While being slowed down, protons may undergo nuclear reactions. For proton energies larger than, typically, 100 MeV, the most violent reactions are called spallation. These account for most of the neutrons produced.

In a crude, black nucleus model, the reaction cross-section reads

°(е )=^^1 — — E (6.7)

with the geometrical cross-section

T = ^(1.3A1/3 + 1)2 barns.

Vc is the coulomb barrier:

1.44Z A + Ap

1.3A1/3 + 1 A

With these expressions it is possible to derive a nuclear range. For high — energy protons, the cross-section reduces to the black nucleus value, and thus the nuclear range reads approximately

Thus, for

• beryllium Rnuc = 35 cm

• lead Rnuc = 16 cm.

The probability that 1 GeV protons suffer nuclear reactions is very high both for beryllium and for lead. The nuclear range is smaller, relative to the electronic range, for light nuclei. On the other hand, the energy depos­ited in the target nucleus following a nuclear encounter is larger for heavy targets. In a simple forward scattering picture (a la Glauber) one expects that the number of target nucleons hit, and hence the energy deposited in the target nucleus, is proportional to the target thickness, i. e. to A1/3. It follows that the ratio of nuclear energy loss to electronic energy loss scales like (A/Z)E0:75.

The over-simplistic considerations we have just made are only intended to give a feeling of the physics of the interaction of high-energy protons with nuclei. It showed that the proton energy should be chosen high enough that nuclear energy losses exceed electronic energy losses. A more detailed treat­ment requires nuclear cascade simulations.