THE EGS5 CODE SYSTEM

At small path lengths t, a very large number of coefficients g l may be required to ensure
convergence of the Legendre series given in Equation 2.362. The default value of the number of
coefficients is set in ELASTINO to 1000, which should yield convergence for reasonably short paths
in most cases. Note that since the time to compute a single distribution function depends on the
square of the number of coefficients, and, typically, a very large number of distributions must be
computed (the distribution function space is three dimensional, depending on material, energy, and
path length, for both positrons and electrons), the total time to compute a full set of distribution
functions for some problems can be very long. To speed the computation, ELASTINO works from
low energy and short paths to higher energies and longer paths, re-setting at each new step the
number of terms to be computed based on the number required at the previous step. Even so,
computation times can be extremely long, even when error tolerances have been set fairly high,
and users should be mindful of this whenever invoking the use of this distribution.
Traditionally, sampling from GS distributions has been either prohibitively expensive (requiring
computation of the slowly converging series) or overly approximate (using very large pre-computed
data tables with limited accuracy). We have developed here new fitting and sampling techniques
that overcome these drawbacks, using a scaling model that have been known for some time [27].
First, the change of variables of Equation 2.363 is performed, and a reduced angle now called
χ = (1 − cos(Θ))/2 is defined. The full range of angles (0 ≤ Θ ≤ π, or 0 ≤ χ ≤ 1) is then broken
into 128 intervals of equal probability, with the 128 th interval further broken down into 32 subintervals
of equal spacing. Because the distribution is so heavily forward peaked, in most cases the
last 32 sub-intervals cover the entire range from approximately π/2 to π, which is why this region
is treated specially. In each of the 287 intervals or sub-intervals, the distribution is parameterized
as
α
f(χ) =
(χ + η) 2 [1 + β(χ − χ −)(χ + − χ)] (2.365)
where α, β and η are parameters of the fit (which can be determined analytically if it is assumed
that the distribution function conforms the shape given above over each given interval) and χ − and
χ + are the endpoints of the interval.
Using the large number of angle bins which we do, this parameterization models the exact form
of the distribution to a very high degree of accuracy, and, in addition, it can be sampled very
quickly. As the 128 bins are equally probable, initial angle intervals can be determined trivially.
When the last (128 th ) interval is selected, a linear table search is used to determine which of the 32
equally spaced sub-intervals to use. This table search is usually very fast because the first several
of the 32 sub-intervals typically cover 90% of the probable angles in the final interval. Once the
correct bin is chosen, the distribution can be sampled quickly. We begin by inverting the first term
of Equation 2.365 to produce a value of χ from
χ = ηζ + χ −(χ + + η)/(χ + − χ − )
, (2.366)
(χ + + η)/(χ + − χ − ) − ζ
and we then use 1 + β(χ − χ − )(χ + − χ) as a rejection function. Rejection efficiencies on the order
of 95% are typical. Energy and pathlength interpolation is done by randomly selecting either the
upper or lower bounding grid point in the tabulated mesh.
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