Overview
After a brief introduction and the presentation of a list of the fundamental electromagnetic processes we first define frequently used quantities, relations and concepts. We then review the processes that are relevant for the development of electromagnetic (photon–electron) cascades such as bremsstrahlung by electrons, pair production by photons, Coulomb scattering, energy loss of electrons by ionization, and Compton and inverse Compton scattering. Subsequently we discuss miscellaneous processes that are of lesser or no relevance for cascade development, such as photonuclear reactions and photon–photon interactions, but also processes that play a significant role for the detection of photons and electrons, like the photo effect, and processes that occur only under extreme conditions, such as the Landau-Pomeranchuk-Migdal effect, magnetic bremsstrahlung, magnetic pair production and the pre-showering effect. These topics are followed by an introduction to cascade theory that is worked out to some degree of detail. The solution of the diffusion equations under different approximations are outlined. The longitudinal shower development profile, the energy spectra and the lateral density distributions of the participating particles and photons are discussed, and the characteristic parameters like the shower age and Molière radius are introduced and defined. A collection of formulae for practical applications is given at the end.
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- 1.
This phenomenon is known as the Greisen-Zatsepin-Kuzmin (GZK) cutoff or effect, discussed later.
- 2.
Rossi and Greisen (1941) have used the simplified expression
$$\chi_{0}^{-1} \ = \ 4\alpha \left ( \frac{N_{A}}{A} \right ) r_{e}^{2} Z^{2} \ln (183\, Z^{-(1/3)}) \ \ [\textrm{g}\,\textrm{cm}^{-2}] \ \ .$$((4.12)) - 3.
A different definition of the critical energy is used by Rossi and Greisen (1941, p. 271).
- 4.
The expression for no screening is (Rossi and Greisen, 1941),
$$\begin{array}{rcl}\varphi(E,v) \mathrm{d}v = 4\alpha \frac{N_{A}}{A} Z^{2}r_{e}^{2} \left ( 1+(1-v)^{2} -\frac{2}{3}(1-v) \right ) \left ( \frac{\mathrm{d}v}{v} \right ) \nonumber \\ & & \nonumber \\ \cdot \left [ \ln \left ( \frac{2E(1-v)}{m_{e} v} \right ) - \frac{1}{2} \right ]\end{array}$$((4.22)) - 5.
The opening angle of the electron pair in pair production and the angles of emission of the photon and electron in the radiation process are irrelevant compared to Coulomb scattering.
- 6.
When an ejected electron has sufficient energy to produce its own trail of ionization it is termed a δ -ray and the ionization which is associated with it is called secondary ionization .
- 7.
For media with \(Z>1\) the thumb rule \(I(Z) \simeq 16(Z)^{0.9}\) eV is frequently used.
- 8.
The compact expression \(\omega_{c} = 2\pi\nu_{\mathrm{crit}} = 3\gamma^{3}_{e}(c/\rho)\) is frequently used in place of Eq. (4.54), where c is the velocity of light [m s−1], γ e the Lorentz factor of the electron and ρ the radius of curvature [m] of the electron orbit.
- 9.
The mean energy of a CMBR photon is \(\langle E_{\mathrm{ph}} \rangle \approx 6\cdot 10^{-4}\) eV. Thus, the condition of expression 4.60 is satisfied for Lorentz factors \(\gamma_{e} < 10^{9}\), corresponding to electron energies \(< 5\cdot 10^{14}\) eV.
- 10.
The author wants to acknowledge the excellent lectures given by Prof. Thomas Erber in electrodynamics which he could enjoy as one of his students at the Illinois Institute of Technology in Chicago during the academic year 1957/1958.
- 11.
The number of low energy photons is a multiple of that of electrons.
- 12.
Geomagnetic interactions, the production of Cherenkov radiation and radio frequency emission cause additional but essentially negligible energy losses.
- 13.
As pointed out by Nishimura (1967, p. 23, 2007, private communication), the importance of the Landau-Rumer theory is that it yields the exact analytic solution under approximation A in the form of complex integrals. The treatments of Bhabha and Heitler, and Carlson and Oppenheimer yield approximate series solutions.
- 14.
Single, plural and multiple Coulomb scattering occur, however, single large-angle scattering events are relatively rare whereas narrow-angle multiple scattering accounts for the bulk of the events.
- 15.
We use N for the total number of particles and photons and N e for the number of electrons (negatrons and positrons) only.
- 16.
- 17.
For further details concerning the Molière unit see Chap. 21.
- 18.
- 19.
For the definition of the Moliére radius see Sect. 4.6.11.
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Grieder, P.K. (2010). Electromagnetic Interactions and Photon–Electron Cascades. In: Extensive Air Showers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76941-5_4
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