Elements of Modern Physics
50
(impact parameter is the distance of the nucleus from the initial direction of
motion of the α particle) of b to b + db be scattered into angles between θ and
θ + dθ (Fig. 2.6). Then, the fraction of α particles scattered into angles between
θ and θ + dθ is
2
dN
bdb
n
N
A
π
=
2
db
tb
d
d
= π ρ
θ
θ
(2.51)
Fig. 2.6 Scattering of α particles by a nucleus of charge Ze.
For evaluating this expression, a relation between b and θ is needed, which
depends on the force experienced by the α particles. Specifically, the force due
to a charge Ze of the nucleus is considered. It is shown in Example 6, that for a
Coulomb potential 2Ze
2
/4πε 0 r, the relation between b and θ is
2
2
0
cot
2
Ze
b
mv
θ
 
=
 
2
πε
 
(2.52)
where m is the mass of α particle and v is its speed. Using this relation in
Eq. (2.51), the fraction of particles scattered into solid angle dΩ = 2π sin θ dθ,
is
2
2
2
4
0
4
sin 2
dN
Ze
d
t
N
mv


Ω
= ρ 

θ
 
πε


 
 
(2.53)
This is the Rutherford formula for the scattering of α particles by nuclei of
change Ze.
The important features to note in this formula are that the fraction is
(i) proportional to the thickness t of the metal foil, (ii) proportional to Z
2
,