Modern Physics

1. The Photoelectric Effect And Compton Scattering
PHOTOELECTRIC EFFECT
Introduction:
When a metal surface is illuminated with light, electrons can be emitted from the surface. This
phenomenon, known as the photoelectric effect was discovered by Heinrich Hertz in 1887 in the
process of his research into electromagnetic radiation. The emitted electrons are called
photoelectrons.
Experimental arrangement:
Photoelectric effect can be studied with the help of the following simple experiment. The apparatus
consists of two photosensitive surface s E and C
enclosed inside a evacuated glass tube. E and C are
connected to variable voltage source Vext through an
ammeter. Vext has positive polarity at E and negative
polarity on C. When E is exposed to light with Vext =
0 a current is found to exist in the circuit A,. this
current is due to the emission of photoelectrons from E
due to illumination by radiation. As Vext is increased the
current in the circuit decreases and becomes zero as
Vext reaches certain value. This reverse potential is
called stopping potential Vs. Here Vs is a measure of
maximum kinetic energy of the photoelectrons as eVs
is the work done against the most energetic
photoelectron to stop it from reaching the collector.
Hence the maximum kinetic energy of the
photoelectrons is
Kmax = eVs
Another distinguisihing factor is that this value of the stopping potential and hence the maximum
kinetic energy of the photoelectron does not depend on the intensity of light.
Work function:
In the classical picture, the surface of the metal is illuminated by an electromagnetic wave of intensity
I. An electron absorbs energy from the wave until the binding energy of the electron to the metal is
exceeded at which point the electron is released. The minimum quantity of energy needed to remove
an electron is called the work function Φ of the material. Table 1 lists some values of the work function
of different materials.
Table 1: Some photoelectric workfunctions
Material Na Al Co Cu Zn Ag Pt Pb
Workfunction
Φ(eV)
2.28 4.08 3.90 4.70 4.31 4.73 6.35 4.14
Predictions of the classical wave theory:
1. The maximum kinetic energy should be proportional to the intensity of the radiation. As the
brightness of the light source is increased, more energy is delivered to the surface and the
electrons should be released with greater kinetic energies.
2. The photoelectric effect should occur for light of any frequency or wavelength. As long as the
light is intense enough to release electrons, the photoelectric effect should occur no matter
what the frequency or wavelength.
Page 1 of 7
Fig: 1 Apparatus for observing
photoelectric effect.

2. 3. The first electrons should be emitted in a time interval of the order of seconds after the
radiation first strikes surface. In the wave theory, the energy of the wave is uniformly
distributed over the wavefront. If the electron absorbs energy directly from the wave, the
amount of energy delivered to any electron is determined by how much radiant energy is
incident on the surface area in which the electron is confined. Assuming this area is about the
size of an atom, a rough calculation leads to an estimate that the time lag between turning on
the light and observing the first photoelectrons should be of the order of seconds.
Example 1:
A potassium foil is at a distance r=3.5 m from an isotropic light source that emits energy at the rate
P=1.5 W. The work function of potassium is 2.2 eV. Suppose that the energy transported by the
incident light were transferred to the target foil continuously and smoothly. How long would it take for
foil to absorb enough energy to eject an electron? Assume that the foil totally absorbs all the energy
reaching it and that the to-be-ejected electron collects energy from a circular patch of the foil whose
radius is 5.0x10-11
m, about that of a typical atom.
Solution:
The time interval ∆t required for the patch to absorb energy ∆E depends on the rate absP at which
the energy is absorbed. Therefore,
absP
E
t
∆
=∆
If the electron is to be ejected from the foil, the least energy ∆E it must gain from the light is equal to
the work function of the potassium. Therefore,
absP
t
φ
=∆
Since the patch is totally absorbing, the rate of absorption absP is equal to the rate arrP at which
energy arrives the patch. Therefore,
arrP
E
t
∆
=∆
Now IAarrP =
where
I = intensity of the light
A = area of the patch
Now, since the source is isotropic, we can write
2
4 r
emitP
I
π
=
Therefore,
hs
mxW
eVJxeVm
AemitP
r
IA
t 314580
100551
1061225344
211
1922
.
]).(][.[
]/.][.[].[
≈====∆
−
−
π
πφπφ
Thus classical physics tells us that we would have to wait more than an hour after turning on the light
source for a photoelectron to be ejected. The actual waiting time is less than 10-9
s. Apparently then an
electron does not gradually absorb energy from the light arriving at the patch containing the electron.
Rather, either the electron does not absorb any energy at all or it absorbs a quantum of energy
instantaneously, by absorbing a photon from the light.
Page 2 of 7

