Diffusion Approximation in .NET

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4 Diffusion Approximation
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when s points into the scattering medium. This is because diffuse intensity comes from scattering and must originate from inside the scattering medium. However, we cannot impose (8.4.37) under the diffusion approximation because the diffusion approximation has an intensity that is close to independent of the direction of S, and the diffusion approximation is an approximation on the diffuse equation. A common approximate boundary condition is as follows. Let n be the normal on the boundary that points inward into the scattering medium, the approximate boundary condition is
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F = Fnn + Ftt
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where the integration is over 271" solid angle that includes those > O. We use (8.4.15) and (8.4.38) and so
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where t is tangential to the boundary. We then have
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r d8(8' n) {U + ~Fnn. 8 + ~Ftt. 8} 471" 471"
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Thus the approximate boundary condition of (8.4.38) is Fn 7I"U + 2 = 0 or
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dS(8' n)U = U
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d8(8' n)(t s)
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n)(n s) =
dO sinO
dcjJcosO = 7I"U
dO sinO
dO sin 0
dcjJ cos O(sin 0 cos cjJ)
dcjJcos 2 0
(8.4.42) (8.4.43)
~U(-) + F(r) . n = 0 2 r 471"
() 8.4.44
for r on the boundary. Note that F(r) can be obtained from U(r) by using (8.4.33). Thus, the diffusion approximation consists of solving (8.4.34) subject to the boundary condition of (8.4.44) and also using (8.4.33) for
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