Consider the scattering of a particle by a central potential. We define the differential scattering cross section
through the following expression:
# of particles scattered into the solid angle dW per unit time = Is(W)dW ,
where I is the intensity of the incident beam, i.e the number of beam particles per unit area per unit time. For a central potential s(W) is independent of f. We write
.
The number of particles scattered through an angle between q and q +dq per unit time is
.
We define the impact parameter b through
,
where M is the angular momentum and v0 is the incident speed at infinite distance. Once E and b are fixed, the scattering angle is uniquely determined.
.
In a central potential the motion is in a plane and M and E are constant.
From
we find
.
Let q be the angle between the incident and the scattered direction and f0 be the angle between r( z=-¥) and rmin. Then
and
determines umax. We have
q =p-2f0 for a repulsive potential,
q =2f0-p for an attractive potential; (or q=p-2f0, q < 0).
If
then
.
.
Rutherford’s formula
The number of particles scattered into a detector is the same in the laboratory and in the CM frame. Therefore
.