374 7 Nuclear Physics – I
Ionization
Bethe’s quantum mechanical formula
−dE/dx=(4πz^2 e^4 n/mv^2 )[ln (2mv^2 /I)−ln(1−β^2 )−β^2 ] (7.22)wheren=number of electrons/cm^3 ,I=ionization potential,v=βcis the particle
velocity and ze is its charge, m is the mass of electron
Note that−dE/dxis independent of the mass of the incident particle (Fig. 7.5).
Fig. 7.5Ionization
(−dE/dx) versus particle
energy
Range–Energy-relation
E=kz^2 nM^1 −nRn (7.23)wherekandnare empirical constants which depend on the nature of the absorber,
Mis the mass of the particle in terms of proton mass.
If two particles of massM 1 andM 2 and atomic numberz 1 andz 2 enter the
absorber with the same velocity then the ratio of their ranges
R 1 /R 2 =(M 1 /M 2 )(z 22 /z 12 ) (7.24)Range in air – Geiger’s rule
R=const.v^3
R= 0. 32 E^3 /^2 (alphas in air) (7.25)
Valid for 4–10 MeVαparticles.Ris in cm andEin MeVThe Bragg–Kleeman rule
IfR 1 ,ρ 1 andA 1 are the range, density and atomic weight in medium 1, the corre-
sponding quantitiesR 2 ,ρ 2 andA 2 in medium 2, then
R 2 /R 1 =(ρ 1 /ρ 2 )(A 2 /A 1 )^1 /^2 (7.26)