Nuclear Instruments and Methods in Physics Research 219 (1984) 441-442 441 North-Holland, Amsterdam Letter to the Editor RANDOM WALK APPROACH FOR NEUTRON DIFFUSION WITH STRONG ABSORPTION Manuel O. CACERES §'* and Horacio WIO §§ Centro Atbmico Bariloche** and lnstituto Balseiro*** 8400.Bariloche, Argentina Received 20 June 1983 We have derived a new diffusion coefficient for situations of high absorption probability. This was done assuming a special random walk structure that leads to a Master Equation in which the principle of detailed balance cannot be fulfilled. In recent times, the importance of a stochastic ap- proach for studying non-equilibrium processes has been stressed. The most fundamental and simple of such processes is the random walk scheme. The Random Walk (RW) method has been extensively used for study- ing different kinds of problems: anomalous dispersion in amorphous solids [1], energy-loss and mass-transfer in heavy-ion collisions [2], electronic energy transfer in molecular crystals [3], migration of molecular excita- tions with traps and emission [4] and so on. The relation between RW and the Fokker-Planck equation (FP) in the continuous limit is very well known, as well as its connection with the diffusion processes [5]. As a matter of fact, it is possible to show that a Master Equation can be reduced to a random walk process by carrying out a mapping from stochastic variables to sites. The deterministic macroscopic equation describing the neu- tron transport process has the form of a linear Boltz- mann equation. This equation, under certain restrictive conditions, can be reduced to a neutron diffusion equa- tion [6]. The strongest restriction is that the absorption probability must be small. However for many neutron physics problems, the instances of high neutron absorp- tion probability is relevant. The relation of the neutron diffusion equation with a RW scheme in the limit of low absorption is already known [7]. In this Letter, we will show that by using a special structure function for the RW processes it is possible to consider situations of strong absorption, finding an equation equivalent to the diffusion equation § Centro Regional Bariloche, Universidad Nacional del Comahue. * Fellow Comisibn Nacional de Energia Atrmica. §§ Member of Carrera del Investigador Cientifico, CON- ICET. ** Comisibn Nacional de Energia At6mica. *** Comisirn Nacional de Energia Atrmica and Universidad Nacional de Cuyo. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) but with a new diffusion coefficient. For a given lattice, let Pn(S) be the probability that the random walker is at position $ after the n th step, then the set { Pn(S)} satisfies the recurrence relation: Pn+1(S ) = Ep( S - S')~n( S'), (1) s" where p(S) represents the translationally-invariant- jumping-probability that any step results in a vector displacement S, and has the normalization condition Ep( S- S') = I. (2) s The vector S is S=ESiei, (3) i where the components S i are integers, and e~ are the primitive translation vectors of the lattice. In what follows we will consider a bidimensional lattice, where only a three step process to first neighbours will take place. This means that the walker can only displace in the upper half plane (y > 0), and after leaving the x-axis it will never come back to it again. This gives the possibility of considering one dimensional diffusion processes on the x-axis, and to equate the displacement out of the axis to absorption processes. Therefore we have for the jumping probabilities t: (P([I,jl-[I-I,jl)=P ~p([l,j] [l+l,j])=q (4) P(S-S')=~p([I,j] [l,j-1])=r [p([l,j] [l',j'])=O otherwise, "t The jumping probability values will be taken as p = q .~s//2.~V, r=~a//~ T where Zs, "~a and ~T(=~s4-~a) are the scattering, absorption and total neutron cross sec- tions, respectively.