THE EGS5 CODE SYSTEM

= ∆E( ∣ ∣∣∣ dE−1 ∣ )∣∣∣f(E 0 )dE−12 dx ∣ + f(E 1 )E 0dx ∣E 1Thus, the random energy hinge step distances are limited by the accuracy which can be achievedin numerically integrating energy dependent quantities of interest using the trapezoid rule. Thislimit suggests a prescription for determining the energy hinge step-sizes in EGS5: we take t asthe longest step-size which assures that Equation 2.379 is accurate to within a given tolerance ɛ fEwhen applied to the integration of the following: the stopping power to compute the energy loss;the scattering power to compute the scattering strength; and the hard collision cross section tocompute the hard collision total scattering probability and mean free path. Thus, in general, if∆F | analis the analytic integral of one of our functions f over t, we wish to satisfyoror∫ E1E 0dEf(E)∣ ∣∣∣ dEdx∫ t0∆F | anal− F (t) ≤ ɛ fE (2.379)[ ]ds f(t)t∣ −anal 2 (f(E 0) + f(E 1 )) ≤ ɛ fE (2.380)∣∣−1 ∣ [∣∣∣anal RC (E 0 ) − R C (E 1 )−2where R C (E) is the CSDA range for an electron with energy E.](f(E 0 ) + f(E 1 )) ≤ ɛ fE (2.381)For the scattering strength, the function f is the scattering power G 1 , and for energy lossf is the stopping power, in which case the analytical expression reduces simply to ∆E. Notethat in the case of the electron mean free path and total scattering probability, the expressionsfor both the analytical function and the random hinge results are somewhat different from theresults described above, as the integrands for those quantities contain the spatial distribution ofthe collision distances. For the random energy hinge methodology, the probability per unit path ofan interaction taking place over a step of length t is given byp(s:h) = Σ 0 e −sΣ 0s ≤ h, (2.382)Σ 1 e −hΣ 0e −(s−h)Σ 1s > hwhere h is the hinge distance, Σ 0 the cross section at the initial energy, and Σ 1 the cross sectionafter the energy hinge. The random hinge mean free path over t is then given asλ Eh (t) =====∫ t1ds s p(s) (2.383)P Eh (t) 0∫1 t ∫ sds s dh p(s:h) p(h)P Eh (t) 0 0∫1 t ∫ s [ Σ0 (t − s)ds s dh e −sΣ 0+ Σ ]1P Eh (t) 0 0 stt e−hΣ 0e −(s−h)Σ 1⎡∫1 tds s ⎣ Σ 0(t − s)e −sΣ 0+ Σ 1e −sΣ 1(1 )− e −s(Σ ⎤0−Σ 1 )⎦P Eh (t) 0 tt(Σ 0 − Σ 1 )⎡1⎣ 1 − e−tΣ 1(1 )( )− e −t(Σ 0−Σ 1 ) (1 + 1 )+ (Σ ⎤0 − Σ 1 ) 1 − e −tΣ 0P Eh (t) Σ 0 (Σ 0 − Σ 1 )tΣ 1 tΣ 2 0 Σ ⎦1109