12 Elastic Scattering : Scattering Cross-Section

Ashok Goyal

1. Introduction

Analysis of data obtained through scattering of particles on fixed or moving targets has emerged as one of the most powerful tools used by particle physicist to unravel the nature of fundamental interactions and the existence of hitherto unknown particles. Large Hadron Collider at CERN (GENEVA) is presently the most powerful machine which proved the existence of Higgs Bosons. Historically Rutherford’s experiments on the scattering of α-particles on gold foil established the Atomic Model with a tiny nucleus where practically the entire mass of the atom is concentrated. Existence of the constituents of hadrons namely, quarks interacting through gluons which carry a new quantum number called ‘colour’ was discovered at the SLAC National Accelerator Laboratory in the electron-proton collision. In this and the next unit we will briefly discuss the kinematics of elastic collision and the concept of scattering cross-section.

2. Elastic Scattering

We will consider elastic scattering between two particles of mass  and  respectively. By elastic scattering is meant that the internal energy of the particle if any remains unchanged. In elastic collision thus, the linear momentum and mechanical energy is conserved. In two body problems it is convenient to define the coordinates of the particle with respect to their centre of mass as follows:

The coordinates of the particle with respect to  M are given by

If we define by  and  the initial velocities of the two particle with respect to the coordinates system fixed at O and  and  the velocities in the CM system, then from equation (12.1) and (12.2) by differentiation with respect to time, we get

where  is the velocity of the CM and  and  are the velocities of  and  in the CM frame.The Laboratory frame is defined as one in which the particle  strikes a stationary target . Thus in the Lab Frame: is the velocity of the  and  and  are final velocities i.e. velocity after the scattering of  and  respectively.

CM Frame:

In the CM frame  and  are the initial and  and  the final velocities of  and  respectively.In the centre of mass frame  which implies =0 i.e. the two particles have equal and opposite momentum in the CM frame.The scattering between the two particles in the Lab and CM frame would look like the following. In the Lab frame, the particle  moving with velocity strikes  at rest. After the scattering  and m2 have velocities  and ,  scatters by an angle  with respect to the initial direction of motion and  scatters by an angle  as shown

Scattering in the Lab Frame.

In the CM frame, the particles  and  approach each other with equal momenta and after scattering go in opposite directions with equal momenta such that total initial and final momenta are zero.

Scattering in the CM Frame

The conservation of linear momentum and kinetic energy implies that the CM velocities of and  before and after the collision are equal

Thus the scattering angle in the Lab and CM frames are approximately equal. In this case the particle  essentially scatters from a fixed scattering centre.

and the scattering angle in the Lab frame is equal to half the CM scattering angle. Further since the maximum value of the scattering angle  in the CM frame is 1800 , the scattering angle in the Lab frame for the case  can not be greater than 900 . There is no back scattering. Let us now look at the scattering of the second particle i.e. of the recoil particle ,  scatters by an angle  which satisfies

The energy expressions in the elastic scattering and the relationship between the expressions in the Lab and CM frame can be easily obtained. We will state the results and leave it as a problem to the done by you. The initial energy of the particles is

3. Scattering Cross Section

In a scattering experiment when the article interactions are known in terms of a force-field, it is possible to predict the scattering outcome in terms of the final energies and scattering angles of the particles. We will now investigate the scattering process in the presence of force field. Let us consider the collision in the Laboratory system in the presence of repulsive force between the particles  M1 and M2.

Particle  approaches particle  such that if no force exists between them, the particle will pass through in a straight line with an impact parameter b. The impact parameter being defined as the distance of closest approach between the two particles. If the particle is moving with an initial velocity , the angular momentum of  about  is

Thus if the force field is known that is of the interaction between the particles is specified,then for a given initial energy and impact parameter, the scattering angle and the final energies are calculable. In actual practice we have a beam of particles of mass  and energy  moving towards a collection of particle of mass  each and which are at rest in the Lab. Frame

The negative sign indicates that the scattering angle decrease with increasing impact parameter.

We can write down the change in the scattering angle moving in a central field from the earlier expression obtained in unit 11 namely

4. Summary

• In elastic scattering both the linear momenta and mechanical energy is conserved.
• In the Centre of Mass frame, the sum of the momenta of the colliding particles is zero both before and after scattering.
• If the colliding particles have equal mass, the scattering angle in the Lab. Frame is half of the scattering angle in the Centre of Mass frame.
• The differential cross-section is defined as the number of particles ( ) scattered into an element of solid angle ( ) scattered into element of solid angle Ω per unit flux of the initial particles = 1Ω Ω.
• In a scattering experiment once the particle interactions are specified in terms of a force field, the scattering outcome in terms of scattering angle and final energies of the particles can be predicted.