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Photons
Concordia College
The Ultraviolet Catastrophe
A blackbody is an idealised object which
absorbs and emits all frequencies.
Classical physics can be used to derive an
equation which describes the intensity of
blackbody radiation as a function of
frequency for a fixed temperature--the
result is known as the Rayleigh-Jeans law.
The Ultraviolet Catastrophe
Although this law worked well for low
frequencies (high wavelength) it did not for
high frequencies (ultraviolet region.)
So shocking was this find, that it was
labelled the Ultraviolet Catastrophe
The horror!
Where did it all go wrong?
The error involved in these shocking
events was the assumption that the
energy levels of the radiation was
continuous.
i.e. any energy level could be obtained.
Max Planck, however, thought differently.
Max Planck
Photons
Max Planck theorised that energy was not
continuous, but came in discrete packets.
For light, we call these photons.
He developed his theory and mathematics
that matched experimental data.
This managed to solve all of our
problems……..
Problem Solved!
………………………….or is it?
New Problems
This created problems more difficult than
the so-called catastrophe.
The implication is that light must be a
particle, if it comes in discrete packets.
But how can we explain all these waverelated phenomena with particles?
Scientists are still working on it…..
Young’s Double Slit
We can arrange Young’s Double slit
experiment so that instead of having a
strong light source, we have a source
which emits just one photon at a
time………….
Look what happens!
Young’s Double Slit
We still get an interference pattern!
Our previous explanation involved one ray
interfering with another.
How can one photon interfere with itself?
Welcome to the strange world of quantum
mechanics……..
Energy of a photon
The energy of a photon is given by:
E  hf
h  6.63 10
34
Js
h is known as Planck’s Constant, named after Max Planck
Photon Energy
This can be re-arranged……
E  hf 
hc

More properties
Another natural consequence of light being a
particle is that it should have similar properties to
particles as well.
One of these is momentum.
It was found that when photons collided with
electrons, they moved in such a way that it
looked like conservation of momentum.
Photon Momentum
Momentum of Photons
The question of giving momentum to
photons, which are massless, seems
difficult.
Einstein came up with the solution.
He said that energy and mass are different
versions of the same thing.
I knew I
shouldn’t have
worn my wife’s
shoes
Einstein
E = mc2
Photon Momentum
We can substitute Einstein’s equation into
our expression for the energy of a photon to
get the ‘mass’ of a photon:
Equating: E  mc2 and E  hf
Give:
hc
mc  hf 
2
Therefore:
h
m
c

Photon Momentum
We know momentum, p =mv
p  mv
For photons, v = c, so….
p  mc
h
p
c
c
p
h

Example
Calculate the energy (in joules and
electron volts) and momentum of each of
the following photons.
a) Orange light of wavelength 600nm
b) An X-ray of wavelength 0.1nm.
The Photoelectric Effect
The photoelectric effect is the ejection of
electrons from the surface of a material
when light of sufficiently high frequency is
incident upon it.
Usually the electrons are emitted from a
metal, and are called photoelectrons.
Example
A laser emitting monochromatic light of
wavelength 780nm, is rated at a power of
0.1mW. How many photons per second is
it emitting?
Photoelectric Effect Setup
Here, a clean metal surface—the cathode—is illuminated with light
from an external source.
If the light causes photoelectrons to be emitted, they will be detected
at the anode. This flow of electrons is called the photoelectric
current, and is registered by a sensitive ammeter.
Photoelectric Effect
The circuit includes a variable voltage supply
which can be used to make the cathode
negative (and the anode positive). When this is
done, the photoelectrons will be helped by the
resulting electric field to the anode, and
a maximum possible current will be measured.
Alternatively, the voltage may be adjusted to
make the cathode positive and the anode
negative—a reverse potential. This
arrangement is used to investigate the kinetic
energy carried by the emitted photoelectrons.
