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11.1 INTRODUCTION
The Maxwell’s equations of electromagnetism and Hertz experiments on
the generation and detection of electromagnetic waves in 1887 strongly
established the wave nature of light. Towards the same period at the end
of 19th century, experimental investigations on conduction of electricity
(electric discharge) through gases at low pressure in a discharge tube led
to many historic discoveries. The discovery of X-rays by Roentgen in 1895,
and of electron by J. J. Thomson in 1897, were important milestones in
the understanding of atomic structure. It was found that at sufficiently
low pressure of about 0.001 mm of mercury column, a discharge took
place between the two electrodes on applying the electric field to the gas
in the discharge tube. A fluorescent glow appeared on the glass opposite
to cathode. The colour of glow of the glass depended on the type of glass,
it being yellowish-green for soda glass. The cause of this fluorescence
was attributed to the radiation which appeared to be coming from the
cathode. These cathode rays were discovered, in 1870, by William
Crookes who later, in 1879, suggested that these rays consisted of streams
of fast moving negatively charged particles. The British physicist
J. J. Thomson (1856-1940) confirmed this hypothesis. By applying
mutually perpendicular electric and magnetic fields across the discharge
tube, J. J. Thomson was the first to determine experimentally the speed
and the specific charge [charge to mass ratio (e/m)] of the cathode ray
Chapter Eleven
DUAL NATURE OF
RADIATION AND
MATTER
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particles. They were found to travel with speeds ranging from about 0.1
to 0.2 times the speed of light (3 ×10
8
m/s). The presently accepted value
of e/m is 1.76 × 10
11
C/kg. Further, the value of e/m was found to be
independent of the nature of the material/metal used as the cathode
(emitter), or the gas introduced in the discharge tube. This observation
suggested the universality of the cathode ray particles.
Around the same time, in 1887, it was found that certain metals, when
irradiated by ultraviolet light, emitted negatively charged particles having
small speeds. Also, certain metals when heated to a high temperature were
found to emit negatively charged particles. The value of e/m of these particles
was found to be the same as that for cathode ray particles. These
observations thus established that all these particles, although produced
under different conditions, were identical in nature. J. J. Thomson, in 1897,
named these particles as electrons, and suggested that they were
fundamental, universal constituents of matter. For his epoch-making
discovery of electron, through his theoretical and experimental
investigations on conduction of electricity by gasses, he was awarded the
Nobel Prize in Physics in 1906. In 1913, the American physicist R. A.
Millikan (1868-1953) performed the pioneering oil-drop experiment for
the precise measurement of the charge on an electron. He found that the
charge on an oil-droplet was always an integral multiple of an elementary
charge, 1.602 × 10
–19
C. Millikan’s experiment established that electric
charge is quantised. From the values of charge (e) and specific charge
(e/m), the mass (m) of the electron could be determined.
11.2 ELECTRON
EMISSION
We know that metals have free electrons (negatively charged particles) that
are responsible for their conductivity. However, the free electrons cannot
normally escape out of the metal surface. If an electron attempts to come
out of the metal, the metal surface acquires a positive charge and pulls the
electron back to the metal. The free electron is thus held inside the metal
surface by the attractive forces of the ions. Consequently, the electron can
come out of the metal surface only if it has got sufficient energy to overcome
the attractive pull. A certain minimum amount of energy is required to be
given to an electron to pull it out from the surface of the metal. This
minimum energy required by an electron to escape from the metal surface
is called the work function of the metal. It is generally denoted by
φ
0
and
measured in eV (electron volt). One electron volt is the energy gained by an
electron when it has been accelerated by a potential difference of 1 volt, so
that 1 eV = 1.602 ×10
–19
J.
This unit of energy is commonly used in atomic and nuclear physics.
The work function (
φ
0
) depends on the properties of the metal and the
nature of its surface. The values of work function of some metals are
given in Table 11.1. These values are approximate as they are very
sensitive to surface impurities.
Note from Table 11.1 that the work function of platinum is the highest
(
φ
0
= 5.65 eV) while it is the lowest (
φ
0
= 2.14 eV) for caesium.
The minimum energy required for the electron emission from the metal
surface can be supplied to the free electrons by any one of the following
physical processes:
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(i) Thermionic emission: By suitably heating, sufficient thermal energy
can be imparted to the free electrons to enable them to come out of the
metal.
(ii) Field emission: By applying a very strong electric field (of the order of
10
8
V m
–1
) to a metal, electrons can be pulled out of the metal, as in a
spark plug.
(iii) Photoelectric emission: When light of suitable frequency illuminates
a metal surface, electrons are emitted from the metal surface. These
photo(light)-generated electrons are called photoelectrons.
11.3 PHOTOELECTRIC EFFECT
11.3.1 Hertz’s observations
The phenomenon of photoelectric emission was discovered in 1887 by
Heinrich Hertz (1857-1894), during his electromagnetic wave experiments.
In his experimental investigation on the production of electromagnetic
waves by means of a spark discharge, Hertz observed that high voltage
sparks across the detector loop were enhanced when the emitter plate
was illuminated by ultraviolet light from an arc lamp.
Light shining on the metal surface somehow facilitated the escape of
free, charged particles which we now know as electrons. When light falls
on a metal surface, some electrons near the surface absorb enough energy
from the incident radiation to overcome the attraction of the positive ions
in the material of the surface. After gaining sufficient energy from the
incident light, the electrons escape from the surface of the metal into the
surrounding space.
11.3.2 Hallwachs’ and Lenard’s observations
Wilhelm Hallwachs and Philipp Lenard investigated the phenomenon of
photoelectric emission in detail during 1886-1902.
Lenard (1862-1947) observed that when ultraviolet radiations were
allowed to fall on the emitter plate of an evacuated glass tube enclosing
two electrodes (metal plates), current flows in the circuit (Fig. 11.1). As
soon as the ultraviolet radiations were stopped, the current flow also
TABLE 11.1 WORK FUNCTIONS OF SOME METALS
Metal Work function Metal Work function
φφ
φφ
φ
οο
οο
ο
(eV)
φφ
φφ
φ
οο
οο
ο
(eV)
Cs 2.14 Al 4.28
K 2.30 Hg 4.49
Na 2.75 Cu 4.65
Ca 3.20 Ag 4.70
Mo 4.17 Ni 5.15
Pb 4.25 Pt 5.65
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stopped. These observations indicate that when ultraviolet radiations fall
on the emitter plate C, electrons are ejected from it which are attracted
towards the positive, collector plate A by the electric field. The electrons
flow through the evacuated glass tube, resulting in the current flow. Thus,
light falling on the surface of the emitter causes current in the external
circuit. Hallwachs and Lenard studied how this photo current varied with
collector plate potential, and with frequency and intensity of incident light.
Hallwachs, in 1888, undertook the study further and connected a
negatively charged zinc plate to an electroscope. He observed that the
zinc plate lost its charge when it was illuminated by ultraviolet light.
Further, the uncharged zinc plate became positively charged when it was
irradiated by ultraviolet light. Positive charge on a positively charged
zinc plate was found to be further enhanced when it was illuminated by
ultraviolet light. From these observations he concluded that negatively
charged particles were emitted from the zinc plate under the action of
ultraviolet light.
After the discovery of the electron in 1897, it became evident that the
incident light causes electrons to be emitted from the emitter plate. Due
to negative charge, the emitted electrons are pushed towards the collector
plate by the electric field. Hallwachs and Lenard also observed that when
ultraviolet light fell on the emitter plate, no electrons were emitted at all
when the frequency of the incident light was smaller than a certain
minimum value, called the threshold frequency. This minimum frequency
depends on the nature of the material of the emitter plate.
It was found that certain metals like zinc, cadmium, magnesium, etc.,
responded only to ultraviolet light, having short wavelength, to cause
electron emission from the surface. However, some alkali metals such as
lithium, sodium, potassium, caesium and rubidium were sensitive
even to visible light. All these photosensitive substances emit electrons
when they are illuminated by light. After the discovery of electrons, these
electrons were termed as photoelectrons. The phenomenon is called
photoelectric effect.
11.4 EXPERIMENTAL STUDY OF PHOTOELECTRIC
EFFECT
Figure 11.1 depicts a schematic view of the arrangement used for the
experimental study of the photoelectric effect. It consists of an evacuated
glass/quartz tube having a thin photosensitive plate C and another metal
plate A. Monochromatic light from the source S of sufficiently short
wavelength passes through the window W and falls on the photosensitive
plate C (emitter). A transparent quartz window is sealed on to the glass
tube, which permits ultraviolet radiation to pass through it and irradiate
the photosensitive plate C. The electrons are emitted by the plate C and
are collected by the plate A (collector), by the electric field created by the
battery. The battery maintains the potential difference between the plates
C and A, that can be varied. The polarity of the plates C and A can be
reversed by a commutator. Thus, the plate A can be maintained at a desired
positive or negative potential with respect to emitter C. When the collector
plate A is positive with respect to the emitter plate C, the electrons are
Simulate experiments on photoelectric effect
http://www.kcvs.ca/site/projects/physics.html
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attracted to it. The emission of electrons causes flow of
electric current in the circuit. The potential difference
between the emitter and collector plates is measured by
a voltmeter (V) whereas the resulting photo current
flowing in the circuit is measured by a microammeter
(µA). The photoelectric current can be increased or
decreased by varying the potential of collector plate A
with respect to the emitter plate C. The intensity and
frequency of the incident light can be varied, as can the
potential difference V between the emitter C and the
collector A.
