THE EGS5 CODE SYSTEM

Assume that an electron starts off from a certain position, which we select as the origin ofour reference frame, moving in the direction of the z-axis. Let f(r, ˆd; t) denote the probabilitydensity of finding the electron at the position r = (x, y, z), moving in the direction given by theunit vector ˆd after having traveled a path length t. The diffusion equation for this problem is givenby Lewis [97] as∫∂f[∂t + ˆd · ∇f = N f(r, ˆd ′ ; t) − f(r, ˆd;] dσ(χ)t) dΩ, (2.354)dΩwhere χ ≡ cos −1 (ˆd · ˆd ′ ) is the scattering angle corresponding to the angular deflection ˆd ′ → ˆd.This equation has to be solved under the boundary condition f(r, ˆd; 0) = (1/π)δ(r)δ(1 − cos Θ),where Θ is the polar angle of the direction ˆd. By expanding f(r, ˆd; t) in spherical harmonics, Lewisobtained general expressions for the angular distribution and for the first moments of the spatialdistribution after a given path length t. The angular distribution is given by∫F GS (Θ; t) ≡ f(r, ˆd;∞∑t) dr =l=02l + 14π exp(−tg l/λ)P l (cos Θ). (2.355)It is worth noticing that F GS (Θ; t)dΩ gives the probability of having a final direction in the solidangle element dΩ around a direction defined by the polar angle Θ. Evidently, the distribution ofEquation 2.355 is symmetrical about the z-axis, i.e., independent of the azimuthal angle of thefinal direction.The result given by Equation 2.355 coincides with the distribution obtained by Goudsmit andSaunderson [63] in a more direct way, which we sketch here to make the physical meaning clearer.Using the Legendre expansion given by Equation 2.350 and a folding property of the Legendrepolynomials, the angular distribution after exactly n collisions is found to bef n (Θ) =∞∑l=02l + 14π (F l) n P l (cos Θ). (2.356)The probability distribution of the number n of collisions after a path length t is Poissonian withmean t/λ, i.e.P (n) = exp(−t/λ) (t/λ)n . (2.357)n!Therefore, the angular distribution after a path length t can be obtained as∞∑∞∑F GS (Θ; t) = P (n)f n (Θ) =n=0l=02l + 14πwhich coincides with expression given in Equation 2.355.From the orthogonality of the Legendre polynomials, it follows thatIn particular, we have[]∞∑ (t/λ) nexp(−t/λ) (F l ) n P l (cos Θ), (2.358)n!n=0∫ 1〈P l (cos Θ)〉 GS ≡ 2π P l (cos Θ)F GS (Θ; t) d(cos Θ) = exp(−tg l /λ). (2.359)−1〈cos Θ〉 GS = exp(−t/λ 1 ) (2.360)93