Boltzmann Approach to Cascade Mixing
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BOLTZMANN APPROACH TO CASCADE MIXING Irwin Manning Naval Research Laboratory, Washington, DC 20375 ABSTRACT The Boltzmann transport equation is used to describe a beam of ions of atomic species 1 (1-atoms) bombarding a target modelled as an amorphous mixture of 2-atoms and 3-atoms. In a manner familiar in nuclear reactor theory, the method of characteristics is used to integrate the resulting transport equations. An exact expression for the migration flux J 3 of 3-atoms is obtained in closed form. This expression can be evaluated in terms of a power series in a distance parameter s. For the case of slowly varying density N 3 of 3-atoms, Fick's law, relating J 3 to the gradient of N 3 , is derived from this expression; it is given by the lowest order term of the power series. J 3 is shown to be proportional to the bombarding flux. Concomittantly, a closed expression for the mixing parameter in Fick's law is obtained, which allows a calculation of this quantity for realistic interatomic potentials. A model Kinchin-Pease displacement cascade is proposed, which is expected to allow a reasonable first approximation calculation of the mixing parameter in Fick's law. It is deduced that the mixing parameter will depend sensitively on the lattice displacement energy. This dependence constitutes a physical mechanism for chemical effect in cascade mixing, as well as for fluence and temperature dependence of cascade mixing. INTRODUCTION Haff and Switkowsky [11have proposed a model for cascade mixing in which atomic migration under ion bombardment derives from a mechanism somewhat similar to that of gaseous diffusion driven by concentration gradients. Matteson [2] has examined this process from the point of view of the theory of random flights. In the present work, we propose to discuss this process on the basis of the Boltzmann transport equation. REVIEW OF LINEAR TRANSPORT THEORY It is assumed that the target can be modelled as being amorphous, and that all atomic interactions take place through uncorrelated binary atomic collisions. It is further assumed that, in all atomic collisions, one of the atoms is at rest in the laboratory frame. Consider the case of a beam of atomic species I bombarding a target of atomic species 2 containing an impurity of atomic species 3, and let qi.(7,V,t)d 3 r d3v be the number of atoms of atomic species a (a - 1,2, or 3) located at time t in space d3r about 7 and having velocity d3v about V. Correspondingly, ja (-,V,t) . idSd 3vdt is the number of a-atoms (particles of atomic species a) with velocity d3v around V crossing an element of area dS with unit normal -d in time dt around t. Let (-" V;()v'p7(;,V',t) d3v'd 3vd 3r dt be the number of a-atoms emitted into the phase space region d3r d3v about 7and V in time dt due to collisions in which a 6-atom of velocity d3v' about V' collides with an a-atom. It follows from these definitions that [3]
If
oo,- (V' - V;7)d 3v - o'9(V';T),
(1)
where o-p is the total cross section for 3-atoms; that is, the reciprocal mean-free path of these atoms. Let
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