Quantum Physics
10 hours
The Quantum Nature of Radiation
13.1 - 13.4
The Photoelectric Effect
• In 1888, Heinrich Hertz
carried out an experiment
to verify Maxwell’s
electromagnetic theory of
radiation. Whilst
performing the
experiment Hertz noted
that a spark was more
easily produced if the
electrodes of the spark
gap were illuminated with
ultra-violet light.
• Hertz paid this fact little
heed, and it was left to one
of his pupils, Wilhelm
Hallswach, to investigate the
effect more thoroughly.
Hallswach noticed that
metal surfaces became
charged when illuminated
with ultra-violet light and
that the surface was always
positively charged. He
concluded therefore that the
ultra-violet light caused
negative charge to be
ejected from the surface in
some manner.
• In 1899 Philipp Lenard,
another assistant of
Hertz, showed that the
negative charge
involved in the photoelectric effect consisted
of particles identical in
every respect to those
isolated by J. J.
Thomson two years
previously, namely,
electrons.
• The figure below shows schematically the sort of
arrangement that might be used to investigate the photoelectric effect in more detail. The tube B is highly
evacuated, and a potential difference of about 10 V is
applied between anode and cathode. The cathode
consists of a small zinc plate, and a quartz window is
arranged in the side of the tube such that the cathode
may be illuminated with ultraviolet light. The current
measured by the micro-ammeter gives a direct measure
of the number of electrons emitted at the cathode.
• When the tube is dark no
electrons are emitted at the
cathode and therefore no
current is recorded. When
ultraviolet light is allowed to
fall on the cathode electrons
are ejected and traverse the
tube to the anode, under the
influence of the anodecathode potential. A small
current is recorded by the
micro-ammeter. The graph to
the right shows a plot of
photoelectric current against
light intensity for a constant
anode-cathode potential. As
you would expect, the graph is
a straight line and doubling
the light intensity doubles the
number of electrons ejected at
the cathode.
• The graph of photoelectric
current against light
frequency, shown bottom
right, is not quite so
obvious. The graph shows
clearly that there is a
frequency of light below
which no electrons are
emitted. This frequency is
called the threshold
frequency, fo. Further
experiment shows that the
value of the threshold
frequency is independent of
the intensity of the light and
also that its value depends
on the nature of the
material of the cathode.
• In terms of wave theory we
would expect photo-emission to
occur for light of any frequency.
For example: consider a very
small portion of the cathode, so
small in fact that it contains only
one electron for photo-emission.
If the incident light is a wave
motion, the energy absorbed by
this small portion of the cathode
and consequently by the electron
will increase uniformly with time.
The amount of energy absorbed
in a given time will depend on the
intensity of the incident light and
not on the frequency. If the light
of a given frequency is made very
very feeble there should be an
appreciable time lag during which
the electron absorbs sufficient
energy to escape from within the
metal. No time lag is ever
observed.
We now have three surprising
observations:
1. The intensity of the incident light does not affect
the energy of the emitted electrons.
2. The electron energy depends on the frequency of
the incident light, and there is a certain minimum
frequency below which no electrons are emitted.
3. Electrons are emitted without a time delay.
Don’t worry class....
Einstein’s got this one.
•
The existence of a threshold
frequency and spontaneous emission
even for light of a very low intensity
cannot be explained in terms of a
wave theory of light. In 1905 Albert
Einstein proposed a daring solution
to the problem. Planck had shown
that radiation is emitted in pulses,
each pulse having an energy hf where
h is a constant known as the Planck
constant and f is the frequency of the
radiation. Why, argued Einstein,
should these pulses spread out as
waves? Perhaps each pulse of
radiation maintains a separate
identity throughout the time of
propagation of the radiation. Instead
of light consisting of a train of waves
we should think of it as consisting of
a hail of discrete energy bundles. On
this basis the significance of light
frequency is not so much the
frequency of a pulsating
electromagnetic field, but a measure
of the energy of each ‘bundle’ or
‘particle of light’. The name given to
these tiny bundles of energy is
photon or quantum of radiation.
• Einstein’s interpretation of a
threshold frequency is that a
photon below this frequency
has insufficient energy to
remove an electron from the
metal.
