A study of an integral equation arising in the moderation of neutrons

A necessary and sufficient condition connecting the soiirce and the scattering kernel is established under which the collision denjBity will be constant for all values of the lethargy. The exponent in the exponential form of the collision density is determined explicitly, for small constant absorption, by making use of infinite determinants. The validity of obtaining approximate solutions of the slowlngdown equation by the process of successive iteration is examined and proven to be universally applicable. The rate of convergence of the iterative procedure is shown to be intimately connected with the "Spinney condition". The iterative procedure is shown to be rapidly convergent when this condition is satisfied but the marked change in the neutron flux distribution below the resonance centre, when the condition is not satisfied, can result in very slow convergence. The iterative procedure of Goldstein and Cohen is extended to a third iteration and the behaviour of the first and second iterations, when the l/E dependence of the asymptotic flux is retained, is examined. The exact solution of the slowing-down equation is formulated and it is shown that this exact solution can be expressed as the successive iteration on the narrow resonance approximation.