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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

m/s). The presently accepted value

of e/m is 1.76 × 10

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

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

This unit of energy is commonly used in atomic and nuclear physics.

The work function (

) 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

(

= 5.65 eV) while it is the lowest (

= 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

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

οο

(eV)

φφ

οο

(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

and fixed

intensity I

. 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

on the plate A. For a particular

frequency of incident radiation, the

minimum negative (retarding) potential V

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

max

= e V

(11.1)

We can now repeat this experiment with incident radiation of the same

frequency but of higher intensity I

and I

> I

). 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

, 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

. 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

> V

if the frequencies are in the

order ν

> ν

. 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

varies linearly with

the frequency of incident radiation for a given

photosensitive material.

(ii) there exists a certain minimum cut-off frequency

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

of incident radiation, lower than the cut-off

frequency

, no photoelectric emission is possible even if the

intensity is large.

This minimum, cut-off frequency

, 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

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 EINSTEIN’S 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

, where h is Planck’s constant and

the

frequency of light. In photoelectric effect, an electron

absorbs a quantum of energy (h

) of radiation. If this

quantum of energy absorbed exceeds the minimum

energy needed for the electron to escape from the metal

surface (work function

), the electron is emitted with

maximum kinetic energy

max

= h

–

(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

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

where

(11.3)

Equation (11.3) shows that the greater the work

function

, the higher the minimum or threshold

frequency ν

needed to emit photoelectrons. Thus,

there exists a threshold frequency

/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

). This explains why, for

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

= h

–

;

for

ν ν

≥

or V

e e

 

−

 

 

(11.4)

This is an important result. It predicts that the V

versus

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

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

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

/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

) and momentum p (= h

/c), and

speed c, the speed of light.

(iii) All photons of light of a particular frequency

, or wavelength

, have

the same energy E (=h

= hc/

) and momentum p (= h

/c= h/

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

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

= ( 6.63 ×10

–34

J s) (6.0 ×10

Hz)

= 3.98 × 10

–19

(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 =

2.0 10 W

3.98 10 J

−

= 5.0 ×10

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

the incident

radiation must be equal to work function

, so that

2.14 eV

6.63 10 J s

−

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

by the

retarding potential V

. 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/

E = (6.63×10

–34

J s) (3×10

m/s)/

–25

1.989 10 Jm

(i) For violet light,

= 390 nm (lower wavelength end)

Incident photon energy, E

–25

–9

1.989 10 Jm

390×10 m

= 5.10 × 10

–19

5.10 10 J

1.6×10 J/eV

= 3.19 eV

(ii) For yellow-green light,

= 550 nm (average wavelength)

Incident photon energy, E

–25

–9

1.989 10 Jm

550×10 m

= 3.62×10

–19

J = 2.26 eV

(iii) For red light,

= 760 nm (higher wavelength end)

Incident photon energy, E

–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

of the material.

Thus, the photoelectric device will operate with violet light (with

E = 3.19 eV) photosensitive material Na (with

= 2.75 eV), K (with

= 2.30 eV) and Cs (with

= 2.14 eV). It will also operate with

yellow-green light (with E = 2.26 eV) for Cs (with

= 2.14 eV) only.

However, it will not operate with red light (with E = 1.64 eV) for any

of these photosensitive materials.

= h

–

or,

= hc/(eV

)

34 8

(6.63 10 Js) (3 10 m/s)

(0.60eV 2.14eV)

−

× × ×

19.89 10 J m

(2.74eV)

−

19.89 10 J m

454 nm

2.74 1.6 10 J

−

= =

× ×

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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

associated with a particle of momentum

p is given as

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

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

/c (11.6)

Therefore,

h c

= =

(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 ),

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

6.63 10 J s

2.40 kg m s

−

= 2.76 × 10

–34

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 =

m v

, so that

p =

2 2

m K m e V

(11.9)

The de Broglie wavelength

of the electron is then

λ =

2 2

h h h

m K m eV

= =

(11.10)

Substituting the numerical values of h, m, e,

we get

1.227

(11.11)

where V is the magnitude of accelerating potential in

volts. For a 120 V accelerating potential, Eq. (11.11) gives

= 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

. 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

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

m/s. Then, momentum

p = m v = 9.11

×10

–31

(kg) × 5.4 × 10

(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

= 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

–

6 63 10 Js

4 50 kg m/s

’ = 1.47 ×10

–34

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,

= h/p

Kinetic energy, K = p

/2m

Then,

/ 2

h mK

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

(

)

is 1836 times massive than an electron and

an α-particle

(

)

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|

∆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

= =

Mass, m = h/

For an electron, mass m

= h/

Now, we have v/v

= 3 and

= 1.813 × 10

–4

Then, mass of the particle, m = m

e e

   

   

   

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

= h /p

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

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

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

= 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

associated with electrons, using

Eq. (11.11), for V = 54 V is given by

= h /p

1 227

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 (

)

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

) 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

= (1/2) m v

max

= K

max

5. Below a certain frequency (threshold frequency)

, 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

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

) and

momentum p (= h/λ), which depend on the frequency (

) 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

max

is equal to

the photon energy (h

) minus the work function

(= h

) of the

target metal:

m v

max

= V

e = h

–

= h (

–

)

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 (

) 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 (

) 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

–1

] J s E = h

constant

Stopping V

[ML

–3

–1

] V e V

= K

max

potential

Work

[ML

–2

] J; eV K

max

= E –

function

Threshold

–1

] Hz

frequency

de Broglie

[L] m

= 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

This is not quite the same as saying that light consists of particles,

each of energy h

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

has physical

significance; its phase velocity v

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

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

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

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

W/m

. 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

Hz. If light

of frequency 8.2 × 10

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

Hz is incident on a metal surface.

Electrons with a maximum speed of 6.0 × 10

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

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

–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

11.17 (a) For what kinetic energy of a neutron will the associated de Broglie

wavelength be 1.40 × 10

–10

(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

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

m s

–1

is subject to a magnetic field of 1.30 × 10

–4

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

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

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

, and the average frequency of white light to

be about 6 × 10

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

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

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:

= 3650 Å, λ

= 4047 Å, λ

= 4358 Å, λ

= 5461 Å, λ

= 6907 Å,

The stopping voltages, respectively, were measured to be:

= 1.28 V, V

= 0.95 V, V

= 0.74 V, V

= 0.16 V, V

= 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

. 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

=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.

= 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

–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

[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?

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

, p =

But while the value of

is physically significant, the value of

(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

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

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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