Question

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.

Answer 1

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.