An energy-adjusted sudden approximation for inelastic scattering

Rotational energy transfer in collisions of nitrogen molecules has been studied theoretically, using the N2–N2 rigid-rotor potential of van der Avoird et al. [J. Chem. Phys. 84, 1629 (1986)]. For benchmarking purposes, converged close coupling (CC) calculations have been carried out to a total energy of about 200 cm−1. Coupled states (CS) approximation calculations have been carried out to a total energy of 680 cm−1, and infinite order sudden (IOS) approximation calculations have also been carried out. The CC and CS cross sections have been obtained both with and without identical molecule exchange symmetry, whereas exchange was neglected in the IOS calculations. The CS results track the CC cross sections rather well: between 113–219 cm−1 the average deviation is 14%, with accuracy improving at higher energy. Comparison between the CS and IOS cross sections at the high energy end of the CS calculations, 500–680 cm−1, shows that IOS is sensitive to the amount of inelasticity and the results for large ΔJ transitions are subject to larger errors. State-to-state cross sections with even and odd exchange symmetry agree to better than 2% and are well represented as a sum of direct and exchange cross sections for distinguishable molecules, an indication of the applicability of a classical treatment for this system. This result, however, does not apply to partial cross sections for given total J, but arises from a near cancellation of the interference terms between even and odd exchange symmetries on summing over partial waves. In order to compare with experimental data for rotational excitation rates of N2 in the n=1 excited vibrational level colliding with ground vibrational level (n=0) bath N2 molecules, it is assumed that exchange scattering between molecules in different vibrational levels is negligible and direct scattering is independent of n so that distinguishable molecule rigid rotor rates may be used. With these assumptions good agreement is obtained. Although the IOS approximation itself is found to provide only moderately accurate values for rate constants, IOS/ECS scaling methods, especially if based on fundamental rates obtained from coupled channel results, are found to provide generally good accuracy.

Test of the infinite-order sudden approximation for electron scattering at intermediate energy

A semiclassical study of vibrational and rotational transition in He+H 2 at suprathermal energies

We have recorded time-domain coherence decay and population decay signals for the pure rotational OCS J=0–1 transition broadened by argon and by helium using pulse methods based on Stark switching of the molecular resonance frequency. The signals have been analyzed to yield the pressure dependence of the two decay rates (1/T2 and 1/T1) and their dependence on the absorber (OCS) speed. For comparison, using a potential which has been used several times in the literature for these systems, IOS (infinite order sudden approximation) scattering results are given. These calculations give substantial agreement on the ratio of the speed dependence of the relaxation rate to the rate itself, but fail to predict the rates quantitatively, especially for helium, or the relationship between the argon and the helium results. Because the IOS is expected to be accurate for these systems, the helium potential is ruled out by these data.

The distorted wave infinite order sudden (DWIOS) approximation for the calculation of vibrationally inelastic molecular collision cross sections

A simple, direct derivation of the rotational infinite order sudden (IOS) approximation in molecular scattering theory is given. Connections between simple scattering amplitude formulas, choice of average partial wave parameter, and magnetic transitions are reviewed. Simple procedures for calculating cross sections for specific transitions are discussed and many older model formulas are given clear derivations. Total (summed over rotation) differential, integral, and transport cross sections, useful in the analysis of many experiments involving nonspherical molecules, are shown to be exceedingly simple: They are just averages over the potential angle of cross sections calculated using simple structureless spherical particle formulas and programs. In the case of vibrationally inelastic scattering, the IOSA, without further approximation, provides a well-defined way to get fully three dimensional cross sections from calculations no more difficult than collinear calculations. Integral, differential, viscosity, and diffusion cross sections for He–CO2 obtained from the IOSA and a realistic intermolecular potential are calculated as an example and compared with experiment. Agreement is good for the complete potential but poor when only its spherical part is used, so that one should never attempt to treat this system with a spherical model. The simplicity and accuracy of the IOSA make it a viable method for routine analysis of experiments involving collisions of nonspherical molecules.

Accurate quantum mechanical close coupling (CC) integral cross sections are reported for rotationally inelastic Ar–N2 collisions at room temperature using an empirical intermolecular potential. These cross sections are used to test the infinite order sudden (IOS) approximation and the results of several methods [classical trajectories (CT), full sudden (SA), generalized phase shift (GPS), and classical limit generalized phase shift (CGPS)] as reported in the preceding paper by Pattengill. The IOS approximation works very well for cross sections involving low rotational states. The CT method works well for all the cross sections examined, justifying the popular belief that the rotational motion of N2 is essentially classical at room temperature.

