Collision Duration Research Articles

Nonlocal kinetic equation and simulations of heavy ion reactions

A kinetic equation which combines the quasiparticle drift of Landau's equation with a dissipation governed by a nonlocal and noninstantaneous scattering integral in the spirit of Enskog corrections is discussed, Numerical values of the off-shell contribution to the Wigner distribution, of the collision duration and of the collision nonlocality are presented for different realistic potentials. On preliminary results we show that simulations of quantum molecular dynamics extended by the nonlocal treatment of collisions leads to a broader proton distribution bringing the theoretical spectra closer towards the experimental values than the local approach.

Local wave theory of elastic-body collision: Plane problem in the ideal-liquid approximation

1. There are several well-known classical approaches to the investigation of elastic-body collision. In elementa~ impact theory, beginning with Newton's work, the basic parameter is the velocity recovery coefficient at impact, which is 1 for absolutely elastic bodies and 0 for absolutely plastic bodies. This approach cannot fully describe the internal state of the bodies. St Venant wave theory considers the longitudinal collision of elastic rods, i.e., one-dimensional objects; an advantage of the theory is the determination of the duration of collision, equal to twice the time for the compression wave to traverse the complete length of the rod. However, the one-dimensional stress----strain state determined in this way cannot be used for adc~iuatr description of the state of bodies of different shape. In addition, attempts to consider oblique or noneentral impact onthe basis of St Venant theory are evidently difficult. Classical local Hertz theory is based on the solution of a static contact problem regarding the action of a point force on the plane boundary of a halfspace, which is extended to the problem of the dynamic contact interaction of elastic bodies in direct central impact. Hertz theory, wifich is essentially quasi-static, is applicable when the contact time of the bodies in collision is considerably greater than the period of the lowest intrinsic oscillations of the bodies. Modification of the local theory by synthesis with St Venant theory has been proposed by Sears, Kil'chevskii, and others. Timoshenko has developed a theory of transverse impact at an elastic beam. Note that the assumption of elastic deformation of the colliding bodies limits the range of relative impact velocities that may be considered. At high velocities, the elastic limit of the material may be exceeded, and the behavior of the body must be described on the basis of more complex theological models, taking account of inelastic deformation. M. A. Lavrent'ev has proposed a hydrodynamic model of collision (interpenetration) of bodies at high velocity. It may be successfully used because the stress ellipsoid resembles a sphere in high-speed collision, i.e., the stress ellipsoid of a low-viscosity liquid. As a rule, numerical methods must be used to consider the collision of bodies with theologically complex behavior. In addition to these and several other approaches in modern continuum mechanics, for three-dimensional bounded by smooth surfaces, we may note an approach associated with the investigation of the collision (interpenetration) of bodies or the impact of a body on a medium in which the behavior of the body or medium is described by the equations of elasticity theory (or an ideal liquid in the case of collision at a liquid surface) and the collision problem is formulated, in the general case, as a nonsteady, often mixed, twoor three-dimensional problem, ff the motion of the bodies is specified only in the initial moment of collision, and their subsequent motion is determined in solving the problem, the boundary of the bodies' contact region is generally mobile and unknown in advance. This relatively complex formulation of the impact problem, which is free from the assumptions of classical appmacbes, has yielded a series of analytical (numerical--analytical) solutions by, among others, Grigolyuk and Gorshkov, Sagomonyan, Pomchikov, Galanov, Borodich, V. N. and V. V. Gavrilenko, Popov, and Tarlakovskii [ 1-3, 81. As a rule, the impact of a solid body or elastic shell on the surface of an infinite elastic or liquid medium is considered in this approach. Inthe present work, an approach to the collision of two elastic bodies such that at least their frontal surface is relatively smooth in the geometric sense (blunt bodies) is developed. The approach is based on research on the impact interaction of bodies of finite size with an infinite acoustic or elastic medium [3-8]. Its development takes into account that the depth of interpenetration of blunt bodies on impact is small in comparison with the clmracteristic dimension of the contact region and, consequently, the corresponding boundary conditions may be formulated at the unperturbed surface of the bodies. In addition, since the surface of a blunt body is only slightly distorted within the contact region, it may be regarded as plane. Finally, the

Extended impact theory of pressure broadening of spectral lines for van der Waals interaction potentials

We study the extension of the impact theory of foreign gas line broadening to the case of non-instantaneous collisions. As has been shown by previous authors, the effect of finite duration of collisions can be taken into account by adding to the usual Lorentzian profile a dispersion type contribution which introduces an asymmetry into the profile. We rederive this result by a new rigorous method and discuss the results in terms of the Weisskopf radius defined for one-term van der Waals interaction potentials. In this way, precise criteria for the validity domain of the theory can be established.

