## 3 Extensive Air Showers: The Heitler-Matthews Model
### Hadronic Showers
The superposition of \(A\) independent showers naturally explains why the shower-to-shower fluctuations are smaller for shower initiated by nuclei as compared to proton showers. By substituting the lower primary energy \((E_{0}/A)\) into the previous expressions and summing \(A\) such showers, we obtain the following relations for the number of electrons and muons in a shower induced by a nucleus \(A\):  
\[N_{e|_{max}}^{A}=A\bigg{(}\frac{E_{0}/A}{\varepsilon_{c}^{em}}\bigg{)}=N_{e|_{ max}}^{p} \tag{13}\]  
\[E_{\rm em}\simeq E_{0}\bigg{[}1-\bigg{(}\frac{2}{3}\bigg{)}^{n}\bigg{]}. \tag{5}\]0From these relations valid _at shower maximum_ follows:  
1. The number of electrons is equal for all primary masses A, that is, is independent of the composition. Therefore, _the shower size \(N_{e|_{max}}^{A}\) can be used as an estimator of the energy;_
2. The number of muons \(N_{\mu|max}^{A}\) increases with the mass of the primary particle with \(A^{1-\beta}\)\(\sim\)\(A^{0.1}\). Accordingly, iron-induced showers contain about 1.5 times as many muons as proton showers with the same energy. In fact, in a shower induced by a nucleus A, due to the smaller energy per nucleon (\(E_{0}/A\)), the secondary pions are less energetic. This favours a pion decay as well as an interaction of heavier nuclei higher in atmosphere, where the air density is smaller. _The number of muons can be used to infer the mass of the primary particle._ Moreover, the evolution of the muon number with energy, \(\mathrm{d}N_{\mu}/\mathrm{d}\) ln\(E\), is a good tracer of changes in the primary composition. In fact, a constant composition gives \(\mathrm{d}N_{\mu}/\mathrm{d}\) ln\(E=\beta\) and any departure from that behavior can be interpreted as a change of the average mass of the primaries, in a similar way as with the elongation rate of the longitudinal development. 3. The muon size grows with primary energy more slowly than proportionally, \(\beta\)\(\sim\)0.90. A large number of ground-based arrays studying the knee energy region are located deep in the atmosphere and do not sample the number of electrons at shower maximum. Therefore, the experimental situation is not ideal because the size, used to recover the energy of the primary particle, is mass-dependent, as discussed in Section 4. Only experiments located at extreme altitude (above 4000 m asl) observe the electrons in the shower maximum region for near-vertical showers with an energy in the PeV range. Deeper in the atmosphere, arrays measure only the attenuated size  
\[E_{\rm em}\simeq E_{0}\bigg{[}1-\bigg{(}\frac{2}{3}\bigg{)}^{n}\bigg{]}. \tag{5}\]1  
where \(\Delta X\) is the distance of the shower maximum from the ground and \(\Lambda\approx 60\) g/cm\({}^{2}\) is the attenuation length of the electron size after the shower maximum. Since heavy nuclei reach the maximum of longitudinal development at smaller depths than light ones, on the ground we have a larger electron number for air showers initiated by light particles. This implies that, due to the steeply falling CR spectrum, showers of equal ln \(N_{e}\) are enriched in light elements.