THE EGS5 CODE SYSTEM

The general formula for the f (i) is (Bethe, Equation 26)f (n) (θ) = (n!) −1For n = 0 this reduces to∫ ∞0[nudu J 0 (θu) × exp(−u 2 /4) 1/4 u 2 ln (u /4)] 2 . (2.282)f (0) (θ) = 2e −θ2 . (2.283)Instead of using the somewhat complicated expressions when n = 1 and 2, we have elected to usea) the numerical values presented in Bethe’s paper (for 29 selected values of θ from 0 to 10), b) thefact that f (i) (θ) behaves as θ −2i−2 for large θ, and c) the fact that f (1) (θ) goes over into the singlescattering law at large θ. That is,lim f (1) (θ)θ 4 = 2. (2.284)θ→∞This also implies thatlim f (2) (θ)θ 4 = 0. (2.285)θ→∞The f (i) (θ) functions are not needed in EGS directly, but rather PEGS needs the f (i) (θ) tocreate data that EGS does use. Letη = 1/θ (2.286)andf (i)η (η) = f (i) (1/η)η −4 = f (i) (θ(η))θ(η) 4 . (2.287)As a result of Equations 2.285 and 2.286 we see that f η (1) (0) = 2 and f η(2) (0) = 0. We now do acubic spline fit to f (i) (θ) for θɛ(0, 10) and f η(i) (η) for ηɛ(0, 5). If we use ˆf (i) (i)(θ) and ˆf η (η) to denotethese fits, then we evaluate the f (i) (θ) as()f (i) (θ) = ˆf (i) 1 (i)(θ) if θ < 10, ˆfθ 4 η (1/θ) . (2.288)Similarly if we want f η(i) (η) for arbitrary η we usef (i)η (η) = (ˆf (i)η (η) if η < 5, 1η 4)ˆf (i) (1/η). (2.289)To complete the mathematical definition of f(Θ) we now give additional formulas for the evaluationof χ c and B. We haveB − ln B = b, (2.290)b = ln Ω 0 , (2.291)Ω 0 = b c t/β 2 , (2.292)b c = ‘6680′ ρZ S e Z E/Z SMe Z X/Z S(2.293)(Note, PEGS computes ˘b c = X 0 b c ),( ) [ ]¯h2(0.885)‘6680 ′ 2= 4πN a = 6702.33, (2.294)m e c 1.167 × 1.1384