THE EGS5 CODE SYSTEM

g 3 (θ) =θ4 ()λf (0) (θ) + f (1) (θ) + f (2) (θ)/Bg 3,Norm. (2.337)When the third sub-distribution function is selected, we first sample η = 1/θ using f η3 (η) andg η3 (η) given byf η3 (η) = 2µ 2 η for ηɛ(0, 1/µ) , (2.338)Then we let θ = 1/η.g η3 (η) =η−4 ()λf (0) (1/η) + f (1) (1/η) + f (2) (1/η)/Bg 3,Norm. (2.339)As presented above, this scheme contains four parameters, λ, µ, g 2,Norm and g 3,Norm ; the lattertwo are so chosen that g 2 (θ) and g η3 (η) have maximum values (over the specified ranges) whichare not greater than 1. The first sub-distribution is the Gaussian (actually exponential in θ 2 )distribution that dominates for large B (thick slabs). The third sub-distribution represents the“ single scattering tail.” The second sub-distribution can be considered as a correction term forcentral θ values. The parameter µ separates the central region from the tail. The parameter λdetermines the admixture of f (0) in the second and third sub-distribution functions. It must be largeenough to ensure that g 2 (θ) and g 3 (θ) are always positive. It will also be noted that α 1 becomesnegative if B < λ so that this case must be specifically treated. After studying the variation of thetheoretical sampling efficiency with the variation of these parameters, the valuesλ = 2, µ = 1, g 2,Norm = 1.80, g 3,Norm = 4.05 (2.340)were chosen. These values do not give the absolute optimum efficiency, but the optimum µ valueswere usually close to one, so we chose µ = 1 for simplicity. λ could not have been chosen muchlower while still maintaining positive rejection functions. Furthermore it was desired to keep λ aslow as possible since this would allow Molière’s distribution to be simulated for as low values ofB as possible. Although Molière’s theory becomes less reliable for B < 4.5, it was felt that it wasprobably as good an estimate as could easily be obtained even in this range.Since α 1 < 0 for B < λ, some modification of the scheme must be devised in this case. What wehave done is to use the computed values of B in computing χ c√B, but for sampling we set ‘1/B’= ‘1/λ’. This has the effect of causing the Gaussian not to be sampled.Our next point is best made by means of Figure 2.7 which is a graph of Equation 2.290, thetranscendental equation relating B and b. It will be observed that when viewed as defining afunction of b the resulting function is double valued. We of course reject the part of the curve forB < 1. We would, however, like to have a value of B for any thickness of transport distance (i.e.,any value of b). In order to obtain a smooth transition to zero thickness we join a straight line fromthe origin, (B = 0, b = 0), to the point on the curve (B = 2, b = 2 − ln 2). B is then determinedby⎧2⎫⎨ 2−ln2b if b < 2 − ln 2, ⎬B = the B > 1 satisfying B − ln B = b,(2.341)⎩⎭if b > 2 − ln2 .For rapid evaluation, B has been fit using a piecewise quadratic fit for bɛ(2, 30); b = 30 correspondingroughly to a thickness of 10 7 radiation lengths, which should be sufficient for any application.89