THE EGS5 CODE SYSTEM

This agrees with formula (10) of Butcher and Messel[39] (except that they use Z AB = Z A + Z B anddo not use Z F in their X 0 definition), since our Z A , Z B , Z P are the same as their a, b, p. Nowignoring the factor preceding the {} in Equation 2.98, noting that we require ˘k > A P (the photonenergy cutoff) and also that energy conservation requires ˘k < Ĕ0 − m, we obtain the factorization(see Equation 2.64)⎛⎞α 1 = ln 2 ⎝ 4 3 + 1[] ⎠ , (2.100)9 ln 183 1 + (Z U if Ĕ 0 > 50, Z P )f 1 (E) = 1 ( ) 1 − Efor E ∈ (0, 1) , (2.101)ln 2 E(g 1 (E) = A ( δ ′ (E) ) () )if EĔ0 ∈ A P , Ĕ 0 − m , 0 , (2.102)α 2 = 1 2 , (2.103)f 2 (E) = 2E for E ∈ (0, 1) , (2.104)(g 2 (E) = B ( δ ′ (E) ) )if EĔ0 ∈ (A P , Ĕ0 − m), 0 . (2.105)We notice that f 2 (E) is properly normalized, but that ( f 1 (E) has ) infinite integral over (0,1). Insteadwe limit the range over which f 1 (E) is sampled to 2 −N Brem, 1 , where N Brem is chosen such thatTo sample f 1 (E) we further factor it to2 −N Brem≤ A PĔ 0< 2 −(N Brem−1). (2.106)f 1 (E) =N∑Bremj=1α 1j f 1j (E) g 1j (E) (2.107)wheref 1j (E) =( 1ln 2 2j−1 if E < 2 −j 1,ln 2The f 1j are properly normalized distributions.α 1j = 1 , (2.108)(1 − E2 j−1 )Eif E ∈(2 −j , 2 −j+1) ), 0 , (2.109)g 1j (E) = 1 . (2.110)We sample f 2 (E) by selecting the larger of two uniform random variables (see Section 2.2);namely,E = max (ζ 1 , ζ 2 ) , (2.111)where ζ 1 and ζ 2 are two random numbers drawn uniformly on the interval (0, 1). To sample f 1 (E),we first select the sub-distribution indexj = Integer Part (N Brem ζ 1 ) + 1 . (2.112)49