We can use the experimental arrangement of
Fig. 11.1 to study the variation of photocurrent with
(a) intensity of radiation, (b) frequency of incident
radiation, (c) the potential difference between the
plates A and C, and (d) the nature of the material
of plate C. Light of different frequencies can be used
by putting appropriate coloured filter or coloured
glass in the path of light falling on the emitter C. The intensity
of light is varied by changing the distance of the light source
from the emitter.
11.4.1 Effect of intensity of light on photocurrent
The collector A is maintained at a positive potential with
respect to emitter C so that electrons ejected from C are
attracted towards collector A. Keeping the frequency of the
incident radiation and the potential fixed, the intensity of light
is varied and the resulting photoelectric current is measured
each time. It is found that the photocurrent increases linearly
with intensity of incident light as shown graphically in
Fig. 11.2. The photocurrent is directly proportional to the
number of photoelectrons emitted per second. This implies
that the number of photoelectrons emitted per second is
directly proportional to the intensity of incident radiation.
11.4.2 Effect of potential on photoelectric current
We first keep the plate A at some positive potential with respect to the
plate C and illuminate the plate C with light of fixed frequency
ν
and fixed
intensity I
1
. We next vary the positive potential of plate A gradually and
measure the resulting photocurrent each time. It is found that the
photoelectric current increases with increase in positive (accelerating)
potential. At some stage, for a certain positive potential of plate A, all the
emitted electrons are collected by the plate A and the photoelectric current
becomes maximum or saturates. If we increase the accelerating potential
of plate A further, the photocurrent does not increase. This maximum
value of the photoelectric current is called saturation current. Saturation
current corresponds to the case when all the photoelectrons emitted by
the emitter plate C reach the collector plate A.
We now apply a negative (retarding) potential to the plate A with respect
to the plate C and make it increasingly negative gradually. When the
FIGURE 11.1 Experimental
arrangement for study of
photoelectric effect.
FIGURE 11.2 Variation of
Photoelectric current with
intensity of light.
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polarity is reversed, the electrons are
repelled and only the sufficiently energetic
electrons are able to reach the collector A.
The photocurrent is found to decrease
rapidly until it drops to zero at a certain
sharply defined, critical value of the negative
potential V
0
on the plate A. For a particular
frequency of incident radiation, the
minimum negative (retarding) potential V
0
given to the plate A for which the
photocurrent stops or becomes zero is
called the cut-off or stopping potential.
The interpretation of the observation in
terms of photoelectrons is straightforward.
All the photoelectrons emitted from the
metal do not have the same energy.
Photoelectric current is zero when the
stopping potential is sufficient to repel even
the most energetic photoelectrons, with the
maximum kinetic energy (K
max
), so that
K
max
= e V
0
(11.1)
We can now repeat this experiment with incident radiation of the same
frequency but of higher intensity I
2
and I
3
(I
3
> I
2
> I
1
). We note that the
saturation currents are now found to be at higher values. This shows
that more electrons are being emitted per second, proportional to the
intensity of incident radiation. But the stopping potential remains the
same as that for the incident radiation of intensity I
1
, as shown graphically
in Fig. 11.3. Thus, for a given frequency of the incident radiation, the
stopping potential is independent of its intensity. In other words, the
maximum kinetic energy of photoelectrons depends on the light source
and the emitter plate material, but is independent of intensity of incident
radiation.
11.4.3 Effect of frequency of incident radiation on stopping
potential
We now study the relation between the
frequency ν of the incident radiation and the
stopping potential V
0
. We suitably adjust the
same intensity of light radiation at various
frequencies and study the variation of
photocurrent with collector plate potential. The
resulting variation is shown in Fig. 11.4. We
obtain different values of stopping potential but
the same value of the saturation current for
incident radiation of different frequencies. The
energy of the emitted electrons depends on the
frequency of the incident radiations. The
stopping potential is more negative for higher
frequencies of incident radiation. Note from
FIGURE 11.3 Variation of photocurrent with
collector plate potential for different
intensity of incident radiation.
FIGURE 11.4 Variation of photoelectric current
with collector plate potential for different
frequencies of incident radiation.
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Fig. 11.4 that the stopping potentials are in the
order V
03
> V
02
> V
01
if the frequencies are in the
order ν
3
> ν
2
> ν
1
. This implies that greater the
frequency of incident light, greater is the
maximum kinetic energy of the photoelectrons.
Consequently, we need greater retarding
potential to stop them completely. If we plot a
graph between the frequency of incident radiation
and the corresponding stopping potential for
different metals we get a straight line,
as shown
in Fig. 11.5
.
The graph shows that
(i) the stopping potential V
0
varies linearly with
the frequency of incident radiation for a given
photosensitive material.
(ii) there exists a certain minimum cut-off frequency
ν
0
for which the
stopping potential is zero.
These observations have two implications:
(i) The maximum kinetic energy of the photoelectrons varies linearly
with the frequency of incident radiation, but is independent of its
intensity.
(ii) For a frequency
ν
of incident radiation, lower than the cut-off
frequency
ν
0
, no photoelectric emission is possible even if the
intensity is large.
This minimum, cut-off frequency
ν
0
, is called the threshold frequency.
It is different for different metals.
Different photosensitive materials respond differently to light. Selenium
is more sensitive than zinc or copper. The same photosensitive substance
gives different response to light of different wavelengths. For example,
ultraviolet light gives rise to photoelectric effect in copper while green or
red light does not.
Note that in all the above experiments, it is found that, if frequency of
the incident radiation exceeds the threshold frequency, the photoelectric
emission starts instantaneously without any apparent time lag, even if
the incident radiation is very dim. It is now known that emission starts in
a time of the order of 10
9
s or less.
We now summarise the experimental features and observations
described in this section.
(i) For a given photosensitive material and frequency of incident radiation
(above the threshold frequency), the photoelectric current is directly
proportional to the intensity of incident light (Fig. 11.2).
(ii) For a given photosensitive material and frequency of incident radiation,
saturation current is found to be proportional to the intensity of
incident radiation whereas the stopping potential is independent of
its intensity (Fig. 11.3).
(iii) For a given photosensitive material, there exists a certain minimum
cut-off frequency of the incident radiation, called the threshold
frequency, below which no emission of photoelectrons takes place,
no matter how intense the incident light is. Above the threshold
frequency, the stopping potential or equivalently the maximum kinetic
FIGURE 11.5 Variation of stopping potential V
0
with frequency ν of incident radiation for a
given photosensitive material.
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energy of the emitted photoelectrons increases linearly with the
frequency of the incident radiation, but is independent of its intensity
(Fig. 11.5).
(iv) The photoelectric emission is an instantaneous process without any
apparent time lag (10
9
s or less), even when the incident radiation is
made exceedingly dim.
11.5 P
HOTOELECTRIC EFFECT AND WAVE THEORY
OF LIGHT
The wave nature of light was well established by the end of the nineteenth
century. The phenomena of interference, diffraction and polarisation were
explained in a natural and satisfactory way by the wave picture of light.
According to this picture, light is an electromagnetic wave consisting of
electric and magnetic fields with continuous distribution of energy over
the region of space over which the wave is extended. Let us now see if this
wave picture of light can explain the observations on photoelectric
emission given in the previous section.
According to the wave picture of light, the free electrons at the surface
of the metal (over which the beam of radiation falls) absorb the radiant
energy continuously. The greater the intensity of radiation, the greater are
the amplitude of electric and magnetic fields.
Consequently, the greater
the intensity, the greater should be the energy absorbed by each electron.
In this picture, the maximum kinetic energy of the photoelectrons on the
surface is then expected to increase with increase in intensity. Also, no
matter what the frequency of radiation is, a sufficiently intense beam of
radiation (over sufficient time) should be able to impart enough energy to
the electrons, so that they exceed the minimum energy needed to escape
from the metal surface . A threshold frequency, therefore, should not exist.
These expectations of the wave theory directly contradict observations (i),
(ii) and (iii) given at the end of
sub-section 11.4.3
.
Further, we should note that in the wave picture, the absorption of
energy by electron takes place continuously over the entire
wavefront of the radiation. Since a large number of electrons absorb energy,
the energy absorbed per electron per unit time turns out to be small.
