Identical particle collisions

It is important to note that the situation simplifies enormously for the case of i identical particle collisions, that is, when B=A [239], Specifically, consider yivfl 

For the case of A + A collisions, this condition is always satisfied. 

This scenario opens up a wide range of possible experimental studies of control bimolecular collisions. Specifically, we need only prepare A and A in a control superposition of two states [e.g., by resonant laser excitation of / A(1))] to produf superposition with r/ A(2)), direct them antiparallel in the laboratory, and vary t coefficients in the superposition to affect the reaction probabilities. Control -originates in quantum interference between two degenerate states associated wiili, r the contributions of 0A(1)) A (2) and I M2))I0A-(1)). This is accompanied. two uncontrolled scattering contributions corresponding to the contributions ol jj 

I0aO))I0a (1)) and I a(2))I0a (2))- Control is achieved by varying the four coefficients ah bj, i = 1,2. Stimulated rapid adiabatic passage (STIRAP) [245, 246] to be discussed in detail in Section 9.1, provides one choice for such state preparation. 

The control approach described above can be generalized to a superposition of N levels in each of the two A and A reactants. Specifically, choosing all kA = kA and with kf = — kA we have 

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Figure 2. Suppression of identical particle collisions. Full squares measured scattering cross-section for Beliaev damping as a function of the excitation wavenumber in units of the inverse healing length. The assumptions of our analysis are tested using hydrodynamic simulations (dashed line), and found to agree with Beliaev damping theory (solid line) and the experimental data. Corrections observed in the hydrodynamic simulation take into account the full inhomogeneity and finite size of the experimental system, and validate the approximations of our analysis.

In order to study the decoherence of quasi-particles within BEC, we use Bragg spectroscopy and Monte Carlo hydrodynamic simulations of the system [Castin 1996], and confirm recent theoretical predictions of the identical particle collision cross-section within a Bose-Einstein condensate. We use computerized tomography [Ozeri 2002] of the experimental images in determining the exact distributions. We then conduct both quantum mechanical and hydrodynamic simulation of the expansion dynamics, to model the distribution of the atoms, and compare theory and experiment [Katz 2002] (see Fig. 2). 

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 highly asymmetrical particles, the probability of collision is greater than that predicted for identical particles. This may be understood by noting that the diffusion coefficient is most influenced by the smaller dimensions of the particles (therefore increased), and the target radius is most influenced by the longer dimension (also increased, relative to the case of symmetrical particles) see Equations (24) and (42). 

The second approximation that we will consider is the three-particle approximation. In the case that three-particle collisions are taken into account we must start with the more general expression (2.34). Using the identity... 

Conceptually similar results were demonstrated by Krutzer et al. [14], who measured the orthokinetic coagulation rate under laminar Couette flow and isotropic turbulent flow (as well as other flow conditions). Despite equal particle collision rates, significance differences were observed in the overall rates indicating different collision efficiencies (higher collision efficiencies were found under a turbulent flow regime). Thus, identical chemical properties of a dispersion do not determine a single collision efficiency the collision efficiency is indeed dependent upon the physical transport occurring in the system. 

To model the particle velocity fluctuation covariances caused by particle-particle collisions and particle interactions with the interstitial gas phase, the concept of kinetic theory of granular flows is adapted (see chap 4). This theory is based on an analogy between the particles and the molecules of dense gases. The particulate phase is thus represented as a population of identical, smooth and inelastic spheres. In order to predict the form of the transport equations for a granular material the classical framework from the kinetic theory of... 

For identical particles, oscillations show up in the total cross section which are sensitive to the repulsive part of the potential. They are produced by head-on collisions for which / = 0 is valid. Expanding the phase shift near this point, one gets for small /... 

The effectiveness of amphipathic polymer molecules in imparting steric stabilization can be understo< if a second identical particle is imagined to approach the one portrayed in Fig. 2.2. The stabilizing moieties must be mutually repulsive if the polymer is to impart stability. In these circumstances, the Brownian collision stresses the stabUizing molecules, which endeavour to escape from the stress zone. This escape can be effected either by desorption from the particle surface or by laterd movement over the particle surface... 

The interaction of electrons with matter is different from that of heavier charged particles for two reasons. One of them is the electron mass which is more than two orders of magnitude lower than that of the second lightest charged particle, the muon this makes photon radiation very important in the stopping power of electrons even at lower energies. The other reason is that at low energies, the interaction with shell electrons dominates and that is collision between identical particles, which has to be taken into account in the calculations. 

Photoassociation experiments involving alkali-metal systems have stimulated considerable interest in chemical reactivity in alkali-metal dimer-alkali-metal atom collisions at ultracold temperatures [14]. Rearrangement collisions in identical particle alkali-metal trimer systems occur without energy barrier, and recent studies have... 