The infinite order sudden (IOS) approximation is extended to calculate rate constants for rotationally inelastic scattering between two H2O molecules. A modified version of the breathing sphere (BS) approximation, which has previously been shown to work reasonably well for N2+N2 rotationally inelastic scattering, is also extended to H2O+H2O scattering. The rate constants obtained using both approximations are used to determine Raman Q-branch spectral linewidths. In order to calculate such linewidths, it is necessary to sum the rate constants for any initial rotational level over all final inelastic rotational levels. While the IOS linewidths are in reasonable agreement with experiment over a range of temperatures, the IOS-BS linewidths are much too small.

Classical trajectory calculations were performed using the Ar-N2 potential surface of Patterngill et al. (1979) for temperatures of 77.3 and 300K. Diffusion and viscosity cross sections, the production and relaxation cross sections governing the viscomagnetic effect and cross sections for both depolarised Rayleigh scattering and the rotational relaxation time were calculated. Comparison was made with classical energy-sudden (ES), classical infinite-order sudden (MM) and quantal infinite-order sudden (IOS) calculations. For both the viscosity and diffusion cross sections the CT and the ES, MM and IOS calculations agree within 3%, except for the IOS results at 77.3K. For the relaxation and DPR cross sections the IOS results are found to differ by about 40-80% from the CT results. The ES result for the DPR cross section is about 20% larger than the CT result, while for the production and relaxation cross sections it is very poor. The authors suggest that the main reason for these discrepancies is the breakdown of the sudden approximation for the non-sudden collisions always arising in thermal averages.

Bearing fault diagnosis is of utmost importance in the maintenance of mechanical equipment. The collected fault vibration signal generally presents a modulated nature due to the special structure and dynamic characteristics of the bearings. This paper introduces a novel demodulation analysis technique via energy separation and local low-rank matrix approximation (LLORMA) to address this type of signal. The amplitude envelope and instantaneous frequency of the signal can be calculated via an energy separation algorithm based on the Teager energy operator. We can confirm the bearing faults by comparing the peak frequencies of the Fourier spectrum of the amplitude envelope and instantaneous frequency with the theoretical bearing fault-related frequencies. However, this algorithm is only suitable for handling single-component signals. In addition, the powerful background noise has a serious effect on the demodulation results. To tackle these problems, a new signal decomposition method based on LLORMA is proposed to decompose the signal into several single-components and eliminate the noise simultaneously. After that, the single-component signal representing the fault characteristics can be identified via the high frequency feature of the modulated signal. The analysis of the simulated signal and the bearing outer race fault signal collected from a bearing-gear fault test rig indicate that the proposed technique has an excellent diagnostic performance for bearing fault signals.

A new approximate method is presented for the rapid calculation of vibrationally–rotationally inelastic molecular collision cross sections. The method treats the coupling between the rotational states, belonging to the same vibrational level, using the sudden approximation. The vibrational states are treated using an adiabatic basis. With this basis it is found that a first order distorted wave approximation gives good agreement with fully converged close coupling calculations. Some previously developed techniques are extended to permit reliable analytic expressions to be formulated for the angle fixed S matrix elements which appear in the theory. The great computational efficiency of the method is based on these analytic formulas which eliminate the need for the numerical solution of differential equations and for the numerical evaluation of difficult integrals. The angle fixed S matrices which result are then treated within the framework of the recently developed sudden approximation to yield inelastic cross sections. The method is applied to the calculation of vibrationally–rotationally inelastic cross sections for the He+H2 system and comparisons are made with close coupling calculations which are fully converged in both rotational and vibrational basis. Analysis of the results shows that for the He+H2 system the sudden approximation fails significantly within the n=1 vibrational manifold at total energies below 1.1425 eV but the validity of the approximation clearly improves with increasing energy. The He+He2 system was used in the present work as it was the only suitable one for which exact calculations were available in the literature for comparison purposes. For heavier systems the sudden approximation will be valid at much lower energies.

