THE EGS5 CODE SYSTEM

That is, using Equations 2.122 and 2.123 in Equation 2.121, we obtain( )/ ∣ f(E) = g h −1 (E) ∣∣dhdy (h−1 (x))∣ (2.125)= g(E/p)/p1=ln 21=ln 2( 1p if E p ∈ (0, 1 2),1 − E/pE/p( 1p if E ∈ (0, p/2), 1 − E/pE· 1( )) 1p if E/p ∈ 2 , 1 )if E ∈ (p/2, p) Q.E.D.Thus we sample f 1j (E) by first sampling E ′ from Equation 2.121 and then letting E = E ′ p.The g(E ′ ) in Equation 2.122 may be decomposed according tog(E ′ ) =2∑i=1α ′ i f ′ i (E′ ) , (2.126)withα ′ 1 = 12 ln 2 , f 1 ′ (E′ ) =(α ′ 2 = 1 − 1 )(, f ′2 ln 22(E ′ 1) =(ln 2) − 1 2( (2 if E ′ ∈ 0, 1 ) ), 02· 1 − E′E ′if E ′ ∈, (2.127)( ) )12 , 1 , 0. (2.128)We sample f 1 ′(E′ ) by letting E ′ = ζ/2, where ζ is a random number uniformly drawn on the interval(0, 1). To sample f 2 ′(E′ ) we let E ′ = 1− 1 2x, x ∈ (0, 1), and sample x from the frequency distributionfunctionh(x) = α ′′ f ′′ (x) g ′′ (x) (2.129)whereα ′′ =1(4 ln 2) − 2 , f ′′ (x) = 2x, g ′′ (x) = 12 − x = 0.5E ′ (2.130)We already know how to sample f ′′ (x) (e.g., see Equation 2.111), so this completes the details ofsampling the bremsstrahlung spectrum.Let us now consider the pair production interaction. The general developments are quite analogousto the bremsstrahlung case and we obtain the following formulas:E ≡ Ĕ˘k , (2.131)where Ĕ is the energy of one of the secondary electrons, ˘k is the incident photon energy,∆ E =1˘kE(1 − E) , (2.132)δ ′ = ∆ C ∆ E , (2.133)51