Neutron Diffusion Theory

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Unformatted text preview: ttering medium. µ= The distance traveled after the second collision is: Z 2 = λs cos(θ1 ). cos(θ 2 ) where the azimuthal angle ( ϕ ) scattering is isotropic. Fig. 1: Geometry for the transport of a particle over three collisions. The average value of Z 2 is: 2 Z 2 = λs cos(θ1 ). cos(θ 2 ) ≈ λs cos(θ1 ) = λs µ 2 (3) In general, for (n-1) collisions: Z n = λs µ n , for n=0, 1, 2, … (4) Since Z n → 0 as n → ∞ , this implies that the neutron could be scattered in either direction with equal probability, at its next collision. This means the neutron forgets its original direction of motion after a succession of collisions which is characteristic of Markov chains, and having been carried a distance of a transport mean free path: ∞ λtr = z 0 + z 1 + L = ∑ λ s µ n n =0 2 = λ s (1 + µ + µ + L) = λs 1− µ (5) , ∀µ < 1 The transport cross section is defined as: Σ tr = 1 λtr = Σ s (1 − µ ) (6) If absorption is present, we generalize the definition of transport cross section to be: Σ tr = Σ a + Σ s (1 − µ ) = Σt − Σ s µ (7) where Σ t is the total macroscopic cross section. 3. FICK’S LAW AND THE DIFFUSION APPROXIMATION The neutron flux ( φ ) and current ( J ) are related in a simple way under certain conditions. This relationship between φ and J is identical in form to a law used in the study of diffusion phenomena in liquids and gases: Fick’s law. In Physical Chemistry, Fick’s law states that: “If the concentration of a solute in one region is greater than in another of a solution, the solute diffuses from the region of higher concentration to the region of lower concentration.” The u...
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This note was uploaded on 02/28/2013 for the course NUC 150 taught by Professor N/a during the Spring '13 term at University of California, Berkeley.

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