Explicit calculations estimate that it can take hours or more for a single
electron to pick up sufficient energy to overcome the work function and
come out of the metal. This conclusion is again in striking contrast to
observation (iv) that the photoelectric emission is instantaneous. In short,
the wave picture is unable to explain the most basic features of
photoelectric emission.
11.6 EINSTEINS PHOTOELECTRIC EQUATION: ENERGY
QUANTUM OF RADIATION
In 1905, Albert Einstein (1879-1955) proposed a radically new picture
of electromagnetic radiation to explain photoelectric effect. In this picture,
photoelectric emission does not take place by continuous absorption of
energy from radiation. Radiation energy is built up of discrete units the
so called quanta of energy of radiation. Each quantum of radiant energy
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has energy h
ν
, where h is Planck’s constant and
ν
the
frequency of light. In photoelectric effect, an electron
absorbs a quantum of energy (h
ν
) of radiation. If this
quantum of energy absorbed exceeds the minimum
energy needed for the electron to escape from the metal
surface (work function
φ
0
), the electron is emitted with
maximum kinetic energy
K
max
= h
ν
φ
0
(11.2)
More tightly bound electrons will emerge with kinetic
energies less than the maximum value. Note that the
intensity of light of a given frequency is determined by
the number of photons incident per second. Increasing
the intensity will increase the number of emitted electrons
per second. However, the maximum kinetic energy of the
emitted photoelectrons is determined by the energy of each
photon.
Equation (11.2) is known as Einstein’s photoelectric
equation. We now see how this equation accounts in a
simple and elegant manner all the observations on
photoelectric effect given at the end of sub-section 11.4.3.
According to Eq. (11.2), K
max
depends linearly on
ν
,
and is independent of intensity of radiation, in
agreement with observation. This has happened
because in Einstein’s picture, photoelectric effect arises
from the absorption of a single quantum of radiation
by a single electron. The intensity of radiation (that is
proportional to the number of energy quanta per unit
area per unit time) is irrelevant to this basic process.
Since K
max
must be non-negative, Eq. (11.2 ) implies
that photoelectric emission is possible only if
h
ν
>
φ
0
or
ν
>
ν
0
,
where
ν
0
=
0
h
φ
(11.3)
Equation (11.3) shows that the greater the work
function
φ
0
, the higher the minimum or threshold
frequency ν
0
needed to emit photoelectrons. Thus,
there exists a threshold frequency
ν
0
(=
φ
0
/h) for the
metal surface, below which no photoelectric emission
is possible, no matter how intense the incident
radiation may be or how long it falls on the surface.
In this picture, intensity of radiation as noted above,
is proportional to the number of energy quanta per
unit area per unit time. The greater the number of
energy quanta available, the greater is the number of
electrons absorbing the energy quanta and greater,
therefore, is the number of electrons coming out of
the metal (for
ν
>
ν
0
). This explains why, for
ν
>
ν
0
,
photoelectric current is proportional to intensity.
ALBERT EINSTEIN (1879 – 1955)
Albert Einstein (1879 –
1955) Einstein, one of the
greatest physicists of all
time, was born in Ulm,
Germany. In 1905, he
published three path-
breaking papers. In the
first paper, he introduced
the notion of light quanta
(now called photons) and
used it to explain the
features of photoelectric
effect. In the second paper,
he developed a theory of
Brownian motion,
confirmed experimentally a
few years later and provided
a convincing evidence of
the atomic picture of matter.
The third paper gave birth
to the special theory of
relativity. In 1916, he
published the general
theory of relativity. Some of
Einstein’s most significant
later contributions are: the
notion of stimulated
emission introduced in an
alternative derivation of
Planck’s blackbody
radiation law, static model
of the universe which
started modern cosmology,
quantum statistics of a gas
of massive bosons, and a
critical analysis of the
foundations of quantum
mechanics. In 1921, he was
awarded the Nobel Prize in
physics for his contribution
to theoretical physics and
the photoelectric effect.
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In Einstein’s picture, the basic elementary process involved in
photoelectric effect is the absorption of a light quantum by an electron.
This process is instantaneous. Thus, whatever may be the intensity
i.e., the number of quanta of radiation per unit area per unit time,
photoelectric emission is instantaneous. Low intensity does not mean
delay in emission, since the basic elementary process is the same.
Intensity only determines how many electrons are able to participate
in the elementary process (absorption of a light quantum by a single
electron) and, therefore, the photoelectric current.
Using Eq. (11.1), the photoelectric equation, Eq. (11.2), can be
written as
e V
0
= h
ν
φ
0
;
for
0
ν ν
or V
0
=
h
e e
φ
ν
(11.4)
This is an important result. It predicts that the V
0
versus
ν
curve is a
straight line with slope = (h/e), independent of the nature of the material.
During 1906-1916, Millikan performed a series of experiments on
photoelectric effect, aimed at disproving Einstein’s photoelectric equation.
He measured the slope of the straight line obtained for sodium, similar to
that shown in Fig. 11.5. Using the known value of e, he determined the
value of Planck’s constant h. This value was close to the value of Planck’s
contant (= 6.626 × 10
–34
J s) determined in an entirely different context.
In this way, in 1916, Millikan proved the validity of Einstein’s photoelectric
equation, instead of disproving it.
The successful explanation of photoelectric effect using the hypothesis
of light quanta and the experimental determination of values of h and
φ
0
,
in agreement with values obtained from other experiments, led to the
acceptance of Einstein’s picture of photoelectric effect. Millikan verified
photoelectric equation with great precision, for a number of alkali metals
over a wide range of radiation frequencies.
11.7 PARTICLE NATURE OF LIGHT: THE PHOTON
Photoelectric effect thus gave evidence to the strange fact that light in
interaction with matter behaved as if it was made of quanta or packets of
energy, each of energy h
ν
.
Is the light quantum of energy to be associated with a particle? Einstein
arrived at the important result, that the light quantum can also be
associated with momentum (h
ν
/c). A definite value of energy as well as
momentum is a strong sign that the light quantum can be associated
with a particle. This particle was later named photon. The particle-like
behaviour of light was further confirmed, in 1924, by the experiment of
A.H. Compton (1892-1962) on scattering of X-rays from electrons. In
1921, Einstein was awarded the Nobel Prize in Physics for his contribution
to theoretical physics and the photoelectric effect. In 1923, Millikan was
awarded the Nobel Prize in physics for his work on the elementary
charge of electricity and on the photoelectric effect.
We can summarise the photon picture of electromagnetic radiation
as follows:
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EXAMPLE
11.1
EXAMPLE
11.2
(i) In interaction of radiation with matter, radiation behaves as if it is
made up of particles called photons.
(ii) Each photon has energy E (=h
ν
) and momentum p (= h
ν
/c), and
speed c, the speed of light.
(iii) All photons of light of a particular frequency
ν
, or wavelength
λ
, have
the same energy E (=h
ν
= hc/
λ
) and momentum p (= h
ν
/c= h/
λ
),
whatever the intensity of radiation may be. By increasing the intensity
of light of given wavelength, there is only an increase in the number of
photons per second crossing a given area, with each photon having
the same energy. Thus, photon energy is independent of intensity of
radiation.
(iv) Photons are electrically neutral and are not deflected by electric and
magnetic fields.
(v) In a photon-particle collision (such as photon-electron collision), the
total energy and total momentum are conserved. However, the number
of photons may not be conserved in a collision. The photon may be
absorbed or a new photon may be created.
Example 11.1 Monochromatic light of frequency 6.0 ×10
14
Hz is
produced by a laser. The power emitted is 2.0 ×10
–3
W. (a) What is the
energy of a photon in the light beam? (b) How many photons per second,
on an average, are emitted by the source?
Solution
(a) Each photon has an energy
E = h
ν
= ( 6.63 ×10
–34
J s) (6.0 ×10
14
Hz)
= 3.98 × 10
–19
J
(b) If N is the number of photons emitted by the source per second,
the power P transmitted in the beam equals N times the energy
per photon E, so that P = N E. Then
N =
3
19
2.0 10 W
3.98 10 J
P
E
×
=
×
= 5.0 ×10
15
photons per second.
Example 11.2 The work function of caesium is 2.14 eV. Find (a) the
threshold frequency for caesium, and (b) the wavelength of the incident
light if the photocurrent is brought to zero by a stopping potential of
0.60 V.
Solution
(a) For the cut-off or threshold frequency, the energy h
ν
0
of
the incident
radiation must be equal to work function
φ
0
, so that
ν
0
=
0
34
2.14 eV
6.63 10 J s
h
φ
=
×
19
14
34
2.14 1.6 10 J
5.16 10 Hz
6.63 10 J s
× ×
= = ×
×
Thus, for frequencies less than this threshold frequency, no
photoelectrons are ejected.
