THE EGS5 CODE SYSTEM

whereX 0 = radiation length (cm),n = electron density (electron/cm 3 ),r 0 = classical electron radius (cm 2 ),m = electron rest energy (MeV),˘k 0 = incident photon energy (MeV),˘k = scattered photon energy (MeV),E = ˘k/˘k 0 ,C 1 = (k ′ 0 )−2 ,k ′ 0 = ˘k 0 /m,C 2 = 1 − 2(1 + k ′ 0 )/(k′ 0 )2 ,C 3 = (1 + 2 k ′ 0)/(k ′ 0) 2 .The Compton cross section integrated over the energy range from ˘k 1 to ˘k 2 can be expressed aswhere∫ ˘k2˘k 1d˘Σ Compt (˘k 0 )d˘kd˘k = X 0nπr 2 0k ′ 0( 1[C 1 − 1 )+E 1 E 2C 2 ln E 2E 1+ E 2 (C 3 + E 2 /2) − E 1 (C 3 + E 1 /2)E 1 = ˘k 1 /˘k 0 ,E 2 = ˘k 2 /˘k 0 .](2.171)The total scattering cross section is obtained from Equation 2.171 with ˘k 1 and ˘k 2 set to theminimum and maximum possible scattered photon energies. To see what these are, we use Equation2.167, noting that m 1 = m 3 = 0, m 4 = m, E 1 = p 1 = ˘k 0 , E 3 = p 3 = ˘k, E 4 = Ĕ, and p 4 = ˘p,and arrive atcos θ = (˘k 0 + m)˘k − ˘k 0 m. (2.172)˘k 0˘kSolving for ˘k we get the well-known formula˘k =The maximum and minimum values of ˘k occur at cos θ = 1, −1, or˘k 01 + (1 − cos θ)˘k 0 /m . (2.173)˘k max = ˘k 0 , (2.174)˘k min =˘k 01 + 2˘k 0 /m . (2.175)63