Behavior of one inelastic ball bouncing repeatedly off the ground

An experimental study of the behavior of one bead bouncing repeatedly off a static flat horizontal surface is presented. We observe that the number of bounces made by the bead is finite. When the duration between two successive bounces becomes of the order of the impact duration, the bead no longer bounces but oscillates on the elastically deformed surface before coming to rest. This transition is explained with a modified Hertz interaction law in which gravity is taken into account during the interaction. For each bounce, measurement of both the duration of collision and the restitution coefficient have been done. The effective restitution coefficient is essentially constant and close to 1 during almost all bounces before decreasing to zero when the impact velocity vanishes. This is due to an interplay between gravity and viscoelastic dissipation.

Formation of correlations and energy-conservation at short times

The formation of correlations due to collisions in an interacting nucleonic system is investigated. Results from one-time kinetic equations are compared with the Kadanoff and Baym two-time equation with collisions included in Born approximation. A reasonable agreement is found for a proposed approximation of the memory effects by a finite duration of collisions. This form of collision integral is in agreement with intuitive estimates from Fermi's golden rule. The formation of correlations and the build up time is calculated analytically for the high temperature and the low temperature limit. Different approximate expressions are compared with the numerical results. We present analytically the time dependent interaction energy and the formation time for Gau\ss{}- and Yukawa type of potentials.

Influence of the finite duration and efficiency of collisions on the orientational relaxation times of non-spherical molecules in the hindered rotation limit

The present work deals with the influence of the efficiency and finite duration of intermolecular interactions on the orientational relaxation times (ORTs) in the hindered rotation limit. To this end, a single-particle non-markovian Zwanzig-type master equation is used. The most general case of asymmetric top molecules is considered and a number of analytical results is derived. A possibility is demonstrated to extract non-trivial information about the peculiarities of molecular rotation and interaction by examining variation of ORTs and ratios of ORTs of different ranks vs. the angular momentum relaxation time. The collision efficiency is proved to influence these dependencies markedly. This effect should be necessarily taken into account in describing the orientational relaxation.

Kinetic theory of band shapes in molecular spectra of gases: Application to band wings

A kinetic theory of spectral band shapes is developed which accounts for the effects of line mixing and finite duration of collisions. The theory is based on a projection operator technique with a nonorthogonal metric. The roles and hierarchy of various approximations commonly employed are analyzed. It is shown that the relaxation operator must be defined before the initial correlations are suppressed so that the double sum rule for the matrix elements of this operator is conserved. This conservation is important for the calculations of the wings of band shapes. Rotovibrational band shapes for some specific cases of Raman and infrared absorption spectra of gases are considered. The expressions obtained relate the band wing profiles with the spectral density of molecular torque arising from binary collisions.

Collisional particle pressure measurements in solid–liquid flows

Experiments were conducted to measure the collisional particle pressure in both cocurrent and countercurrent flows of liquid–solid mixtures. The collisional particle pressure, or granular pressure, is the additional pressure exerted on the containing walls of a particulate system due to the particle collisions. The present experiments involve both a liquid-fluidized bed using glass, plastic or steel spheres and a vertical gravity-driven flow using glass spheres. The particle pressure was measured using a high-frequency-response flush-mounted pressure transducer. Detailed recordings were made of many different particle collisions with the active face of this transducer. The solids fraction of the flowing mixtures was measured using an impedance volume fraction meter. Results show that the magnitude of the measured particle pressure increases from low concentrations (<10% solid volume fraction), reaches a maximum for intermediate values of solid fraction (30–40%), and decreases again for more concentrated mixtures (>40%). The measured collisional particle pressure appears to scale with the particle dynamic pressure based on the particle density and terminal velocity. Results were obtained and compared for a range of particle sizes, as well as for two different test section diameters.In addition, a detailed analysis of the collisions was performed that included the probability density functions for the collision duration and collision impulse. Two distinct contributions to the collisional particle pressure were identified: one contribution from direct contact of particles with the pressure transducer, and the second one resulting from particle collisions in the bulk that are transmitted through the liquid to the pressure transducer.

