Resonance disadvantage factor

We have used here Eq. (10.131) for Pe., and Mr Vrpy When these expressions are compared with (10.38) and (10.39), it is seen that the NR approximation for leads rather easily to a statement in the standard form. Note, however, that because of the flat flux assumption the present derivation does not contain the resonance disadvantage factor fr. This quantity is customarily computed using a one-velocity model to represent the entire fast-neutron population. It is well recognized that this point of view is crude and somewhat unclear. On the one hand, when used with the NR approximation, it may be observed that only one collision is required to remove a resonance neutron from the vicinity of a resonance thus the spatial distribution would be very nearly uniform and isotropic. On the other hand, if the NRIA approximation is valid, then the absorptions are necessarily very strong and the use of a disadvantage factor based on diffusion theory is not well justified. For these reasons it has been omitted in this treatment as an unwarranted refinement not in keeping with the precision of the over-all model. ... 

The ordinary two region formula for the resonance absorption factor must be modified to include the fact that there are two metal surfaces exposed to good moderating materials, and that the disadvantage factor is rather more complicated than before. It may be shown that the proper generalization of for the three region problem is... 

The higher terms in the expansion of qi represent transients which die away much more rapidly than the fundamental, i = 1, Hence the fact that di (a constant) is a good approximation to the true average disadvantage factor means that the neutron distribution reaches its stationary shape very little below the top of the resonance band. The actual computation of the dj, or equivalently, the computation of A,, is performed by solving the characteristic equation (29). The form of this equation will depend on the particular function Zj, and these functions depend, in turn, on the geometry of the system. 

In the case of the resonance absorption, p2 is very far from 1 (about. 16) and the above approximation becomes very poor. It continues to be possible, however, to use (11a) for both (/C2 2) and ( o o) and obtain in this way a manageable expression for the disadvantage factor d. One obtains... 

Table 3 gives the disadvantage factors for the resonance absorption in an oxide sphere of density 6. 

Table 3. Disadvantage factor for the resonance absorption in oxide spheres of density 6...

The best method for analyzing the errors and possibly for correcting them would seem to be by a Monte Carlo analysis of selected cases. Methods for achieving this have been developed by Richtmyer, et al. [23]. Before such calculations have been made with sufficient accuracy, it would seem best not to introduce any special disadvantage factors. The successful application of the old method is attributable to the fact that the available data were analyzed in terms of a theory which used disadvantage factors, and thus led to larger values of the resonance integrals than later measurements. 

Either ratio, Ra or pa, can be related to the resonance escape probability or the conversion ratio. Microscopic (intracellular) flux distributions can be measured, and values of disadvantage factors and thermal utilisation can be obtained from them, b lattices of slightly enriched uranium rods in ordinary water it has also been possible to determine, indirectly, the migration area or age, and thus to obtain an experimental link between the macroscopic and microscopic properties of such lattices. 

The unit-cell method offers a relatively simple computational procedure for determining the various factors in /c . As already demonstrated, the calculation of these factors in each case reduces ultimately to the determination of the neutron spatial distributions for the entire energy range. For the thermal utilization we require the thermal-flux distribution and in particular the thermal-flux depression (i.e., the thermal disadvantage factor). The resonance-escape probability, on the other hand, requires a knowledge of the spatial distribution of resonance neutrons, and finally, of course, the calculation of the fast effect involves, essentially, the determination of the spatial distribution of successive (cascading) generations of fission neutrons. 

Corrections to the coupling-constant measurements because of quadrupolar relaxation by deuterium were employed, and one disadvantage of this indirect method is that the 3/Dccc values measured may be too small to be resolved, since they are a factor of 6.5 smaller than the corresponding 3/hccc values.92 In sugars, the 13C resonance of a deuterium-substituted carbon may disappear,93 due either to quadrupolar relaxation by deuterium, and/or to saturation caused by lengthening of the 13C 7, on replacement of dipolar relaxation by a proton with the weaker effect of a deuteron. 

---