Momentum conservation collision

A minima] set of symmetric binary and triple collision rules conserving both momentum and particle number-... 

Whatever scheme we choose for treating head-on collisions, however, unless we also have triple collisions, spurious conservations laws are inevitable. In addition to the total particle numbei and momentum, it is easy to see that head-on collisions also conserve the difference in particle number in opposite directions-, that is, the difference in particle numbers in directions c and c +s- A simple way to fix this problem is to introduce a triple-collision of the form (cp, Ck+2, k+4) ( k+i, Ck+s, Ck+5)... 

It is easy to verify that multiparticle collisions conserve mass, momentum, and energy in every cell. Mass conservation is obvious. Momentum and energy conservation are also easily established. For momentum conservation in cell E, we have... 

Although this collision rule conserves momentum and energy, in contrast to the original version of MPC dynamics, phase space volumes are not preserved. This feature arises from the fact that the collision probability depends on AV so that different system states are mapped onto the same state. Consequently, it is important to check the consistency of the results in numerical simulations to ensure that this does not lead to artifacts. 

Collision The particles at all lattice sites undergo a collision that conserves the total number of particles and the total momentum at each site. The collision rules may or may not be deterministic. 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

This model consists of a one-dimensional chain of elastically colliding particles with alternate masses m and M. In order to prevent total momentum conservation we confine the motion of particles of mass M (bars) inside separate cells. Schematically the model is shown in Fig.4 particles with mass m move horizontally and collide with bars of mass M which, besides suffering collisions with the particles, are elastically reflected back at the edges of their cells. In between collisions, particles and bars move freely. 

In order to discuss the fundamental problems that are connected with the bound states in kinetic theory, we first restrict ourselves to systems with two-particle bound states only. The states of the two-particle system are determined by Eq. (2.12). Furthermore, we remark that to describe the formation of two-particle bound states by a collision, at least three particles are necessary in order to fulfill energy and momentum conservation. Thus, it is necessary to consider the quantum mechanics of three-particle systems. 

The molecular beam deflection method is shown schematically in Figure 3-17 (Buck et al. 1985). It is based on momentum transfer between clusters entrained in a molecular beam and rare gas atoms which are the constituents of a second molecular beam at 90° to the cluster beam. Collisions between the rare gas atoms and the clusters under single-collision conditions deflect a small percentage of the clusters from their original path. The maximum deflection angle depends on the mass of the cluster. For example, binary clusters may be deflected into a broad range of angles with a well defined upper limit set by the momentum conservation... 

Figure 3-17. Schematic of the molecular beam deflection method. Under single-collision conditions, collisions between clusters and He atoms deflect the clusters to angles whose maxima are determined by momentum conservation.

Figure 4 Results from classical trajectory calculations for in-plane scattering of Ar from Ag(l 11) with an incidence angle of 40° measured with respect to the surface normal. In the panels a and c results for the relative final energy Ef/Ei are shown, where E is the initial energy. Lines indicate the energy transfer computed with the cube model (parallel momentum conservation) and a binary collision model. In panels b and d angular distributions are shown. Calculations for 0.1, 1,10 and lOOeV are shown. The panels a and b are calculated for a zero temperature, static lattice panels c and d for Ts = 600 K. From Lahaye et al. [43].

Mass effects are also important in placing limits on product rotational excitation because of the constraints imposed by angular momentum conservation. The initial angular momentum in a reactive collision comes from the reagent rotational angular momentum and the orbital angular... 

Of course, due to momentum conservation in the elastic Coulomb collisions, one has ... 

It is quite possible to take as an extreme case that a molecule may stick to the walls on a collision, later to be evaporated. This will not violate the momentum conservation principle if the evaporating molecule carries away, on the average, as much momentum as the original molecule brought. 

Although the model is idealized, the principles of momentum conservation which were used to calculate the transfer properties apply equally well to more realistic models including quantum-mechanical treatments and the results are the same, the chief differences being in the effective molecular diameters and the possibility of inelastic collisions. 

The process of ion scattering is illustrated schematically in Figure 5. Because collision times are very short (10 to 10 s), the interactions can be approximated as elastic binary collisions [28] between the incident ion and a single surface atom (i.e., with an effective mass equal to the atomic mass). Diffraction effects are negligible. The basic equation in ISS, using energy and momentum conservation, is... 

Nevertheless, the given momentum flux formula (2.368) is not useful before the unknown average velocity after the collisions v( has been determined. For elastic molecular collisions this velocity can be calculated, in an averaged sense, from the classical momentum conservation law and the definition of the center of mass velocity as elucidated in the following. 

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