Download Physics 334 Modern Physics

Physics 334
Modern Physics
Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were
based on the textbook “Modern Physics” by Thornton and Rex. I have replaced some images from the adopted
text “Modern Physics” by Tipler and Llewellyn. Others images are from a variety of sources (PowerPoint clip art,
Wikipedia encyclopedia etc) and were part of original lectures. Contributions are noted wherever possible in the
PowerPoint file. The PDF handouts are intended for my Modern Physics class.
1
CHAPTER 3
Quantization of Charge, Light, and Energy
3-1 Quantization of Electric Charge
3-2 Blackbody Radiation
3-2.1 The Photon
3-3 The Photoelectric Effect
3-4 X Rays and the Compton Effect
3-4.1 Pair Production and Pair Annihilation
Max Karl Ernst Ludwig Planck
(1858-1947)
We have no right to assume that any physical laws exist, or if they have
existed up until now, or that they will continue to exist in a similar
manner in the future.
An important scientific innovation rarely makes its way by gradually
winning over and converting its opponents. What does happen is that
the opponents gradually die out.
- Max Planck
Photo: http://www.spaceandmotion.com/quantum-theory-max-planck-quotes.htm
2
3.1: Discovery of the X-Ray and the Electron
In the 1890s scientists and
engineers were familiar with
“cathode rays.” These rays
were generated from one of
the metal plates in an
evacuated tube with a large
electric potential across it.
Wilhelm Röntgen
(1845-1923)
J. J. Thomson
(1856-1940)
It was surmised that cathode rays had something to do with atoms.
It was known that cathode rays could penetrate matter and were
deflected by magnetic and electric fields.
Rontgen: www.aip.org/history/ electron/jjsound.htm
Thomson: www.haverford.edu/.../ songs/ionsmine.htm
3
Observation of X Rays
Wilhelm Röntgen studied the effects
of cathode rays passing through
various materials. He noticed that a
phosphorescent screen near the
tube glowed during some of these
experiments. These new rays were
unaffected by magnetic fields and
penetrated materials more than
cathode rays.
He called them x rays and deduced
that they were produced by the
cathode rays bombarding the glass
walls of his vacuum tube.
Wilhelm Röntgen
www.peoples.ru/science/ physics/rontgen/
4
Röntgen’s
X-Ray Tube
Röntgen constructed an x-ray tube by
allowing cathode rays to impact the glass
wall of the tube and produced x rays. He
used x rays to make a shadowgram the
bones of a hand on a phosphorescent
screen.
X-ray photo: http://pt.wikipedia.org/wiki/Imagem:Roentgen-x-ray-von-kollikershand.jpg
5
3-1 Quantization of Electric Charge
Thomson used an evacuated cathode-ray tube to show that the
cathode rays were negatively charged particles (electrons) by
deflecting them in electric and magnetic fields.
Thomson’s method of measuring the ratio of the electron’s charge to
mass was to send electrons through a region containing a magnetic
field perpendicular to an electric field.
6
Calculation of e/m
Exercise 1: An electron is moving under the influence of electric and
magnetic fields. Show that the charge to mass ratio is given by;
q
E
=
m RB 2
The B filed can be computed by
measuring the current using an
ammeter, the electric field can be
computed by measuring the
voltage and the radius R can be
measured experimentally by a
rod
11
q / m = 0.7 × 10 C / kg
This is independent of the nature of gas or metal for the Cathode.
Lorentz called the charge electron e as one unit of negative charge
7
Determination of
Electron Charge
Millikan’s oil-drop experiment
Robert Andrews Millikan
(1868 – 1953)
Millikan was able to
show that electrons
had a particular
charge.
http://imglib.lbl.gov/ImgLib/COLLECTIONS/BERKELEYLAB/images/pg08_Millikan.lowres.jpeg
8
Calculation of the oil drop charge
Exercise 2: For Millikan’s experiment derive an expression for the
charge of an electron as a function of mdrop, acceleration due to
gravity g, plate separation d and voltage V. Then using Stoke’s law to
determine the terminal velocity and thus the mass of the drop,
calculate the charge of an electron.
e = 1.602 x 10-19 C
9
3-2 Blackbody Radiation
When matter is heated, it emits
radiation.
A blackbody is a cavity with a
material that only emits thermal
radiation. Incoming radiation is
absorbed in the cavity.