3. Experimental characteristics of photoelectric effect:
Photo electrons are emitted from metal surfaces when exposed to light of suitable frequency.
Experiment reveals the following characteristics of the photoelectric effect.
1. The photoelectric effect does not occur at all
if the frequency of the light source is below
a certain value. This value, which is
characteristic of the kind of metal surface
used in the experiment, is called the cutoff
frequency νC. Above νC, any light source, no
matter how weak, will cause emission of
photoelectrons; below νC, no light source, no
matter how strong, will cause the emission
of photoelectrons. This experimental result
also disagrees with the predictions of the
wave theory.
2. The photoelectron current i.e. the number of
photoelectrons emitted from a surface is directly
proportional to the intensity of the light. Increasing the
intensity of light increase the photo current only; the
stopping potential remains the same.
3. The maximum kinetic energy [determined from the
stopping potential] is totally independent of the
intensity of the light source. Fig. 2 shows a
representation of the experimental results. Doubling
the intensity of the source leaves the stopping
potential unchanged, indicating no change in the
kinetic energy of the electrons. This experimental
result disagrees with the wave theory, which predicts
that the maximum kinetic energy should depend on
the intensity of the light. When the intensity is
doubled, the current is doubled but the stopping
potential sV remains the same. The stopping
potential and hence the maximum kinetic energy of the photoelectrons varies linearly with
the frequency of the light [Fig. 3].
4. The first photoelectrons are emitted virtually instantaneously [within 10-9
s] after the light
source is turned on. The wave theory predicts a measurable time delay, so this result also
disagrees with the wave theory.
All of these four experimental results suggest that the classical wave theory completely fails to
account for the photoelectric effect.
Einstein’s theory of the photoelectric effect:
A successful theory of the photoelectric effect was developed by Einstein in 1905. Based on Planck’s
ideas, Einstein postulated that a beam of light consists of small packages of energy called photons or
quanta. The energy of a photon associated with an electromagnetic wave of frequency ν is
][, 1νhE =
where h is Planck’s constant. The photon energy can also be related to the wavelength of the
electromagnetic wave by substituting ν=c/λ which gives
][, 2
λ
hc
E =
Since the photons travel with the electromagnetic wave at the speed of light, they must obey the
relativistic relationship
Page 3 of 7
Vs
CurrentI
Fig 2: the photo current I as a
function of potential difference V
for two different values of intensity
of light
Kmax
Fig 3: Variation of Kmax
with
frequency ν
νc

4. ][, 3
c
E
p =
Combining [2] and [3], we get
λ
h
p = . Like other particles, photons carry linear momentum as well as
energy.
In Einstein’s interpretation, a photoelectron is released as a result of an encounter with a single
photon. The entire energy of the photon is delivered instantaneously to a single photoelectron. If the
photon energy hν is greater than the work function φ of the material, the photoelectron will be
released. If the photon energy is smaller than the work function, the photoelectric effect will not occur.
This explanation thus accounts for two of the failures of the wave theory: the existence of the cutoff
frequency and the lack of any measurable time delay.
If the photon energy exceeds the work function, the excess energy appears as the kinetic energy of
the electron:
][,max 4φν −= hK
From [4], we can write
][,max 5φν += Kh
Since, chνφ = , we can write [5] as
][,max 6chhK νν −=
The intensity of light source does not appear in this expression. Doubling the intensity of the light
source means that twice as many photoelectrons are released, but they all have precisely the same
maximum kinetic energy.
The maximum kinetic energy corresponds to the release of the least tightly electron. Some electrons
may lose energy through interactions with other electrons in the material and emerge with smaller
kinetic energy.
The photon theory appears to explain all of the
observed features of the photoelectric effect.
The most detailed test of the theory was done
by Robert Milikan in 1915. Milikan measured
the maximum kinetic energy [stopping
potential] for different frequencies of the light
and obtained a plot of the equation
φν −= hKmax
A sample of his results are shown in Fig. 3.
From the slope of the line, Milikan obtained a
value for Planck’s constant
sJxh .. 34
10576 −
=
In part for his detailed experiments on the photoelectric effect, Milikan was awarded the 1913 Nobel
prize in physics. Einstein was awarded the 1921 Nobel prize for his photon theory as applied to the
photoelectric effect.
The value of Planck’s constant has been measured to great precision in a variety of experiments. The
presently accepted value is sJxh .. 34
1062607556 −
=
Example 2:
Page 4 of 7
Fig: 3 Milikan’s result for the photoelectric effect in
sodium.

