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| Geant 3.15 | GEANT User's Guide | PHYS325 ##
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Author(s) : M.S. Dixit Submitted: 17.06.85
Origin : Same Revised: 11.01.93
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|CALL GMULOF |
+--------------------------------+
GMULOF calculates the table of steps corresponding to the energies in the array ELOW for electrons and muons. It is called at the initialization time by the routine GPHYSI. For hadrons, this quantity is computed during the tracking in the routine GTHADR.
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| CALL GMOLI (AC,ZC,WMAT,NLM,DENS,OMC,CHC) |
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GMOLI calculates the two material-dependent constants (OMC and CHC) that are needed later for simulation of the process. It is called at initialization time by the routine GPROBI which initializes some material constants and computes the probabilities for various processes.
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|CALL GMULTS |
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GMULTS is a steering routine for the multiple scattering. It decides if
the two material-dependent constants computed at the initialization by
GMULOF have to be corrected because of their small dependence on beta, and
selects Moli`ere theory (GMOLIE) or single Coulomb scattering (GMCOUL)
depending of the range of validity of the theories. It updates the
direction of the particle in the variable VECT (in COMMON GCTRAK). It is
called by the tracking routines GTELEC, GTHADR and GTMUON.
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|CALL GMOLIO (AC,ZC,WMAT,NLM,DENS,BETA2,CHARG2,OMC) |
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GMOLIO re-evaluates the two material-dependent constants which have previously been defined in GMOLI. It is called by GMULTS, if needed.
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| CALL GMOLIE (OMEGA,BETA2,DIN) |
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GMOLIE computes the multiple scattering angle according to Moli`ere theory corrected for finite angle scattering. It is called by the multiple scattering steering routine GMULTS, from which it gets the constants OMEGA and BETA2 as an input and gives the new direction cosines (DIN) as an output.
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| CALL GMOL4 (Y,X,VAL,ARG,EPS,IER) |
+----------------------------------------------------+
GMOL4 inverts the integral of Moli`ere distribution function using 4-point continued fraction interpolation. It is called by GMOLIE.
Moli`ere theory corrected for finite angle scattering (sintau!= tau) as described by Bethe [bib-BET1], [bib-SCOT] is used to calculate the effect of multiple Coulomb scattering on the charged particle trajectory. The angular distribution is given by
f(tau) taudtau = sqrt(((sintau)/(tau))) f(eta)etadeta, (1)
where
(0) (1) -1 (2) -2
f(eta) = f (eta) +f (eta)B +f (eta)B
eta = ((tau)/ (chi sqrt(B))).
c
eta is called the reduced angle and chi the critical scattering angle c
defined as
2 4 2 2 4 2 2 2 4
chi =((4pie Z Z N rhot)/(WE beta )) =chi Z ((t)/(E beta ))(2)
c inc s Av cc inc
with
2 -2
chi =((4pie Z N rho)/(W)) (3)
s Av
cc
and
Z incident particle charge
inc
Z defined below
s
N Avogadro's number
Av
rho density
W molecular weight
E particle energy MeV
t total path length in the scatterer (and not its thickness).
B is defined by the equation
B- lnB= lnOmega ,
0
2 2c-1 2 2 2 2c-1
where Omega is the total number of collisions: Omega = ((chi )/(e chi )) =b Z ((t)/ (beta e ))(4)
0 0 c alpha c inc
Here c is Euler's constant and chi the atomic electron screening alpha angle given by
(Z -Z )/Z
2 2 4 2 2 2 x E s
chi =((m e 1.13)/(p h (0.885) ))e (5)
alpha e
and
4 2 4
b =((4pie N rhoZ )/(WE beta )) (6)
c Av s
Z , Z and Z are defined by s E x
N
Z = sum n Z (Z +1)
s n =1 i i i
i
-2/3
Z = sum n Z (Z + 1)ln Z
E i i i i
2 2
Z = sum n Z (Z + 1)ln[1 +3.34(((ZZ )/(137beta ))) ] (7)
x i i i inc
n are the numbers of atoms of atomic number Z in the molecule/mixture. i i The component distribution functions are given by
2
(0) -eta 2
f (eta) = 2e = 2D (1,1,- eta )
0
(1) 2
f (eta) = 2D (2,1,- eta )
(2) 1 2
f (eta) = D (3,1,- eta ),
2
n n where D (a,b,z) =d / da [Gamma(a)M(a,b, z)] with M(a,b,z) the Kummers n hypergeometric function.
(1) (2) Integrals of the functions f and f needed for the Monte Carlo can be written down directly in terms of the D functions
1 (1) 2 2 2
int etaf etadeta = D (2, 1,-eta )- D (1,1,- eta )-D (1, 1,-eta )
eta 1 1 0
and
1 (2) 2 2 2
int etaf etadeta = ((1)/(2))D (3,1,- eta )-D (2,1, -eta )- D (2,1,- eta )
eta 2 2 1
We factorize the variable chi into two parts. The first part depends on c the incident particle energy-momentum and the path length in the medium. The second part is a constant of the medium for a given incident particle.
2 The atomic electron screening angle chi has a small dependence on alpha
2 beta via the variable Z . The quantities chi ,chi and Omega are x c alpha 0 evaluated at tracking time with the routines GMULTS and GMOLIO, when necessary.
Moli`ere distribution is for the total scattering angle. A similar expression may be written down for the lateral displacement of the scattered particles. However, the problem of joint angle lateral displacement in the Moli`ere approximation has not been solved, and, for small step size, lateral displacement is of second order and may be neglected.
Restrictions on the step taken arise from:
2 4 2 2 2 2 2
t =((E beta )/(chi Z ln[((b )/ (chi ))E beta ]))
Bethe c
cc inc cc
For electrons and muons this constraint on the step-length is tabulated at
initialization time in the routine
GMULOF [PHYS201]. For hadrons
this formula can be approximated as:
4 2 2
t '(((1000)/ (14.1)) ((E beta )/(Z ))) X ,
Bethe inc 0
where E is in GeV and X is the radiation length in centimeters.
0
A path length correction may be applied in an approximate manner. We have from the Fermi-Eyges theory [bib-EGS3]
t 2
t =S +((1)/ (2))int tau (t) (8)
0
where
2 #
tau (t) the mean square angle of scattering
S straight line step size
t the actual path length
We have further:
2 2
tau (t) =(0. 0212((Z )/(pbeta))) ((t)/ (X )) (9)
inc 0
X is the radiation length. From (8) and (9) we get 0
2 -4 2 2 2
S =t- Kt with K = 1.12x 10 ((Z )/ (p beta X )) (10)
0
inc
Equation (10) may be used to make the path length correction. Solving equation (10) with respect to t implies that:
2 2 2
S< =((1)/(4K)) i.e. S< =2232((X p beta )/(Z )) (11)
0
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 (11).