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| Geant 3.15 | GEANT User's Guide | PHYS332 ##
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Author(s) : G.N.Patrick Submitted: 10.04.86
Origin : G.N.Patrick, R.Brun Revised: 19.12.92
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| CALL GLANDO (IMODE,STEP,Z,A,RHO,P,E,XMASS,DE,IFLAG) |
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GLANDO samples the fluctuation around the average value of the ionization loss. It uses the following input and output:
Input: IMODE= 2 automatic selection of relevant distribution
= 3 Landau (RANLAN) sampling used
= 4 Landau (GENLAN) sampling used
= 5 Vavilov (DINVAV) sampling used
= 6 Gaussian sampling
STEP = current step-length (cm)
Z = atomic number of absorber
A = atomic weight of absorber3
RHO = density of absorber (g/cm )
P = momentum of incident particle (GeV)
E = energy of incident particle (GeV)
XMASS= mass of incident particle (GeV)
Output: DE = DE/DX - (GeV)
IFLAG= 3 Landau (RANLAN) sampling used
= 4 Landau (GENLAN) sampling used
= 5 Vavilov (DINVAV) sampling used
= 6 Gaussian sampling used
= 30 Vavilov arguments out of range
GLANDO is automatically called by the tracking routines GTELEC, GTHADR and GTMUON when the LOSS data record argument ILOSS is set > 1. The various samplings can be selected by means of this data record since it has a direct correspondence with the IMODE argument above. The default is ILOSS=2 (automatic selection of distribution).
VALUE = GLANDR(X)
GLANDR samples from the restricted Landau distribution. It is a copy of the CERN library routine RANLAN. It not used at the moment.
VALUE = GLANDG(YRAN)
GLANDG samples from the Landau distribution. It is a copy of the CERN library routine GENLAN. GLANDG is called from GLANDO.
VALUE = GVAVIV(RKAPPA,BETA2,RAN)
GVAVIV samples from the Vavilov distribution. It has been extracted from the CERN library routines VAVCOE and VAVFSM. It is called from GLANDO.
Due to the statistical nature of ionization energy loss, large fluctuations can occur in the amount of energy deposited by a particle traversing an absorber element. As recently reviewed [bib-PATR], continuous processes such as multiple scattering and energy loss play a dominant role in the longitudinal and lateral development of electromagnetic hadronic showers, and in the case of sampling calorimeters the measured resolution can be significantly degraded by such fluctuations in their active layers. The description of ionization fluctuations is characterized by the significance parameter, kappa, which is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single collision with an atomic electron
kappa =((xi)/ (E )) (1)
max
E is the maximum transferable energy in a single collison with an max
atomic electron.
2 2 2
E =((2m beta gamma )/(1 +((2gammam )/(m )) +(((m )/ (m ))) )), (2)
max e e x e x
where gamma= E/m (E energy of the incident particle, m mass of the x x 2 2 incident particle), beta = 1-1/gamma and m is the electron mass. e
xi comes from the Rutherford scattering cross section and is defined as
2 4 2 2 2 2
xi =((2piz e N Zrhodeltax)/(m beta c A)) =153. 4((z )/(beta ))((Z)/ (A))rhodeltax (keV), (3)
A e
where
z = charge of the incident particle N = Avogadro's number A Z = atomic number of the material A = atomic weight of the material rho = density deltax = thickness of the material
For a given absorber, kappa can therefore tend towards large values if deltax is large and/or if beta is small. Likewise, kappa can tend towards zero if deltax is small and/or if beta approaches 1. There are therefore two basic regimes which occur in the description of ionization fluctuations :
For a particle of mass m traversing a thickness of material deltax, the x Landau probability distribution may be written in terms of the universal Landau [bib-LAND] function phi(lambda) as (see also [bib-KOL3]):
f(epsilon, deltax) = ((1)/(xi))phi(lambda) (4)
where
c+i1
phi(lambda) = ((1)/ (2pii))c-i1exp (u lnu+ lambdau)du c real
lambda = ((epsilon- )/(xi))-1 +gamma-ln((xi)/ (E ))
max
gamma = 0. 577215... (Euler's constant)
= average energy loss
epsilon = actual energy loss
After Landau's work, Vavilov [bib-VAVI] derived a more accurate straggling distribution by introducing the kinematic limit on the maximum transferable energy in a single collision, rather than using E = 1. max From [bib-SCH2] one has:
2
f(epsilon, deltas) = ((1)/(xi))phi (lambda ,kappa, beta ) (5)
v v
where
2 c+i1 lambdas
phi (lambda , kappa,beta ) = ((1)/ (2pii))int phi(s)e ds c real
v v c-i1
2
phi(s) = exp [kappa(1 +beta gamma)] exp [psi(s)],
2 -s/kappa
psi(s) = s lnkappa+ (s+ beta kappa)[ln(s/kappa) +E (s/kappa)]- kappae ,
1
and
1 -1 -t
E (z) = inz t e dt (the exponential integral)
1
gamma = 0. 577216... (Euler's constant)
2
lambda = kappa[1/ xi(epsilon-)-1- beta + gamma]
v
The Vavilov parameters are simply related to the Landau parameter by lambda= lambda /kappa-ln kappa. It can be shown that as kappa->0, the v Vavilov distribution approaches that of Landau and for kappa>=10 the Vavilov distribution tends to a Gaussian distribution [bib-SCHO].
Various conflicting forms have been proposed for Gaussian straggling functions, but most of these appear to have little theoretical or experimental basis. However, as noted by Schorr [bib-SCH1] it has been demonstrated by Seltzer and Berger [bib-SELT], [bib-BER1], [bib-BER2] that for kappa>=10.0 the Vavilov distribution can be replaced by a Gaussian of the form :
2 2 2 2
f(epsilon, deltas)# ((1)/(xi))sqrt((())/(2pi))kappa(1- beta /2)exp [(epsilon- ) ((kappa)/(2))xi (1- beta /2)] (6)
(7)
thus implying
mean =
2 2
sigma = xi(1- beta /2)E
max
We have utilised two routines from the CERN Program Library to sample random numbers X from the Landau distribution :
X = RANLAN(RNDM)
CALL GENLAN(X)
It should be noted that over the years considerable confusion has arisen over the precise form and features of the Landau distribution. As an example, phi(lambda) has been quoted as having its maximum at lambda= 0.225 (Fano [bib-FANO]) and lambda=-0. 05 (Landau [2]), whereas the true value is in fact lambda=-0. 222782... (Koelbig, Schorr [bib-KOL2]). For the Vavilov distribution we have used the function GVAVIV by Rotondi, Montagna and Koelbig [bib-ROTO].
The Landau formalism makes two restrictive assumptions :
As noted by Nelson et al [bib-EGS4], one should ideally simulate a restricted straggling distribution in much the same way that the restricted Bethe-Bloch formula is used for mean energy loss. This is because high-energy transfers are often simulated discretely (and therefore separately) in the form of delta-ray production, and not as continuous energy loss. Such a distribution has been implemented by L.Urban in the routine GLANDZ. See [PHYS334] for more information.