5. The work-function for tungsten metal is 4.52 eV. (a) What is the cut-off wavelength λ c for tungsten. (b)
What is the maximum K. E of the electron when radiation of wavelength 198 nm is used? (c) What is
the stopping potential in this case?
Solution:
(a) We know that the stopping potential can be written as
c
c
hc
h
λ
νφ ==
Therefore, we can write the cutoff wavelength cλ as
nm
eV
nmeVhc
c 274
524
1240
===
.
.
φ
λ
(b) Maximum kinetic energy of the photoelectron can be written as
eVeV
nm
nmeVhc
hvhEK c 741524
198
1240
..
.
. max =−=−=−= φ
λ
ν
(c) The stopping potential Vs is given by V
e
eV
e
EK
Vs 741
741
.
.. max ===
COMPTON SCATTERING
A phenomenon called Compton scattering, first explained by the American physicist A. H. Compton,
provides additional direct confirmation of the quantum nature of x-rays. When x-rays strike matter,
some of the radiation is scattered, just as visible light falling on a rough surface undergoes diffusion
reflection. Compton and others discovered that some of the scattered has smaller frequency [longer
wavelength] than the incident radiation and that the change in wavelength depends on the angle
through which the radiation is scattered. Specifically, if the scattered radiation emerges at an angle ϕ
with respect to the incident direction, and if λ and λ’
are the wavelengths of the incident and scattered
radiation respectively, we find that
],cos[ ϕλλ −=−′ 1
0cm
h
where, m0 is the electron rest mass. The quantity
cm
h
0
has the unit of length and its value is:
mx
smxkgx
sJx
cm
h 12
831
34
0
104262
109982101099
106266 −
−
−
== .
].][.[
..
Page 5 of 7
Fig. 4: A Compton effect experiment

6. Fig. 5: (a) An incident photon with wavelength λ, energy E
and momentum p

approaching an electron e at rest.
(b) The incident photon is scattered through an angle φ and
the struck electron recoils.
(c) Vector diagram for the conservation of momentum.
The Compton scattering is simply the elastic scattering of a photon by an electron, in which both
energy and momentum is conserved.
Since the recoiling electron gains some kinetic energy, the energy E’ of the scattered photon is less
than Ei , and since the wavelength of a photon is inversely proportional to its energy, the wavelength λ’
of the scattered photon is larger than λ.
We shall now derive the relation
],cos[ ϕλλ −=−′ 1
0cm
h
The electron recoil energy may be in the relativistic range, so we have to use the relativistic energy
momentum relations. The incident photon has momentum p

with magnitude p and energy pc. The
scattered photon has momentum p

′ with magnitude p’ and energy p’c. The electron is at rest initially,
so its initial momentum is zero and its initial energy is its rest energy 2
0cm . The final electron
momentum P

has magnitude P and the final electron energy E is given by
,][][ 222
0
2
PccmE +=
Then the conservation of energy gives us the relation
][,][][
][,
,
aPccm
Ecmcppcor
Ecpcmpc
222
0
222
0
2
0
+=
=+′−
+′=+
We may eliminate the electron momentum P

from this
equation by using momentum conservation:
Ppp

+′=
][,, bppPor

′−=
By taking the scalar product of each side of [b], we get
ϕcosppppP ′−′+= 2222
We now substitute this expression into [a] and after
simplification get
We
now
substitute λλ hphp =′=′ , into [c] and get
]cos[ ϕλλ −=−′ 1
0cm
h
Page 6 of 7
a. Compton shift =∆λ ]cos[ ϕλλ −=−′ 1
0cm
h
is independent of the wavelength λof the
incident x-rays.
b. Fractional wavelength shift =
λ
λ∆
c. Compton wavelength = tcons
cm
h
c tan==
0
λ
d. Fractional energy loss =
λλ
λ
λ
λλ
λ
λλ
ν
νν
∆+
∆
=
′
−′
=
′−
=
′−
=
′−
c
cc
h
hh
E
EE
][,cos c
p
cm
p
cm
ϕ−=−
′
100
p
p′ P

7. Photon-atom interactions:
[1] Elastic scattering:
The energy of the incoming photon is too small to excite the atom to an excited state, so the atom
remains in its ground state and the photon is said to be scattered. Since the incoming and scattered
photons have the same energy, the scattering is said to be elastic.
[2] Inelastic scattering:
Inelastic scattering occurs when the incident photon has enough energy to excite the atom to one of its
excited states. The energy of the scattered photon is less than that of the incident photon by ∆E, the
difference between the energy of the ground state and the energy of the excited state. Inelastic
scattering of light from molecules was first observed by C. V. Raman and is often called Raman
scattering.
[3] Fluorescence:
When the energy of the incident photon is great enough to excite to one of its higher excited states,
the atom then loses its energy by spontaneous emission as it makes one or more transitions to lower
energy states. A common example occurs when the atom is excited by ultraviolet light and emits
visible light as it returns to its ground state. This process is called fluorescence. Since the lifetime of a
typical excited atomic state is of the order of 10-8
s, this process appears to occur instantaneously.
However, some excited states have much longer lifetimes-of the order of milliseconds or even
minutes. Such a state is called a metastable state. Phosphorescent materials have very long-lived
metastable states and so emit light long after the original excitation.
[4] Photoelectric effect:
In this process, the absorption of the photon ionizes the atom by causing the emission of an electron.
[5] Compton scattering:
This occurs if the energy of the incident photon is much greater than the ionization energy.
Example 3: Compare Compton scattering for x-rays (λ ≈ 20 pm) and visible light (λ ≈ 500 nm) at a
particular angle of scattering. Which has greater [a] Compton shift, [b] fractional wavelength shift, [c]
fractional photon energy change and [d] energy imparted to the electron.
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