Compton effect
Lecture notes for the exercises class Struttura della Materia
Oct. 12, 2001
Key concepts
-
Electromagnetic waves scatter elastically and inelastically against "free"
electrons.
-
The inelastic scattering is known as
Compton scattering.
(the elastic part is called Rayleigh scattering, and would not exist at all
if the electrons were actually free, i.e. not bound to some atom).
-
The "surprising" fact about Compton scattering is that it is successfully
described in a relativistic collision formalism treating the
electromagnetic field as point particles (photons) of energy hν =
hc/
λ and of
momentum h
ν/c = h/λ.
-
The relativistic scattering problem is fully described by equating the
total energy-momentum 4-vector before and after the scattering.
Call θ the angle between the 3-momentum of the outgoing and incoming
photons, and φ the angle between the 3-momentum of the outgoing
electron and that of and incoming photons.
By solving a system of 3 coupled equations, one finds that the photon
wavelength after the scattering changes as
λ'
-
λ
=
λc
(1-cos θ),
where
λc
= h/mec = 2.42631 pm.
Exercises
The following exercises should be solved to check one's own understanding
of the subject and in training to pass successfully the written test.
-
Verify that (in the notation above)
cot(θ/2) = (1 + λc/λ) tan(φ)
Hint: substitute tan(φ) into the right term, obtaining it
from the space components of the momentum conservation eq., to get
sin θ/(1 - cos θ), then use standard trigonometric
manipulation...
-
A
γ ray of
wavelength 6.2 pm is incident on an electron initially at rest. The
electron is observed to recoil with kinetic energy 60 keV.
Calculate the energy of the scattered
γ ray (in
keV) and determine the direction in which it is scattered.
RESULT:
139.975 keV; 1.66628 rad = 95.47 degree.
Trivial(?) questions
-
Is Compton effect easier to observe with infrared, visible, ultraviolet or
X-ray light? Why?
-
Why is the Compton shift independent of the scattering material?
Comments and debugging are welcome!