CERN Program Library Long Writeup W5013 - CERNLIB ...

2.3 Sampling of the distribution function
The component distribution functions (2) are given by
f r (0) (η) = 2e −η2 = 2D 0 (1, 1, −η 2 )
f r (1) (η) = 2D 1 (2, 1, −η 2 )
f r (2) (η) = D 2 (3, 1, −η 2 ),
where D n (a, b, z) =d n [Γ(a)M(a, b, z)]/da n , with M(a, b, z) the Kummers hypergeometric function.
Integrals of the functions f r (1) and f r
(2) needed for the Monte Carlo can be written down directly in terms of
the D functions
∫ ∞
ηf r (1) ηdη = D 1 (2, 1, −η 2 ) − D 1 (1, 1, −η 2 ) − D 0 (1, 1, −η 2 )
η
∫ ∞
η
ηf (2)
r ηdη = 1 2 D 2(3, 1, −η 2 ) − D 2 (2, 1, −η 2 ) − D 1 (2, 1, −η 2 )
We note that the first term is just the Gaussian part and it dominates for large B, that is, for large number of
scatters, as it has to be expected. Recalling the definition of η in (2) we can say that the half-width of the
Gaussian part of the Molière distribution is σ 2 = χ 2 cB/2. Routine GMOLIE performs the sampling of the
Molière distribution via the following steps:
1. the value of B is calculated recursively. If we set f(B) =B with f(B) =lnΩ 0 +lnB the relation
used is B n = B n−1 +∆B = B n−1 +(f(B n−1 ) − B n−1 )/(1 − 1/B n−1 ). Convergence is assumed
when |∆B| ≤10 −4 .
2. a random number r 1 is sampled and a table of four values F r (η i ) is built with i = j − 2,j− 1,j,j+1
and F r (η j ) ≥ r 1 . F is the distribution function derived from (2):
F r (η) =
=
∫ η
0
∫ η
0
f r (t) dt =
∫ η
∫ η
f r
(0) (t) dt + B −1 0
= F (0)
r (η)+B −1 F (1)
40 values of the functions F (n)
r
0
f r (0) (t)+f r (1) (t)B −1 + f r
(2) (t)B −2 dt
∫ η
f r
(1) (t) dt + B −2 0
r (η)+B −2 F r
(2) (η)
are tabulated.
f r
(2) (t) dt
3. using a four-points continued-fraction interpolation method (GMOL4) a table of ηi 2 is interpolated using
the values of r 1 and F tabulated in the previous step. This is similar to the inversion of the distribution
function, but instead of obtaining directly the random variable, we interpolate a table of its squares.
This provides a better fit to the inverse function;
4. the value of θ = χ c
√
Bη 2 is calculated;
5. a random number r 2 is extracted, and the rejection function g(θ) = √ sin θ/θ is computed. If r 2 >
g(θ) resampling begins from step 2, otherwise the value is accepted.
The Molière distribution gives the space scattering angle. A similar expression may be written for the
lateral displacement of the scattered particles. However, the problem of joint angle lateral displacement in
the Molière approximation has not been solved, and, for small step size, lateral displacement is of second
order and may be neglected. This introduces a problem when large step sizes are taken. The step size can
be artificially limited via the use of the variable STEMAX, which is an argument to the routine GSTMED. For
more information on this point the user is invited to consult chapter [CONS200].
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