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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)
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.”
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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.
- Spring '13