On the critical radius of reflected spheres

Abstract

In a recent issue of Kerntechnik, we studied the critical size of slabs and spheres for one-speed neutron transport by the PN method [1]. For the reflected sphere part of this article, we compared our results to those of the work by Sahni and Sjostrand (1997) by the integral transport method [2] as only available source in the literature. We showed that if we choose N 1⁄4 9 in the approximation, to generate close results to those of the integral transport method was possible. Two method results were given together in Table 7 of Ref. [1]. If we then plot the variation of the critial radius ðr0Þ as a function of the reflection coefficient (R), we get Fig. 2 of Ref. [1]. Recently, Sahni and Sjostrand (2003) extended their work in Ref. [3] and showed the expected variation of r0 with R. They tabulated their integral transport method results and claimed that our results were unphysical. We admit the fact that the problem was related to choosing the order of approximation in this specific point. Our recent study showed that we were facing one of the interesting shortcoming of the PN method. If we examine the variation of r0 vs. R, choosing low N is definitely a cause of accuracy loss, when R is small. Choosing large N, on the other hand, leads us to incorrect results when R is close to unity. For a given c, to examine the variation of the critical radius as a function of R with a fixed N is always a cause of the problem. We therefore thank the authors of Ref. [1] for bringing this point to the attention of the nuclear community. In support of our claim, we performed the following study. Without any modification or improvement we used our computer program which had generated the results of Ref. [1], to obtain the results given in Table 1. In order to do so we changed appropriately the order of approximation which was shown in parenthesis in the second and fourth colums of the table. It is shown that to generate the expected results, gradually lowering order of approximation is necessary as the reflection coefficient approaches to unity. As pointed out by Aronson (1984), the PN method results first improve then diverge as one increases the order N of the approximation in spherical geometry. However, we are here challenged even by a further drawback of this method. That is, to obtain correct r0 vs. R variation with any fixed N seems to be not possible in the PN method.