diff --git a/docs/source/methods/geometry.rst b/docs/source/methods/geometry.rst index 93dec2df4..65f5e9fe3 100644 --- a/docs/source/methods/geometry.rst +++ b/docs/source/methods/geometry.rst @@ -901,8 +901,39 @@ Dxy + Eyz + Fxz + Gx + Hy + Jz + K = 0`. Thus, the gradient to the surface is ------------------------------ White Boundary Conditions ------------------------------ +The white boundary condition is usually applied in deterministic code, where the particle +will hit the surface and travel back with isotropic angular distribution. Essentially, the +change of particle's direction will comply to the cosine distribution instead of uniform +distribution. +The probability distribution function (pdf) for the reflected direction can be expressed +as follows, +.. math:: + : label: white-reflection-pdf +where mu is the cosine of the polar angle between reflected direction and the normal to the +surface; and theta is the azimuthal angle. +Thus, the cosine of the polar angle can extracted like this, +.. math:: + : label: white-reflection-cosine + +and the azimuthal angle is uniform in the range of 2*PI, +.. math:: + : label: white-reflection-unifrm + +Then, cosine can be obtained by analytical inversion of cumulative probability distribution (cdf) +like this, +.. math:: + : label: white-reflection-sqrt-prn + +Eventually, the final reflected direction vector can be computed via the rotation of normal to +the surface like this, +.. math:: + : label: white-reflection-rotation + +The white boundary condition can apply to any kind of surface, as long as the normal to the surface +is known as mentioned above in :ref:`reflection`. + .. _constructive solid geometry: http://en.wikipedia.org/wiki/Constructive_solid_geometry .. _surfaces: http://en.wikipedia.org/wiki/Surface .. _MCNP: http://mcnp.lanl.gov