Lecture PowerPoints
Chapter 27
Physics: Principles with
Applications, 7th edition
Giancoli
© 2014 Pearson Education, Inc.
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Chapter 27
Early Quantum Theory and
Models of the Atom
© 2014 Pearson Education, Inc.
Contents of Chapter 27
•  Discovery and Properties of the Electron
•  Blackbody Radiation; Planck’s Quantum Hypothesis
•  Photon Theory of Light and the Photoelectric Effect
•  Energy, Mass, and Momentum of a Photon
•  Compton Effect
•  Photon Interactions; Pair Production
© 2014 Pearson Education, Inc.
Contents of Chapter 27
•  Wave-Particle Duality; the Principle of Complementarity
•  Wave Nature of Matter
•  Electron Microscopes
•  Early Models of the Atom
•  Atomic Spectra: Key to the Structure of the Atom
•  The Bohr Model
•  de Broglie’s Hypothesis Applied to Atoms
© 2014 Pearson Education, Inc.
27-1 Discovery and Properties of the Electron
In the late 19th century, discharge tubes were made that emitted “cathode
rays.” These rays are emitted at the cathode, or negative terminal. In 1879
Crookes had proposed that the cathode rays were 'radiant matter', negatively
charged particles that were repelled from the negatively charged cathode
and attracted to the positively charged anode. The nature of the cathode rays
was controversial. Although Thomson thought the rays must be particles,
many Europeans thought they were waves. In Germany Hertz had observed
the rays passing through thin sheets of gold. It seemed impossible that
particles could pass through solid matter.
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27-1 Discovery and Properties of the Electron
J.J. Thomson realized that he could deflect the cathode rays in an electric field
produced by a pair of metal plates. One of the plates was negatively charged, while the
other was positively charged. The cathode rays moved towards the positively charged
plate so the rays must be negatively charged.
A current in a coil of wire produces a magnetic field. Two coils will produce a uniform
magnetic field. A beam of charged particles passing through the magnetic field will be
bent.
The magnetic field produces a force which deflects the cathode rays. Thomson
positioned the coils so that the deflection was in the opposite direction to the deflection
produced by the electric field. By adjusting the strengths of the fields the rays could be
deflected, in one direction by the electric field, and back an equal amount by the
magnetic field. The forces were balanced.
27-1 Discovery and Properties of the Electron
By accelerating the rays through a known potential and then
measuring the radius of their path in a known magnetic field, the
charge to mass ratio could be measured:
(27-1)
The result is e/m = 1.76 × 1011 C/kg.
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27-1 Discovery and Properties of the Electron
Cathode rays were soon called electrons.
Millikan devised an experiment to measure the charge
on the electron by measuring the electric field needed
to suspend an oil droplet of known mass between parallel plates.
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27-1 Discovery and Properties of the Electron
The mass and charge of each droplet were measured; careful
analysis of the data showed that the charge was always an integral
multiple of a smallest charge, e.
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27-1 Discovery and Properties of the Electron
The currently accepted value of e is:
e = 1.602 × 10−19 C
Knowing e allows the electron mass to be calculated:
me = 9.11 × 10−31 kg
© 2014 Pearson Education, Inc.
27-2 Blackbody Radiation;
Planck’s Quantum Hypothesis
We shall now turn to another puzzle confronting physicists at the turn of the
century (1900): just how do heated bodies radiate? There was a general
understanding of the mechanism involved—heat was known to cause the
molecules and atoms of a solid to vibrate. What is meant by the phrase
“black body” radiation? The point is that the radiation from a heated body
depends to some extent on the body being heated.
All objects emit radiation whose total intensity is proportional to the fourth
power of their temperature. This is called thermal radiation; a blackbody is
an object that emits thermal radiation only.
The spectrum of blackbody radiation has been measured; it is found that the
frequency of peak intensity increases linearly with temperature.
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27-2 Blackbody Radiation;
Planck’s Quantum Hypothesis
This figure shows
blackbody radiation curves
for three different
temperatures. Note that
frequency increases to the
left.
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27-2 Blackbody Radiation;
Planck’s Quantum Hypothesis
This spectrum could not be reproduced using19th-century physics.
A solution was proposed by Max Planck in 1900:
The energy of atomic oscillations within atoms cannot have an
arbitrary value; it is related to the frequency:
E = hf
The constant h is now called Planck’s constant.