Blackbody radiation is theoretically interesting because the
radiation properties of the blackbody are independent of the
particular material. Physicists can study the properties of intensity
versus wavelength at fixed temperatures.
10
Wien’s Displacement Law
The spectral intensity R(λ,Τ ) is the total power radiated per unit area
per unit wavelength at a given temperature.
Wien’s displacement law: The maximum of the spectrum shifts to
smaller wavelengths as the temperature is increased.
λmT = constant=2.898 ×10-3 m.K
11
Stefan-Boltzmann Law
The total power radiated increases with the temperature:
R = εσ T 4
This is known as the Stefan-Boltzmann law, with the constant σ
experimentally measured to be 5.6705 × 10−8 W / (m2 · K4).
The emissivity є (є = 1 for an idealized blackbody) is simply the
ratio of the emissive power of an object to that of an ideal
blackbody and is always less than 1.
12
Example – Wein’s Displacement Law
How Hot is a Star? Measurement of the wavelength at which
spectral distribution R(λ) from the Sun is maximum is found to be at
500nm, how hot is the surface of the Sun?
13
Example – Stefan Boltzmann Law
How Big is a Star? Measurement of the wavelength at which
spectral distribution R(λ) from a certain star is maximum indicates
that the star’s surface temperature is 3000K. If the star is also
found to radiate 100 times the power Psun radiated by the Sun, how
big is the star? Take the Sun’s surface temperature as 5800 K.
14
Rayleigh-Jeans Formula
Lord Rayleigh used the classical
theories of electromagnetism and
thermodynamics to show that the
blackbody spectral distribution
should be:
u (λ ) = 8π kT λ −4
u (λ ) = 8π kT λ −4
Integrating u (λ ) from 0 → ∞, we find
∞
∫ u(λ )d λ → 0 when λ → 0
0
It approaches the data at longer wavelengths, but it deviates badly at
short wavelengths. This problem for small wavelengths became
known as the ultraviolet catastrophe and was one of the
outstanding exceptions that classical physics could not explain.
15
Planck’s Radiation Law
Planck assumed that the radiation in the cavity was emitted (and
absorbed) by some sort of “oscillators.” He used Boltzman’s
statistical methods to arrive at the following formula that fit the
blackbody radiation data.
8π hcλ −5
u (λ ) = hc λ kT
e
−1
Planck’s radiation law
Planck made two modifications to the classical theory:
The oscillators (of electromagnetic origin) can only have certain
discrete energies, En = nhf, where n is an integer, ν is the frequency,
and h is called Planck’s constant: h = 6.6261 × 10−34 J·s.
The oscillators can absorb or emit energy in discrete multiples of the
fundamental quantum of energy given by:
∆E = hf
16
Planck’s Radiation Law
Exercise 3 Derive Planck’s radiation law.
Exercise 4 Show that Stefan’s Boltzmann law, Wein’s displacement
law and Rayleigh-Jean’s law can be derived from Planck’s law
Application: The Big Bang theory
predicts black body radiation. This
radiation was discovered in 1965
by A. Penzias & R. Wilson. Cosmic
Background Explorer (COBE) and
Wilkinson Microwave Anisotropy
Probe (WMAP) detected this
radiation field at 2.725± 0.001 °K
This data supports the Big
Bang Theory
17
Example Planck’s Radiation Law
Exercise 5 What is the average energy of an oscillator that has a
frequency given by hf=kT according to Planck’s calculations?
18
What is a Photon?
Planck introduced the idea of a photon or quanta. A cavity emits
radiation by way of quanta. How does the radiation travel in
space? We think that radiation is a wave phenomenon however
the energy content is delivered to atoms in concentrated groups
of waves (quanta).
19
Properties of Photon
If a photon is to be considered as a particle we must be able to
describe its mass, momentum, energy, statistics etc.
