Electromagnetic Showers in the Atmosphere

We illustrate the concepts of a simplified electromagnetic cascade (initiated by a photon or an electron) model illustrated in the book of W. Heitler (ed. 1954).

The model is illustrated in the figure below, as reproduced in Matthiew's paper and discussed in an excellent review paper on EAS (extensive air showers).

The toy model of Heitler for an electromagnetic shower initiated by a high-energy photon. The shower is divided several interaction steps n. Photons are drawn with wavy lines. Electrons and positrons are drawn with solid lines. From Matthieu's paper.

The model supposes that electromagnetic interactions, namely electron-position pair production for photons and bremsstrahlung radiation for electrons and positrons, occur after a typical distance $d=X_0 \ln(2)$ , with $X_0$ = radiation length, which in the atmosphere has the value $X_0 = 37~{\rm g \cdot cm^{-2}}$ . Since $X_0 \sim \frac{7}{9} L_p$, we consider the radiation length and the interaction length for pair production about equal. Following this, after n steps, the distance X is given by$X = n X_0 \ln 2$ and the number of particles can be expressed in terms of the distance traveled as:

$$N(x) = 2^{\frac{x}{X_0 \ln 2}} = e^{\ln2^{\frac{x}{X_0\ln 2}}} = e^{\frac{x}{X_0 \ln 2} ln2} = e^{x/X_0}$$

The cascade process stops when the secondary particle energy becomes lower than the critical energy, which in the atmosphere is $E_c = 85$ MeV. Supposing that at each step the energy is partitioned between the 2 produced particles, their energy at step n is :

$$E_n (X) = \frac{E_0}{N(x)} = \frac{E_0}{2^n}$$

where $E_0$ is the energy of the primary particle and the maximum number of steps corresponding to the step where the maximum number of particles of the shower is achieved is:

$$E_{n_{max}} = E_c \Rightarrow n_{max} = \ln 2 \left(\frac{E_0}{E_c }\right)$$

The corresponding distance travelled and number of particles are:

$$X_{max} = X_0 + n_{max} X_0 \ln(2) = X_0 \ln \left(\frac{E_0}{E_c }\right) \\ N_{max} = X_0 + \frac{E_0}{E_c}$$

where we sum the logarithmic development to the distance of the first interaction of the primary. In its simplicity, the Heitler model reproduces the following important features:

• The total number of particle $N_{max}$ is proportional to the primary particle energy $E_0$.

• The evolution of the depth of the maximum of the shower (measured in $\rm g/cm^2$ ) is logarithmic with energy: $X_{max} = X _0 ​+​ X_0 \ln(E _0/E_c​)$;

• The elongation rate in air is $\Lambda = \frac{d X_{max}}{d\log_{10}E_0} = 2.3X_0 \sim 85~{\rm g \cdot cm^{-2}}$, which provides the rate of the increase of the maximum depth of the shower with energy, namely it increases by $85 \, {\textrm g \cdot cm^{-2}}$ for an increase of one energy decade.

Despite the model is highly simplified, e.g. the attenuation of particles is not taken into account, it predictes quite accurately $X_{max}$ . The total number of electrons is overestimated by factors of 2-3 in this model. In the Heitler’s model the ratio of electrons to photons is 2 while simulations and direct cascade measurements in EAS arrays show a ratio of the order of 1/6. This is in particular due to the facts that multiple photons are emitted during bremsstrahlung and that electrons lose energy much faster than photons do.

The length scale of the lateral distribution of low-energy particles in a shower is characterized by the Molière unit, $r_M = (21 \rm MeV/E_c) X_0 \sim 9.3 \rm g cm^{-2}$.