The energy loss due to electronic collisions is given by Bethe’s formula which reads, to a very good approximation,

with A, Z and p respectively the mass number, the charge and the mass density of the target nucleus and Zp, ft and E respectively the charge, velocity and energy of the projectile. The constant D = 0.3071 MeV cm2/g; I is the average ionization potential of target atoms, with approximate value I(Z) = 16Z0 9 eV. Also mec2 = 0.511 MeV.

Bethe’s formula does not let us obtain an analytic expression of the projectile range. A common approximation, which allows reasonable proton range estimates, is

Note that the stopping power per atom is proportional to Z, and indepen­dent of A and p. For a 1 GeV proton one gets an electronic range

A

Rel(E = 1 GeV) = 205 . (6.6)

PZ

Taking the examples of beryllium and lead we get, for a 1 GeV proton,

• for beryllium: Rel(E = 1 GeV) = 250 cm

• for lead Rel(E = 1 GeV) = 45 cm.