STOCHASTIC PROBLEMS IN TRANSPORT THEORY.

§ 1. I n t r o d u c t t o n T h e multiple Compton scattering of low energy (Ä ^O -öM ev) photons has been studied analytically by Chandrasekhar (1948) and O’Rourke (1952, 1953) using the Gaussian quadrature and spherical harmonic approximations to Boltzmann’s equation. The problem is somewhat simplified at low photon energies because the Klein-Nishina differential cross section (i.e. the cross section for scattering through a given angle per unit solid angle) is then effectively energy-independent and is given, in Thomson units per electron, b y : <*(0) = 77T ( l+ c o s 20) (1) where 9 is the angle of deflection resulting from a collision and / e 2 \ 2 1 Thomson unit= — ( —-) =0-665barns. 3 \mc2/ The mean free path d of the photons is related to a{8) by the equation: d -i = ivJa(6»)dG, (2) where N is the number of electrons per unit volume. We shall use d as the unit of length throughout. Given that a collision occurs, the probability that a photon is scattered through an angle 6 into the element dQ. of solid angle is given by -^—(1 +cos20)dQ. 1 677* If we make the assumption that the momenta of the scattering electrons are small so that Compton’s relation is applicable, then the wavelength ! Present address: Applied Mathematics Division, Argonne National Laboratory, Argonne Illinois. 516 P. J. Brockwell on the of each scattered photon increases according to the equation : A' = A + 1 — cos0. ( 3) where A and A' are the wavelengths in Compton units (1 Compton unit = h/mc = 0-02426 A) before and after a collision and 9 is the angle of scattering. For scattering by plane infinite slabs both Chandrasekhar and O’Rourke derive the spectral distribution of forward scattered radiation integrated over all forward directions of emergence. They consider only the isotropic approximation to (1), i.e. o(9) = (1/477), but O’Rourke (1953) indicates how the method can also be applied when the more accurate cross section (1) is used. In the present paper we shall assume that cr(9) is given by eqn. (1) and show how the method of Brockwell (1964), in which the set of all directions in space is approximated by a set of 30 directions corresponding to 30 points uniformly spaced on the unit sphere, can be used to derive the angular distribution of the photons emerging from the slab as well as the spectral distribution of the radiation at each angle of emergence. This method has already been applied in the above-mentioned paper to the well-known Milne problem of isotropic scattering in a half-space and the agreement with the exact solution was found to be very good. Chandrasekhar’s method for the Compton scattering problem depends on the replacement of the radiation field intensity 7(a;,p,A') by the first two terms of a power series in (A' — A), i.e. The resulting approximation to the Boltzmann equation is then solved by Gaussian quadrature. This power series approximation is of course invalid for a monochromatic source of radiation and, as Chandrasekhar (1948) points out, his results (and also those of O’Rourke (1952)) which are derived formally for a monochromatic source are of physical significance only when integrated with respect to a relatively smooth spectral distribu tion of incident radiation. The method to be used in the present paper does not suffer from this limitation. Assuming a monochromatic source of radiation of wave length A0 we are led quite naturally to determine the distribution function of the emergent wavelength A as the sum of two components, an absolutely continuous component due to the scattered radiation and a simple dis continuity at A0 whose magnitude is equal to the probability tt0 that an incident photon passes through the slab without being scattered. For monochromatic radiation incident isotropically on one face of a slab of thickness l we determine in §3 the angular distribution of the transmitted radiation and the proportion of incident radiation transmitted by the slab. In §4 we determine the proportion 770 of incident radiation transmitted with no scattering and the probability density £(A) (normalized I ( x , p , A ) p , A) -f(A A) (x , p , A). £(A) rlA = 1) of the increase in wavelength of the transmitted radiation which is scattered at least once. Multiple Compton Scattering of Low Energy Gamma Radiation 517 Like Chandrasekhar and O’Rourke we shall neglect photoelectric absorption; however if we assume a constant probability of absorption at any collision the problem can be solved in exactly the same way as explained in Brockwell (1964), §7. A further advantage of the method to be described is that it can he used to determine not only the overall emergent spectral intensities but also the intensities in particular directions of emergence. § 2. A ppl ic a t io n o f t h e I cosahedral A ppr o x im a tio n Using the method of Brockwell (1964) we replace the set of all directions in space by 30 directions corresponding to points uniformly spaced on the unit sphere. For a detailed description of this procedure the reader is referred to the above paper. The directions of motion then fall into classes numbered 1,2, . . . , 10 such that the angles they make with the positive X-axis have cosines y x, /x2, . . . , p,10 respectively, where P l= l > / b i = M 5 > \ / 5 + 1 Ma 4 > ^ 7 =