Chapter 10: Wave Properties of

Particles

Particles such as electrons may demonstrate wave

properties under certain conditions. The electron

microscope uses these properties to produce magnified

images of minute objects that could not be produced by

optical microscope.

Overview

Wave Properties

of Particles

Electron

Diffraction

De Broglie

Wavelength

10.1 The de Broglie Wavelength

State wave-particle duality.

Use de Broglie wavelength,

Learning Objectives

p

h

The de Broglie Wavelength

Wave-particle duality is the phenomenon where under

certain circumstances a particle exhibits wave

properties, and under other conditions a wave exhibits

properties of a particle. But we cannot observe both

aspect of its behaviour simultaneously.

According to the Planck’s quantum theory, a photon

of electromagnetic radiation of wavelength λ has energy:

hchfE (1)

The de Broglie Wavelength

According to Einstein’s theory of special relativity, the

energy equivalent E of a mass m is given by

Since momentum p = mc, the equation can also be

written as E = pc.

By equating (1) and (2):

2mcE (2)

pchc

p

h

De Broglie

Wavelength Properties

of wave Properties

of particle

The de Broglie Wavelength

Evidences to show duality of light:

Light

can behave as

Particle Wave

Photoelectric Effect Young’s Double Slit

experiment

Compton effect Diffraction grating

experiment

The de Broglie Wavelength

Evidences to show duality of particle:

Particle can

behave as a wave

Electron Diffraction

(Davisson-Germer

Experiment)

Example 1

Calculate the de Broglie wavelength for :

a. A car of mass 2×103 kg moving at 50 m s -1

b. An electron of mass 9.11×10-31 kg moving at 1×108 m s-1

(Given the speed of photon in the vacuum, c = 3.0×108 m s-1

and Planck constant, h = 6.63×10-34 J s)

Example 1 – Solution

Example 2

In a photoelectric effect experiment, a light source of

wavelength 500 nm is incident on a potassium surface. Find

the momentum and energy of a photon used.

(Given the speed of photon in the vacuum, c = 3.0×108 m s-1

and Planck constant, h = 6.63×10-34 J s)

Example 2 – Solution

Davisson-Germer Experiment

Electron diffraction tube

Davisson-Germer Experiment

In 1927, two physicists C.J Davission and L. H Germer

carried out electron diffraction experiment to prove the

de Broglie relationship.

A graphite film is used as a target.

A beam of electrons in a cathode-ray tube is

accelerated by the applied voltage towards a graphite

film.

The beam of electrons is diffracted after passing

through the graphite film.

A diffraction pattern is observed on the fluorescence

screen.

Davisson-Germer Experiment

This shows that a beam of fast moving particles

(electrons) behaves as a wave, exhibiting diffraction – a

wave property.

Davisson and Germer discovered that if the velocity of

electrons is increased, the rings are seen to become

narrower showing that the wavelength of electrons

decreases with increasing velocity as predicted by de

Broglie relationship.

,

mv

h ,v

Davisson-Germer Experiment

The velocity of electrons can be determined from the

accelerating voltage (voltage between anode and

cathode):

By substituting equation above into de Broglie relation:

KU

2

2

1mveV

m

eVv

2

meV

h

2

Example 3

An electron is accelerated from rest through a

potential difference of 1200 V. Calculate its de

Broglie wavelength.

(Given c = 3.00108 m s1, h = 6.631034 J s, me = 9.111031 kg and e = 1.601019 C)

Example 3 – Solution

Example 4

An electron and a proton have the same kinetic

energy. Determine the ratio of the de Broglie

wavelength of the electron to that of the proton.

Example 4 – Solution

Electron Microscope

A practical device that relies on the wave properties of electrons is electron microscope.

It is similar to optical compound microscope in many aspects.

The advantage of the electron microscope over the optical microscope is the resolving power of the electron microscope is much higher than that of an optical microscope.

◦ The resolving power is inversely proportional to the wavelength - a smaller wavelength means greater resolving power, or the ability to see details.

Electron Microscope

This is because the electrons can be accelerated to a very high kinetic energy (KE) giving them a very short wavelength λ typically 100 times shorter than those of visible light.

As a result, electron microscopes are able to distinguish details about 100 times smaller.

◦ Thus, an electron microscope can distinguish clearly 2 points separated by a distance which is of the order of nanometer.

◦ But a compound microscope can only distinguish clearly 2 points separated by a distance which is of order of micrometer.

Electron Microscope There are two types of electron microscopes:

◦ Transmission – produces a two-dimensional image.

◦ Scanning – produces images with a three-

dimensional quality.

Wave Behaviour of Electron in an

Electron Microscope 1. In the electron microscope, electrons are produced by

the electron gun.

2. Electrons are accelerated by voltages on the order of

105 V have wavelengths on the order of 0.004 nm.

3. Electrons are deflected by the “magnetic lens” to form

a parallel beam which then incident on the object.

4. The “magnetic lens” is actually magnetic fields that

exert forces on the electrons to bring them to a focus.

The fields are produced by carefully designed current-

carrying coils of wire.

Wave Behaviour of Electron in an

Electron Microscope 5. When the object is struck by the electrons, more

penetrate in some parts than in others, depending on

the thickness and density of the part.

6. The image is formed on a fluorescent screen. The

image is brightest where most electrons have been

transmitted. The object must be very thin, otherwise

too much electron scattering occurs and no image

form.

Example 5

Why can an electron microscope resolve smaller

objects than a light microscope?

Example 5 – Solution

An electron microscope resolve smaller objects than a light

microscope because the electrons can be accelerated to a

very high kinetic energy (KE) giving them a very short

wavelength λ typically 100 times shorter than those of

visible light.

Since the resolving power is inversely proportional to

the wavelength,

wavelength ↓, resolving power ↑

Therefore electron microscopes are able to distinguish

details about 100 times smaller than optical microscope.