(b) Photocurrent reduces to zero, when maximum kinetic energy of
the emitted photoelectrons equals the potential energy e V
0
by the
retarding potential V
0
. Einstein’s Photoelectric equation is
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EXAMPLE 11.3 EXAMPLE 11.2
Example 11.3 The wavelength of light in the visible region is about
390 nm for violet colour, about 550 nm (average wavelength) for yellow-
green colour and about 760 nm for red colour.
(a) What are the energies of photons in (eV) at the (i) violet end, (ii)
average wavelength, yellow-green colour, and (iii) red end of the
visible spectrum? (Take h = 6.63×10
–34
J s and 1 eV = 1.6×10
–19
J.)
(b) From which of the photosensitive materials with work functions
listed in Table 11.1 and using the results of (i), (ii) and (iii) of (a),
can you build a photoelectric device that operates with visible
light?
Solution
(a) Energy of the incident photon, E = hν = hc/
λ
E = (6.63×10
–34
J s) (3×10
8
m/s)/
λ
–25
1.989 10 Jm
λ
×
=
(i) For violet light,
λ
1
= 390 nm (lower wavelength end)
Incident photon energy, E
1
=
–25
–9
1.989 10 Jm
390×10 m
×
= 5.10 × 10
–19
J
–19
–19
5.10 10 J
1.6×10 J/eV
×
=
= 3.19 eV
(ii) For yellow-green light,
λ
2
= 550 nm (average wavelength)
Incident photon energy, E
2
=
–25
–9
1.989 10 Jm
550×10 m
×
= 3.62×10
–19
J = 2.26 eV
(iii) For red light,
λ
3
= 760 nm (higher wavelength end)
Incident photon energy, E
3
=
–25
–9
1.989 10 Jm
760×10 m
×
= 2.62×10
–19
J = 1.64 eV
(b) For a photoelectric device to operate, we require incident light energy
E to be equal to or greater than the work function
φ
0
of the material.
Thus, the photoelectric device will operate with violet light (with
E = 3.19 eV) photosensitive material Na (with
φ
0
= 2.75 eV), K (with
φ
0
= 2.30 eV) and Cs (with
φ
0
= 2.14 eV). It will also operate with
yellow-green light (with E = 2.26 eV) for Cs (with
φ
0
= 2.14 eV) only.
However, it will not operate with red light (with E = 1.64 eV) for any
of these photosensitive materials.
eV
0
= h
ν
φ
0
=
hc
λ
φ
0
or,
λ
= hc/(eV
0
+
φ
0
)
34 8
(6.63 10 Js) (3 10 m/s)
(0.60eV 2.14eV)
× × ×
=
+
26
19.89 10 J m
(2.74eV)
×
=
26
19
19.89 10 J m
454 nm
2.74 1.6 10 J
λ
×
= =
× ×
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11.8 WAVE NATURE OF MATTER
The dual (wave-particle) nature of light (electromagnetic radiation, in
general) comes out clearly from what we have learnt in this and the
preceding chapters. The wave nature of light shows up in the phenomena
of interference, diffraction and polarisation. On the other hand, in
photoelectric effect and Compton effect which involve energy and
momentum transfer, radiation behaves as if it is made up of a bunch of
particles – the photons. Whether a particle or wave description is best
suited for understanding an experiment depends on the nature of the
experiment. For example, in the familiar phenomenon of seeing an object
by our eye, both descriptions are important. The gathering and focussing
mechanism of light by the eye-lens is well described in the wave picture.
But its absorption by the rods and cones (of the retina) requires the photon
picture of light.
A natural question arises: If radiation has a dual (wave-particle) nature,
might not the particles of nature (the electrons, protons, etc.) also exhibit
wave-like character? In 1924, the French physicist Louis Victor de Broglie
(pronounced as de Broy) (1892-1987) put forward the bold hypothesis
that moving particles of matter should display wave-like properties under
suitable conditions. He reasoned that nature was symmetrical and that
the two basic physical entities – matter and energy, must have symmetrical
character. If radiation shows dual aspects, so should matter. De Br
oglie
proposed that the wave length
λ
associated with a particle of momentum
p is given as
λ
=
h h
p m v
=
(11.5)
where m is the mass of the particle and v its speed. Equation (11.5) is
known as the de Broglie relation and the wavelength
λ
of the matter
wave is called de Broglie wavelength. The dual aspect of matter is evident
in the de Broglie relation. On the left hand side of Eq. (11.5), λ is the
attribute of a wave while on the right hand side the momentum p is a
typical attribute of a particle. Planck’s constant h relates the two
attributes.
Equation (11.5) for a material particle is basically a hypothesis whose
validity can be tested only by experiment. However, it is interesting to see
that it is satisfied also by a photon. For a photon, as we have seen,
p = h
ν
/c (11.6)
Therefore,
h c
p
λ
ν
= =
(11.7)
That is, the de Broglie wavelength of a photon given by Eq. (11.5) equals
the wavelength of electromagnetic radiation of which the photon is a
quantum of energy and momentum.
Clearly, from Eq. (11.5 ),
λ
is smaller for a heavier particle (large m) or
more energetic particle (large v). For example, the de Broglie wavelength
of a ball of mass 0.12 kg moving with a speed of 20 m s
–1
is easily
calculated:
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p = m v = 0.12 kg × 20 m s
–1
= 2.40 kg m s
–1
λ
=
h
p
=
34
1
6.63 10 J s
2.40 kg m s
×
= 2.76 × 10
–34
m
PHOTOCELL
A photocell is a technological application of the photoelectric effect. It is a device whose
electrical properties are affected by light. It is also sometimes called an electric eye. A photocell
consists of a semi-cylindrical photo-sensitive metal plate C (emitter) and a wire loop A
(collector) supported in an evacuated glass or quartz bulb. It is connected to the external
circuit having a high-tension battery B and microammeter (µA) as shown in the Figure.
Sometimes, instead of the plate C, a thin layer of photosensitive material is pasted on the
inside of the bulb. A part of the bulb is left clean for the light to enter it.
When light of suitable wavelength falls on the
emitter C, photoelectrons are emitted. These
photoelectrons are drawn to the collector A.
Photocurrent of the order of a few microampere
can be normally obtained from a photo cell.
A photocell converts a change in intensity of
illumination into a change in photocurrent. This
current can be used to operate control systems
and in light measuring devices. A photocell of lead
sulphide sensitive to infrared radiation is used
in electronic ignition circuits.
In scientific work, photo cells are used
whenever it is necessary to measure the intensity
of light. Light meters in photographic cameras
make use of photo cells to measure the intensity
of incident light. The photocells, inserted in the
door light electric circuit, are used as automatic
door opener. A person approaching a doorway
may interrupt a light beam which is incident on
a photocell. The abrupt change in photocurrent
may be used to start a motor which opens the
door or rings an alarm. They are used in the
control of a counting device which records every interruption of the light beam caused by a
person or object passing across the beam. So photocells help count the persons entering an
auditorium, provided they enter the hall one by one. They are used for detection of traffic
law defaulters: an alarm may be sounded whenever a beam of (invisible) radiation is
intercepted.
In burglar alarm, (invisible) ultraviolet light is continuously made to fall on a photocell
installed at the doorway. A person entering the door interrupts the beam falling on the
photocell. The abrupt change in photocurrent is used to start an electric bell ringing. In fire
alarm, a number of photocells are installed at suitable places in a building. In the event of
breaking out of fire, light radiations fall upon the photocell. This completes the electric
circuit through an electric bell or a siren which starts operating as a warning signal.
Photocells are used in the reproduction of sound in motion pictures and in the television
camera for scanning and telecasting scenes. They are used in industries for detecting minor
flaws or holes in metal sheets.
A photo cell
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This wavelength is so small that it is beyond any
measurement. This is the reason why macroscopic objects
in our daily life do not show wave-like properties. On the
other hand, in the sub-atomic domain, the wave character
of particles is significant and measurable.
Consider an electron (mass m, charge e) accelerated
from rest through a potential V. The kinetic energy K
of the electron equals the work done (eV ) on it by the
electric field:
K =
e V (11.8)
Now, K =
1
2
m v
2
=
2
2
p
m
, so that
p =
2 2
m K m e V
=
(11.9)
The de Broglie wavelength
λ
of the electron is then
λ =
2 2
h h h
p
m K m eV
= =
(11.10)
Substituting the numerical values of h, m, e,
we get
1.227
nm
V
λ
=
(11.11)
where V is the magnitude of accelerating potential in
volts. For a 120 V accelerating potential, Eq. (11.11) gives
λ
= 0.112 nm. This wavelength is of the same order as
the spacing between the atomic planes in crystals. This
suggests that matter waves associated with an electron could be verified
by crystal diffraction experiments analogous to X-ray diffraction. We
describe the experimental verification of the de Broglie hypothesis in the
next section. In 1929, de Broglie was awarded the Nobel Prize in Physics
for his discovery of the wave nature of electrons.
