Highly energetic hadrons also create a particle cascade when passing through absorber material. These hadron showers are substantially more complex than the electromagnetic ones which were discussed in the last section.
Due to the higher mass of charged hadrons their energy loss due to Bremsstrahlung is irrelevant, and thus they do not directly induce electromagnetic cascades. Instead hadrons are capable of interacting with the absorber nuclei via the strong force. This is especially true for neutral particles for which, apart from seldom weak interactions, the strong force presents the only way of ionization.
The strong interaction between the hadronic projectile and the absorber nucleus enables a big set of different processes, most of which will lead to some kind of break up of the absorber nucleus. Such a break-up is called nuclear spallation[30].
Nuclear Spallation
Nuclear spallation can be divided into two phases.
In the first phase, sometimes called cascade phase, the incident hadron undergoes a series of quasi-free collisions with the nucleons, transferring part of its energy. The struck nucleons themselves collide with other nucleons, causing an intra-nuclear cascade.
At each of these collision the transferred energy can be used to create new hadronic particles if the projectile’s energy is sufficient (pion production threshold is atO(1 GeV)). Most of these particles are light mesons such as π or K, but baryons such as p, n or Λ can be created as well, especially if the projectile is a baryon, too (baryon number
conservation).
These particles are created in partonic interactions, i.e. only a parton of the projectile interacts with one parton of the nucleus, filling up the missing quarks to built the new particles by taking them from the “sea”. As the remaining quarks are not participating, they are called spectator quarks. The spectator quarks of the projectile carry a significant part of its momentum and thus the new particle consisting of these spectator quarks does, too. Therefore this high energetic particle it is called leading particle.
The newly created particles and the nucleons with sufficient energy escape the nucleus in the same general direction as the projectile. Their energy is typically in the order of a few GeV and above. The intra-nuclear cascade is a pure QCD process and thus very fast (O(10−22s)[31]).
In the second step of the nuclear spallation, which is called evaporation phase, the intermediate nucleus de-excites by emitting free nucleons (n, p), photons (γ) and sometimes heavier nucleons aggregates (α-particles, . . . ). Unlike the intra-nuclear cascade, the emissions are independent on the direction of the incident hadron and are distributed isotropically. These de-excitation emissions are only possible if the available energy of the nucleus is large enough to overcome the nucleons binding energy. Note that as protons have to overcome the Coulomb-Barrier on top, the number of created neutrons is typically higher than the number of protons. This is especially true for materials with a large atomic numberZ, where the Coulomb-Barrier is larger. The escaping neutrons and protons have a typical energy of 1−10 MeV. The nuclear de-excitation is considerably slower than the intra-nuclear cascade and takes about
O(10−13−10−18s)[31].
Note that for the release of nucleons in both phases energy is necessary to overcome the binding energy barrier. This energy is thus lost for ionization of the target material and hence is sometimes called invisible energy. It can make up of up to 30% of the entire non-EM component of the shower.
The Hadronic Cascade
The cascade is formed mainly by the secondary hadrons created during the intra-nuclear cascade, as they carry enough energy to induce further reactions.
One of these reactions is the induction of further spallation processes, creating additional secondary particles. Similar to the radiation length in electromagnetic cascades, the nuclear interaction lengthλI denotes the typical distance these hadron fly before the interaction. It is defined as the pathx after which the numberN of particles without interaction drops to the 1/e part of the number of incident particles N0.
N(x) =N0·e
−x
λI (3.5)
As the cross section, i.e. the probability for such an interaction, is different for protons and pions, the nuclear interaction length has different values for pions. Hence a special pion interaction lengthλπ is introduced. An example scheme of such a cascade is shown in Figure 3.6.
