## 3 Extensive Air Showers: The Heitler-Matthews Model
### Energy and Mass  
The relevance of muon measurements to the question of the primary composition has been first remarked by the Institute for Nuclear Studies (INS) group in Tokyo [42]. They were the first group to point out the key information that the mass of the primary particle could be derived from a study of plots of muon versus electron number. Due to the intuitive relation between shower to shower fluctuations and primary mass, the study of fluctuations in the muon number distributions was historically the first method employed to study the primary CR mass composition [42; 43; 44]. The narrowing of the distribution of \(N_{\mu}/N_{e}\) was considered to be due partly to the change in the composition of primary particles with energy [43]. On general grounds, the elemental composition can be investigated if the total size and the muon component depend differently from the primary energy. If we assume that their dependences from the energy of a primary proton \(E_{0}\) can be described as  
\[N_{e}\propto E_{0}^{\beta_{e}},\qquad\qquad N_{\mu}\propto E_{0}^{\beta_{\mu}}, \tag{18}\]  
for a nucleus \(A\), we have  
\[N_{e}\propto\left(\frac{E_{0}}{A}\right)^{\beta_{e}}\cdot A,\qquad\qquad N_{ \mu}\propto\left(\frac{E_{0}}{A}\right)^{\beta_{\mu}}\cdot A, \tag{19}\]  
with a mass-number dependency of the type \(1-\beta_{e}\) and \(1-\beta_{\mu}\), respectively. The relation \(N_{\mu}/N_{e}\) can be easily deduced  
\[N_{\mu}\propto N_{e}^{\beta_{\mu}/\beta_{e}}A^{1-(\beta_{\mu}/\beta_{e})}. \tag{20}\]  
In 1962, [PERSON], [PERSON], and [PERSON] working at the MIT Volcano Ranch Station observed, for the first time, a muon/electron correlation: \(N_{\mu}{\sim}A^{1-\alpha}\cdot(N_{e})^{\alpha}\), thus establishing that the muon size is a mass-sensitive observable [45]. The Equation (13) can be transformed to obtain the energy \(E_{0}\) to be introduced in the relation (14) to obtain \(\beta_{\mu}/\beta_{e}\)\(\sim\)0.86. The muon size for a given mass \(A\) as a function of the total size \(N_{e}\) is then  
\[N_{\mu}\propto N_{e}^{0.86}A^{0.14}. \tag{21}\]  
In a similar way, we can obtain the muon size as a function of the total size for a given primary energy \(E_{0}\). The Equation (13) is transformed to obtain the mass \(A\) which in turn is introduced in the relation (14)  
\[N_{\mu}\propto\left(\frac{E_{0}}{1\ {\rm PeV}}\right)^{3.17}N_{e}^{-2.17}. \tag{22}\]  
In experiments with ground-based arrays the reconstructed number of muons and electrons are plotted in a \(\ln N_{\mu}-\ln N_{e}\) plane to recover the energy and mass of the primary particle. This diagram, when combined with detailed shower simulations, proved to be a powerful tool for extracting information on primary mass. Therefore, it is interesting to study the electron-to-muon ratio at shower maximum  
\[\frac{N_{e}}{N_{\mu}}\approx 35.1\cdot\left(\frac{E_{0}}{A}\right)^{0.15}. \tag{23}\]  
with the energy in PeV. This ratio depends on the energy per nucleon \(E_{0}/A\) of the primary particle, thus showing that \(N_{e}/N_{\mu}\) can be used to infer the mass of the primary particle if the energy is measured with a different, independent observable. We can use the relation (23) to investigate the sensitivity of EAS arrays to the primary mass A [38,46]  
\[\lg\left(\frac{N_{e}}{N_{\mu}}\right)=1.54+0.15\cdot\lg\left(\frac{E_{0}}{1\ {\rm PeV}} \right)-0.065\cdot\ln A=C-0.065\cdot\ln A \tag{24}\]  
If the energy is reconstructed from another independent observable, the mass of the primary CR can be determined by measuring the ratio \(N_{e}/N_{\mu}\). Therefore, the relative error on the electron-to-muon ratio is  
\[\frac{\Delta(N_{e}/N_{\mu})}{N_{e}/N_{\mu}}\sim 0.15\left[\frac{\Delta E_{0}}{E_ {0}}+\frac{\Delta A}{A}\right]\sim 0.15\left[\frac{\Delta A}{A}\right] \tag{25}\]  
with the consequence that to measure the elemental composition with a resolution of one unit in \(\ln A\) the relative error on \(N_{e}/N_{\mu}\) must be \(\approx\)15%. A resolution of one unit in \(\ln A\) in principle allows to reconstruct 4 (or 5?)