The matter–wave picture elegantly incorporated the Heisenberg’s
uncertainty principle. According to the principle, it is not possible to
measure both the position and momentum of an electron (or any other
particle) at the same time exactly. There is always some uncertainty (x)
in the specification of position and some uncertainty (p) in the
specification of momentum. The product of x and p is of the order of ħ*
(with ħ = h/2π), i.e.,
x p ħ (11.12)
Equation (11.12) allows the possibility that x is zero; but then p
must be infinite in order that the product is non-zero. Similarly, if p is
zero, x must be infinite. Ordinarily, both x and p are non-zero such
that their product is of the order of ħ.
Now, if an electron has a definite momentum p, (i.e.p = 0), by the de
Broglie relation, it has a definite wavelength
λ
. A wave of definite (single)
LOUIS VICTOR DE BROGLIE (1892 – 1987)
Louis Victor de Broglie
(1892 – 1987) French
physicist who put forth
revolutionary idea of wave
nature of matter. This idea
was developed by Erwin
Schródinger into a full-
fledged theory of quantum
mechanics commonly
known as wave mechanics.
In 1929, he was awarded the
Nobel Prize in Physics for his
discovery of the wave nature
of electrons.
* A more rigorous treatment gives x p
ħ/2.
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wavelength extends all over space. By Born’s
probability interpretation this means that the
electron is not localised in any finite region of
space. That is, its position uncertainty is infinite
(x → ∞), which is consistent with the
uncertainty principle.
In general, the matter wave associated with
the electron is not extended all over space. It is
a wave packet extending over some finite region
of space. In that case x is not infinite but has
some finite value depending on the extension
of the wave packet. Also, you must appreciate
that a wave packet of finite extension does not
have a single wavelength. It is built up of
wavelengths spread around some central
wavelength.
By de Broglie’s relation, then, the
momentum of the electron will also have a
spread – an uncertainty p. This is as expected
from the uncertainty principle. It can be shown
that the wave packet description together with
de Broglie relation and Born’s probability
interpretation reproduce the Heisenberg’s
uncertainty principle exactly.
In Chapter 12, the de Broglie relation will
be seen to justify Bohr’s postulate on
quantisation of angular momentum of electron
in an atom.
Figure 11.6 shows a schematic diagram of
(a) a localised wave packet, and (b) an extended
wave with fixed wavelength.
Example 11.4 What is the de Broglie wavelength associated with (a) an
electron moving with a speed of 5.4×10
6
m/s, and (b) a ball of mass 150
g travelling at 30.0 m/s?
Solution
(a) For the electron:
Mass m = 9.11×10
–31
kg, speed v = 5.4
×10
6
m/s. Then, momentum
p = m v = 9.11
×10
–31
(kg) × 5.4 × 10
6
(m/s)
p = 4.92 × 10
–24
kg m/s
de Broglie wavelength, λ = h/p
=
.
.
–34
–24
6 63 10 J s
4 92 10 kg m/s
×
×
λ
= 0.135 nm
(b) For the ball:
Mass m = 0.150 kg, speed v = 30.0 m/s.
Then momentum p = m v = 0.150 (kg) × 30.0 (m/s)
p= 4.50 kg m/s
de Broglie wavelength λ = h/p.
FIGURE 11.6 (a) The wave packet description of
an electron. The wave packet corresponds to a
spread of wavelength around some central
wavelength (and hence by de Broglie relation,
a spread in momentum). Consequently, it is
associated with an uncertainty in position
(x) and an uncertainty in momentum (p).
(b) The matter wave corresponding to a
definite momentum of an electron
extends all over space. In this case,
p = 0 and x .
EXAMPLE 11.4
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EXAMPLE 11.6
EXAMPLE
11.4
.
.
34
6 63 10 Js
4 50 kg m/s
×
=
×
λ
= 1.47 ×10
–34
m
The de Broglie wavelength of electron is comparable with X-ray
wavelengths. However, for the ball it is about 10
–19
times the size of
the proton, quite beyond experimental measurement.
Example 11.5 An electron, an α-particle, and a proton have the same
kinetic energy. Which of these particles has the shortest de Broglie
wavelength?
Solution
For a particle, de Broglie wavelength,
λ
= h/p
Kinetic energy, K = p
2
/2m
Then,
/ 2
h mK
λ
=
For the same kinetic energy K, the de Broglie wavelength associated
with the particle is inversely proportional to the square root of their
masses. A proton
(
)
1
1
H
is 1836 times massive than an electron and
an α-particle
(
)
4
2
He
four times that of a proton.
Hence, α – particle has the shortest de Broglie wavelength.
EXAMPLE 11.5
PROBABILITY INTERPRETATION TO MATTER WAVES
It is worth pausing here to reflect on just what a matter wave associated with a particle,
say, an electron, means. Actually, a truly satisfactory physical understanding of the
dual nature of matter and radiation has not emerged so far. The great founders of
quantum mechanics (Niels Bohr, Albert Einstein, and many others) struggled with this
and related concepts for long. Still the deep physical interpretation of quantum
mechanics continues to be an area of active research. Despite this, the concept of
matter wave has been mathematically introduced in modern quantum mechanics with
great success. An important milestone in this connection was when Max Born (1882-
1970) suggested a probability interpretation to the matter wave amplitude. According
to this, the intensity (square of the amplitude) of the matter wave at a point determines
the probability density of the particle at that point. Probability density means probability
per unit volume. Thus, if A is the amplitude of the wave at a point, |A|
2
V is the
probability of the particle being found in a small volume V around that point. Thus,
if the intensity of matter wave is large in a certain region, there is a greater probability
of the particle being found there than where the intensity is small.
Example 11.6 A particle is moving three times as fast as an electron.
The ratio of the de Broglie wavelength of the particle to that of the
electron is 1.813 × 10
–4
. Calculate the particle’s mass and identify the
particle.
Solution
de Broglie wavelength of a moving particle, having mass m and
velocity v:
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h h
p mv
λ
= =
Mass, m = h/
λ
v
For an electron, mass m
e
= h/
λ
e
v
e
Now, we have v/v
e
= 3 and
λ
/
λ
e
= 1.813 × 10
4
Then, mass of the particle, m = m
e
e e
v
v
λ
λ
m = (9.11×10
–31
kg) × (1/3) × (1/1.813 × 10
–4
)
m = 1.675 × 10
–27
kg.
Thus, the particle, with this mass could be a proton or a neutron.
Example 11.7 What is the de Broglie wavelength associated with an
electron, accelerated through a potential differnece of 100 volts?
Solution Accelerating potential V = 100 V. The de Broglie wavelength
λ
is
λ
= h /p
.
1 227
V
=
nm
λ
.
1 227
100
=
nm = 0.123 nm
The de Broglie wavelength associated with an electron in this case is of
the order of X-ray wavelengths.
11.9 DAVISSON AND GERMER EXPERIMENT
The wave nature of electrons was first experimentally verified by C.J.
Davisson and L.H. Germer in 1927 and independently by G.P. Thomson,
in 1928, who observed
diffraction effects with beams of
electrons scattered by crystals.
Davisson and Thomson shared
the Nobel Prize in 1937 for their
experimental discovery of
diffraction of electrons by
crystals.
The experimental arrange-
ment used by Davisson and
Germer is schematically shown
in Fig. 11.7. It consists of an
electron gun which comprises of
a tungsten filament F, coated
with barium oxide and heated
by a low voltage power supply
(L.T. or battery). Electrons
emitted by the filament are
accelerated to a desired velocity
FIGURE 11.7 Davisson-Germer electron
diffraction arrangement.
EXAMPLE 11.6
EXAMPLE 11.7
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by applying suitable potential/voltage from a high voltage power supply
(H.T. or battery). They are made to pass through a cylinder with fine
holes along its axis, producing a fine collimated beam. The beam is made
to fall on the surface of a nickel crystal. The electrons are scattered in all
directions by the atoms of the crystal. The intensity of the electron beam,
scattered in a given direction, is measured by the electron detector
(collector). The detector can be moved on a circular scale and is connected
to a sensitive galvanometer, which records the current. The deflection of
the galvanometer is proportional to the intensity of the electron beam
entering the collector. The apparatus is enclosed in an evacuated chamber.
By moving the detector on the circular scale at different positions, the
intensity of the scattered electron beam is measured for different values
of angle of scattering
θ
which is the angle between the incident and the
scattered electron beams. The variation of the intensity (I) of the scattered
electrons with the angle of scattering
θ
is obtained for different accelerating
voltages.
