The Theory of Cascade Showers in Heavy Elements

The one-dimensional equations of the cascade theory of showers are solved, account being taken of the variation with energy of the cross section for pair production, as well, of course, as ionization and bremsstrahlung effects. This modification is necessary for the discussion of cosmic-ray showers and bursts in elements of high atomic number since, for such elements, there is a region of energy (10 Mev-200 Mev for Pb) in which the pair production process is important while the cross section usually assumed for the process is greatly in error. Introducing the usual units $t$ of length and $\beta$ of energy, both characteristic of the material traversed $(t,\beta =240\mathrm{} \mathrm{m}, 95 \mathrm{Mev}\mathrm{and} 0.4 \mathrm{cm}, 6.5 \mathrm{Mev}$ for air and lead, respectively), one finds that the maximum number of particles arising in lead from an incident particle of energy $E_0=\beta e^{\epsilon }$ varies between $\frac{E_0}{11\beta }$ and $\frac{E_0}{10\beta }$ as $E_0$ varies between ${10}^9$ ev and ${10}^{11}$ ev. For a given $\epsilon$, the maximum number of particles occurs at a distance (in units of $t$) slightly less for lead than for air, and for large distances the effect of the variation with energy of the pair production cross section is to increase the number of particles and $\gamma$- rays to be expected.