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C. W. Fabjan and D. Fournier
by ionization and excitation rather than by particle production. Photons propagate
somewhat deeper into the material, being ultimately absorbed primarily via the
photoelectric process.
Given the large number of particles (electrons, positrons, photons) present in a
high energy electromagnetic cascade (more than one thousand for a 10 GeV electron
or photon in lead), global variables have been sought to describe the average shower
behaviour. Scale variables, such as X 0 as unit length, can be used to parameterize the
radiation effects. However, since energy losses by dE/dx and by radiation depend in
a different way on material characteristics, one should not expect perfect ‘scaling’.
Analytical Description
In an analytical description [6] a first simplification consists in ‘factorizing’ the
longitudinal development and the lateral spread of showers, with the assumption
that the lateral excursion of electrons and photons around the direction of the initial
particle does not affect the longitudinal behaviour and in particular the ‘total track
length’ (see below).
As for any statistical process the first goal is to obtain analytical expressions
for average quantities. Particularly relevant (for a shower of initial energy E 0 ) are:
c(E 0 ,E,t) the average number of electrons plus positrons with energy between E
and E + dE at depth t (expressed in radiation length), and the integral distribution
C(E 0 ,E,t) =
E
0 c(E 0 ,E’,t)dE’; n(E 0 ,E,t) and N(E 0 ,E,t) are the corresponding
functions for photons.
Using the probability distribution functions of the physical effects driving the
shower evolution (Bremsstrahlung, Compton, dE/dx, pair production) one can write
and solve [15, 16] ‘evolution equations’ correlating C(E 0 ,E,t) and N(E 0 ,E,t). In
the so called ‘approximation B’ of Rossi, the energy loss of electrons by dE/dx
is taken as constant, and the pair production and bremsstrahlung cross-sections are
approximated by their asymptotic expression.
As an illustration, Fig. 6.8 shows the number of electrons and positrons as a
function of depth, in a shower initiated by an electron of energy E 0 , and by a
photon of energy E 0 in units of the “Rossi critical energy ε 0 ” (see Sect. 6.2.1).
These distributions are integrated over E from 0 to the maximum possible. The area
under the curves is to a good approximation equal to E 0 /ε 0 , in accordance with the
physical meaning of ε 0 . The two sets of curves also show that a photon initiated
shower is shifted on average by about 1 X 0 to larger depths compared to an electron
(or positron) initiated one.
The total track length T T L =
∞
0 C(E 0 , 0, t)dt the energy transferred to the
calorimeter medium by dE/dx, the source of the calorimeter signal.
Results from Monte Carlo Simulations
While analytical descriptions are useful guidelines, many applications require the
use of Monte-Carlo (MC) simulations reproducing step by step, in a statistical
manner, the physical effects governing the shower formation. For several decades,
the standard simulation code for electromagnetic cascades has been EGS4 [17]. A
recent alternative is encoded in the Geant4 framework [18].
C. W. Fabjan and D. Fournier
by ionization and excitation rather than by particle production. Photons propagate
somewhat deeper into the material, being ultimately absorbed primarily via the
photoelectric process.
Given the large number of particles (electrons, positrons, photons) present in a
high energy electromagnetic cascade (more than one thousand for a 10 GeV electron
or photon in lead), global variables have been sought to describe the average shower
behaviour. Scale variables, such as X 0 as unit length, can be used to parameterize the
radiation effects. However, since energy losses by dE/dx and by radiation depend in
a different way on material characteristics, one should not expect perfect ‘scaling’.
Analytical Description
In an analytical description [6] a first simplification consists in ‘factorizing’ the
longitudinal development and the lateral spread of showers, with the assumption
that the lateral excursion of electrons and photons around the direction of the initial
particle does not affect the longitudinal behaviour and in particular the ‘total track
length’ (see below).
As for any statistical process the first goal is to obtain analytical expressions
for average quantities. Particularly relevant (for a shower of initial energy E 0 ) are:
c(E 0 ,E,t) the average number of electrons plus positrons with energy between E
and E + dE at depth t (expressed in radiation length), and the integral distribution
C(E 0 ,E,t) =
E
0 c(E 0 ,E’,t)dE’; n(E 0 ,E,t) and N(E 0 ,E,t) are the corresponding
functions for photons.
Using the probability distribution functions of the physical effects driving the
shower evolution (Bremsstrahlung, Compton, dE/dx, pair production) one can write
and solve [15, 16] ‘evolution equations’ correlating C(E 0 ,E,t) and N(E 0 ,E,t). In
the so called ‘approximation B’ of Rossi, the energy loss of electrons by dE/dx
is taken as constant, and the pair production and bremsstrahlung cross-sections are
approximated by their asymptotic expression.
As an illustration, Fig. 6.8 shows the number of electrons and positrons as a
function of depth, in a shower initiated by an electron of energy E 0 , and by a
photon of energy E 0 in units of the “Rossi critical energy ε 0 ” (see Sect. 6.2.1).
These distributions are integrated over E from 0 to the maximum possible. The area
under the curves is to a good approximation equal to E 0 /ε 0 , in accordance with the
physical meaning of ε 0 . The two sets of curves also show that a photon initiated
shower is shifted on average by about 1 X 0 to larger depths compared to an electron
(or positron) initiated one.
The total track length T T L =
∞
0 C(E 0 , 0, t)dt the energy transferred to the
calorimeter medium by dE/dx, the source of the calorimeter signal.
Results from Monte Carlo Simulations
While analytical descriptions are useful guidelines, many applications require the
use of Monte-Carlo (MC) simulations reproducing step by step, in a statistical
manner, the physical effects governing the shower formation. For several decades,
the standard simulation code for electromagnetic cascades has been EGS4 [17]. A
recent alternative is encoded in the Geant4 framework [18].