Module 5 – Radiation and Matter
I.
Sources of Electromagnetic Radiation
Electromagnetic radiation is produced when a charge accelerates.
II.
Bremsstrahlung (German for Breaking) X-Rays
A.
Bremsstrahlung x-rays are the electromagnetic radiation produced when a charged
particle slows down due to collisions with the atoms in the material.
B.
A continuous distribution of wavelengths due to the statistical nature of the
interaction between the incoming charged particle and the atoms of the target
material.
Intensity
Characteristic
X-rays
Cut-off
Wavelength
c
Wavelength
C.
The minimum wavelength (maximum frequency) occurs when ALL of the
particle's Kinetic Energy is given off in a single photon.
D.
Sharp x-ray peaks may also be visible. These are from characteristic x-rays (xrays emitted when bound electrons move from a high energy state to a lower
energy state in the target. These are atomic finger prints. We will talk more about
this later in the course.
E.
X-Ray Machine Equation
V
Target
q
KEi = 0 J
From PHYS1224 and PHYS2424, we know that
ΔKE  KEf  KEi   q Δ V
KE f  q V
For the cut-off wavelength, we have that all of the particles energy is given off to
one single photon
KE f  hc
Using Einstein's energy-wavelength relation for a photon, we obtain the final
equation that
λ min  h c  1240 eV  nm
qV
qV
III.
Linear Momentum of a Photon
Because a photon has no rest mass (energy), you CAN NOT use the formula


pMv
Mo
1  v 
c
2

v
This is true for all particles with zero rest mass!!
A.
Linear Momentum and Energy Formula
p
E
c
Proof:
E o  p c   E 2
“Einstein’s Energy-Momentum Relation”
p c2  E 2
“Since Eo is zero”
2
2
pcE
p
E
c
Q.E.D.
B.
Linear Momentum - Frequency
p
hν
c
Proof:
E
c
p
Eh ν
“From part A”
“Einstein’s Energy Relation for Photons”
hν
c
p
Q.E.D.
C.
Linear Momentum - Wavelength
p
h
λ
Proof:
p
hν
c
νλc
ν
c
λ
p
hc
cλ
p
h
λ
Q.E.D.
“Form Part B”
“Frequency-Wavelength Relationship for a wave”
“Substituting for frequency”
D.
Linear Momentum - Wave Number
p  k
where  
2π
h
and k 
{from PHYS2424}
2π
λ
This formula is the key to quantum mechanics. Louis de Broglie says that it
applies to everything including you!
Proof:
p
h
λ
p
h  2π 


λ  2π 
p
h  2π 
 
2π  λ 
“Result from part C”
p  k
IV.
Compton Effect
A.
The elastic collision between a photon and a free electron.
B.
The classical wave picture of light fails to explain the Compton effect.
C.
The Compton effect provides experimental evidence for Einstein's particle nature
of light assumption and relativity.
D.
All collisions conserve linear momentum and angular momentum
E.
Elastic Collisions also conserve energy.
F.
Experiment
E', p', ', and '
E, p, , and 

-
x

KE
Linear Momentum Equations:
x:
p  p e Cos θ   p Cos  
y:
0   p e Sin θ   p Sin  
Energy Equation
E  E  K
The following equation can be obtained from the energy and linear momentum equations
after a lot of algebra (see the Textbook):
h
λ  λ  λ c 1 cos  where λ c 
 2.46 pm
mo c
If we divide both sides of the equation by hc, we can rewrite the equation in the
following useful form
1  1  1 1 cos  
E E E o
where Eo is the rest energy of an electron.
G.
Useful Facts
1.
λ  λ
"always"
2.
E  E
"always"
3.
K  E - E
4.
"energy difference of photons is gained by electron"
Maximum energy is transferred to the electron when the photon is scattered 180
degrees.
λ  λ  λ c 1  cos (180) 2 λ c
λ max  λ  2 λ c
also
1  1  1 (1  cos(180) )  2
Eo
E E E o
1  1  2  2E  E o
E Eo
E E E o
E E
E  o
2 E  Eo
Eo E
2 E 2  EE o - E o E
2 E2
K max  E - E  E 


