The Multiplicative Theory of Showers as Applied to Large Bursts of Cosmic-Ray Ionization

In order to decide whether the multiplicative theory of showers offers an adequate explanation of the experimental data on large showers, or whether it is necessary to suppose that they are produced by some other process, such as has been outlined by Heisenberg, the following calculations are undertaken. The energy spectrum of the electrons and photons incident upon a small thickness of material is calculated from the observed frequency distribution in size of bursts of ionization produced by large showers, the effect of fluctuations being taken into account. The energy distribution obtained is of the form $E^{-\alpha }$, where $\alpha$ is 2.6 for energies of the order of ${10}^9$ volts, and decreases slowly with increasing energy, in agreement with the energy distribution calculated by Heitler to explain the variation of cosmic-ray intensity with altitude. The maximum number of electrons necessary to give the observed frequency of bursts is less than 0.5 percent of the total number of particles observed in a cloud chamber at sea level in the energy range between ${10}^9$ and ${10}^{10}$ volts, which is in good accord with the cloud chamber observations. This calculated incident energy spectrum is utilized to calculate the number and frequency distribution of large bursts for large thicknesses of material. These calculated values differ considerably from the experimental ones, but this difference is probably to be ascribed to the effect of the penetrating cosmic rays, and is not to be regarded as evidence of a breakdown of the cascade theory. The experiments on the absorption of a shower are also shown to be in harmony with the theoretical estimates. It is concluded that no mechanism involving the production of many shower particles in a single act need be invoked to explain the occurrence of large showers, but that the ordinary multiplicative processes are entirely adequate when proper account is taken of the fluctuations.