase atom–diatom collisions. This enables the wave packet propagation for the relative motion to be reduced from three dimensions to one. The approach utilizes a close coupling expansion of the wave packet into subpackets labeled by quantum numbers for total angular momentum J, z-component of angular momentum m0, rotor angular momentum j, and orbital angular momentum l. The number of coupled subpackets is less than the number for the plane wave boundary condition CCWP method when J<jmax and they are equal when J≥jmax. The present method requires solving for the time evolution of such coupled subpackets for 0≤J≤jmax +lmax, where lmax is the largest orbital angular momentum for which significant scattering occurs. However, the number of grid points required in the fast Fourier transform portion of the evolution of the wave packet will be far fewer since only a 1D FFT transform is required in the present version of the CCWP-J. All the other attractive features of the CCWP method are common to both the total angular momentum and plane wave representation versions of the CCWP; namely, results are obtained over the range of energies included in the initial packet, the labor scales as the number of rotor states squared, and standard approximation methods may be used in conjunction with the formalism. We also present the l-labeled coupled states or centrifugal sudden wave packet (CSWP) formalism as an example approximate version of this approach. The CCWP-J method is illustrated by application to a model atom–diatom collision problem. The extension to treat collisions involving a vibrating rotor is given in an Appendix. Finally, we compare features of the CCWP-J and CSWP with standard close coupling, CS approximation, and the plane wave basis CCWP methods.

On the choice of phase in the CS approximation: Integral equation approach

As usually presented, the centrifugal sudden approximation examines how each individual angular momentum component of the scattering wave function and scattering matrix element is to be estimated. Here the three-dimensional structure of the scattering wave function is emphasized and any decisions about associating exact and approximate partial waves with correcting phase factors are put off till the implications on the three-dimensional wave function have been made clear. It is found that no correcting phase factor needs to be applied to the CS S matrix when estimating the exact S matrix, while at the same time, the asymptotic behavior of the exact and approximating wave functions has the same form. Direct and operator methods of estimating the transition matrix confirm this conclusion. A suggested modification of how the sudden approximations are to be implemented allows the weak potential limit of the resulting scattering amplitudes to reduce to the Born approximation.

Electronic fine structure transitions and rotational excitation in NO rare gas collisions

The excitation of rotationally cold NO in the ( j=1/2,2Π1/2) state by collisions with different rare gases is studied for multiplet conserving and multiplet changing transitions. In the crossed beam experiment we use jet cooling to prepare the cold NO and LIF to measure the state distribution of the collisionally excited NO in the scattering center. The measured integral state to state cross sections are compared to theory. The calculations are performed in the CS approximation and based on recent advances in treating open shell molecules. For multiplet conserving transitions the agreement between experiment and theory is good. Although for the multiplet changing collisions the general structure is in qualitative agreement, the strength of the Ω=1/2→3/2 transitions is underestimated in the theory.

