General theory of elastic moderation in infinite homogeneous media

A formula that allows the exact determination of the non-trivial root $\text{ξ}{}^{\mathrm{\ast }}$ arising in the solution $\text{q(u) = ξ}{}^{\mathrm{\ast }}\text{(u)F(u)}$ to the slowing down equation is given. A general approximation scheme for obtaining approximations to $\text{ξ}{}^{\mathrm{\ast }}$ is derived. The classical approximations of Fermi and of Greuling and Goertzel are the two lowest order approximations of this scheme. A method of consistent approximation, as opposed to the usual inconsistent method of which the Fermi and GG are particular examples, is also introduced. It is shown that this consistent approach greatly improves on the inconsistent one. A new physical interpretation of $\text{ξ}{}^{\mathrm{\ast }}$ is discussed.