In: Physics
A solid sphere of radius R is found at the origin. Point mass
particles m have elastic collisions with the sphere.
Find a relationship between the (scattering) angle and the impact
parameter to calculate the differential (cross-section) and the
total differential.
The point mass particles m have elastic collisions with the sphere as given can be represented as shown in fig.1 below. R is the radius of the solid sphere.
Now using geometrical interpretation and
here is the
scattering angle.
Therefore b can be written as
This above equation is basically needed to calculate the scattering
angle based on the impact parameter in 2-Dimension.
But for 3-Dimension is replaced by
and measurement
will be done in steradiun unit.
the basic calculation in between 2D angle (degree) and the 3D solid
angle (steradiun) are:
where s is the curvature of the circumference of
the circle and r is the radius of that
circle.
on the other hand,
where A is the surface area of the sphere and
r is the radius of the sphere.
Now as we know that
is
the small volume in the spherical coordinate system
and
Therefore
The total cross sectional area can be found by integrating a
differentiatd small area over all the
solid angle for 3D as given below
if we change the impact parameter slightly it will be like
Using Taylor's series approximation the above equation can be
written as
Therefore,
Now the differentiated cross section is shown in fig.2
Now, as
Integrating this equation for all ()
In this method neglecting the -ve sign the magnitude of the
integration is nothing but the total cross section area.
The total cross sectional area can be found also by integrating
over the complete solid angle as given
below:
Here
is the differential scattering cross section.
Ongoing further calculation the total scattering cross section will
be as
calculated before.