The Two-group Model

The analytical statement of the two-region system is as follows. Consider first the diffusion equations for the fast and thermal neutrons in the core  

The quantity which appears in (o) is to be chosen so that is a good estimate of the slowing-down density of fast neutrons out of the fast group at r. The source of fast neutrons is taken to be the neutrons produced by fissions caused by thermal neutrons vXf l 2 r). The parameters Di and 2) are the thermal-diffusion coefficient and thermal-absorption cross section in the core, respectively, computed in the usual way. The source term for the thermal neutron equation (h) is taken to be the number of neutrons slowing down out of the fast group, reduced by the number lost in actual absorption during slowing down. We define 

The parameter 2 corresponds to 2 for the core. Both of these quantities are called the fast removal cross sections. Note that the form (c) explicitly assumes a nonmultiplying reflector material, since a zero source of fast neutrons has been taken in the reflector. As in the core, and 25, are thermal properties of the reflector calculated in the usual way  

If we take the ratio of these two equations, the constants t i and 4 2 cancel out, and the resulting expression may be written 

A comparison of this relation and the appropriate criticality equation for this system which was obtained in Chap. 4, Eq. (4.274), wherein 6=1, reveals that the quantity pc may be correctly identified as the resonance-escape probability to thermal. This was previously indicated. 

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Because the time scale of the Raman scattering event ( 3.3 x 10-14s for a vibration with wavenumber shift 1000 cm-1 excited in the visible) is much shorter than that of the fastest conformational fluctuations in biomolecules, the ROA spectrum is a superposition of snapshot spectra from all the distinct chiral conformers present in the sample. Together with the dependence of ROA on chirality, this leads to an enhanced sensitivity to the dynamic aspects of biomolecular structure. The two-group model provides a qualitative explanation since it predicts ROA intensities that depend on absolute chirality in the form of a sin x dependence... 

One advantage of the bond polarizability theory is that, since it is based on a decomposition of the molecule into bonds or groups that can support local internal vibrational coordinates, it can be applied to idealized normal modes containing just a few internal coordinates and so can provide conceptual models of the generation of ROA by some simple chiral structures. Indeed, as mentioned above, the bond polarizability theory actually developed out of a synthesis of the two-group model and the inertial model, both of which have been applied in detail to a number of simple chiral structures 3 5). 

The distribution in the moderator is calculated,as in the two group model, by ELSSuming that a neutron is absorbed when it is thrown out of the resonance energy range. The absorption cross section for this process is... 

The second basically different approach to the multiregion problem is the two-group model. This model attempts to describe the distribution of neutrons in both space and energy by separating all neutrons into thermal neutrons and nonthermal (fast) neutrons. The spatial distribution of neutrons in each group is given by a one-velocity diffusion relation in much the same manner as in the multigroup treatment however, the... 

Our primary interest in this calculation is the determination of the critical mass of the hot clean reactor and the radial distribution of the fast and thermal flux throughout the core and reflector. An accurate analysis of this system must necessarily take into account the completely reflected cylindrical geometry shown in Fig. 8.216. However, since this would entail a somewhat involved calculation, we will approximate the actual configuration by an equivalent reflected sphere of the same composition. This will reduce our computation appreciably and yet not obscure any of the essential steps in the application of the two-group model. A study of the effect of the corners in the completely reflected cylinder will be deferred until the next section. 

The application of the two-group model to each of the elementary reactors in the iterative sequence described above requires the solution of the criticality equation (8.169). For this calculation it is convenient to write this relation in the form... 

For many practical applications, such as preliminary studies, the primary information required is a reliable estimate of the critical mass. Since the two-group model is well suited to this purpose, it is clear that a formulation of the method which eliminates a large portion of the computational labor would be most useful. Such a method has been devised by R. P. Feynman and T. A. Welton. The Fe5mman-Welton method draws directly from the results of the one-velocity integral theory. The method effectively separates the space and energy dependence of the neutron-flux function and describes the spatial distribution of each... 

The present treatment of the problem of computing the effectiveness of a control rod is carried out on the basis of two analytical models. In the application of the first, the one-velocity model, attention is directed toward displaying some of the elementary physical ideas involved in the general problem. The model is used here primarily for purposes of illustration it is not intended that it be considered an accurate tool for computing control-rod effectiveness. The second method utilizes the two-group model, and it is expected that the results obtained from this formulation will be useful for the solution of practical problems. 

We begin the analysis with a summary of the general solutions derived for the two-group model. The appropriate differential equations for the fast flux ( i) and the thermal flux in the multiplying region of the reactor are given by (8.147). In the present problem it is convenient to write the solutions for these functions in the form [cf. Eqs. (8.164)]... 

The problem of determining the criticality condition and the flux distribution in the two-group model for a ring of gray rods has been considered by Avery. The method used is a generalization of the Nordheim-Scalettar approach and may be applied when either the rods or the reactor, or both, are multiplying media. 

The two-group model is a fairly good representation of the critical equation for a heavy water system. Hence, the critical equation can be written as... 

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