The experiment was performed by varying the accelarating voltage
from 44 V to 68 V. It was noticed that a strong peak appeared in the
intensity (I) of the scattered electron for an accelarating voltage of 54V at
a scattering angle
θ
= 50
°
The appearance of the peak in a particular direction is due to the
constructive interference of electrons scattered from different layers of the
regularly spaced atoms of the crystals. From the electron diffraction
measurements, the wavelength of matter waves was found to be
0.165 nm.
The de Broglie wavelength
λ
associated with electrons, using
Eq. (11.11), for V = 54 V is given by
λ
= h /p
1 227
V
=
nm
λ
1 227
54
=
nm = 0.167 nm
Thus, there is an excellent agreement between the theoretical value
and the experimentally obtained value of de Broglie wavelength. Davisson-
Germer experiment thus strikingly confirms the wave nature of electrons
and the de Broglie relation. More recently, in 1989, the wave nature of a
beam of electrons was experimentally demonstrated in a double-slit
experiment, similar to that used for the wave nature of light. Also, in an
experiment in 1994, interference fringes were obtained with the beams of
iodine molecules, which are about a million times more massive than
electrons.
The de Broglie hypothesis has been basic to the development of modern
quantum mechanics. It has also led to the field of electron optics. The
wave properties of electrons have been utilised in the design of electron
microscope which is a great improvement, with higher resolution, over
the optical microscope.
Development of electron microscope
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SUMMARY
1. The minimum energy needed by an electron to come out from a metal
surface is called the work function of the metal. Energy (greater than
the work function (
φ
ο
)
required for electron emission from the metal
surface can be supplied by suitably heating or applying strong electric
field or irradiating it by light of suitable frequency.
2. Photoelectric effect is the phenomenon of emission of electrons by metals
when illuminated by light of suitable frequency. Certain metals respond
to ultraviolet light while others are sensitive even to the visible light.
Photoelectric effect involves conversion of light energy into electrical
energy. It follows the law of conservation of energy. The photoelectric
emission is an instantaneous process and possesses certain special
features.
3. Photoelectric current depends on (i) the intensity of incident light, (ii)
the potential difference applied between the two electrodes, and (iii)
the nature of the emitter material.
4. The stopping potential (V
o
) depends on (i) the frequency of incident
light, and (ii) the nature of the emitter material. For a given frequency
of incident light, it is independent of its intensity. The stopping potential
is directly related to the maximum kinetic energy of electrons emitted:
e V
0
= (1/2) m v
2
max
= K
max
.
5. Below a certain frequency (threshold frequency)
ν
0
, characteristic of
the metal, no photoelectric emission takes place, no matter how large
the intensity may be.
6. The classical wave theory could not explain the main features of
photoelectric effect. Its picture of continuous absorption of energy from
radiation could not explain the independence of K
max
on intensity, the
existence of
ν
o
and the instantaneous nature of the process. Einstein
explained these features on the basis of photon picture of light.
According to this, light is composed of discrete packets of energy called
quanta or photons. Each photon carries an energy E (= h
ν
) and
momentum p (= h/λ), which depend on the frequency (
ν
) of incident
light and not on its intensity. Photoelectric emission from the metal
surface occurs due to absorption of a photon by an electron.
7. Einstein’s photoelectric equation is in accordance with the energy
conservation law as applied to the photon absorption by an electron in
the metal. The maximum kinetic energy (1/2)m v
2
max
is equal to
the photon energy (h
ν
) minus the work function
φ
0
(= h
ν
0
) of the
target metal:
1
2
m v
2
max
= V
0
e = h
ν
φ
0
= h (
ν
ν
0
)
This photoelectric equation explains all the features of the photoelectric
effect. Millikan’s first precise measurements confirmed the Einstein’s
photoelectric equation and obtained an accurate value of Planck’s
constant h. This led to the acceptance of particle or photon description
(nature) of electromagnetic radiation, introduced by Einstein.
8. Radiation has dual nature: wave and particle. The nature of experiment
determines whether a wave or particle description is best suited for
understanding the experimental result. Reasoning that radiation and
matter should be symmetrical in nature, Louis Victor de Broglie
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attributed a wave-like character to matter (material particles). The waves
associated with the moving material particles are called matter waves
or de Broglie waves.
9. The de Broglie wavelength (
λ
) associated with a moving particle is related
to its momentum p as: λ
= h/p. The dualism of matter is inherent in the
de Broglie relation which contains a wave concept (
λ
) and a particle concept
(p). The de Broglie wavelength is independent of the charge and nature of
the material particle. It is significantly measurable (of the order of the
atomic-planes spacing in crystals) only in case of sub-atomic particles
like electrons, protons, etc. (due to smallness of their masses and hence,
momenta). However, it is indeed very small, quite beyond measurement,
in case of macroscopic objects, commonly encountered in everyday life.
10. Electron diffraction experiments by Davisson and Germer, and by G. P.
Thomson, as well as many later experiments, have verified and confirmed
the wave-nature of electrons. The de Broglie hypothesis of matter waves
supports the Bohr’s concept of stationary orbits.
Physical Symbol Dimensions Unit Remarks
Quantity
Planck’s h [ML
2
T
–1
] J s E = h
ν
constant
Stopping V
0
[ML
2
T
–3
A
–1
] V e V
0
= K
max
potential
Work
φ
0
[ML
2
T
–2
] J; eV K
max
= E
φ
0
function
Threshold
ν
0
[T
–1
] Hz
ν
0
=
φ
0
/h
frequency
de Broglie
λ
[L] m
λ
= h/p
wavelength
POINTS TO PONDER
1. Free electrons in a metal are free in the sense that they move inside the
metal in a constant potential (This is only an approximation). They are
not free to move out of the metal. They need additional energy to get
out of the metal.
2. Free electrons in a metal do not all have the same energy. Like molecules
in a gas jar, the electrons have a certain energy distribution at a given
temperature. This distribution is different from the usual Maxwell’s
distribution that you have learnt in the study of kinetic theory of gases.
You will learn about it in later courses, but the difference has to do
with the fact that electrons obey Pauli’s exclusion principle.
3. Because of the energy distribution of free electrons in a metal, the
energy required by an electron to come out of the metal is different for
different electrons. Electrons with higher energy require less additional
energy to come out of the metal than those with lower energies. Work
function is the least energy required by an electron to come out of the
metal.
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4. Observations on photoelectric effect imply that in the event of matter-
light interaction, absorption of energy takes place in discrete units of h
ν
.
This is not quite the same as saying that light consists of particles,
each of energy h
ν
.
5. Observations on the stopping potential (its independence of intensity
and dependence on frequency) are the crucial discriminator between
the wave-picture and photon-picture of photoelectric effect.
6. The wavelength of a matter wave given by
h
p
λ
=
has physical
significance; its phase velocity v
p
has no physical significance. However,
the group velocity of the matter wave is physically meaningful and
equals the velocity of the particle.
EXERCISES
11.1 Find the
(a) maximum frequency, and
(b) minimum wavelength of X-rays produced by 30 kV electrons.
11.2 The work function of caesium metal is 2.14 eV. When light of
frequency 6 ×10
14
Hz is incident on the metal surface, photoemission
of electrons occurs. What is the
(a) maximum kinetic energy of the emitted electrons,
(b) Stopping potential, and
(c) maximum speed of the emitted photoelectrons?
11.3 The photoelectric cut-off voltage in a certain experiment is 1.5 V.
What is the maximum kinetic energy of photoelectrons emitted?
11.4 Monochromatic light of wavelength 632.8 nm is produced by a
helium-neon laser. The power emitted is 9.42 mW.
(a) Find the energy and momentum of each photon in the light beam,
(b) How many photons per second, on the average, arrive at a target
irradiated by this beam? (Assume the beam to have uniform
cross-section which is less than the target area), and
(c) How fast does a hydrogen atom have to travel in order to have
the same momentum as that of the photon?
11.5 The energy flux of sunlight reaching the surface of the earth is
1.388 × 10
3
W/m
2
. How many photons (nearly) per square metre are
incident on the Earth per second? Assume that the photons in the
sunlight have an average wavelength of 550 nm.
11.6 In an experiment on photoelectric effect, the slope of the cut-off
voltage versus frequency of incident light is found to be 4.12 × 10
–15
V s.
Calculate the value of Planck’s constant.
11.7 A 100W sodium lamp radiates energy uniformly in all directions.
The lamp is located at the centre of a large sphere that absorbs all
the sodium light which is incident on it. The wavelength of the
sodium light is 589 nm. (a) What is the energy per photon associated
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with the sodium light? (b) At what rate are the photons delivered to
the sphere?
11.8 The threshold frequency for a certain metal is 3.3 × 10
14
Hz. If light
of frequency 8.2 × 10
14
Hz is incident on the metal, predict the cut-
off voltage for the photoelectric emission.
11.9 The work function for a certain metal is 4.2 eV. Will this metal give
photoelectric emission for incident radiation of wavelength 330 nm?
11.10 Light of frequency 7.21 × 10
14
Hz is incident on a metal surface.
