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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.
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 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)
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 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.
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 – 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.