Neutron Diffusion Theory

# Where tr is the transport mean free path defined

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Unformatted text preview: the flux is mostly from scattering collisions. Sources can be present. Because of the attenuation factor, however, a small number of the source neutrons will contribute to the flux if they are more than a few mean free paths from sources. 3. Anisotropic LAB system scattering: Isotropic scattering in the LAB system occurs at low energies, but is not true in general. However, Fick’s law is still valid with moderate anisotropy in scattering, if a modified form of the diffusion coefficient is used, based on transport theory. Such an expression for D is given by: D= 1 ⎛ ⎞ 4 Σa + L⎟ 3Σ s (1 − µ )⎜1 − ⎜ ⎟ 5 Σt ⎝ ⎠ (23) If Σ a &lt;&lt; Σ t Eq, 23 reduces to: D= since: 1 3Σ s (1 − µ ) = Σ s (1 − µ ) = Σ tr = λ 1 = tr , 3 3Σ tr 1 λ tr (24) . where λtr is the transport mean free path defined earlier. 4. Highly absorbing media: The flux was expanded in a Taylor’s series and was assumed slowly varying. The flux however changes rapidly in strongly absorbing media. Thus Fick’s law applies to systems in which: Σ a &lt;&lt; Σ s When absorption is present, D should be computed using Eq. 23 and not Eqs. 24 or 21. 5. Proximity to interfaces: The assumption of a uniform medium was used in the derivation of Fick’s law. At the boundary between two media of different scattering properties, Fick’s law is still valid, provided that the sharp change does not lead to a rapidly varying flux, which invalidates the Taylor’s series expansions of the flux in the derivation. In fact, the second order derivatives, either integrate to zero, or cancel out in their contribution to J z . The third derivatives do in fact contribute to the integral of J . Thus Fick’s law woul...
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