Particle collisions

This chapter deals with qnantal and semiclassical theory of heavy-particle and electron-atom collisions. Basic and nsefnl fonnnlae for cross sections, rates and associated quantities are presented. A consistent description of the mathematics and vocabnlary of scattering is provided. Topics covered inclnde collisions, rate coefficients, qnantal transition rates and cross sections. Bom cross sections, qnantal potential scattering, collisions between identical particles, qnantal inelastic heavy-particle collisions, electron-atom inelastic collisions, semiclassical inelastic scattering and long-range interactions. 

For electronic transitions in electron-atom and heavy-particle collisions at high unpact energies, the major contribution to inelastic cross sections arises from scattering in the forward direction. The trajectories implicit in the action phases and set of coupled equations can be taken as rectilinear. The integral representation... 

Although reduction or elimination of the repulsion barrier is a necessary prerequisite of successful flocculation, the actual flocculation in such a destabilized suspension is effected by particle—particle collisions. Depending on the mechanism that induces the collisions, the flocculation process may be either perikinetic or orthokinetic. 

Another type of flocculation results from particle—particle collisions caused by differential settlement. This effect is quite pronounced in full-size plants where large rapidly falling particles capture small particles that settle more slowly. 

Two and Three Particle Collision Operator for the FHP LG Let us look more closely at the form of the LG collision operator for a hexagonal lattice. Conceptually, it is constructed in almost the same manner as its continuous counterpart. In particular, we must examine, at each site, the gain and loss of particles along a given direction. 

Let fp x,t) be the probability that a particle moving into the direction cp is at the site positioned at x and time t, and consider, first of all, only two-particle collisions. A particle may scatter into the direction cp only if it comes from a binary collision involving... 

Handling the triple-collision gain-lo.ss term in the same way, the complete two- and three- particle collision term for the Cp direction can be written in the following form [hciss88b] ... 

Fig. 12.13 Particle collisions in Toffoli s deterministic diffusion model ([toff89],[toff90b]) see text.

There are two features of this example that are rather common. First, none of the steps in the reaction mechanism requires the collision of more than two particles. Most chemical reactions proceed by sequences of steps, each involving only two-particle collisions. Second, the overall or net reaction does not show the mechanism. In general, the mechanism of a reaction cannot be deduced from the net equation for the reaction , the various steps by which atoms are rearranged and recombined must be determined through experiment. 

Two-Particle Collisions.—One of the basic assumptions in the derivation of the Boltzmann equation is that the gas being described is sufficiently dilute so that only two-particle collisions are of importance. The mechanics of a two-body encounter will thus be described in order... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

Because the total collision number, Cj, is the sum of the number of particle-particle collisions, Cp, and the number of particle-wall collisions, C, we can define the particle-wall collision ratio, M, as below. 

It is clear that if S> Xj, particle-particle collision happens most probably and X(9,P) = Xh. If Sparticle-wall collision happens within the travel distance of <5/cos 9. Then, the local free path of the test particle, X, is the average oiX 9,0) over the whole ranges of 6 and j3. That is. 

A cell (a container with an impeller or an aeration device, capable of keeping the solids in suspension and providing aeration for frequent air bubble-particle collisions). 

A. Malevanets and R. Kapral, Mesoscopic multi-particle collision model for fluid flow and molecular dynamics, in Novel Methods in Soft Matter Simulations, M. Karttunen, I. Vattulainen, and A. Lukkarinen (eds.), Springer-Verlag, Berlin, 2003, p. 113. 

M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper, Low-Reynolds-number hydrodynamics of complex fluids by multi-particle-collision dynamics, Europhys. Lett. 68, 106... 

A. Lamuraand G. Gompper, Numerical study of the flow around a cylinder using multi-particle collision dynamics, Eur. Phys. J. 9, All (2002). 

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