The neutron distribution in an infinite medium around an infinitely long black cylinder is investigated. The medium satisfies the conditions typical for Milne's problems. By the method of expanding the distribution function in terms of the spherical harmonics, we transform the Boltzmann transport equation for our case [eq. (1)] into an infinite set of ordinary differential equations for the infinite number of coefficients of expansion feq. (10)]. We obtain approximate solutions-P1, P3, and P5 approximations [expressions (21), (25)and (26)] by retaining only the first two, six and twelve terms of the expansion.Numerical results of the calculation of the extrapolated length λ for various values of aare shown in Table 2 and plotted in the figure. Both λ and a are expressed in units of l-themean free path of neutrons in the medium.For comparison, we plotted also the curve given by Davison (curve D in our figure). His curve is based on his approximate solutions of the Peierls integral equation for the two limit cases a?1 and a?1 and an interpolation for intermediate values of a according to the result of P3 approximation. It appears from the figure that for large a the result of P5 approximation is already very near to the curve D and that it seems more reasonable to lower a little the part of the curve D near a=1 in order to conform better with the tendency of the curve P5 (as, e.g., shown by the dotted line).In the Appendix, some relations concerning spherical harmonics needed in the text are obtained.
下载:
