6 Calorimetry
225
active parts of the calorimeter in their specific ways (see Sect. 6.2.7). Contributions
from neutrons and photons from nuclear reactions, which have consequences for
the performance of these instruments, are also shown in Fig. 6.18. The total energy
carried by photons from nuclear reactions is substantial: only a fraction, however,
will be recorded in practical instruments, as most of these photons are emitted
with a considerable time delay (~1 μs). The event-by-event fluctuations in the
invisible energy dominate the fluctuations in the detector signal, and hence the
energy resolution. The road to high-performance hadronic calorimetry has been
opened by understanding how to compensate for these invisible energy fluctuations
[35].
6.2.6 Hadronic Shower Profile
The total cross section for hadrons is only weakly energy dependent in the
range of few to several hundred GeV, relevant for calorimetry. For protons, the
total pp. cross section σ tot is approximately 39 mb. For pion-proton collisions
σ tot (πp) = 2/3 σ tot (pp) is naively expected, i.e. 26 mb, compared to the measured
value of σ tot (π + p) ≈ 23 mb. For hadronic calorimetry the inelastic cross sections,
σ inel (pA) or σ inel (πA), determine the value of the corresponding interaction
length, λ int = A/N A σ inel (hadron, A). On geometrical grounds σ inel (hadron, A) is
expected to scale as A 2/3 σ inel (hadron, p), close to the measured approximate scaling
A 0.71 σ inel (hadron, p) and therefore λ int ≈ A 0.29 /{N A σ inel (hadron, p)} [g cm −2 ].
This characteristic length λ int is the mean free path of high energy hadrons
between hadronic collisions and sets the scale for the longitudinal hadronic shower
profile. The probability P(z) for a hadron traversing a distance z without undergoing
an interaction is therefore P(z) = exp. (−z/λ int ). The equivalence with the characteristic distance X 0 for the electromagnetic cascade is evident. In analogy to the
parameterization of electromagnetic showers the longitudinal profile of hadronic
showers can be parameterized in the form.
dE/dx = c
w{x/X 0 }
α−1 exp (−bx/X 0 ) + (1 − w) (x/λ)
α−1 exp (−dx/λ)
.
(6.24)
The overall normalization is given by c; α, b, d, w are free parameters and x
denotes the distance from the shower origin [36].
Longitudinal pion-induced shower profiles are shown in Fig. 6.19 for different
energies together with the analytical shower fits. The longitudinal energy deposit
rises to a maximum, followed by a slow decrease due to the predominantly lowenergy, neutron-rich part of the cascade. Proton-induced showers show a slightly
different longitudinal shape due to the differences in the first few initial collisions.
Shower profiles in different materials, when expressed as a function of λ int exhibit
approximate scaling in λ int , in analogy to approximate scaling of electromagnetic
225
active parts of the calorimeter in their specific ways (see Sect. 6.2.7). Contributions
from neutrons and photons from nuclear reactions, which have consequences for
the performance of these instruments, are also shown in Fig. 6.18. The total energy
carried by photons from nuclear reactions is substantial: only a fraction, however,
will be recorded in practical instruments, as most of these photons are emitted
with a considerable time delay (~1 μs). The event-by-event fluctuations in the
invisible energy dominate the fluctuations in the detector signal, and hence the
energy resolution. The road to high-performance hadronic calorimetry has been
opened by understanding how to compensate for these invisible energy fluctuations
[35].
6.2.6 Hadronic Shower Profile
The total cross section for hadrons is only weakly energy dependent in the
range of few to several hundred GeV, relevant for calorimetry. For protons, the
total pp. cross section σ tot is approximately 39 mb. For pion-proton collisions
σ tot (πp) = 2/3 σ tot (pp) is naively expected, i.e. 26 mb, compared to the measured
value of σ tot (π + p) ≈ 23 mb. For hadronic calorimetry the inelastic cross sections,
σ inel (pA) or σ inel (πA), determine the value of the corresponding interaction
length, λ int = A/N A σ inel (hadron, A). On geometrical grounds σ inel (hadron, A) is
expected to scale as A 2/3 σ inel (hadron, p), close to the measured approximate scaling
A 0.71 σ inel (hadron, p) and therefore λ int ≈ A 0.29 /{N A σ inel (hadron, p)} [g cm −2 ].
This characteristic length λ int is the mean free path of high energy hadrons
between hadronic collisions and sets the scale for the longitudinal hadronic shower
profile. The probability P(z) for a hadron traversing a distance z without undergoing
an interaction is therefore P(z) = exp. (−z/λ int ). The equivalence with the characteristic distance X 0 for the electromagnetic cascade is evident. In analogy to the
parameterization of electromagnetic showers the longitudinal profile of hadronic
showers can be parameterized in the form.
dE/dx = c
w{x/X 0 }
α−1 exp (−bx/X 0 ) + (1 − w) (x/λ)
α−1 exp (−dx/λ)
.
(6.24)
The overall normalization is given by c; α, b, d, w are free parameters and x
denotes the distance from the shower origin [36].
Longitudinal pion-induced shower profiles are shown in Fig. 6.19 for different
energies together with the analytical shower fits. The longitudinal energy deposit
rises to a maximum, followed by a slow decrease due to the predominantly lowenergy, neutron-rich part of the cascade. Proton-induced showers show a slightly
different longitudinal shape due to the differences in the first few initial collisions.
Shower profiles in different materials, when expressed as a function of λ int exhibit
approximate scaling in λ int , in analogy to approximate scaling of electromagnetic