2 E  Eo
2 E  Eo
2 E  Eo
V.
Pair Production and Annihilation
Einstein's famous energy equation E = mc2 states that energy and mass are
equivalent. Thus, a photon can also be converted into mass! This process is called
pair production.
Pair Production
Positron
Heavy
Nucleus
Electron
Recoils
1)
A positron (antiparticle of electron) and electron are created. They require
a rest energy of 2 x 0.511 MeV = 1.022 MeV for their creation. Any
excess energy is carried off as kinetic energy.
2)
The process can only occur near a heavy nucleus. This is because the
nucleus is required to conserve conservation of linear momentum. The
nucleus carries away momentum but no appreciable energy.
3)
The positron and electron are emitted at 180 degrees with respect to each
other. The energy of the positron is slightly different than the energy of the
electron due to the different Coulombic interaction with the heavy
nucleus.
Annihilation
An electron can also combine with a positron to convert matter into energy. In
this process the electron and positron are converted into two 0.511 MeV photons
that are emitted 180 degrees apart. Although it is possible for more than 2 photons
to be emitted, statistically 2 photons dominate the process (see Eisberg and
Resnick).
Positron
Electron
It is difficult to obtain and trap antimatter in a laboratory. When a positron is
created through pair production in a gamma detector, it will soon lose its kinetic
energy due to electromagnetic interactions. The positron will then annihilate with
an electron inside the detector creating two annihilation gamma rays.
By using ingenuity, physicists have successfully trapped anti-particles. Rubia and
Van de Meer won the Nobel prize for finding intermediate vector bosons that
mediate the weak nuclear force using the collision of a proton and anti-proton
beam. PAS (positron annihilation spectroscopy) is a powerful materials analysis
technique for studying defects in semiconductors using positron sources.
VI.
Gamma Ray Spectroscopy Using a Scintillator Detector
Electrical circuits deal with three four physical quantities: voltage, current,
charge, and time. Thus, it is necessary for us to convert other physics quantities
like energy if we are to take advantage of modern electronics for analysis and
detection. Let us consider the problem of detecting a 4.4 MeV gamma ray emitted
from a carbon nucleus as shown below:
Voltage
Out
Photomultiplier
Scintillator Crystal
Photocathode
High
Voltage
Input
Photocathode:
The photocathode converts very low energy light photons to electric currents by
the photoelectric process. However, the incoming gamma ray has a very poor
probability of interacting with the photocathode.
Scintillator Crystal:
To improve the detector's detection efficiency, a crystal containing a high Z
element like I or Bi is used to convert the high energy photon into lower energy
photons. The incoming photon can give energy to the electrons in the crystal in
one of three ways: photoelectric, compton, or pair production. The electrons and
positrons will then give their energy to the atoms of the crystal through
electromagnetic interactions. The atoms of the crystal then release their energy as
low energy photons to the photocathode.
Photomultiplier:
As described in the book, the photomultiplier is used to amplify the electron
current from the photocathode so that it will be large enough to distinguish from
electrical noise.
Carbon Gamma Ray Spectrum (NaI Detector)
Intensity
VII.
Energy
X-rays & Gamma Rays Interacting With Matter
1.
An x-ray is a short wavelength photon emitted by an electron or charges particle.
A gamma ray is a photon emitted from the nucleus of an atom. Generally gamma
rays are more energetic, but this doesn’t have to be the case.
2.
X-rays generally interact with matter through the photoelectric effect if the energy
is a few keV or less. The probability for this interaction (cross section) depends
5
on the atomic number of the target to the fifth power. σ  Z
3.
X-rays and gamma rays with energies from 10 keV to a few MeV primarily
interact with matter through Compton scattering.
4.
Gamma rays and x-rays with energies more than a few MeV primarily interact
with matter through pair production.
5.
The intensity of a beam of photons passing through a material decreases
exponentially with thickness of material.
I  I o e  x
x
Io
I
Because the photons interact with a greater probability at the surface, they are less
efficient for fighting deep tumors and cause more side effects.