Electrons with a maximum speed of 6.0 × 10
5
m/s are ejected from
the surface. What is the threshold frequency for photoemission of
electrons?
11.11 Light of wavelength 488 nm is produced by an argon laser which is
used in the photoelectric effect. When light from this spectral line is
incident on the emitter, the stopping (cut-off) potential of
photoelectrons is 0.38 V. Find the work function of the material
from which the emitter is made.
11.12 Calculate the
(a) momentum, and
(b) de Broglie wavelength of the electrons accelerated through a
potential difference of 56 V.
11.13 What is the
(a) momentum,
(b) speed, and
(c) de Broglie wavelength of an electron with kinetic energy of
120 eV.
11.14 The wavelength of light from the spectral emission line of sodium is
589 nm. Find the kinetic energy at which
(a) an electron, and
(b) a neutron, would have the same de Broglie wavelength.
11.15 What is the de Broglie wavelength of
(a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km/s,
(b) a ball of mass 0.060 kg moving at a speed of 1.0 m/s, and
(c) a dust particle of mass 1.0 × 10
–9
kg drifting with a speed of
2.2 m/s?
11.16 An electron and a photon each have a wavelength of 1.00 nm. Find
(a) their momenta,
(b) the energy of the photon, and
(c) the kinetic energy of electron.
11.17 (a) For what kinetic energy of a neutron will the associated de Broglie
wavelength be 1.40 × 10
–10
m?
(b) Also find the de Broglie wavelength of a neutron, in thermal
equilibrium with matter, having an average kinetic energy of
(3/2) k
T at 300 K.
11.18 Show that the wavelength of electromagnetic radiation is equal to
the de Broglie wavelength of its quantum (photon).
11.19 What is the de Broglie wavelength of a nitrogen molecule in air at
300 K? Assume that the molecule is moving with the root-mean-
square speed of molecules at this temperature. (Atomic mass of
nitrogen = 14.0076 u)
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ADDITIONAL EXERCISES
11.20 (a) Estimate the speed with which electrons emitted from a heated
emitter of an evacuated tube impinge on the collector maintained
at a potential difference of 500 V with respect to the emitter.
Ignore the small initial speeds of the electrons. The
specific charge of the electron, i.e., its e/m is given to be
1.76 × 10
11
C kg
–1
.
(b) Use the same formula you employ in (a) to obtain electron speed
for an collector potential of 10 MV. Do you see what is wrong ? In
what way is the formula to be modified?
11.21 (a) A monoenergetic electron beam with electron speed of
5.20 × 10
6
m s
–1
is subject to a magnetic field of 1.30 × 10
–4
T
normal to the beam velocity. What is the radius of the circle traced
by the beam, given e/m for electron equals 1.76 × 10
11
C kg
–1
.
(b) Is the formula you employ in (a) valid for calculating radius of
the path of a 20 MeV electron beam? If not, in what way is it
modified?
[Note: Exercises 11.20(b) and 11.21(b) take you to relativistic
mechanics which is beyond the scope of this book. They have been
inserted here simply to emphasise the point that the formulas you
use in part (a) of the exercises are not valid at very high speeds or
energies. See answers at the end to know what ‘very high speed or
energy’ means.]
11.22 An electron gun with its collector at a potential of 100 V fires out
electrons in a spherical bulb containing hydrogen gas at low
pressure (10
–2
mm of Hg). A magnetic field of 2.83 × 10
–4
T curves
the path of the electrons in a circular orbit of radius 12.0 cm. (The
path can be viewed because the gas ions in the path focus the beam
by attracting electrons, and emitting light by electron capture; this
method is known as the ‘fine beam tube’ method.) Determine
e/m from the data.
11.23 (a) An X-ray tube produces a continuous spectrum of radiation with
its short wavelength end at 0.45 Å. What is the maximum energy
of a photon in the radiation?
(b) From your answer to (a), guess what order of accelerating voltage
(for electrons) is required in such a tube?
11.24 In an accelerator experiment on high-energy collisions of electrons
with positrons, a certain event is interpreted as annihilation of an
electron-positron pair of total energy 10.2 BeV into two γ-rays of
equal energy. What is the wavelength associated with each γ-ray?
(1BeV = 10
9
eV)
11.25 Estimating the following two numbers should be interesting. The
first number will tell you why radio engineers do not need to worry
much about photons! The second number tells you why our eye can
never ‘count photons’, even in barely detectable light.
(a) The number of photons emitted per second by a Medium wave
transmitter of 10 kW power, emitting radiowaves of wavelength
500 m.
(b) The number of photons entering the pupil of our eye per second
corresponding to the minimum intensity of white light that we
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humans can perceive (10
–10
W m
–2
). Take the area of the pupil
to be about 0.4 cm
2
, and the average frequency of white light to
be about 6 × 10
14
Hz.
11.26 Ultraviolet light of wavelength 2271 Å from a 100 W mercury source
irradiates a photo-cell made of molybdenum metal. If the stopping
potential is –1.3 V, estimate the work function of the metal. How
would the photo-cell respond to a high intensity (10
5
W m
–2
) red
light of wavelength 6328 Å produced by a He-Ne laser?
11.27 Monochromatic radiation of wavelength 640.2 nm (1nm = 10
–9
m)
from a neon lamp irradiates photosensitive material made of caesium
on tungsten. The stopping voltage is measured to be 0.54 V. The
source is replaced by an iron source and its 427.2 nm line irradiates
the same photo-cell. Predict the new stopping voltage.
11.28 A mercury lamp is a convenient source for studying frequency
dependence of photoelectric emission, since it gives a number of
spectral lines ranging from the UV to the red end of the visible
spectrum. In our experiment with rubidium photo-cell, the following
lines from a mercury source were used:
λ
1
= 3650 Å, λ
2
= 4047 Å, λ
3
= 4358 Å, λ
4
= 5461 Å, λ
5
= 6907 Å,
The stopping voltages, respectively, were measured to be:
V
01
= 1.28 V, V
02
= 0.95 V, V
03
= 0.74 V, V
04
= 0.16 V, V
05
= 0 V
Determine the value of Planck’s constant h, the threshold frequency
and work function for the material.
[Note: You will notice that to get h from the data, you will need to
know e (which you can take to be 1.6 × 10
–19
C). Experiments of this
kind on Na, Li, K, etc. were performed by Millikan, who, using his
own value of e (from the oil-drop experiment) confirmed Einstein’s
photoelectric equation and at the same time gave an independent
estimate of the value of h.]
11.29 The work function for the following metals is given:
Na: 2.75 eV; K: 2.30 eV; Mo: 4.17 eV; Ni: 5.15 eV. Which of these
metals will not give photoelectric emission for a radiation of
wavelength 3300 Å from a He-Cd laser placed 1 m away from the
photocell? What happens if the laser is brought nearer and placed
50 cm away?
11.30 Light of intensity 10
–5
W m
–2
falls on a sodium photo-cell of surface
area 2 cm
2
. Assuming that the top 5 layers of sodium absorb the
incident energy, estimate time required for photoelectric emission
in the wave-picture of radiation. The work function for the metal is
given to be about 2 eV. What is the implication of your answer?
11.31 Crystal diffraction experiments can be performed using X-rays, or
electrons accelerated through appropriate voltage. Which probe has
greater energy? (For quantitative comparison, take the wavelength
of the probe equal to 1 Å, which is of the order of inter-atomic spacing
in the lattice) (m
e
=9.11 × 10
–31
kg).
11.32 (a) Obtain the de Broglie wavelength of a neutron of kinetic energy
150 eV. As you have seen in Exercise 11.31, an electron beam of
this energy is suitable for crystal diffraction experiments. Would
a neutron beam of the same energy be equally suitable? Explain.
(m
n
= 1.675 × 10
–27
kg)
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(b) Obtain the de Broglie wavelength associated with thermal
neutrons at room temperature (27 °C). Hence explain why a fast
neutron beam needs to be thermalised with the environment
before it can be used for neutron diffraction experiments.
11.33 An electron microscope uses electrons accelerated by a voltage of
50 kV. Determine the de Broglie wavelength associated with the
electrons. If other factors (such as numerical aperture, etc.) are
taken to be roughly the same, how does the resolving power of an
electron microscope compare with that of an optical microscope
which uses yellow light?
11.34 The wavelength of a probe is roughly a measure of the size of a
structure that it can probe in some detail. The quark structure
of protons and neutrons appears at the minute length-scale of
10
–15
m or less. This structure was first probed in early 1970’s using
high energy electron beams produced by a linear accelerator at
Stanford, USA. Guess what might have been the order of energy of
these electron beams. (Rest mass energy of electron = 0.511 MeV.)
11.35 Find the typical de Broglie wavelength associated with a He atom in
helium gas at room temperature (27 °C) and 1 atm pressure; and
compare it with the mean separation between two atoms under these
conditions.