Measuring the Kinetic Energy
- Fix the frequency above the threshold
frequency for electrons to be emitted
- Apply an increasing reverse potential
difference. As the reverse voltage
increases, the current decreases
- A point will arrive when there is no
measured current. This occurs at the
stopping potential, Vs.
Measuring the Kinetic Energy
The kinetic energy of the electrons can
now be measured using: E = eΔV
i.e.
Kmax = eVs
Kinetic Energy of Emitted Electrons
The maximum kinetic energy can be found
to depend on two factors:
- the frequency of the radiation (E=hf)
- the surface from which electrons are
emitted
Free electrons
Conductors have free (de-localised)
electrons.
These are not bound to a specific atom,
but to the conductor as a whole.
It is these electrons which make electric
current possible.
Photoelectric Effect explained
ln order to explain the photoelectric effect, Einstein used
the photon concept that Planck had developed.
He considered that, within the metal, each electron was
bound to the metal by a different amount of energy. (At
that time he knew nothing of electron shells.)
Some electrons required substantial amounts of energy
to become free, while others required less energy.
Einstein was able to represent this situation using a
‘potential well’.
Photoelectric Effect explained
If the y-axis represents
the total energy of the
electrons, the electron
will be bound to the metal
where the energy is
negative, will be freed
where the energy is
positive, and this energy
is kinetic energy.
An electron with zero energy
would be free, but have no
speed.
Photoelectric Effect Explained
- If the frequency is below the threshold frequency, even
the least bound electron will not be emitted. This is
independent of the intensity. E.g. if the minimum energy
is 4.0eV, you can’t get two photons of energy 2.1eV to
emit the electron.
- When the frequency is above the threshold frequency,
the photon can be absorbed, giving the electron more
energy, and it becomes free. E.g. An electron which
requires 3.2eV can absorb a 7.0eV photon, and be
emitted with an energy of 7.0eV – 3.2 eV = 3.8eV
Photoelectric Effect Explained
- Since the photon is absorbed, the
electrons are emitted instantaneously
(almost).
(This is a strong argument for the particle
theory of light)
- We call the minimum energy required to
free the electron the ‘Work Function’
Work Function Values
Metal
Work Function
(eV)
Metal
Work Function
(ev)
Na
2.46
Pt
6.35
Al
4.08
Pb
4.14
Cu
4.70
Fe
4.50
Zn
4.31
Cs
1.90
Ag
4.73
K
2.24
W
4.58
Mo
4.2
Photoelectric Effect Equation
- Radiation of sufficiently high frequency, f,
is incident on a metal surface of work
function W.
- This photon, of energy E, is absorbed by
the electron
- This electron thus leaves with kinetic
energy Kmax = E - W
Photoelectric Effect Equation
K max  hf  W
Or if you like electron volts……
h
K max (eV )  f  W (eV )
e
Photoelectric Effect Graph
Example
Calculate:
a) the threshold frequency of silver if its work
function is 4.73eV;
b) the work function of barium, if its threshold
frequency is 5.985×1014Hz.
Example
The work function of aluminium is 4.08eV.
Determine
a) whether light of frequency 9×1014Hz will
cause electrons to be emitted;
b) the maximum kinetic energy of the emitted
electrons if the metal is irradiated with ultraviolet light of wavelength 2.0×10-7m.
Example
The work function W of zinc is 4.3l eV. A
zinc surface is irradiated with lights of
different frequency. Sketch a graph of the
maximum kinetic energy of emitted
electrons against the frequency of light.
Example
The stopping potentials for electrons emitted from a metal by light of four different
wavelengths are given in the table below
Draw a graph of the maximum kinetic energy of the emitted electrons against
frequency.
From your graph determine the following:
a) Planck’s constant
b) The work function of the metal
c) The threshold frequency of the metal
d) The maximum kinetic energy of electrons emitted by light of frequency
8.0 ×1014Hz.
X-Rays
If electromagnetic radiation can give electrons
kinetic energy, the converse should also be true.
That is, a loss of kinetic energy should produce
electromagnetic radiation.