The smallest amount of energy possible hf is called the quantum
of energy
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27-2 Blackbody Radiation;
Planck’s Quantum Hypothesis
Planck found the value of his constant by fitting blackbody
curves:
h = 6.626 × 10−34 J⋅s
Planck’s proposal was that the energy of an oscillation had to be
an integral multiple of hf. This is called the quantization of energy.
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Example
•  Estimate the peal wavelength of light emitted from the pupil of
the human eye assuming normal body temperature (~37°C).
•  Solution: T = 273 + 37 = 310 K.
−3
2.9
×
10
m⋅ K
−3
λP T = 2.9 × 10 m⋅ K ⇒ λP =
= 9.35 × 10 −6 m
310K
€
© 2014 Pearson Education, Inc.
27-3 Photon Theory of Light and the
Photoelectric Effect
Einstein suggested that, given the success of Planck’s theory, light
must exist as a set of small energy packets:
(27-4)
These tiny packets, or particles, are called photons.
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27-3 Photon Theory of Light and the
Photoelectric Effect
The photoelectric effect:
If light strikes a metal, electrons
are emitted. The effect does not
occur if the frequency of the
light is too low; the kinetic
energy of the electrons increases
with frequency.
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27-3 Photon Theory of Light and the
Photoelectric Effect
If light is a wave, theory predicts for the photoelectric effect:
1.  Number of electrons and their energy should increase
with intensity
2.  Frequency should not matter
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27-3 Photon Theory of Light and the
Photoelectric Effect
If light is particles (photons), theory predicts for the photoelectric
effect:
•  Increasing intensity increases number of electrons but not
energy
•  Above a minimum energy required to break atomic bond,
kinetic energy will increase linearly with frequency
•  There is a cutoff frequency below which no electrons will be
emitted, regardless of intensity
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27-3 Photon Theory of Light and the
Photoelectric Effect
The particle theory assumes that an electron absorbs a single photon.
Plotting the kinetic energy vs. frequency:
This shows clear
agreement with the
particle theory of light,
not wave theory
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27-3 Photon Theory of Light and the
Photoelectric Effect
The photoelectric effect is how “electric eye” detectors work. It is
also used for movie film soundtracks.
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27-4 Energy, Mass, and
Momentum of a Photon
Photons must travel at the speed of light. Looking at the
relativistic equation for momentum, it is clear that this can only
happen if its rest mass is zero.
We already know that the energy is hf; we can put this in the
relativistic energy-momentum relation and find the momentum:
(27-6)
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Example
•  About 0.1 eV is necessary to break a hydrogen bond in a
protein molecule. Calculate the minimum frequency and
maximum wavelength of a photon that can do this.
•  Solution:
E min 0.1 eV ⋅ 1.6 × 10 −19 J / eV
13
E min = hf min ⇒ f min =
=
=
2.4
×
10
Hz;
−34
h
6.63 × 10 J⋅ s
c
3.0 × 10 8 m / s
−5
λmax =
=
=
1.24
×
10
m
13
f min
2.4 × 10 Hz
€
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27-5 Compton Effect
Compton did experiments in which he scattered X-rays from different
materials. He found that the scattered X-rays had a slightly longer
wavelength than the incident ones, and that the wavelength depended
on the scattering angle:
(27-7)
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27-6 Photon Interactions; Pair Production
Photons passing through matter can undergo the following
interactions:
1.  Photoelectric effect: photon is completely absorbed,
electron is ejected
2.  Photon may be totally absorbed by electron, but not
have enough energy to eject it; the electron moves into
an excited state
3.  The photon can scatter from an atom and lose some
energy
4.  The photon can produce an electron-positron pair.
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27-6 Photon Interactions; Pair Production
In pair production, energy, electric charge, and momentum must
all be conserved.
Energy will be conserved
through the mass and
kinetic energy of the
electron and positron;
their opposite charges
conserve charge; and the
interaction must take
place in the electromagnetic field of a
nucleus, which can
contribute momentum.
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27-7 Wave-Particle Duality;
the Principle of Complementarity
We have phenomena such as diffraction and interference that show
that light is a wave, and phenomena such as the photoelectric effect
and the Compton effect that show that it is a particle.
Which is it?
This question has no answer; we must accept the dual waveparticle nature of light.
The principle of complementarity states that both the wave
and particle aspects of light are fundamental to its nature.
Indeed, waves and particles are just our interpretation of how light
behaves.
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27-8 Wave Nature of Matter
Just as light sometimes behaves as a particle, matter sometimes
behaves like a wave.
The wavelength of a particle of matter is:
(27-8)
This wavelength is extraordinarily small for most objects. The
wave nature of matter becomes more important for very light
particles such as the electron.
