CERN Program Library Long Writeup W5013 - CERNLIB ...

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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
X 0 is the radiation length. From (7) and (8) we get S = t − Kt 2 with K =1.12 × 10 −4 Z 2 inc p 2 β 2 X 0 (9) Equation (9) may be used to make the path length correction. Solving equation (9) with respect to t implies that, in order to obtain real solutions: S ≤ 1 4K i.e. S ≤ 2232 X 0p 2 β 2 Z 2 inc (10) This condition provides an additional constraint to the maximum step length for multiple scattering (variable TMXCOR in routine GMULOF). The corrected step can be approximated as: t ≈ S(1 + KS)=S(1 + CORR) where CORR ≤ 0.25 due to condition (10). PHYS325 – 8 249
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CERN Program Library Long Writeup W
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Chapter 1: Catalog of Geant section
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IOPA500 KINE001 KINE100 KINE199 KIN
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Geant 3.21 GEANT User’s Guide AAA
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The routines made available for GEA
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AAAA Bibliography [1] H.J. Klein an
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2 Event simulation framework The fr
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GUDIGI GUOUT GTRIGC UGLAST GLAST GU
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Particles JPART Materials JMATE/JTM
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3.4 User applications KEY N I VAR S
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SUBROUTINE UGEOM + (’TGT ’,2,
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LQ(JHEAD-1) user link IQ(JHEAD+1) I
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