130_notes.dvi

30 Scattering

This material is covered inGasiorowicz Chapter 23.

Scattering of one object from another is perhaps our best way ofobserving and learning about the
microscopic world. Indeed it is the scattering of light from objects and the subsequent detection of
the scattered light with our eyes that gives us the best informationabout the macroscopic world.
We can learn the shapes of objects as well as some color propertiessimply by observing scattered
light.

There is a limit to what we can learn with visible light. In Quantum Mechanics we know that
we cannot discern details of microscopic systems (like atoms) that are smaller than the wavelength
of the particle we are scattering. Since theminimum wavelength of visible light is about
0.25 microns, we cannot see atoms or anything smaller even with the use of optical microscopes.
The physics of atoms, nuclei, subatomic particles, and the fundamental particles and interactions in
nature must be studied by scattering particles of higher energy than the photons of visible light.

Scattering is also something that we are familiar with from our every day experience. For example,
billiard balls scatter from each other in a predictable way. We can fairlyeasily calculate how billiard
balls would scatter if the collisions were elastic but with some energy loss and the possibility of
transfer of energy to spin, the calculation becomes more difficult.

Let us take the macroscopic example ofBBs scattering from billiard ballsas an example to
study. We will motivate some of the terminology used in scattering macroscopically. Assume we fire
a BB at a billiard ball. If we miss the BB does not scatter. If we hit, the BB bounces off the ball
and goes off in a direction different from the original direction. Assume our aim is bad and that the
BB has a uniform probability distribution over the area around the billiard ball. The area of the
projection of the billiard ball into two dimensions is justπR^2 ifRis the radius of the billiard ball.
Assume the BB is much smaller so that its radius can be neglected for now.

We can then say something about the probability for a scattering tooccur if we know the area of
the projection of the billiard ball and number of BBs per unit area that we shot.

Nscat=

N

A

πR^2

WhereNis the number of BBs we shot,Ais the area over which they are spread, andRis the
radius of the billiard ball.

In normal scattering experiments, we have a beam of particles andwe know the number of particles
per second. We measure the number of scatters per second so wejust divide the above equation by
the time periodTto get rates.

Ratescat=

N

AT

πR^2 = (Incident Flux)(cross section)

Theincident fluxis the number of particles per unit area per unit time in the beam. This isa
well defined quantity in quantum mechanics,|~j|. Thecross sectionσis the projected area of the
billiard ball in this case. It may be more complicated in other cases. Forexample, if we do not
neglect the radiusrof the BB, the cross section for scattering is

σ=π(R+r)^2.