A calculation of the neutron energy spectrum produced by a pulsed source in a heavy moderator, assuming a constant mean free path

The energy distribution of neutrons from a pulsed source in a moderator of mass number M ⪢ 1 is shown, on the assumption of a constant mean free path l, to be $\text{n(z) =}\text{const.}\text{×}\text{exp}\text{{}\text{1}\text{2}\text{(M + 1)ƒ}{}_{\mathrm{-1}}\text{(z) + ƒ}{}_0\text{(z) + 2(M + 1)}{}^{\mathrm{-1}}\text{ƒ}{}_1\text{(z) + …}}$ at energies small in comparison with the source energy. z = 1(M +1)/vt, with v the neutron velocity and t the slowing-down time. $\text{ƒ;}{}_{\mathrm{-1}}\text{(z), ƒ}{}_0\text{(z)}\text{and}\text{ƒ}{}_1\text{(z)}$ are given in integral form, together with analytical expressions valid near the maximum, and asymptotic expansions; detailed tables of the functions are also provided. It is demonstrated by numerical calculation that a knowledge of these three functions is sufficient to allow the neutron spectrum to be determined even in deuterium. The situation in a moderator composed of a number of different types of nuclei is considered. In this case the problem is solved by developing a method for solving integral and integrodifferential equations whose kernel K(x, y) is significantly different from zero only where |x − y|/|x + y| is very small.