Charge transfer due to collisions of ground state ${\mathrm{O}}^{3+}(2{s}^{2}2p{\phantom{\rule{0.2em}{0ex}}}^{2}{P}^{o})$ ions with molecular hydrogen are investigated using the quantum-mechanical molecular-orbital (QMO) coupled-channel method. The QMO calculations utilize $ab$ initio adiabatic potentials and nonadiabatic radial coupling matrix elements obtained with the spin-coupled valence-bond approach for a representative range of orientation angles and diatom internuclear separations. Vibrationally resolved cross sections for nondissociative single electron capture are obtained for energies between 0.1 eV∕u and 10 keV∕u for ${\mathrm{H}}_{2}$ in its ground vibrational level using the infinite order sudden approximation (IOSA). Two further approximations are considered in which the electronic radial couplings are assumed to be independent of the diatom stretching. In the first case, vibrational motion is taken into account by multiplying the electronic radial couplings by Franck-Condon (FC) ionization factors while in the second, vibrational motion is completely neglected. We refer to these two approaches as the vibrational sudden approximation (VSA) and the electronic approximation (EA), respectively. In the latter, the resulting cross sections for electronic transitions are multiplied by FC factors to obtain relative vibrationally resolved cross sections which are independent of the collision energy (the centroid approximation). Comparison with existing experimental data for total and electronic state-selective cross sections shows best agreement with IOSA and VSA, but discrepancies for EA. The triplet-singlet electronic cross section ratio reveals a departure at low collision energies from the statistical value.

Differential and integral cross sections for the rotational excitation of Na2 in collisions with He at 0.1 eV are calculated in the coupled states and the rotationally sudden approximation. The final rotational state distribution at fixed scattering angle exhibits an interference structure similar to the usual rainbow pattern in potential scattering, namely, a broad main rainbow maximum for high Δj transitions and ’’supernumerary’’ oscillations on the bright side of the rotational rainbow for lower Δj transitions. This rainbow effect is explained in the stationary phase limit of the sudden approximation by interferences between’’trajectories’’ from different atom–molecule orientations. It is shown that the oscillations in the rotational distribution at fixed angle are equivalent to the oscillations vs scattering angle for a single rotational transition as reported by Schinke [Chem. Phys. 34, 65 (1978)]. The sudden cross sections are in satisfactory accord with the coupled-states results, when the average rotational state ? is choosen to reflect energy conservation.

Vibrationally and rotationally elastic and inelastic collisions between He and I/sub 2/ in the total energy range of 0.044-0.5 eV are treated under the vibrational/rotational infinite-order sudden approximation (VRIOSA). In this method, one simultaneously makes the vibrational energy sudden, rotational energy sudden, and centrifugal sudden approximations and generates a completely uncoupled set of equations. The solution of these equations is discussed and the integral and differential cross sections, the integral and differential cross sections summed over final rotational states, the integral cross sections summed over final rotational and vibrational states, and a type of cross section which recently has been measured experimentally are presented and compared with vibrational close-coupled infinite-order sudden approximation results. The dependence of the accuracy of the VRIOSA upon the choice of the arbitrary vibrational quantum number parameters, n vector, is also investigated. Setting n vector equal to the final vibrational quantum number gives good results for the integral and experimental cross sections, while picking n vector equal to the average of the initial and final vibrational quantum numbers gives good results for the differential cross sections. The present comparisons involving a full three-dimensional system and quantities, such as the differential cross section, which are sensitive tomore » the phase of the S matrix extend earlier, more limited studies and show that the VRIOSA should be considered convenient and reliable for light atom-heavy diatomic molecule collisions at total energies up to a few electronvolts.« less

An exponential form of time-dependent perturbation theory (the Magnus approximation) is developed for rotationally inelastic scattering. A phase-shift matrix is calculated as an integral in time over the anisotropic part of the potential. The trajectory used for this integral is specified by the diagonal part of the potential matrix and the arithmetic average of the initial and final velocities and the average orbital angular momentum. The exponential of the phase-shift matrix gives the scattering matrix and the various cross sections. A special representation is used where the orbital angular momentum is either treated classically or may be frozen out to yield the orbital sudden approximation. Calculations on Ar+N2 and Ar+TIF show that the theory generally gives very good agreement with accurate calculations, even where the orbital sudden approximation (coupled-states) results are seriously in error.