THE EGS5 CODE SYSTEM

where P Eh (t) is the probability that there is a hard collision of any kind over t, p(h) is uniform andgiven by 1/s, with the probability for a given s that we have yet to encounter the hinge given by(t − s)/t and the probability that we are past the hinge given by s/t. P Eh (t) is given by∫ t∫ sP Eh (t) = ds dh p(s:h) p(h) (2.384)0 0∫ t ∫ s [ Σ0 (t − s)= ds dh e −sΣ 0+ Σ ]10 0 stt e−hΣ 0e −(s−h)Σ 1⎡∫ t= ds ⎣ Σ 0(t − s)e −sΣ 0+ Σ 1e −sΣ 1(1 )− e −s(Σ ⎤0−Σ 1 )⎦0 tt(Σ 0 − Σ 1 )= 1 − e−tΣ 1(1 )− e −t(Σ 0−Σ 1 )t(Σ 0 − Σ 1 )Note that the distribution of collision distances, p(s), for the random energy hinge can be seen fromthe integrand in the above expressions to bep(s) = Σ 0(t − s)e −sΣ 0+ Σ 1e −sΣ 1(1 )− e −s(Σ 0−Σ 1 )(2.385)tt(Σ 0 − Σ 1 )In the exact case for electrons passing through media with varying cross sections, we have, ofcourse,λ(t) = 1 ∫ t{ ∫ s }ds s Σ(s) exp − ds ′ Σ(s ′ )(2.386)P (t) 00with the expression for P (t), the probability of any scatter,∫ t{ ∫ s }P (t) = ds Σ(s) exp − ds ′ Σ(s ′ )00Expressed in terms of energy loss steps rather than distance these become∫ E1{λ(t) = dE (R C (E 0 ) − R C (E))dE−1∫ E∣ ∣∣∣ ∣ dx ∣ Σ(E) exp − dE ′ dE ′E 0dx ∣E 0−1Σ(E ′ )}(2.387)(2.388)andP (t) =∫ E1E 0dEdE∣ dx ∣−1Σ(E) exp{−∫ E∣ ∣∣∣dE ′ dE ′E 0dx}−1∣ Σ(E ′ )(2.389)Note that in the above, we have described energy hinge steps in both terms of the change inenergy loss (from E 0 to E 1 ) and also in terms of distance traveled t, as convenient. In EGS5 weuse a simple prescription for relating the two and for switching back and forth. For a given initialenergy E 0 and a pathlength t, E 1 is given as E 0 − ∆E(t) with ∆E(t) computed as follows. A tableof electron CSDA ranges R C (E) is constructed as a function of energy asR C (E) =∫ E0110dE ′ ∣ ∣∣∣ dEdx−1∣ . (2.390)