German Physicist
Wilhelm Roentgen
discovered X-rays
using this principle.
How X-Rays are produced
(Coolidge Tube 1913)
What’s happening in the
Coolidge Tube?
- A high potential difference is applied over
the cathode and anode
- The voltage source for heater heats the
cathode, providing energy to the electrons,
allowing them to leave more easily
- These electrons are torn off of the
cathode towards the anode
What’s happening in the
Coolidge Tube?
- The anode is a good conductor of heat,
and this heat is conducted to the cooling
fins
- The face of the anode has a hard metal
of tungsten. This is where the electrons
strike.
- As the electrons enter the anode, the
electrons encounter the positive nuclei of
the metal
What’s happening in the
Coolidge Tube?
- As the electrons come close the nuclei, they
are attracted to them due to their positive charge
- As they are attracted towards the nuclei, they
no longer want to travel in their initial direction,
so they actually slow down
- Since they slow down, they accelerate and we
know that accelerating charges give off
electromagnetic radiation
- The type of radiation in this case turns out to
be X-Rays
X-Ray Spectrum
(Bremsstrahlung Radiation)
-The spectrum shows a continuous
range of frequencies (this graph
shows wavelength) up to a
maximum
- Each voltage has a cut off
frequency
- As the voltage increases the
maximum frequency (minimum
wavelength) increases
- As voltage increases, so does the
intensity of the emitted X-Rays
Intensity
X-Ray Spectrum
(Bremsstrahlung Radiation)
The spikes occur if the
potential difference is very
high. The location of the
spikes depends on what metal
is used as the anode.
frequency
Intensity
They correspond to certain
electron transitions in metals
wavelength
Why continuous?
- Atoms consist mainly of empty space
- As the electrons enter the cathode they are attracted to towards
the nuclei, but will rarely collide with a nuclei
- This means that there a large number of possible paths of electron
deflection
- Each of these paths correspond to a different acceleration, and
hence a different frequency of electromagnetic radiation
Maximum Energy of Photons
The energy of the electrons is given by:
K  eV
The maximum energy of photons is therefore:
Emax  eV
So….
hf max  eV
f max
eV

h
X-Rays in Medicine
Example
X-rays are generated in an X-ray tube with
an operating voltage of 50kV.
a) The maximum energy of the emitted photons;
b) The maximum frequency of the emitted
photons.
Example
An X-ray tube emits radiation with a
minimum wavelength of 0.03nm. What is
the operating voltage of the tube?
Uses of X-Rays in Medicine
- Since X-rays have such high energy, they are
able to penetrate matter quite well
- When X-rays are incident on a photographic
plate, they can expose the film and create an
image
- This is what happens when we take X-rays
when bones break etc.
- As the X-rays pass through the body, some are
absorbed. Therefore the intensity varies, forming
an image
What affects the exposure?
- The degree of absorption (attenuation) of the X-rays
- The penetrating power of the X-rays
- The exposure time
Absorption (Attenuation)
The denser the material, the
greater the attenuation. Here the
bone has the greater attenuation,
and shows up best.
A similar effect happens with thick
materials.
Elements with high atomic
numbers tend to have a greater
attenuation
Penetrating Power
- Highly penetrating X-rays are described
by ‘hardness’
- Hard X-rays are typically 100-150kV
- Soft X-rays are typically around 50kV
- Hard X-rays are used to take images of
bones and similar materials
- Soft X-rays are used for images of
tissues and organs
Exposure Time
- Unfortunately, X-rays are harmful to living
tissue.
- Lead shielding is used to protect you and the
radiographer
- It is best to keep exposure to a minimum
- The exposure depends on :
- the hardness. (harder = more
dangerous)
- the intensity (high intensity = more
dangerous)
- exposure time
Determining Exposure Time
- Depends on medical condition
- Shorter exposure time means less
chance of injury, but
- Shorter exposure time may not result in
the appropriate clarity of the image
X-rays are set to an appropriate voltage
(to get hardness) and an appropriate
current (to get intensity)