Abstract
By introducing a cutoff on the cumulative measure of a force, a unified kinetic theory is developed for both rigid-sphere and inverse-square force laws. The difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and -3 for inverse-square force law interactions. The quantities governed by γ include the specific reaction rates, kernels, collision frequencies, arbitrarily high orders of transition moments, arbitrarily high orders of Fokker-Planck expansion (also called Kramers-Moyal expansion) coefficients, and arbitrarily high orders of energy exchange rates. The cutoff constants are shown to be incomplete gamma functions of different orders. The widely used cutoff constant in plasma physics (usually known as Coulomb logarithm) is found to be exactly the zeroth order of the incomplete gamma function. The well known Arrhenius reaction rate formula comes from the first order of the incomplete gamma functions, while the negative first order can be used for fitting the fusion reaction rate between deuterium and tritium.
Highlights
The concept of collision strength, CF, has several applications in physics and chemistry
In the collision theory for chemical reaction rates,[1] it is assumed that only strong collisions are capable of chemical reactions; there won’t be a chemical reaction if CF is below a critical value called the activation energy
The most widely used cutoff variable for the collision operators is based on the scattering angle, θ, but the divergence problem can still exist after the scattering-angle cutoff has been made.[5]
Summary
The concept of collision strength, CF, has several applications in physics and chemistry. The purposes of this paper are: to give an explicit definition of CF; to show that, for rigid-sphere and inverse-square force laws, σ is a function of CF; and, to provide expressions for measurable quantities in terms of CF. The value of this approach is that it provides a consistent way of treating weak collision events. The new functions qn(k)(ymin, u) introduced here and the associated functions qn(k)(u)[22] allow the previous expressions to be extended and optimized These functions make it possible, for the first time, to calculate transition moments and energy exchange rates to arbitrarily high order. As an illustration of the usefulness of this approach, the specific reaction rates for Coulomb interactions are used to fit the fusion reaction rate between deuterium and tritium; satisfactory agreement is obtained with empirical data over a large temperature region, 5 kev ≤ T ≤ 1000 kev
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