CERN Program Library Long Writeup W5013 - CERNLIB ...

3 Path and step length
3.1 Restrictions on the step length
Restrictions on the length of the step arise from:
1. the number of scatters Ω 0 ≥ 20 to stay within the multiple scattering regime. When Ω 0 < 20, an
appropriate number of single scatterings is used. See routine GMCOUL [PHYS328].
2. χ 2 cB ≤ 1 i.e. the width of the Gaussian part of the distribution should be less than one radian. This
condition induces a maximum step length for the multiple scattering called t Bethe . In order to find this
value we write the limiting condition as χ 2 c(t Bethe )B(t Bethe )=1, that is B(t Bethe )=1/χ 2 c(t Bethe ).
Now using equation (5) we take the exponential of both members B(t Bethe )Ω 0 (t Bethe )=exp(1/χ 2 c (t Bethe)).
Dividing the two last equalities we obtain:
Ω 0 (t Bethe )= b cZ 2 inc
β 2 t Bethe = χ 2 c (t Bethe)exp(1/χ 2 c (t Bethe))
= χ2 cc Z2 inc t Bethe exp [(E 2 β 4 )/(χ 2 cc Z2 inc t Bethe)]
E 2 β 4
exp [(E 2 β 4 )/(χ 2 cc Z2 inc t Bethe)] = b cE 2 β 2
χ 2 cc
E 2 β 4
t Bethe =
χ 2 cc Zinc 2 ln [b cE 2 β 2 /χ 2 cc]
For electrons and muons this constraint on the step-length is tabulated at initialisation time in the
routine GMULOF [PHYS201]. For hadrons this formula can be approximated as:
(
)
1 E 2 2
β
t Gauss ≈
14.110 −3 X 0 ,
Z inc
where E is in GeV and X 0 is the radiation length in centimeters and the formula has been taken
from the Gaussian approximation to multiple scattering (see [PHYS320]). This condition is more
restrictive, because it is equivalent to require that the width of the Gaussian part of the distribution be
less than 0.5, but it has been found that the two conditions are numerically equivalent;
3. limitation due to the path length correction algorithm used (see below).
3.2 Path Length Correction
A path length correction may be applied in an approximate manner. We have from the Fermi-Eyges theory
[15]
where
t = S + 1 2
¯θ 2 (t) ¯
S
t
We have further:
¯θ 2 (t) ¯
(
=
∫ t
0
¯θ 2 ¯ (t) dt (7)
the mean square angle of scattering;
step size along a straight line;
actual path length.
0.0212 Z inc
pβ
) 2
t
X 0
(8)
248 PHYS325 – 7