13 Scattering From Central-Force Fields

Ashok Goyal

1. Introduction

In the previous unit we considered elastic collision between two particles in the C.M. frame and in the Laboratory frame. We saw that a lot of information can be obtained from the consideration of laws of energy and momentum conservation. We defined scattering cross – section and expressed it in the presence of a force – field. In this unit we will obtain the transformation of scattering cross – sections for some specific central force fields including constant potential well. We will derive expression for the Rutherford scattering cross–section.

2. Transformation from the Lab to C.M. Frame

The number of particles scattered into a unit solid angel is invariant and thus remains the same irrespective of the frame of reference used. In the C.M. frames, we have

3. Rutherford Scattering

One of the most important scattering formulae in physics is the Rutherford scattering formula.Rutherford scattering involves scattering of charged particles in an electrostatic field. Historically Rutherford scattering was performed by colliding a beam of positively charged α-particles on to electrostatic field provided by the positively charged gold nuclei. The potential for the case can be written as

This is the celebrated Rutherford Scattering formula and says that the scattering cross-section in the Centre of Mass frame varies as inverse fourth power of . The scattering cross-section is independent of the fact whether the force is attractive or repulsive since it is proportional to .

4. Scattering by a Hard Sphere

A beam of particles is incident on a perfectly elastic rigid sphere of radius R. The particles will rebound from the surface of the sphere such that the angle of incidence is equal to the angle of reflection. The force-field potential in this case can be considered as

An incident particle of mass  travels in a straight line and enters the potential well with a velocity .Inside the well it moves under the force field and let its velocity be inside.By the law of conservation of energy, the sum of the kinetic and potential energies inside and outside the well are equal. Thus

Where E is the energy of the incident particle at infinity. The problem has spherical symmetry i.e. the potential energy depends only on the coordinate r. Thus the momentum at all points on the spherical surface of any radius has its component along the surface same on either side of the well. Thus if the incident particles enters the well at an angle α and is refracted by an angle where the angles are subtend by the particles with respect to normal at the surface, then

6. Summary

• For the elastic scattering of a particle frame is half of the scattering angle in of mass the CM 1 on frame for 2 , the = 1 scattering angle in the Lab 2.
• For scattering from a fixed centre, the CM scattering angle is equal to the Lab scattering angle.
• Rutherford scattering is independent of the fact whether the force is attractive or repulsive.