In the case of aqueous suspensions of oxide materials, the stability is mainly determined by the van-der-Waals and double layer interactions (DLI) between the particles. The former is always attractive in the event of particle collision. Whether the DLI is attractive or repulsive depends on the signs of the surface charges, the absolute values of the zeta-potential, and the regulation capacity of the double layer. When all particles are identically charged, the DLI is repulsive. The most influential factors on the stability are the pH value, the total electrolyte concentration, and the valency of the ions. The pH range in which a binary suspension remains stable is typically different to the respective stability ranges of the two particle components. 

Interestingly, positronium is not stable. If the spins are aligned Ps annihilates in approximately 1.4 X 10 sec by emitting 3 photons[42]. If the spins are opposed 2 photons are emitted[43] in about 1.25 X 10 °sec. We assume that the collision process occurs on a shorter time scale than annihilation. Scattering energies will be kept below the three body break up, or ionization, energy and only the J = 0 partial wave will be examined. Further, since there are no identical particles spin will be ignored. 

Let us first consider some of the basic concepts that are needed to describe the collision of two ground state atoms. We initially consider the collision of two distinguishable, structureless particles a and b with interaction potential Vg R) moving with relative momentum k, where R is the vector connecting a and b. We will generalize below to the cases of identical particles and particles with internal structure. The collision energy is... 

This also works for the case of elastic scattering of identical particles in identical quantum states K aa aa must be multiplied by a factor of 2 to get the rate of momentum transfer (k scatters to kg k) since two atoms scatter per collision event. Gao ° has also described the formal theory for collisions of cold atoms taking into account identical particle symmetry. 

These reactants possess anisotropic reactivity the reaction occurs at a certain mutual orientation, which does not take place at each collision. In the first approximation, these reactants can be considered as spheres each of which has a small reaction spot. The reaction occurs when the spheres collide by their spots. The reaction occurs without an activation energy. The size of the spot in the form of a circumference on the sphere-reactant can be characterized by the angle q, the relative surface area of the spot on the sphere is equal to sin (0/2), and the probability of collision with the favorable mutual orientation of two identical particles is sin (0/2). This is the geometric steric factor = sin (0/2) or - sin (0A/2)sin (0B/2) if the sizes of the spots differ for A and B. After collision the particle-reactants exist near each other for some time and turn relatively to each other. The rate of turn of the particles depends, naturally, on viscosity because the coefficient of rotational diffusion depends on viscosity... 

The RMD simulations used dynamical data generated by three molecular dynamics runs of 10 collisions each. The runs involved 100 identical particles, in a box of side 16o, with periodic boundary conditions. The dynamical initial conditions were identical except for the assignment of particle velocity directions using a different set of random numbers in each run. If =... 

Robert Brown s observation that small pollen grains are in perpetual irregular motion on the surface of water laid dormant for nearly eight decades, before Einstein explained it as the result of particle collisions with solvent molecules (Fig. 11.6). Of course, even Einstein was not entirely certain that what he was modeling was what Brown had observed. In 1905, Einstein wrote It is possible that the motions to be discussed here are identical with the so-called Brownian molecular motion however, the data available to me on the latter are so imprecise that I could not form a judgment on the question. It was actually Jean Perrin who accurately measured Brownian motion and connected Einstein s theory to Brown s observations in 1909. 

The identity of target elements is established by the energy of the scattered particles after an elastic collision. The number of atoms per unit area, N, is found from the number of detected particles (called the yield, Y) for a given number Q of particles incident on the target. The connection is given by the scattering cross section a(9) by... 

Reactive scattering or a chemical reaction is characterized by a rearrangement of the component particles within the collision system, thereby resulting in a change of the physical and chemical identity of the original collision reactants A + B into different collision products C + D. Total mass is conserved. The reaction is exothemiic when rel(CD) > (AB) and is endothermic when rel(CD) < (AB). A threshold energy is required for the endothemiic reaction. 

Consider a one-dimensional lattice populated by particles moving either to the left or right with unit velocity. When two particles collide head-on at a site, they reverse direction (figure 12.13-a). If the particles are all identical, head-on collisions are indistinguishable from there being no interactions are at all (figure 12.13-b). 

Two identical flasks are each filled with a gas at 0°C. One flask contains l mol CO, and the other 1 mol Ne. In which flask do the particles have more collisions per second with the walls of the container ... 

This equation says that a nitrogen nucleus is composed of seven protons and seven neutrons. An alpha particle, which is identical to a helium ion, has two protons and two neutrons. A highly energetic collision fuses the two nuclei. The result is a rare isotope of oxygen with eight protons and nine neutrons. The leftover proton is ejected. And that proton is what Rutherford detected. 

Solitary waves, especially in shallow water, have been studied for many years[24]. They have the interesting property of interacting with other solitary waves and to separate afterwards as if there had been no interaction at all. This persistence of the wave led to the name soliton, to emphasize the particle-like character of these waves which seem to retain their identities in a collision. 

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