Statistical theory of nuclear reactions

Theoretical reaction cross sections. Consider the reaction  

In light nuclei it is often possible to follow the decay of compound nuclei in resonance reactions for the spacing of levels can be wide compared with the resolving power of existing apparatus. For heavy nuclei, discrete levels in the compound nucleus usually can be detected at the present time only by the interaction of slow neutrons with nuclei. This subject is discussed elsewhere in this series. We are concerned here with compound nuclei in which a great many levels are excited. The theoretical discussion therefore necessitates a. statistical approach. Thus if the probability of decay of the compound nucleus C in the direction (7.1) is represented hy F B,b), and if Fq is the sum of the probabilities of all possible reactions, then the cross section for the reaction aA- bB is  

In this section we are interested only in the value of the reaction cross section a/, the probability ratio in (7.2) is discussed in Sect. 8. The reaction cross sections for neutrons and for charged particles are very different for the latter are affected by the Coulomb barrier. We shall deal with them separately. 

Reaction cross sections for charged particles. For charged particles the calculation of the transmission coefficients is much more complicated. The theory is sketched in the preceding article. If the Coulomb barrier is represented by B, where B=ZZ e jR, where Z and Z are the charges on the incident particle and on the target nucleus respectively, in units of the electronic charge, e, an approximate expression for Ti is  

Here Bi is the barrier height for particles with angular momentum I, 

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Statistical theory of nuclear reactions, sta-tistische Theorie der Kernreaktionen 17, 58, 105, 204, 212f. 

The microscopic theories of a decay can be divided into several groups. The statistical models were developed in analogy with the statistical theories of nuclear reactions. The a decay is a surface phenomenon, and reliable information on the nuclear surface is available mostly from... 

Library of Congress Cataloging in Publication Data. Christov, St. G. Collision theory and statistical theory of chemical reactions. (Lecture notes in chemistry 18) Bibliography p. Includes index. 1. Chemical reaction, Rate of. 2. Collisions (Nuclear physics) I. Title. QD502.K47 541.3 94 80-18112... 

The mode of decay then must depend only on the angular momentum and the energy contributed by the incident particle, i.e. the sum of its kinetic energy and its binding energy to the compound nucleus. This we may call the Bohr hypothesis. To calculate the probability of a particular reaction it is necessary to make an additional assumption. The simplest is to assume that the emission of the different constituents of the nucleus, neutrons, protons, and more complex bodies like a-particles, may be treated on an equal footing and that the probability of emission of a specified particle may be calculated on a statistical basis. This is the statistical assumption. The theory of nuclear reactions founded on this basis is called the statistical theory. 

This chapter concerns the energetics of charge-transfer (CT) reactions. We will not discuss subjects dealing with nuclear dynamical effects on CT kinetics. " The more specialized topic of employing the liquid-state theories to calculate the solvation component of the reorganization parameters is not considered here. We concentrate instead on the general procedure of the statistical mechanical analysis of the activation barrier to CT, as well as on its connection to optical spectroscopy. Since the very beginning of ET research, steady-state optical spectroscopy has been the major source of reliable information about the activation barrier and preexponential factor for the ET rate. The main focus in this chapter is therefore on the connection between the statistical analysis of the reaction activation barrier to the steady-state optical band shape. 

The essential nature of this relationship is clear statistical theories are based on a number of simplifying assumptions consistent with chaotic behavior. Specifically,2 any such theory must satisfy microscopic reversibility and the condition of zero relevance. The latter condition requires that the final state be independent of all initial conditions other than conserved quantities, that is, from the viewpoint of classical mechanics, that the system display the relaxation characteristic of chaotic motion. We note, for reference, that this minimal set of requirements allows for the construction of a large number of theories,3 the most prominant of which are the RRK.M theory of uni-molecular dissociation4 and the phase space theory of bimolecular reactions.5 Such theories have analogues, and in some cases their origins are in other areas such as nuclear physics.6... 

The first attempts to account for the behaviour of nuclear reactions were made with the use of an optical model. The failure of this model to account for the sharp resonances was the starting point for Bohr s compound nucleus hypothesis. In recent years, the inadequacy of the statistical theory which is based on it has renewed interest in the optical model, in particular, by Feshbach, Porter and Weisskopf. ... 

The emission of atomic and nuclear radiation obeys the rules of quantum theory. As a result of this, one can only talk about the probability that a reaction will take place or that a particle will be emitted. If one attempts to measure the number of particles emitted by a nuclear reaction, that number is not constant in time it has a statistical uncertainty because of the probabilistic nature of the phenomenon under study. 

The situation is quite different in bimolecular reactions with an activation energy (E >0). In particular, the "diatomic" model is certainly a bad approximation for radical-radical rebinding along a double bond in which the maximum of the effective potential (35 IV) lies near the saddle-point of the potential energy surface /141/, In this case no central forces govern the nuclear motion hence, the total angular momentum is not a constant, which means that the reaction cannot be rotationally adiabatic. Therefore, in this situation the statistical theory cannot correctly reproduce the results of the simple collision theory. 

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