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Example 2
•  Visible light incident on a diffraction grating with slot spacing of 0.01mm
has the first maximum at an angle of 3.6° from the central peak. If electrons
could be diffracted by the same grating, what electron velocity would
produce the same diffraction pattern as the visible light?
•  Solution: First maximum means that we have constructive interference here:
d sin θ h
h
d sin θ = nλ ⇒ λ =
= =
⇒
n
p mv
6.63 × 10 J⋅ s)⋅ 1
(
hn
v=
=
= 1159 m / s
md sin θ (9.11 × 10 kg)(0.01 × 10 m)( sin 3.6°)
−34
−31
€
© 2014 Pearson Education, Inc.
−3
27-8 Wave Nature of Matter
Electron wavelengths can
easily be on the order of
10−10 m; electrons can be
diffracted by crystals just
as X-rays can.
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Examples
(1) Calculate the wavelength of a 0.21-kg ball traveling at 0.1 m/s.
Solution:
h
h
6.63 × 10 −34 J⋅ s
λ= =
=
= 3.2 × 10 −32 m
p mv (0.21kg)(0.1m / s)
Extremely small! Cannot detect it at all in everyday’s life.
€
(2) What is the wavelength of an electron (m=9.11×10−31 kg)
traveling at 8.5×104 m/s?
h
h
6.63 × 10 −34 J⋅ s
−9
λ= =
=
=
8.6
×
10
m
−31
4
p mv (9.11 × 10 kg)(8.5 × 10 m / s)
€
Waves with such wavelength can be observed in diffraction
experiments
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27-9 Electron Microscopes
The wavelength of electrons will vary with energy,
but is still quite short. This makes electrons useful for imaging—
remember that the smallest object that can
be resolved is about one wavelength. Electrons used
in electron microscopes have wavelengths of about 0.004 nm.
Example: What voltage is needed to produce electron wavelength of
0.06 nm?
Solution: The kinetic energy is acquired by electrostatic potential energy.
mv 2 p 2
h
h2
KE =
=
; p = ⇒ KE =
2
2m
λ
2mλ2
KE = eV (electron is accerated by electric field) ⇒
2
6.63 × 10 J⋅ s)
(
h
h
= eV ⇒ V =
=
= 419 V
2mλ2
2meλ2 2(9.11 × 10 −31 kg)(1.6 × 10 −19 C )(0.06 × 10 −9 m) 2
2
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2
−34
27-9 Electron Microscopes
Transmission electron microscope
—the electrons are focused by
magnetic coils
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27-9 Electron Microscopes
Scanning electron microscope
—the electron beam is
scanned back and forth across
the object to be imaged
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27-9 Electron Microscopes
Scanning tunneling microscope—up and down motion of the
probe keeps the current constant.
Plotting that motion
produces an image of
surface.
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the
27-10 Early Models of the Atom
It was known in the late 19th century that atoms were electrically
neutral, but that they could become charged, implying that there
were positive and negative charges and that some of them could
be removed.
One popular atomic model was
the “plum-pudding” model. This
model had the atom
consisting of a bulk positive
charge, with negative
electrons buried throughout.
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27-10 Early Models of the Atom
Around 1911, Rutherford did an experiment that showed that the
positively charged nucleus must be extremely small compared to
the rest of the atom. He scattered alpha particles—helium nuclei—
from a metal foil and observed the scattering angle. He found that
some of the angles were far larger than the plum-pudding model
would allow.
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27-10 Early Models of the Atom
The only way to account for the large angles was to assume that all
the positive charge was contained within a tiny volume.
Therefore, Rutherford’s model of the atom is mostly empty space:
Now we know that the
radius of the nucleus is
1/10000 that of the atom.
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27-11 Atomic Spectra: Key to the Structure
of the Atom
A very thin gas heated in a discharge tube emits light only at
characteristic frequencies.
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27-11 Atomic Spectra: Key to the Structure
of the Atom
An atomic spectrum is a line spectrum—only certain frequencies
appear. If white light passes through such a gas, it absorbs at those
same frequencies.
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27-11 Atomic Spectra: Key to the Structure
of the Atom
The wavelengths of electrons emitted from hydrogen have a
regular pattern:
(27-9)
This is called the Balmer series. R is the Rydberg constant:
R = 1.0974 × 107 m−1
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27-11 Atomic Spectra: Key to the Structure
of the Atom
Other series include the Lyman series:
And the Paschen series:
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27-11 Atomic Spectra: Key to the Structure
of the Atom
A portion of the complete spectrum of hydrogen is shown here.
