## 3 Extensive Air Showers: The Heitler-Matthews Model  
A general idea of the main characteristics of EAS and of how different nuclei produce showers with different properties can be obtained from some relatively simple arguments, as suggested by [PERSON] [32] and [PERSON] [33]. This toy model is useful to show how different observables depend on the primary mass and energy, and why certain techniques have historically been used to study elemental composition or to reconstruct the energy spectrum. Nevertheless, detailed Monte Carlo simulations must be used to describe quantitatively all the characteristics of these random processes, with particular care to the role of shower fluctuations. In a nutshell, the collision of a primary CR with a nucleus of the atmosphere produces one large nuclear fragment and many charged and neutral pions (with a smaller number of kaons) (Figure 3) [34]. A significant fraction of the total energy is carried away by a single _\"leading\"_ particle. This energy is unavailable immediately for new particle production. Roughly speaking, half of the energy of the primary particle is transferred to the nuclear fragment and the other half is taken by the pions (and kaons). The fraction of energy transferred to the new shower particles is referred as _inelasticity_. Accurate description of the leading particles is crucial because these high-energy nucleons feed energy deeper into the atmosphere. Approximately equal number of positive, negative and neutral pions are produced. The e.m. component, the most intense of an EAS, is produced by the photons coming from the decay of the neutral pions. At each interaction before the charged pions decay, nearly a third of the hadronic component energy is released into the e.m. one. As the number of particles increases, the energy per particle decreases. They will also scatter, losing energy, and many will range-out. Thus, the number of particles (or, with less ambiguities in the definition, the quantity of energy transferred to secondaries and eventually released into the atmosphere) will reach a maximum at some depth \(X_{max}\) which is a function of energy, of the nature of the primary particle and of the details of the interactions of the particles in the cascade. After that, the energy/particle is so degraded (will be below some _\"critical energy\"_) that energy losses dominate over particle multiplication process, and the shower \"size\" will decrease as a function of depth: it grows 'old'. Once the pions have reached an energy which is low enough, they will decay into muons and neutrinos (\(\pi^{+}\to\mu^{+}\
u_{\mu}\) or \(\pi^{-}\to\mu^{-}\bar{\
u}_{\mu}\)). The resulting muons propagate unimpeded to the ground. The muon cascade grows and maximizes, but the decay is slower as a consequence of the relative stability of the muon and small energy losses by ionization and pair production. These are the most common processes, but not at all the only ones. As an example, successive hadronic interactions of the primary CR, interactions/decays of kaons and muon decays, multiple scattering and production angles must also be considered (see, for example, the the Ref. [16]). Only detailed simulations with Monte Carlo methods are able to describe all the characteristics of these random processes. Historically, one of the main problems in analyzing data from shower arrays was related to the fact that each experiment used its own simulation of shower development and detectors. This made difficult the comparison of the results and the understanding of their differences. Starting in the 1990s, all experiments began to use the same Monte Carlo simulation code CORSIKA [35], a framework containing different hadronic interaction models to describe the shower development in the atmosphere, and the software GEANT [36] to simulate the detectors operated in the arrays. Over the years, other simulation codes have been developed, in particular to describe the development of showers at ultra-high energies, such as AIRES [37].