Another possible reaction is the decay of the hadron. Charged pions for example can decay via the weak processπ±
→W±
0 1 2 3 λi π0 hadron (p, π±, K,. . . ) π± µ± νµ electromagnetic π,n,Λ,K,p,. . . subshower nuclear interaction scattering / elastic neutron neutron capture
Figure 3.6: Example scheme of the development of a hadronic shower starting with a hadron. The high energetic fast spallation processes are marked by black dots, while the slower neutron capture and scattering is shown in red. For the sake of visibility not all particles are drawn. Especially the neutron capture is only shown at one of the spallation processes.
and a muon which performs MIP-like ionization before escaping the absorber material (lower part in Figure 3.6) without contributing any further to the development of the
hadronic cascade.
Electromagnetic Subshower
A special case of decay processes within a shower is the decay of neutral pionsπ0. Their
average life time is extremely short (8.52×10−17s [14]), such that their mean path
before decay ofcτ = 25.5 nm is too short for inducing new spallation processes. Instead, it decays into a pair of electron and positron, which then starts an electromagnetic subshower. This is indicated in the top part of the first interaction shown in Figure 3.6. These showers mark the electromagnetic component of hadronic showers.
In most materials the radiation length is much smaller than the nuclear interaction length. Thus electromagnetic cascades are much denser compared to the sparse nuclear interactions.
As the particles involved in an EM shower cannot induce further spallation processes, their creation is in this sense a one-way street. At each nuclear interaction there is a non-zero probability to produce aπ0. Thus, with a rising number of nuclear interactions
the overall probability to createπ0 rises and with it the electromagnetic fraction. The
number of hadronic interactions rises with the energy of the primary particle, and thus the electromagnetic fraction of the cascade rises, too.
Note that the creation ofπ0 is a statistical process. It might happen at every nuclear
interaction or even not at all. Hence the electromagnetic fraction varies heavily from event to event. In combination with the large number of possible processes that can happen within a hadronic cascade in general, not two hadronic showers look alike, thus making predictions on individual shower development impossible.
Time Development: Late Neutron Component
Both the intra-nuclear cascade and the particle de-excitation are depositing the energy almost instantaneously.
However, the evaporating neutrons of the de-excitation phase undergo different kind of interactions with the nuclei of the absorber material and thus introduce a significant delay in the ionization. The interactions of these “late neutrons”, which have an energy of a few MeV, are:
• Inelastic scattering. This is the low energy variant of spallation without the creation of additional hadrons via the intra-nuclear cascade. Instead the excited nucleus emits photons or, depending on the type of material, even nucleons or
α-particles. This process is valid for neutron energies above 1 MeV.
• Elastic scattering. For neutron energies in the region of keV to MeV the elastic scattering of the neutron on the absorber nuclei is the dominating process. At each collision the neutron loses a part of its energy. As the scattering is elastic, the average amount of energy lost per collision is dependent on the mass difference of the two particles. While it is small for collisions of neutrons with heavy nuclei, the average transferred energy is maximal for collisions with hydrogen (on average 50% energy loss per collision). At each collision the neutron loses energy and gets slower. However, at the same time the cross-section for elastic scattering rises, decreasing the mean free path length. Hence, the time between two collisions is approximately constant. All collisions together sum up to a typical time length of 10 ns.
• Neutron capture. If the energy of the neutron is insufficient for performing elastic scattering, it can be captured by a nucleus. The time scale for this process is in the order of µs, which is far above the typical integration time of detectors at a collider experiment. Note that during this process the binding energy of the neutron is released (O(MeV)), reclaiming the invisible energy component
θC
wavefront
particle with velocity vp =βc > c n vc= c n A B C
Figure 3.7: Creation of a wavefront of cherenkov radiation by a charged particle passing through material with a speed vp greater than the speed of light in the same material
vc.
from above. But as the time constant of neutron capture is way beyond typical recording times at particle detectors, this energy is still invisible.
The amount of late energy contributions thus increases in hadronic cascades with a large number of evaporation neutrons. This number is dependent on the absorber material and increases for elements with a large atomic number Z.