impacts on the area A, P 2 the probability that there is a target particle in region A, and
P int the probability that occur the specific interaction being studied.
If the distributions of the particles are sufficiently sparse, P 1 and P 2 are proportional to the average number N
0
i (i ¼ 1, 2) of particles in this area:
P 1 ¼ N
0
1
=N1 ¼ n 1 A=N1 1 : incident
ð
Þ
P 2 ¼ N
0
2
=N2 ¼ n 2 A=N2
2 : target particle
ð
Þ
where n 1 and n 2 represent the surface densities in a projection transversal to the axis
of the beam.
The two particle will certainly collide when b
R 1 + R 2 and
σ tot ¼ AP int ¼ π(R 1 + R 2 )
2 . This result is valid for any reference system and can
be used for a geometrical interpretation of the concept of cross section. When
R 1 ( R 2 , then σ tot % πR 2
2 , that is the projection of body 2 on a plane transversal
to the direction of the collision, independently by the shape of body 2.
The histogram in Fig. 10.1b represents the angular distribution of the results of
collisions in the second example we discussed; the ith bin of the histogram contains
the number N i of the outcomes for which the scattering angle is included between the
extremes (θ, θ + Δθ) of the bin itself. Thus the cross section for the collisions with a
scattering angle at such an interval is: σ i ¼ AP i % AN i /(N i A) ¼ N i /n i . This value
depends on the bin width Δθ.
To avoid such dependency, the differential cross section is introduced:
dσ
dθ
¼ lim
Δθ!0
σ
Δθ
¼ lim
Δθ!0
N i
h i
n i
1
Δθ
It is convenient to define differential cross section with respect to the solid angle:
dσ
dΩ
¼ lim
ΔΩ!0
N i
h i
n i
1
ΔΩ
ð3Þ
where ΔΩ ¼ ΔcosθΔϕ and ϕ is the azimuthal scattering angle, with respect to the
axis of the beam.
The differential section for the scattering of the material point on a sphere with
radius R can be calculated considering the number dN 1 of incident bodies which
have a collision parameter included between b and b + db: dN 1 ¼ n 1 (2πbdb), all
scattered at the same angle θ, which is linked to b by: b ¼ Rcos(θ/2). Substituting in
the equation of the cross section, we derive:
dσ
dΩ
¼
R
2
4
which is independent from θ and obviously coincides with the area of the section of
the sphere.
108
M. Michelini
P int the probability that occur the specific interaction being studied.
If the distributions of the particles are sufficiently sparse, P 1 and P 2 are proportional to the average number N
0
i (i ¼ 1, 2) of particles in this area:
P 1 ¼ N
0
1
=N1 ¼ n 1 A=N1 1 : incident
ð
Þ
P 2 ¼ N
0
2
=N2 ¼ n 2 A=N2
2 : target particle
ð
Þ
where n 1 and n 2 represent the surface densities in a projection transversal to the axis
of the beam.
The two particle will certainly collide when b
R 1 + R 2 and
σ tot ¼ AP int ¼ π(R 1 + R 2 )
2 . This result is valid for any reference system and can
be used for a geometrical interpretation of the concept of cross section. When
R 1 ( R 2 , then σ tot % πR 2
2 , that is the projection of body 2 on a plane transversal
to the direction of the collision, independently by the shape of body 2.
The histogram in Fig. 10.1b represents the angular distribution of the results of
collisions in the second example we discussed; the ith bin of the histogram contains
the number N i of the outcomes for which the scattering angle is included between the
extremes (θ, θ + Δθ) of the bin itself. Thus the cross section for the collisions with a
scattering angle at such an interval is: σ i ¼ AP i % AN i /(N i A) ¼ N i /n i . This value
depends on the bin width Δθ.
To avoid such dependency, the differential cross section is introduced:
dσ
dθ
¼ lim
Δθ!0
σ
Δθ
¼ lim
Δθ!0
N i
h i
n i
1
Δθ
It is convenient to define differential cross section with respect to the solid angle:
dσ
dΩ
¼ lim
ΔΩ!0
N i
h i
n i
1
ΔΩ
ð3Þ
where ΔΩ ¼ ΔcosθΔϕ and ϕ is the azimuthal scattering angle, with respect to the
axis of the beam.
The differential section for the scattering of the material point on a sphere with
radius R can be calculated considering the number dN 1 of incident bodies which
have a collision parameter included between b and b + db: dN 1 ¼ n 1 (2πbdb), all
scattered at the same angle θ, which is linked to b by: b ¼ Rcos(θ/2). Substituting in
the equation of the cross section, we derive:
dσ
dΩ
¼
R
2
4
which is independent from θ and obviously coincides with the area of the section of
the sphere.
108
M. Michelini