Collision duration in the elastic regime

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Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν3 and 3ν3 bands of CO2 perturbed by Ar up to 1000 bar

We present high-density experimental and theoretical results on CO2Ar gas-phase absorption in the ν3 and 3ν3 infrared bands. Measurements have been made at room temperature for pressures up to 1000 bar in both the central and wing regions of the bands. A non-linear perturber density dependence of the absorption, clearly shown in the far wing, is attributed to the finite volume of the molecules. Furthermore, experiments show vibrational dephasing and narrowing effects. We have performed line-mixing computations based on the Energy Corrected Sudden approximation (ECS impact model). Significant discrepancies between experimental and calculated spectra appear when pressure increases. We then tested the influence of the finite duration of collision by using interpolations between ECS and quasi-static calculations, and we have evaluated the sensitivity of the band profiles to the interbranch mixing effects. Finally, an effective width is used in order to take other effects into account.

Time domain modeling of spectral collapse in high density molecular gases

In many cases, the widely used matrix inversion approach to describe the spectral interference in collisionally perturbed molecular spectra is not feasible if the particular molecular interactions do not allow the sudden impact approximation (infinitely short collision duration). To overcome this problem, we present a time domain model that describes collisional broadening and narrowing phenomena without requiring the sudden approximation. The key element of the model is a Monte Carlo type sampling process to quantify the temporal autocorrelation of the molecular dipole moment. The spectrum is then obtained numerically via fast Fourier transform. The model does not require a frequency-dependent relaxation operator; the finite collision duration is simply an adjustable parameter in the time domain process. Our approach, which is generally applicable to any set of transition lines, is derived from concepts of both conventional quantum-mechanical and semiclassical theory of line interference. Coherent transfer effects from rotationally inelastic collisions are described as randomly occurring events which affect frequency, amplitude, and phase of the sampled oscillation. Effects of vibrational dephasing are included as well. To demonstrate its feasibility, we apply the model here to the 2.7 μ absorption spectrum of carbon dioxide diluted in high density air (ρ=43–485 amagat, T=297–754 K). The successful modeling of the experimental data, especially the full collapse of P and R branches at ultrahigh densities, accounts for interbranch mixing and for incoherent effects. The calculations make extensive use of the new Hitran (HITEMP) molecular database. Results include revised estimates for the collision duration of CO2 with nitrogen and oxygen at room temperature.

Two types of ball-effector interaction and their relative contribution to soccer kicking

The study attempts a closer examination of the ball-foot interaction at the instant of soccer kicking using high speed videotaping. This allows a detailed biomechanical analysis of the collision. A central question is whether the ball-foot interaction can be modeled as an elastic impact, or as a mixture of ‘impact-like’ and ‘throwing-like’ patterns. The findings prove that the theory of conservation of momentum alone is not adequate for modeling kicking: additional momentum and mechanical energy is supplied to the system during the collision phase because of the muscle work. This is due to the relatively long duration of collision (about 16 ms) and the large ball-foot displacement (about 26.0 cm). More than 50% of the ball's speed and at least 30% of its kinetic energy is imparted to the ball without any contribution of the potential energy of the ball deformation.

Drift-velocity degradation caused by an electric field during collision in one-dimensional quantum wires.

Electron transport in quasi-one-dimensional quantum wires under high electric fields is investigated with the Monte Carlo method. The intracollisional field effect that is associated with the energy change of the electron due to the high electric field during the finite collision duration is taken into account. It is shown that the electron drift velocity, contrary to previous predictions made with semiclassical treatments, is greatly reduced under moderately strong electric fields (\ensuremath{\gtrsim} hundreds of V/cm). This holds true even if the electron confinement is not strictly strong in thick quantum wires so that multisubband scatterings are involved in determining the electron transport properties.

Line mixing in the infrared spectra of simple gases at moderate and high densities

A review of papers in the field of the line-mixing effect on IR absorption band shapes is presented, with most attention being paid to work carried out at the Molecular Spectroscopy Laboratory of St. Petersburg University. After general formulation of the problem, the empirical methods of rotational relaxation matrix evaluation are considered, and the strong collision model is discussed in more detail. The method of far band wing calculation, which takes into consideration both line mixing and the finite collision duration effect, shows the important role of line mixing in far wing shape formation. In the Markov approximation, line mixing is considered semiclassically. The classical distribution of the angular momentum alteration probability in collisions is used for reconstruction of Γ matrices which describe line broadening, line shifting and line mixing effects in the discrete rotational state representation. This method is shown to give acceptable results for the prediction of line mixing effects for central regions of Σ-Σ bands of diatomic and linear molecules.

Collision-duration time for optical-phonon emission in semiconductors.