The lines cannot be explained by the Rutherford theory.
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27-12 The Bohr Atom
Bohr proposed that the possible energy states for atomic electrons
were quantized—only certain values were possible. Then the
spectrum could be explained as transitions from one level to
another.
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27-12 The Bohr Atom
Bohr found that the angular momentum was quantized:
(27-11)
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27-12 The Bohr Atom
An electron is held in orbit by the Coulomb force:
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27-12 The Bohr Atom
Using the Coulomb force, we can calculate the radii of the orbits:
(27-12)
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27-12 The Bohr Atom
The lowest energy level is
called the ground state; the
others are excited states.
Bohr assumed that
electrons in fixed orbits do
not radiate light. He was
not able to say how an
electron moved when it
made a transition from one
energy level to another.
The idea of electron orbits
was rejected. Today
electrons are thought as
forming “probability
clouds”.
© 2014 Pearson Education, Inc.
Example
•  Electrons accelerated from rest
by a potential difference of
12.3 V pass through a gas of
hydrogen atoms at room
temperature. What
wavelengths of light will be
emitted?
•  Solution: the potential
difference gives the electrons a
kinetic energy of 12.3 eV, so it
is possible to provide this
much energy to the hydrogen
atom through collisions. From
the ground state, the maximum
energy of the atom is -13.6 eV + 12.3 eV = −1.3 eV. From the energy level diagram,
we see that this means that the atom could be excited to the n = 3 state, so the
possible transitions when the atom returns to the ground state are n=3 to n=2,
n=3 to n=1, and n=2 to n=1.
Example (continued)
•  So we will calculate the wavelengths using equation
1
1
=
E n − E n −1 )
(
λ hc
€
h = 4.136 × 10 −15 eV ⋅ s; c = 3 × 10 8 m / s
hc
1240eV ⋅ nm
λ3→1 =
=
= 102 nm
E 3 − E1 [ −1.5eV − ( −13.6eV )]
hc
1240eV ⋅ nm
λ2→1 =
=
= 122 nm
E 2 − E1 [ −3.4eV − ( −13.6eV )]
λ3→ 2
hc
1240eV ⋅ nm
=
=
= 650 nm
E 3 − E 2 [ −1.5eV − ( −3.4eV )]
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27-12 The Bohr Atom
The correspondence principle applies here as well—when the
differences between quantum levels are small compared to the
energies, they should be imperceptible.
Correspondence principle means that when you move from
quantum world to macro-world, the theory must be able to predict
classical results.
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27-13 de Broglie’s Hypothesis Applied
to Atoms
De Broglie’s hypothesis is the one associating a wavelength with
the momentum of a particle. He proposed that only those orbits
where the wave would be a circular standing wave will occur. This
yields the same relation that Bohr had proposed.
In addition, it makes more reasonable the fact that the electrons do
not radiate, as one would otherwise expect from an accelerating
charge.
.
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27-13 de Broglie’s Hypothesis Applied
to Atoms
De Broglie argued that the electron
wave was a circular standing wave
that closes on itself. If this does not
happen, destructive interference takes
place as the wave travels around the
loop, and the wave quickly dies out.
For constructive interference we need
2πrn=nλ; λ=h/mvmvrn=nh/(2π) –
exactly what Bohr proposed!
These are circular standing waves for
n = 2, 3, and 5.
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Example 3
•  Construct the energy-level diagram for doubly ionized lithium,
Li2+.
•  Solution: Lithium nucleus has 3 protons (positively charged
particles), so Z=3. Energy levels are given as
Z2
32
122.4 eV
E n = ( −13.6 eV ) 2 = ( −13.6 eV )⋅ 2 = −
n
n
n2
E1 = −122.4 eV ; E 2 = −30.6 eV ; E 3 = −13.6 eV ;
€
Summary of Chapter 27
•  Planck’s hypothesis: molecular oscillation energies are
quantized
•  Light can be considered to consist of photons, each of energy
•  Photoelectric effect: incident photons knock electrons out of
material
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Summary of Chapter 27
•  Compton effect and pair production also support photon theory
•  Wave-particle duality—both light and matter have both wave
and particle properties
•  Wavelength of an object with mass:
© 2014 Pearson Education, Inc.
Summary of Chapter 27
•  Principle of complementarity: both wave and particle properties
are necessary for complete understanding
•  Rutherford showed that atom has tiny nucleus
•  Line spectra are explained by electrons having only certain
specific orbits
•  Ground state has the lowest energy; the others are called
excited states
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