Compton scattering is interesting as to completely explain the
effect requires
quantum physics and
relativity.
When
Arthur Holly Compton first conducted his experiments into this
effect in 1923 at
Washington University,
St. Louis relativity and
quantum physics in particular were fairly new theories, perhaps not completely accepted...
The experiment's carried out by Compton however
proved that light must be
quantised and exist as '
particles' called
photons.
A diagram of the
apparatus he used is shown below :
Detector
______
 
__..__
/
/ Scattered XRays
   /
   ***/
XRays~~~~~~~~~~~~~~~~~~>***>
   ***
   Graphite
Collimating Target
Slits
The
Xray beam hits the
graphite target,
electrons in the
sample
scatter the the photons in all directions. Both the
intensity and
the
wavelength of these xrays are measured at various
angles.
The results he obtained were that the scattered xrays had
two
different wavelengths and different intensities. Shown below is a
diagram of the result for an angle of 135
^{o}, relative to the
incident beam. You can see that the
wavelength of the xray has been
shifted upwards by about 5
pico meters. In fact Compton found, the
larger the scattering angle, the larger the
shift in
frequency.
For phi (scattering angle) = 135^{o}

I :
N :
T : : :
E : : :
N : : : :
S : : : :
I : : : :
T : : : :
Y ..: :..: :..
_________________________
70 75
Wavelength in Picometers
This result
cannot be explained if the xray is treated as a
wave; if you did, then the wave would hit the electrons, causing them to
oscillate at the frequency of the beam. The electrons would act like an
antenna and
radiate the
energy they
absorbed from the beam
at the
same frequency as the incident beam.
If you treat (as Compton did) the xrays and electrons as being particles,
then they
interact through
collisons, the scattering being like
snooker balls colliding at various angles. As the electron therefore
picks up
kinetic energy from the photon, the photon must loose kinetic
energy and therefore must in turn have a longer wavelength. Which is of
course exactly what he observed....
To model the system accurately, you have to take relativity into account
as the electrons may be hit so hard they are accelerated to near the
speed of light.
Now comes the maths of the derivation of the final equation...
Using the principle of the
conservation of energy the system consists
of three components, the incoming xray radiation, (energy
hf), the
outgoing xray radiation, (
hf') and the electron which has the
energy given by the term on the right, after the '+'.
hf=hf' + mc^{2}(1/sqrt(1(v/c)^{2}) 1)
Where
h is
Plank's constant,
f the frequency, m
the
mass of the electron, v the
velocity of the electron and c the
speed of light.
You can subsitute c/
lambda for
hf (
lambda being the
wavelength
of the photons), which gives :
h/lambda=h/lambda' + mc^{2}(1
/sqrt(1(v/c)^{2}) 1)
The
momentum,
p of the electron is given by :
p =
mv/sqrt(1(v/c)
^{2})
Now the collision of the photon with the electron looks a bit like this
:
y Photon (lambda')
 /
 / ^
 / :Angle phi
/ v
~~~~~~~~>* x
Photon \ ^
(lambda)  \ :Angle PHI
 \ v
 \
* Electron
The electron is scattered off at angle
PHI, and the photon with
angle
phi and wavelength
lambda'. Using a vector based
description of the
conservation of
momentum for this system gives :
For the x component....
(h/lambda) = (h/lambda') cos phi+
mv/sqrt(1(v/c)
^{2}) cos
PHI
....and for the y component....
0 = (h/lambda') sin phi 
mv/sqrt(1(v/c)
^{2}) sin
PHI
Because we are actually only interested in the photon, v and
PHI
which deal with the electron can be eliminated, which leads to the
following expression for the Compton shift :
h
delta lambda =  (1  cos phi)
mc
If you examine the above
equation you can see that the shift
only
depends on the scattering angle, and not on the energy of the incident
photon. All the above depends of the electrons in the sample being
'
free', that is able to be knocked out of the sample. If the electron
is too tightly
bound, then you're really trying to move not just the
electron, but the whole
carbon atom, which is 22,000 times heavier.
This makes the Compton shift due to
bound electons too small to see.
Which explains why there is two peaks in the
spectrum above, one due
to free electrons, and one due to bound electrons.
The huge
EMP pulse generated by a
nuclear bomb, which can knock
out
electronic equipment is caused by the Compton effect. When the
bomb denonates, radiation is generated across a huge
range of frequencies, all the way up to
gamma rays. The collisons
between these high energy photons with electrons in the atmosphere cause
a huge movement of charge, which in turn set up large
electromagnetic
fields.