NUCLEAR INSTRUMENTS AND METHODS 148 (1978) 585-589 ; © NORTH-HOLLAND PUBLISHING CO. STATIONARY AND NONSTATIONARY FLOW OF ULTRACOLD NEUTRONS ALONG A GUIDE TUBE IN THE FRAME OF THE DIFFUSION APPROXIMATION V. K. IGNATOVICH and G. I. TEREKHOV* Joint Institute for Nuclear Research, Laboratory of Neutron Physics, Dubna, U.S.S.R. Received 6 May 1977 and in revised form 13 July 1977 The present paper considers the cases of stationary and nonstationary diffusion of ultracold neutrons (UCN) in a guide tube with and without UCN losses in collisions with the walls and under different conditions of UCN reflection at the ends of the guide tube. 1. Introduction In the present paper an attempt is made to give a qualitative interpretation of the results of an ex- periment 1) on propagation of ultracold neutrons (UCN) along a guide tube. In the experiment the guide tube of length L had a shutter at a distance L1 from the UCN converter. At the moment t = 0 the shutter is opened and the counting rate is measured in dependence on the time at the guide tube exit. The experiment was performed in two ways. Firstly the section of the guide tube be- tween the shutter and the detector (L-L1 long) was a nickel coated glass tube with well polished walls, then this section was replaced by a nickel coated copper tube of the same quality as the rest part of the guide tube. The results turned out to be strongly dependent on the quality of polishing of the surface. In the present work an attempt is made to interpret these results with the help of the diffusion model in order to find out whether it is possible to employ such experiments for the determination of the probability of non-specular reflection and the Coefficient of losses per single collision with the wall. The diffusion model has already been applied in refs. 2-4 to the case of stationary propagation of UCN. In ref. 4 Monte Carlo calculations were used to consider the UCN transmission along the guide tube in dependence on the law of UCN ref- lection from the walls. Quantitative Monte Carlo calculations were also performed in refs. 5 and 6 and experimental determination of the diffusion coefficients is reported in refs. 3 and 7. Section 2 of the present paper is devoted to the consideration of stationary diffusion of UCN un- der different conditions of reflection at the neu- The Research Institute of Atomic Reactors, Dimitrovgrad, U.S.S.R. tron guide ends both with and without taking into account UCN losses per reflection from the wall. In section 3 a nonstationary problem correspond- ing to the experiment of ref. 1 is solved, the so- lution being also given for two cases: with and without taking into account the UCN losses in the walls. Comparing theory with experiment one may deduce that the diffusion model can be successful- ly applied to the investigation of the UCN prop- agation and for the determination from these ex- periments performed under appropriate conditions of such parameters as the probability of non-spec- ular reflection 8) and of UCN losses. 2. Stationary diffusion of UCN along the guide tube 2.1. DIFFUSION WITHOUX LOSSES IN THE WALLS AND WITHOUT REFLECTIONS AT THE GUIDE TUBE ENDS The propagation of neutrons can be described by a flux which under diffusion approximation can be taken as: J (z) = - D' dn(z)/dz , (la) where D is the diffusion coefficient, z is the dis - tance from the convertor taken along the guide tube axis, n(z) is the density of neutrons at the point z. In the absence of UCN losses in the walls n (z) sa- tisfies the following equation: d 2 n(z)/dz 2 = 0. (2a) In order to solve this equation boundary condi- tions are necessary. Under the assumption that neutrons leaving the guide tube do not return into it we have the following boundary conditions: J+(O) = n(O) v/4 - ½D dn(z)/dzl~=o Jo, ' J_(L) = n(L) v/4 + ½D dn z)/dzlz=r, = 0, (3a)