Neutron Diffusion Theory

Notice that the gradient operator turns the neutron

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Unformatted text preview: 3 Σ t2 ∂x 0 1 Σ s ∂φ 3 Σt2 ∂y 0 1 Σ s ∂φ 3 Σ t2 ∂z 0 (19) Substituting into Eq. 8, we get the expression for the current density after dropping the evaluation at the origin notation, since the origin of the coordinates is arbitrary: 1 Σ s ⎛ ∂φ ˆ ∂φ ˆ ∂φ ˆ ⎞ i+ j+ k 3 Σt2 ⎜ ∂x ∂y ∂z ⎟ ⎝ ⎠ 1 Σs 1 Σs ∇φ = − J =− gradφ 3 Σt2 3 Σt2 J =− (20) We define the “diffusion coefficient”: D= Σs 3Σ t2 (21) Thus Fick’s law for neutron diffusion is given by: J = − D ∇φ (22) It states that the current density vector is proportional to the negative gradient of the flux, and establishes a relationship between them under the enunciated assumptions. Notice that the gradient operator turns the neutron flux, which is a scalar quantity into the neutron current, which is a vector quantity. 4. LIMITATIONS OF DIFFUSION THEORY Fick’s law expresses the fact that if the gradient of the flux is negative, then the current density is positive. This means that the particles will diffuse from the region of higher flux to the region of lower flux through collisions in the medium. Fick’s law imposes some limitations on the problems solved, because of the inherent assumptions, and it becomes invalid, needing corrections, under the following conditions: 1. Closeness to boundaries: The derivation assumed an infinite medium. For a finite medium, Fick’s law is valid only at points which are more than a few mean free paths from the edges of the medium. This is so, since the exponential term dies off quickly with distance, and only points of a few mean free paths from the point where the flux is computed make a significant contribution to the integral. 2. Proximity to sources or sinks: It was assumed that the contribution to...
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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 Berkeley.

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