Collision probability theory

Neuhauser, D., Baer, M., Judson, R.S. and Kotiri, D.J. (1990) A time-dependent wave packet approach to atom diatom reactive collision probabilities - theory and application to H -P H2 (J=0) system,, 7. Chem. Phys. 93, 312-322. 

Neuhauser D, Baer M, Judson RS, Kouri DJ (1990) A time-dependent wave packet approach to atomdiatom reactive collision probabilities theory and application to the H -F H2 (/ = 0) system. J Chem Phys 93(1) 312... 

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

For polyatomic molecules equation (18) is employed with the vibrational matrix elements modified as described above. For vibrational exchange, in equation (18) the single vibrational matrix element is replaced by the product of the squares of the matrix elements for each molecule. In general, the theory leads to collision probabilities which are in good agreement with experiment. 

If one hopes to develop detailed, predictive models of plasmas, microscopic information such as electron-molecule collision probabilities clearly is needed. But why obtain that information from theory The short answer is that experimental data are often absent and—given the difficulty of the measurements and the paucity of research groups conducting them—in many cases are likely to remain so indefinitely. A longer answer would add that, as both theoretical methods and computer hardware improve, theory is, at least in some areas, becoming competitive with experiment in terms of accuracy and time to solution. 

The conditions in which slow reactions of relative simplicity become accessible to precise measurement are not normally obvious, and have to be discovered. Even when they have been found, the phenomena which become apparent would be, in the eyes of many, little more than curiosities. Nevertheless, the development of any phenomenon in time has a fascination of its own, and the laws which it follows have an attraction to those interested in the quantitative aspect of things. The application of the so-called law of mass action led to the idea of reaction order, and provided a basis for a rational classification of slow chemical changes. Examples of reactions of different orders were sought and found, and indeed the existence of this convenient system of grouping not infrequently led to the oversimplification of the real relations. But the obvious molecular explanation of the order in terms of collision probability did not fail to arouse interest in the statistical theory of reaction rates. Even so, an unconscious tendency to compare chemical changes with phenomena of viscous flow or movement under friction persisted, terms such as chemical resistance were endowed with a fictitious significance, and catalysts were likened to lubricants. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

This section presents perturbation theory expressions and adjoint functions that correspond to the collision probability, flux, birth-rate density, and fission density formulations [see also reference (54)]. The functional relation between different first-order approximations of perturbation theory in integral and in integrodifferential formulations is established. Specifically, the approximation of the integrodifferential formulation that is equivalent, in accuracy, to each of the first-order approximations of the integral theory formulations is identified. The physical meaning of the adjoint functions corresponding to each of the transport theory formulations and their interrelation are also discussed. 

For the collision probabilities, transport theory gives a relatively simple answer [11]. Both Pc and Pio can be expressed with the help of the escape probability for a flat source, Po ... 

In this paper, we review three developments of perturbation theory which have considerably broadened the scope and role it has to play in reactor physics. These three topics are (1) computing characteristics in subcritical systems, (2) perturbation theory for collision probability methods, and (3) computation of ratios in critical systems. 

Conventional perturbation theory will be of no help in the first-order approximation, since not only would the adjoint equation be space independent in the unperturbed homogeneous problem, but no distinction is made between the collisions with different nuclides in respect to the subsequent contribution to the reactivity. Thus, conventional first-order perturbation theory shows no heterogeneity effect. But we shall see that the collision probability method leads to a importance that varies with nuclides in the homogeneous problem and admits the computation of a finite heterogeneity effect even to first order. 

So far, we have discussed the extension of classical perturbation theory to the collision probability model and the generalization of perturbation theory to arbitrary characteristics in steady-state but subcritical systems having a source. Even this is not the most general characterization we might require of a reactor. We are, in addition, concerned to compute arbitrary ratios in a critical system. These might (typically) be the breeding ratio, although several others are of interest. 

Parameter Calculation and Establishment of Relationships. The use of molecular modeling tools not being evident for nonexperts in the field, alternative tools can be applied for the assessment of values for rate coefficients, preexponential factors, and/or activation energies (22). Collision rate theory provides a simple description of a kinetic process. It counts the number of collisions, Zab, between the reacting species A and B in a bimolecular reaction or between the reacting species and the surface in the case of an adsorption step and applies a reaction probability factor, Prxn, to account for the fact that not every collision leads to a chemical reaction. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

In impact theory the result of a collision is described by the probability /(/, /)dJ of finding angular momentum J after the collision, if it was equal to / before. The probability is normalized to 1, i.e. / /(/, /)d/=l. The equilibrium Boltzmann distribution over J is... 

The curve marked ion-dipole is based on the classical cross-section corresponding to trajectories which lead to intimate encounters (9, 13). The measured cross-sections differ more dramatically from the predictions of this theory than previously measured cross-sections for exothermic reactions (7). The fast fall-off of the cross-section at high energy is quite close to the theoretical prediction (E 5 5) (2) based on the assumption of a direct, impulsive collision and calculation of the probability that two particles out of three will stick together. The meaning of this is not clear, however, since neither the relative masses of the particles nor the energy is consistent with this theoretical assumption. This behavior is, however, probably understandable in terms of competition of different exit channels on the basis of available phase space (24). 

This study has made no substantial improvement in the original theory of distorted waves. By evaluating the vibrational transition probability explicitly for the inverse (12-6-4) power potential, however, we were able to study some interesting aspects of the ion-molecule collisions. We summarize them here. 

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