The time required to emit an optical (polar and intervalley) phonon by a nearly-free electron in a semiconductor is evaluated using a nonequilibrium Green's-function formalism. The leading idea of the work is that the so-called "collision duration" is related to the time required to build up correlation between the initial and the final state, and then to destroy this correlation as the collision is completed. The use of the nonequilibrium Green's-function formalism gives us the possibility to evaluate explicitly the effects of the correlations in time. Our approach is based on two crucial assumptions: we build the self-energy from only the polarization field of the polar-optical phonon; that is, the self-energy is a function of a single time and position, and we introduce the electron correlation function between the initial and the final states, written in terms of a generalized less-than Green's function in the momentum variables. We derive an analytical expression for the probability for a carrier to end up in a final state k as a consequence of the emission of a phonon as a function of time. We find that the probability rises to the "Fermi golden rule" result within a few femtoseconds. If the total lifetime broadening of the initial state is comparable to the scattering time, the probability oscillates as it approaches the asymptotic value. For larger initial-state broadening (due to more scattering processes), these oscillations disappear.

Single cytoplasmic dynein molecule movements: characterization and comparison with kinesin

Cytoplasmic dynein is a major microtubule motor for minus-end directed movements including retrograde axonal transport. To better understand the mechanism by which cytoplasmic dynein converts ATP energy into motility, we have analyzed the nanometer-level displacements of latex beads coated with low numbers of cytoplasmic dynein molecules. Cytoplasmic dynein-coated beads exhibited greater lateral movements among microtubule protofilaments (ave. 5.1 times/microns of displacement) compared with kinesin (ave. 0.9 times/micron). In addition, dynein moved rearward up to 100 nm over several hundred milliseconds, often in correlation with off-axis movements from one protofilament to another. We suggest that single molecules of cytoplasmic dynein move the beads because 1) there is a linear dependence of bead motility on dynein/bead ratio, 2) the binding of beads to microtubules studied by laser tweezers is best fit by a first-order Poisson, and 3) the run length histogram of dynein beads follows a first-order decay. At the cellular level, the greater disorder of cytoplasmic dynein movements may facilitate transport by decreasing the duration of collisions between kinesin and cytoplasmic dynein-powered vesicles.

A classical trajectory study of O + O 2 collision

Classical trajectory calculations have been performed for the O + O 2 collision at different initial relative translational energies. Several modes of collision have been identified by the close examination of the plot of the number of trajectories on collision duration (τ)-the number of exchanges of the shortest interatomic distance (NESID) plane. The nature of each collision mode is explained in terms of collision dynamics. Also, the collision duration distribution shows peaks contributed by different collision modes.

Minimal separation distance in energy transferring collisions. Trajectory calculations

Classical trajectory calculations of toluene-Ar collisions were performed. The energy transferred in up, down and all collisions is reported as a function of the minimal distance, MD, of approach between the Ar atom and the nearest atom to it in the toluene molecule. There is a distribution of MD at which energy transfer takes place and most energy transfer occurs in the range 0.29–0.35 nm. Increasing the temperature shifts the average to lower values. The collision duration and the MD are independent. The average MD is 0.308 nm at 300 K and 0.299 nm at 1500 K. Supercollisions were found but no correlation could be made between their duration and their MD.

Theory of ultrafast phenomena in laser-excited semiconductors

The study of semiconductor carriers excited by femtosecond laser excitation has proved to be one of the most effective methods of characterization of semiconductors, as well as providing a probe for new physical effects. On the femtosecond time-scale, however, it is not expected that the semiclassical Boltzmann equation will be correct, and a more basic quantum mechanical formulation is required. The formulation of equivalent transport equations from the Schrodinger equation and its variants will be briefly reviewed. Results, using an ensemble Monte Carlo method, for electron-electron and electron-phonon interactions will be presented to show how these studies can be used to investigate the strength of the electron-phonon interaction and how the finite collision duration affects such studies.

Classical Cascade Calculations on the Analysis of Surface Bonding Structures by Secondary Particle Mass Spectrometry

AbstractThe application of secondary particle mass spectrometry to analyze the atomic bonding configuration on a surface was investigated with computer calculations of particle sputtering from an Ar‐bombarded surface of Ag{111}. In contrast to the prediction from the present sputtering theory, better resolved structure in the angular distribution was not observed for particles of high energy sputtered from the surface. These particles ejected to a wide angle and their angular distribution may not reflect well the atomic bonding geometry on the surface. The analytical capability of secondary particle mass spectrometry to determine surface structures can be significantly improved by selecting for detection only those particles that have prolonged collisions. The preferred directions of ejection and the relative sputtering intensity in the distribution of azimuthal angle between major open channels on the surface vary insignificantly with the duration of collision in the regime of long collisions. The results provide the first evidence that at small energies of sputtering the particles emitted from the surface may contain information about only the top two surface layers. Secondary particle mass spectrometry can thus be extremely surface sensitive for analysis of properties of solid surfaces.