11.36 Compute the typical de Broglie wavelength of an electron in a metal
at 27 °C and compare it with the mean separation between two
electrons in a metal which is given to be about 2 × 10
–10
m.
[Note: Exercises 11.35 and 11.36 reveal that while the wave-packets
associated with gaseous molecules under ordinary conditions are
non-overlapping, the electron wave-packets in a metal strongly
overlap with one another. This suggests that whereas molecules in
an ordinary gas can be distinguished apart, electrons in a metal
cannot be distintguished apart from one another. This
indistinguishibility has many fundamental implications which you
will explore in more advanced Physics courses.]
11.37 Answer the following questions:
(a) Quarks inside protons and neutrons are thought to carry
fractional charges [(+2/3)e ; (–1/3)e]. Why do they not show up
in Millikan’s oil-drop experiment?
(b) What is so special about the combination e/m? Why do we not
simply talk of e and m separately?
(c) Why should gases be insulators at ordinary pressures and start
conducting at very low pressures?
(d) Every metal has a definite work function. Why do all
photoelectrons not come out with the same energy if incident
radiation is monochromatic? Why is there an energy distribution
of photoelectrons?
(e) The energy and momentum of an electron are related to the
frequency and wavelength of the associated matter wave by the
relations:
E = h
ν
, p =
λ
h
But while the value of
λ
is physically significant, the value of
ν
(and therefore, the value of the phase speed
ν λ
) has no physical
significance. Why?
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APPENDIX
11.1 The history of wave-particle flip-flop
What is light? This question has haunted mankind for a long time. But systematic experiments were done by
scientists since the dawn of the scientific and industrial era, about four centuries ago. Around the same time,
theoretical models about what light is made of were developed. While building a model in any branch of
science, it is essential to see that it is able to explain all the experimental observations existing at that time.
It is therefore appropriate to summarize some observations about light that were known in the seventeenth
century.
The properties of light known at that time included (a) rectilinear propagation of light, (b) reflection from
plane and curved surfaces, (c) refraction at the boundary of two media, (d) dispersion into various colours, (e)
high speed. Appropriate laws were formulated for the first four phenomena. For example, Snell formulated his
laws of refraction in 1621. Several scientists right from the days of Galileo had tried to measure the speed of
light. But they had not been able to do so. They had only concluded that it was higher than the limit of their
measurement.
Two models of light were also proposed in the seventeenth century. Descartes, in early decades of seventeenth
century, proposed that light consists of particles, while Huygens, around 1650-60, proposed that light consists
of waves. Descartes proposal was merely a philosophical model, devoid of any experiments or scientific
arguments. Newton soon after, around 1660-70, extended Descartes particle model, known as corpuscular
theory
, built it up as a scientific theory, and explained various known properties with it. These models, light
as waves and as particles, in a sense, are quite opposite of each other. But both models could explain all the
known properties of light. There was nothing to choose between them.
The history of the development of these models over the next few centuries is interesting. Bartholinus, in
1669, discovered double refraction of light in some crystals, and Huygens, in 1678, was quick to explain it on
the basis of his wave theory of light. In spite of this, for over one hundred years, Newton’s particle model was
firmly believed and preferred over the wave model. This was partly because of its simplicity and partly because
of Newton’s influence on contemporary physics.
Then in 1801, Young performed his double-slit experiment and observed interference fringes. This
phenomenon could be explained only by wave theory. It was realized that diffraction was also another
phenomenon which could be explained only by wave theory. In fact, it was a natural consequence of Huygens
idea of secondary wavelets emanating from every point in the path of light. These experiments could not be
explained by assuming that light consists of particles. Another phenomenon of polarisation was discovered
around 1810, and this too could be naturally explained by the wave theory. Thus wave theory of Huygens
came to the forefront and Newton’s particle theory went into the background. This situation again continued
for almost a century.
Better experiments were performed in the nineteenth century to determine the speed of light. With more
accurate experiments, a value of 3×10
8
m/s for speed of light in vacuum was arrived at. Around 1860, Maxwell
proposed his equations of electromagnetism and it was realized that all electromagnetic phenomena known at
that time could be explained by Maxwell’s four equations. Soon Maxwell showed that electric and magnetic
fields could propagate through empty space (vacuum) in the form of electromagnetic waves. He calculated the
speed of these waves and arrived at a theoretical value of 2.998×10
8
m/s. The close agreement of this value
with the experimental value suggested that light consists of electromagnetic waves. In 1887 Hertz demonstrated
the generation and detection of such waves. This established the wave theory of light on a firm footing. We
might say that while eighteenth century belonged to the particle model, the nineteenth century belonged to
the wave model of light.
Vast amounts of experiments were done during the period 1850-1900 on heat and related phenomena, an
altogether different area of physics. Theories and models like kinetic theory and thermodynamics were developed
which quite successfully explained the various phenomena, except one.
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Every body at any temperature emits radiation of all wavelengths. It also absorbs radiation falling on it.
A body which absorbs all the radiation falling on it is called a black body. It is an ideal concept in physics, like
concepts of a point mass or uniform motion. A graph of the intensity of radiation emitted by a black body
versus wavelength is called the black body spectrum. No theory in those days could explain the complete black
body spectrum!
In 1900, Planck hit upon a novel idea. If we assume, he said, that radiation is emitted in packets of energy
instead of continuously as in a wave, then we can explain the black body spectrum. Planck himself regarded
these quanta, or packets, as a property of emission and absorption, rather than that of light. He derived a
formula which agreed with the entire spectrum. This was a confusing mixture of wave and particle pictures –
radiation is emitted as a particle, it travels as a wave, and is again absorbed as a particle! Moreover, this put
physicists in a dilemma. Should we again accept the particle picture of light just to explain one phenomenon?
Then what happens to the phenomena of interference and diffraction which cannot be explained by the
particle model?
But soon in 1905, Einstein explained the photoelectric effect by assuming the particle picture of light.
In 1907, Debye explained the low temperature specific heats of solids by using the particle picture for lattice
vibrations in a crystalline solid. Both these phenomena belonging to widely diverse areas of physics could be
explained only by the particle model and not by the wave model. In 1923, Compton’s x-ray scattering experiments
from atoms also went in favour of the particle picture. This increased the dilemma further.
Thus by 1923, physicists faced with the following situation. (a) There were some phenomena like rectilinear
propagation, reflection, refraction, which could be explained by either particle model or by wave model. (b)
There were some phenomena such as diffraction and interference which could be explained only by the wave
model but not by the particle model. (c) There were some phenomena such as black body radiation, photoelectric
effect, and Compton scattering which could be explained only by the particle model but not by the wave model.
Somebody in those days aptly remarked that light behaves as a particle on Mondays, Wednesdays and Fridays,
and as a wave on Tuesdays, Thursdays and Saturdays, and we don’t talk of light on Sundays!
In 1924, de Broglie proposed his theory of wave-particle duality in which he said that not only photons
of light but also ‘particles’ of matter such as electrons and atoms possess a dual character, sometimes
behaving like a particle and sometimes as a wave. He gave a formula connecting their mass, velocity, momentum
(particle characteristics), with their wavelength and frequency (wave characteristics)! In 1927 Thomson, and
Davisson and Germer, in separate experiments, showed that electrons did behave like waves with a wavelength
which agreed with that given by de Broglie’s formula. Their experiment was on diffraction of electrons through
crystalline solids, in which the regular arrangement of atoms acted like a grating. Very soon, diffraction
experiments with other ‘particles’ such as neutrons and protons were performed and these too confirmed with
de Broglie’s formula. This confirmed wave-particle duality as an established principle of physics. Here was a
principle, physicists thought, which explained all the phenomena mentioned above not only for light but also
for the so-called particles.
But there was no basic theoretical foundation for wave-particle duality. De Broglie’s proposal was
merely a qualitative argument based on symmetry of nature. Wave-particle duality was at best a principle, not
an outcome of a sound fundamental theory. It is true that all experiments whatever agreed with de Broglie
formula. But physics does not work that way. On the one hand, it needs experimental confirmation, while on
the other hand, it also needs sound theoretical basis for the models proposed. This was developed over the
next two decades. Dirac developed his theory of radiation in about 1928, and Heisenberg and Pauli gave it a
firm footing by 1930. Tomonaga, Schwinger, and Feynman, in late 1940s, produced further refinements and
cleared the theory of inconsistencies which were noticed. All these theories mainly put wave-particle duality
on a theoretical footing.
Although the story continues, it grows more and more complex and beyond the scope of this note. But
we have here the essential structure of what happened, and let us be satisfied with it at the moment. Now it
is regarded as a natural consequence of present theories of physics that electromagnetic radiation as well as
particles of matter exhibit both wave and particle properties in different experiments, and sometimes even in
the different parts of the same experiment.
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