Reactor general ------ equation—Boltzmann

The generalized perturbation theory expressions presented in this section for systems described by the homogeneous Boltzmann equation (excluding Section V,B,2) are in the form proposed by Stacey (40, 41). Had we assumed that the overall alteration in the reactor retains criticality, we would have achieved the Usachev-Gandini version of GPT. Stacey s version is often associated (41, 46, 48, 62) with the variational perturbation theory as distinguished from the GPT of Usachev-Gandini. Does the variational approach provide a different perturbation theory than the GPT derived (35,39) from physical considerations Is one of these versions of perturbation theory more general or more accurate than the other What does the term GPT stand for ... 

Only when a complete description including all seven variables is required is it necessary to solve the Boltzmann equation in all its generalities, Frequently, simplifying assumptions and limiting conditions can be imposed which reduce the integrodifTerential equation to more tractable form. Thus much of the subject of reactor analysis is devoted to the development and the application of simplified analytical models which define, within the limits of engineering needs, the nuclear characteristics of the reactor complex. 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the... 

This equation relates the reaction rate (cross section) of a material at the temperature to its cross section at a lower temperature Tn- As in all the preceding analyses, it is assumed that the nuclei are distributed according to the Maxwell-Boltzmann relation (4.198). The above equation may be used then to compute the cross-section curves for a reactor operating at any temperature from the known cross-section data of the reactor material which have been determined at some temperature Tn. Note that U = Tn then Eq. (4.233) reduces to an identity [see also (4.225)]. In general, the indicated integration must be carried out in detail. There is one special case, however, which leads to an especially simple result. If the measured cross-section curve varies as l/v, which is the case for many absorbers in the low-energy range, then it is easily shown that the reaction rate is independent of the moderator temperature. For example, if we take aa v) = Co/y, where Co is some constant, then from (4.233)... 

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