• The minimum energy required
to remove an electron from
the surface of a metal is called
the work-function of the
metal. The electrons in the
metal surface will have varying
kinetic energy and so at a
frequency above the threshold
frequency the ejected
electrons will also have widely
varying energies. However,
according to Einstein’s theory
there will be a definite upper
limit to the energy that a
photo-electron can have.
•
•
•
Suppose we have an electron in the
metal surface that needs just φ units of
energy to be ejected, where φ is the
work function of the metal. A photon
of energy hf strikes this electron and so
the electron absorbs hf units of energy.
If hf ≥ φ the electron will be ejected
from the metal, and if energy is to be
conserved it will gain an amount of
kinetic energy EK given by
EK = hf - φ
Since φ is the minimum amount of
energy required to eject an electron
from the surface, it follows that the
above electron will have the maximum
possible kinetic energy. We can
therefore write that
Ek max = hf - φ
or
Ek max = hf – hf0
Where f0 is the threshold frequency.
Either of the above equations is
referred to as the Einstein
Photoelectric Equation.
It is worth noting that Einstein received the Nobel prize for
Physics in 1921 for “his contributions to mathematical
physics and especially for his discovery of the law of the
photoelectric effect”.
Graphic Representation of Einstein’s
Photoelectric Equation
Ek max = hf - φ
or
Ek max = hf – hf0
Millikan’s Experimental Verification of
Einstein’s Photoelectric Equation
•
In 1916 Robert Millikan verified the Einstein
photoelectric equation using apparatus similar to that
shown in the figure below. Millikan reversed the
potential difference between the anode and cathode
such that the anode was now negatively charged.
Electrons emiited by light shone onto the cathode now
face a ‘potential barrier’ and will only reach the anode
if they have a certain amount of energy. The situation is
analogous to a car freewheeling along the flat and
meeting a hill. The car will reach the top of the hill only
if its kinetic energy is greater than or equal to its
potential energy at the top of the hill.
• For the electron, if the potential difference between
cathode and anode is Vs (‘stopping potential’) then it
will reach the anode only if its kinetic energy is
equal to or greater than Vse where e is the electron
charge. In this situation, the Einstein equation
becomes:
Vse = hf – hf0
• Millikan recorded values of the stopping potential
for different frequencies of the light incident on the
cathode.
• For the electron, if the potential difference between cathode and anode is
Vs (‘stopping potential’) then it will reach the anode only if its kinetic
energy is equal to or greater than Vse where e is the electron charge. In this
situation, the Einstein equation becomes:
Vse = hf – hf0
• Millikan recorded values of the stopping potential for different frequencies
of the light incident on the cathode.
• For Einstein’s theory of the photoelectric effect to be correct, a plot of
stopping potential against frequency should produce a straight line whose
gradient equals h/e. A value for the Planck constant had been previously
determined using measurements from the spectra associated with hot
objects.
• The results of Millikan’s experiment yielded the same value and the
photoelectric effect is regarded as the method by which the value of the
Planck constant is measured. The modern accepted value is 6.2660693 ×
10-34 J s. The intercept on the frequency axis is the threshold frequency and
intercept of the Vs axis is numerically equal to the work function measured
in electron-volt.
Summary
The fact that the photoelectric effect gives convincing evidence for the particle nature
of light, raises the question as to whether light consists of waves or particles. If
particulate in nature, how do we explain such phenomena as interference and
diffraction?
Examples
1. Calculate the energy of a photon in light of
wavelength 120 nm.
2. The photoelectric work function of potassium
is 2.0 eV. Calculate the threshold frequency
of potassium.
Problems:
1. Use data from example 2 to calculate the
maximum kinetic energy in electron-volts of
electrons emitted from the surface of
potassium when illuminated with light of
wavelength 120 nm.
2. State and explain two observations
associated with the photoelectric effect that
cannot be explained by the Classical theory
of electromagnetic radiation.
3. In an experiment to measure the Planck constant, light of different
frequencies f was shone on to the surface of silver and the stopping
potential Vs for the emitted electrons was measured.
The results are shown below. Uncertainties in the data are not shown.
Vs / V
f / 1015 Hz
1.2
0.25
1.6
1.7
2.0
3.3
2.5
5.6
3.0
7.7
3.2
8.4
Plot a graph to show the variation of Vs with f. Draw a line of best-fit for
the data points.
Use the graph to determine
(i) a value of the Planck constant
(ii) the work function of silver in electron-volt.