1. Lecture 5. Particle Properties of Waves (cont’d)
Outline:
Compton Effect
Pair production
Doppler Effect (photon-based derivation)
Photons in Gravitational Field
Numerous processes of interaction of photons with matter:
Rayleigh Scattering
Raman Scattering
Photoelectric Effect
Compton Scattering
Pair Production
etc., etc., etc.
Compton Scattering

2. Photoelectric Effect vs. Compton Effect
Photoelectric Effect: absorption of a photon by an “electron-metal” system, demonstration of Eph=hf (but not pph=ħk, recall why).
after
before
Compton Effect: scattering of a photon by a “free” electron, demonstration of both Eph=hf and pph=ħk. t

3. target (light atoms, e.g. graphite)
Compton Effect (1922)
X-ray detector
X-ray tube
The spectrum of scattered X-rays contains not only the original line, but also a shifted line. The wavelength shift (Compton shift)  depends on  but does not depend on the target material and the initial X-ray wavelength . e-
target (light atoms, e.g. graphite)
The effect could not be explained within the framework of classical physics. Compton (and Debye) successfully explained the effect by considering a single act of elastic collision between an X-ray photon and a free electron.
To observe the effect, the energy transfer from a photon to an electron should be relatively large for two reasons: (a) the wavelength shift should be measurable, and (b) to consider electrons bound in atoms as free particles, the transferred energy should exceed by far the ionization energy of light atoms (e.g.,13.6eV for hydrogen atoms).
For a large energy transfer in an elastic collision between a photon and an electron, the photon energy (and, thus, its “mass” Eph/c2) should be large (not necessarily comparable to me0c2, but a reasonable fraction of it).

4. Elastic collisions (non-relativistic treatment)
“lab” RF=“rest-of-M-before-collision” RF (K)
Energy transfer
in elastic collisions:
center-of-mass RF (K’)
(We need to calculate the “recoil” speed of M after the collision, )
before
time before collision
after
CoM
CoM
In the center-of-mass RF:
relative velocity of K’ with respect to K: t K K’

5. Elastic collisions (non-relativistic treatment, cont’d)
Back to the “lab” RF:
after
Max energy transfer (percent-wise) – if M = m
before
after
Conclusion: for a large energy transfer (and, thus, a measurable wavelength shift), the photon energy should be much higher than that in the visible range – X-rays.

6. Compton Effect (cont’d)
Energy and momentum conservation:
- Compton wavelength of the electron

7. Compton Effect (cont’d)
The Compton wavelength of a scattering particle is inversely proportional to its mass – e.g., for a proton,
- a photon with Compton wavelength would have an energy = the rest energy of the scattering particle
The Compton wavelength of a particle is the length scale at which relativistic quantum field theory becomes crucial for its accurate description.
Experimental consequences:
- indeed, the wavelength shift is independent of the target material and the initial photon wavelength.
- cannot be observed in the visible range (insufficient accuracy and non-free electrons)
Non-shifted spectrum line – due to scattering by atoms (mat>>me, Cat<< Ce)
The increase of intensity for larger  - more electrons can be considered as free (E increases with )

8. Problem
An X-ray photon with =60pm is scattered over 1500 by a target electron.
Find the change of its wavelength.
Find the angle between the directions of motion of the recoil electron and the incident photon.
Find the energy of the recoil electron.
(a)
momentum conservation:
(b)
along y axis:
along x axis:
(c)

9. Problem
An X-ray photon with wavelength 0.8nm is scattered by an electron at rest. After the scattering the electron recoils with a speed equal to 1.4106m/s (non-relativistic case).
NOTE: IN THE FOLLOWING, DO THE CALCULATIONS TO AT LEAST 4-DECIMAL PLACE ACCURACY
(a) Calculate the energy of the scattered photon in units eV (use conservation of energy).
m/s
(b) Calculate the Compton shift in the photon’s wavelength, in meters.
(c) Calculate the angle through which the photon was scattered.
(a)
(b)
(c)

10. Energy transfer in Compton Scattering
The maximum energy transferred from a photon to an electron:
Let’s calculate the max energy that a 20keV-photon can transfer to an electron:
- much greater than the ionization energy for light elements.

11. The range of photon energies for Compton Effect observation
“free” electrons
e- - e+ pair production
Atomic number of the target
Photon energy (MeV)
Pair Production: (conversion of e.m. energy into the rest energy)
the nucleus takes care of momentum conservation, the recoil energy of a (heavy) nucleus – negligibly small
The Compton wavelength can be thought of as a fundamental limitation on measuring the position of a particle by shining light on it. Measuring the position accurately requires light of shorter wavelength, and, thus, higher energy. If the photon energy exceeds 2mc2, it would have enough energy to create a pair of similar particles, which would make difficult to find the original particle's location (more on this issue – after discussion of the uncertainty principle).

12. Problem
Show that a photon after a head-on collision with an electron cannot create an electron-positron pair, no matter how high was its initial energy.
On the other hand, the threshold for pair production:
Q.E.D.

13. Inverse process does not require a nucleus
e- - e+ annihilation e-
Inverse process does not require a nucleus e+
Beiser, A positron with a kinetic energy of 2MeV collides with an electron at rest and the two particles are annihilated. Two photons are produced; one moves in the same direction as the incoming positron and the other moves in the opposite direction. Find the energies of the photons.
energy conservation: e+ e-
before
after
momentum conservation:

14. Doppler Effect (re-derivation using the concept of phonons)
Let’s consider the source of photons that moves toward the observer with speed V.
“source”
“observer”
the source is macroscopic, its recoil energy is negligibly small
In the RF where the source was at rest before emission:
(“primed” – after emission)
again,
the source is macroscopic, so K  K’
In the RF where the observer is at rest:
energy of the source before emission
energy of the source+photon after emission

15. Gravitational Red Shift
The photon “inertial mass”:
A photon is not accelerated by gravitational field (it always travels with the speed of light), rather, the photon’s energy changes by the amount that corresponds to the change of the potential energy of a mass m in the gravitational field. For a uniform field:
In the early 60's, Pound, Rebka, and Snyder at Harvard measured the shift using a tower just 22.6 meters tall:
hf=14.4 keV
The tool to measure this tiny shift: Mössbauer effect . The result of measurements was within 1% of the predicted shift!
The red shift of a photon that escapes from a white dwarf of a mass M (~ the solar mass) and radius R (~ the Earth’s radius):

16. HW 3 Homework #3: Beiser Ch. 2, Problems 11,12,14,15,26,29,32,34,39,53
Apply relativistic approach only when it is necessary!