A coupled states distorted wave study of the O(3P)+H2 (D2, HD, DH) reaction

In this paper, the coupled states distorted wave (CSDW) method is used to study the quantum reactive collision dynamics of O(3P)+H2 → OH+H and its D2, HD, and DH counterparts. The potential surface used is the sum of a LEPS potential (due to Johnson and Winter) and a correction factor which raises the barrier for H+OH → HO+H exchange to a realistic value. Full basis set convergence of the CSDW transition probabilities is established at low energies where tunneling dominates the dynamics, which means that the calculated cross sections should be exact except for errors introduced by the CS approximation, and the latter are expected to be less than 30%. The results presented for all four isotopes include: reaction probabilities as a function of energy E and total angular momentum J, total and state to state integral cross sections (including an analysis of product state distributions), and thermal and state resolved rate constants. Comparison of the results with those of several previous dynamical calculations on the same or similar surfaces is made and the accuracy of the approximations made in those calculations is assessed. For example, the product rotational distributions predicted by vibrationally adiabatic distorted wave theory are found to be quite close to what we calculate, although the absolute magnitudes of the cross sections are quite different. Comparison with the results of quasiclassical trajectory calculations indicates good agreement of the reactive cross sections well above the classical threshold, but not of the rate constants (because of tunneling) or of isotope ratios. Wigner corrected conventional transition state theory is very inaccurate in predicting rate constants, but a method which uses collinear exact quantum (CEQ) transmission coefficients to correct transition state theory does quite well. Variational transition state theory estimates of the rate constants and isotope ratios are also quite good, with the CSDW results generally bracketed by results obtained using the least action ground state (LAG) and small curvature ground state (SCTSAG) tunneling approximations. Comparison with experimental rate constants and isotope ratios is studied, and we find that the CSDW results are just outside the experimental error bars in all cases. The present results on the corrected Johnson and Winter surface are found to be slightly less accurate than the best estimates of rate constants and isotope ratios obtained using the modified POLCI surface of Walch and co-workers, suggesting that the latter surface is more accurate.

Vibrational relaxation of CO(n=1) in collisions with He

We present a full ab initio study of vibrational relaxation of CO(n=1) in collisions with 3He and 4He. The vibrational coordinate dependence of the interaction potential is calculated directly. Dynamical calculations are performed within the IOS and the CS approximation. The IOS cross sections agree very well with the corresponding CS results for both He isotopes, however, the ratio IOS/CS shows a different energy behavior for 3He and 4He, respectively. The CS rates for 3He agree extremely well with the experimental relaxation rates. Although only the reduced mass is changed for 4He the agreement with experiment is less favorable in this case. It is explicitly demonstrated that the often used procedure to extract the vibrational coordinate dependence of the interaction potential from a dumbbell fit of the rigid-rotor surface is wrong for He–CO. For 3He the corresponding cross sections are 3–10 times smaller than those obtained form the ab initio potential although the vibrationally elastic parts of the interaction potential are identical.

The Enskog theory for multicomponent mixtures. II. Mutual diffusion

We present a detailed description of the mutual diffusion coefficients of binary and ternary dense fluid mixtures of hard spheres, as given by the revised Enskog theory (RET) of van Beijeren and Ernst and the standard Enskog theory (SET) of Tham and Gubbins. The formulas for the diffusion coefficients [see part I of this series, J. Chem. Phys. 78, 2746 (1983)] involve the contact values of the equilibrium pair distribution functions and the chemical potentials, for which the Carnahan–Starling approximation is used. The formulas, which were obtained by making an expansion in Sonine polynomials, are evaluated up to the third order and the convergence of the Sonine polynomial expansion is discussed. Except at low densities, the SET cannot be used to describe diffusion in hard-sphere mixtures. We find, using the CS approximation, that the SET gives values for the direct mutual diffusion coefficients that become negative at very high densities. A study is made of the dependence of the RET mutual diffusion coefficients on a variety of the parameters that determine the mixture (density, composition, molecular masses, and diameters) and a summary of the trends found is given. In particular, the influence of the addition of a third component on the mutual diffusion of two other components is discussed.

Theoretical investigation of rotational rainbows in K + N2 and K + CO collisions

The appearance of rotational rainbows in K + N2 and K + CO collisions is investigated using ab initio SCF potential surfaces. The scattering calculations are performed within the semiclassical coupled-states (SCCS) approximation which is a combination of the semiclassical IOS and the CS approximation. We demonstrate that this method is ideally suited to study rotational excitation in impulsive collisions. The results are compared with the experimental data of Beck and co-workers and satisfactory agreement is obtained for K + N2. For K + CO we have the interesting situation that the more inelastic rainbow is nicely reproduced while the other occurring at lower transitions is poorly described. It is shown that failures of the potential surface are responsible for this artifact. The isotope effect observed experimentally in K + C???O and K + C???O collisions is accurately reproduced and an interpretation in terms of the potential surface is given.