Energy of a photon
=
× -34
=
× -15
E
hf ; h=6.626
10 J .s
4.136
10 eV .s
E = Energy, f = frequency
limiting value 1/2hf otherwise integral multiple of hf
Interaction of photons with matter
Complete absorption or partial absorption with the photon
adjusting its frequency to remain as particle
Intensity of photon
Intensity ∝ number of photons
intensity has nothing to do with the energy of photons
20
Properties of Photons
Constant h of photon
h defines the smallest quantum angular momentum of a particle
∴ E = hf = mc 2 ⇒ m =
hf
hf
hf
and
p
c
=
=
c2
c2
c
Exercise 6 Show that h has units of angular momentum
Mass and momentum of photon
photons move with velocity v=c
E = hf = mc 2 =
m0
2
1− v c
2
c 2 ⇒ m0 =
(
)
1 − v2 c2 •
hf
=0
c2
Photons have no rest mass m0
Photon is not a material particle since rest mass is zero, it is a wave
structure that behaves like a particle
21
Properties of Photons
Charge of a photon
photons do not carry charge, however they can eject charge
particles from matter when they impinge on atoms
Photon Statistics
Consider the radiation as a gas of photon. Photons move randomly
like molecules in a gas and have wide range of energies but same
velocity
Statistics of photons is described by Bose-Einstein. We can talk
about intensity and temperature in the same way as density and
temperature.
22
3-3 Photoelectric Effect
Methods of electron emission:
Thermionic emission: Applying
heat allows electrons to gain
enough energy to escape.
Secondary emission: The electron gains enough energy by transfer
from another high-speed particle that strikes the material from outside.
Field emission: A strong external electric field pulls the electron out of
the material.
Photoelectric effect: Incident light (electromagnetic radiation) shining
on the material transfers energy to the electrons, allowing them to
escape. We call the ejected electrons photoelectrons.
Image adapted from
http://sol.sci.uop.edu/~jfalward/particlesandwaves/particlesandwaves.html
23
Photo-electric Effect
Experimental Setup
Image adapted from http://www.chem1.com/acad/webtut/atomic/qprimer/#Q8
24
Photo-electric effect
observations
Electron
kinetic
energy
The kinetic energy of
the photoelectrons is
independent of the
light intensity.
The kinetic energy of
the photoelectrons, for
a given emitting
material, depends only
on the frequency of
the light.
Classically, the kinetic
energy of the
photoelectrons should
increase with the light
intensity and not
depend on the
frequency.
Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png
25
Photoelectric effect
observations
Electron
kinetic
energy
There was a threshold
frequency of the light,
below which no
photoelectrons were
ejected (related to the
work function φ of the
emitter material).
The existence of a threshold frequency is completely inexplicable in
classical theory.
Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png
26
Photoelectric effect
observations
(number of
electrons)
When photoelectrons
are produced, their
number is proportional
to the intensity of light.
Also, the photoelectrons
are emitted almost
instantly following
illumination of the
photocathode,
independent of the
intensity of the light.
Classical theory predicted that, for
extremely low light intensities, a long
time would elapse before any one
electron could obtain sufficient
energy to escape. We observe,
however, that the photoelectrons are
ejected almost immediately.
Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png
27
Einstein’s Theory: Photons
Einstein suggested that the electro-magnetic radiation field is
quantized into particles called photons. Each photon has the energy
quantum:
E = hf
where f is the frequency of the light and h is Planck’s constant.
Alternatively,
E = hω
where:
h = h 2π
28
Einstein’s Theory
Conservation of energy yields:
1
hf = φ + mv 2
2
where φ is the work function of the metal (potential energy to be
overcome before an electron could escape).
In reality, the data were a bit more
complex. Because the electron’s energy
can be reduced by the emitter material,
consider fmax (not f):
1

eV0 =  mv 2  = hf − φ
2
 max
29
Example – Photoelectric Effect
An experiment shows that when electromagnetic radiation of
wavelength 270 nm falls on an aluminum surface, photoelectrons
are emitted. The most energetic of these are stopped by a potential
difference of 0.46 volts. Use this information to calculate the work
function of aluminum in electron volts.
30
Example – Photoelectric Effect
The threshold wavelength of potassium is 558 nm. What is the work
function for potassium? What is the stopping potential when light of
400 nm is incident on potassium?
31
Example – Photoelectric Effect
Light of wavelength 400 nm and intensity 10-2 W/m2 is incident on
potassium. Estimate the time lag expected classically.
32
3-4 X-Ray Production: Theory
An energetic electron
passing through matter will
radiate photons and lose kinetic
energy, called bremsstrahlung.
Since momentum is conserved,
the nucleus absorbs very little
energy, and it can be ignored.
The final energy of the electron is
determined from the conservation
of energy to be:
Ei
Ef
hν
E f = Ei − hf
33
3-4 X-Ray Production: Theory
If photons can transfer energy to electrons, can
part or all of the kinetic energy of electron be
converted into photons?
