Activity 1 - Compton Scattering
This example should predominantly use maths and physics that is familiar to you, and give
an example of the power of vectors in physics as well as being an interesting real world
example of quantum physics.
We will use energy conservation, momentum conservation,
,
which you should be familiar with, and the relativistic equation for the energy of a particle,
which may be new to you:
,
√
which replaces our classical kinetic energy equation. We don’t need to know where this
comes from for this problem, but if you are interested we point you to:
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html#c4 which gives a good
explanation.
Compton scattering is where high energy photons are scattered by electrons, and lose some
of their energy (changing their wavelength) in the process. It turns out that classically
photons can be scattered, but they don’t lose energy. To explain the observed change in
wavelength, we need to go quantum!
We want to find an expression for the change in wavelength:
We will treat this
as a derivation, not a worked example: you are not expected to be able to derive this
yourself. If you want to try however it would be excellent practice, and we encourage you to
give it a go, but not to get disheartened if you don’t quite make it.
eWe have a vector equation for momentum:
⃗⃗⃗⃗
⃗⃗⃗⃗⃗ ⃗⃗⃗⃗
(1)
and a scalar energy equation:
(2)
√
We note that the electron is stationary on the left hand side of the equation but moving on
the right hand side after the photon (light packet) has hit.
Here and throughout these pages we use the notation:
.
We now subtract ⃗⃗⃗⃗⃗ from both sides of the momentum equation (1), and take the vector dot
product of each side with itself:
⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
⃗⃗⃗⃗ ⃗⃗⃗⃗
⃗⃗⃗⃗ ⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
⃗⃗⃗⃗ ⃗⃗⃗⃗ ,
The rest of the problem is just algebra! We square both sides of the energy equation (2):
Substitute in
from the momentum equation and use the de Broglie relation to substitute in
the photon momenta:
,
Make sure you work through this algebra yourself.
We could stop here, but we wanted the answer in terms of wavelengths, so finally:
You see that just using physics you already know we are able to derive very interesting and
unexpected results, which explain real experiments.
Some background to Compton Scattering and a Java applet: http://www.ndted.org/EducationResources/CommunityCollege/Radiography/Physics/comptonscattering.ht
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