THE EGS5 CODE SYSTEM

g 3 (θ) =
θ4 (
)
λf (0) (θ) + f (1) (θ) + f (2) (θ)/B
g 3,Norm
. (2.337)
When the third sub-distribution function is selected, we first sample η = 1/θ using f η3 (η) and
g η3 (η) given by
f 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)/B
g 3,Norm
. (2.339)
As presented above, this scheme contains four parameters, λ, µ, g 2,Norm and g 3,Norm ; the latter
two are so chosen that g 2 (θ) and g η3 (η) have maximum values (over the specified ranges) which
are 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 for
central θ values. The parameter µ separates the central region from the tail. The parameter λ
determines the admixture of f (0) in the second and third sub-distribution functions. It must be large
enough to ensure that g 2 (θ) and g 3 (θ) are always positive. It will also be noted that α 1 becomes
negative if B < λ so that this case must be specifically treated. After studying the variation of the
theoretical sampling efficiency with the variation of these parameters, the values
λ = 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 values
were usually close to one, so we chose µ = 1 for simplicity. λ could not have been chosen much
lower while still maintaining positive rejection functions. Furthermore it was desired to keep λ as
low as possible since this would allow Molière’s distribution to be simulated for as low values of
B as possible. Although Molière’s theory becomes less reliable for B < 4.5, it was felt that it was
probably 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 we
have 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, the
transcendental equation relating B and b. It will be observed that when viewed as defining a
function of b the resulting function is double valued. We of course reject the part of the curve for
B < 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 from
the origin, (B = 0, b = 0), to the point on the curve (B = 2, b = 2 − ln 2). B is then determined
by
⎧
2
⎫
⎨ 2−ln2
b 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 corresponding
roughly to a thickness of 10 7 radiation lengths, which should be sufficient for any application.
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