Ei
“The Inverse photoelectric effect”
Ef
This was discovered before the work of Planck
and Einstein
hν
34
Observation of X Rays
1895: Wilhelm Röntgen studied the
effects of cathode rays passing
through various materials. He
noticed that a phosphorescent
screen near the tube glowed during
some of these experiments. These
new rays were unaffected by
magnetic fields and penetrated
materials more than cathode rays.
He called them x rays and deduced
that they were produced by the
cathode rays bombarding the glass
walls of his vacuum tube.
Wilhelm Röntgen
www.peoples.ru/science/ physics/rontgen/
35
X-Ray
Production:
Experiment
Current passing through a filament produces copious numbers of
electrons by thermionic emission. If one focuses these electrons by
a cathode structure into a beam and accelerates them by potential
differences of thousands of volts until they impinge on a metal
anode surface, they produce x rays by bremsstrahlung as they stop
in the anode material.
36
Electromagnetic theory Predicts X-Ray
Accelerated charges produce electromagnetic waves, when fast
moving electrons are brought to rest, they are certainly accelerated
1906: Barkla found that X-Ray show Polarization, this establishing
that X-Rays are waves.
X-Rays have wavelength range 0.1 nm – 100 nm
Even though classical theory predicts x-ray’s, the experimental data
is not explainable.
37
Two distinctive features
1. Some targets have enhanced
peaks. For example
Molybdenum shows two
peaks at specific
wavelengths.
a) This is due to
rearrangement of electrons
of the target material after
bombardment.
b) X-ray ‘s have a continuous
spectrum
2. No matter what the target ,
the threshold wavelength
depends on the accelerating
potential
38
Inverse Photoelectric Effect
Since the work function of the target
is of the order of few eV, whereas
the accelerating potential is
thousand of eV
1

eV0 =  mv 2  = hf − φ
2
 max
eVo = hf max =
From photoelectric effect
hc
λmin
Duane-Hunt rule
39
X-Ray Scattering
Max von Laue suggested that if x-rays were a form of electromagnetic
radiation, interference effects should be observed.
Crystals act as three-dimensional gratings, scattering the waves and
producing observable interference effects.
40
Bragg’s Law
William Lawrence Bragg
interpreted the x-ray
scattering as the reflection of
the incident x-ray beam from
a unique set of planes of
atoms within the crystal.
There are two conditions for
constructive interference of
the scattered x rays:
1) The angle of incidence must equal the
angle of reflection of the outgoing wave.
2) The difference in path lengths must be
an integral number of wavelengths.
Bragg’s Law: nλ = 2d sin θ (n = integer)
41
The Bragg Spectrometer
A Bragg spectrometer scatters x
rays from crystals. The intensity of
the diffracted beam is determined
as a function of scattering angle
by rotating the crystal and the
detector.
When a beam of x rays passes
through a powdered crystal, the
dots become a series of rings.
42
Examples
Exercise 1: What is the shortest wavelength present in a radiation if the
electrons are accelerated to 50,000 volts?
Exercise 2:The spacing of one set of crystals planes in common salt is
d=0.282nm. A monochromatic beam of X-rays produces a Bragg
maximum when its glancing angle is 7 degrees. Assuming that this is
the first order maximum (n=1), find the wavelength of the X-rays, what
is the minimum possible accelerating voltage Vo that produced the Xrays?
43
Compton Effect
If photons behave like particles than collisions between a
photon and a particle can be studied just like billiard ball
collisions.
hc / λ
Classical theory predicts that the scattered frequency of
the photon should be the same as the incident photon.
When a photon enters matter, it can interact with one of
the electrons. The laws of conservation of energy and
momentum apply, as in any elastic collision between two
particles. The momentum of a particle moving at the
speed of light is:
p=
Ee
hc / λ ′
E hν h
=
=
c
c λ
The electron energy is:
This yields the change in
wavelength of the scattered
photon, known as the Compton
effect:
∆λ = λ2 − λ1 =
h
(1 − cos θ )
mc
44
Compton Effect
Intensity versus wavelength for Compton scattering at several
angles. The first peak results from photons of the original
wavelength that scattered by tightly bound electrons, which have
an effective mass equal t that of the atom. The separation in
wavelength of the peaks is given by the Compton scattering
equation
45