The Physics of Spallation Processes

3.2. CALCULATIONS OF HADRONIC SHOWERS 33further hadronic interactions once the π ± are created. At still higher energies, well above10 GeV, other meson production channels open up which in addition deplete the cascadeof energy 2 . Using Monte-Carlo methods, the spatial distribution of such hadronic showersinside a massive cylindrical Pb-target will be discussed in the following.3.2 Calculations of hadronic showersIn the case of thick targets, the reaction scenario includes secondary and higher-orderreactions induced by the reaction products themselves and, therefore, the calculationsmust include a 3-dimensional simulation of inter-nuclear cascades. Such a 3-dimensionaldescription of the propagation of the inter-nuclear cascade and the transport of particles inthick targets is a rather complex problem that involves various boundary conditions. Thisissue is addressed in the following, where the propagation of various species of particletypes (p, n) is considered separately in longitudinal and radial directions. The energylosses of high-energy particles (≥ 1 GeV) traveling through matter are mainly determinedby the production of secondary particles and not due to electronic stopping which isdominating at lower bombarding energies. Thus, the main feature of the cascade is aninitial increase of the particle intensity with depth and time. As already mentioned, ifthe energy of the produced secondary particles is high enough, they in turn knock outadditional particles. There exists however a physical limit for the development of furthercascades, because the initial energy of the primary particle is distributed among theproduced particles. Therefore, the multiplicities tend to decrease again during the cascadeprocess and fade away because the average energy of the cascade particles decreases anda greater fraction of the individual particle energy is now dissipated by ionization losses.At the end of the inter-nuclear cascade process, subsequent emission of many low energyparticles, mainly neutrons, takes place, known as evaporation process [Wei40].The development of electromagnetic showers with their principal production processes -bremsstrahlung for electrons and positrons, pair production for photons, becoming energydependent above 1 GeV - are well described by quantum electro dynamics (QED-theory)over a wide energy range. In HERMES this is taken into account by the EGS4 code aswill be discussed in section 4.3. The complexity and entanglement of all intra- and internuclearcascade processes finally causing the production of neutrons requires a complexrecord keeping of all particles actually participating in terms of energy, direction andlocation. The simulated propagation of the three-dimensional hadronic showers followingthe bombardment of cylindrical lead targets of 35 cm x 15 cm (for the length and diameter,respectively) by 0.4, 1.2 and 2.5 GeV protons is illustrated in the contour plots of Fig. 3.3.In the HETC+MORSE (cf.sect. 4.3) Monte-Carlo calculations to produce the datafor Fig. 3.3 the cylindrical target is divided into cylindrical zones of 0.5 cm in radial (r)and 1 cm (z) in longitudinal direction and the tracklength flux 3 [Clo88] is represented.The symmetry axis of the cylinder is oriented in z direction and pointing downstream theproton beam. The tracklength flux of neutrons (left) and protons (right) reflects the radialand longitudinal propagation of particles involved in the intra- and internuclear cascades2 an effect being responsible for the decrease of the number of neutrons plotted in Fig. 7.9 of sect. 7.1.13 tracklength flux as defined in equation 4.2 of sect. 4.1