Extension of the method of the small angle approximation: open detector Page: 5 of 14
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4
entire hierarchy of approximations which encompasses the range of the
ratio o^/o0^ and the error involved in each step of the hierarchy may
be estimated.
2. FORMALISM.
We shall use a Green's function formalism to combine the small angle
approximation and the diffusion approximation. This method not only
yields a useful physical insight into the scattering process, but provides
us in the end with a set of radiative transfer equations to solve. These
equations may then be solved using standard techniques. Thus a practical
scheme for the combination of the small angle and diffusion
approximations is obtained.
Let I(r,Q) be the specific intensity of the radiation being propagated
through a scattering medium whose total cross-section (absorption ♦
scattering) is Of. If there were no scattering processes, then the loss in
l(r,Q) at r, in the direction Q is given by9:
QV l(r.Q) = - ot l(r.Q) ♦ Q(r.Q) (2.1)
where Q(r,Q) is taken to be a gaussian, mono-directional source of
radiation, located at z=z8.
Q(r,Q) = q0 8^p-p8)S(Z"2s) &(P”Ps) V (2.1a)
MP’Ps)= (#2/7T) exp(-#2(p-p8)2) (2.1b)
where p is the azimuthal angle, and % is the initial width of the beam.
If the radiation were to reappear due to some scattering process then
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Chitanvis, S.M. & Gerstl, S.A.W. Extension of the method of the small angle approximation: open detector, article, August 18, 1986; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1072804/m1/5/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.