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Physics 334Modern 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.

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3-1 Quantization of Electric Charge3-2 Blackbody Radiation

3-2.1 The Photon

3-3 The Photoelectric Effect3-4 X Rays and the Compton Effect

3-4.1 Pair Production and Pair Annihilation

CHAPTER 3Quantization of Charge, Light, and Energy

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

Max Karl Ernst Ludwig Planck

(1858-1947)

Photo: http://www.spaceandmotion.com/quantum-theory-max-planck-quotes.htm

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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.

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.

J. J. Thomson

(1856-1940)

Wilhelm Röntgen

(1845-1923)

Rontgen: www.aip.org/history/ electron/jjsound.htm

Thomson: www.haverford.edu/.../ songs/ionsmine.htm

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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/

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Röntgen’sX-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-kollikers-

hand.jpg

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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.

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Exercise 1: An electron is moving under the influence of electric and magnetic fields. Show that the charge to mass ratio is given by;

Calculation of e/m

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q E

m RB=

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/ 0.7 10 /q m 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

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Millikan’s oil-drop experiment

Determination of Electron Charge

Robert Andrews Millikan

(1868 – 1953)

Millikan was able to show that electrons had a particular charge.

http://imglib.lbl.gov/ImgLib/COLLECTIONS/BERKELEY-

LAB/images/pg08_Millikan.lowres.jpeg

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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

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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.

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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.

-3constant=2.898 10 .mT m Kλ = ×

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Stefan-Boltzmann Law

The total power radiated increases with the temperature:

This is known as the Stefan-Boltzmann law, with the constant σexperimentally 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.

4R Tεσ=

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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?

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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.

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Rayleigh-Jeans Formula

Lord Rayleigh used the classical theories of electromagnetism and thermodynamics to show that the blackbody spectral distribution should be:

4( ) 8u kTλ π λ −=

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.

4

0

( ) 8

Integrating ( ) from 0 , we find

( ) 0 when 0

u kT

u

u d

λ π λ

λ

λ λ λ

−

∞

=

→ ∞

→ →∫

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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:

Planck’s radiation law

Planck’s Radiation Law

Planck assumed that the radiation in the cavity was emitted (andabsorbed) by some sort of “oscillators.” He used Boltzman’sstatistical methods to arrive at the following formula that fit the blackbody radiation data.

58( )

1hc kT

hcu

e λ

π λλ

−

=−

E hf∆ =

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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

Planck’s Radiation LawExercise 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

This data supports the Big Bang Theory

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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?

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What is a Photon?

Planck introduced the idea of a photon or quanta. A cavity emitsradiation 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).

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Properties of Photon

If a photon is to be considered as a particle we must be able todescribe its mass, momentum, energy, statistics etc.

� Energy of a photon

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 has nothing to do with the energy of photons

-34 -15; h=6.626 10 . 4.136 10 .

,

E hf J s eV s

E Energy f frequency

= × = ×

= =

Intensity number of photons∝

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Properties of Photons� Constant h of photon

h defines the smallest quantum angular momentum of a particle

Exercise 6 Show that h has units of angular momentum

� Mass and momentum of photonphotons move with velocity v=c

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

2

2 2 and

hf hf hfE hf mc m p c

c c c∴ = = ⇒ = = =

( )2 2 2 200 22 2

1 01

m hfE hf mc c m v c

cv c= = = ⇒ = − • =

−

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Properties of Photons

� Charge of a photonphotons do not carry charge, however they can eject charge particles from matter when they impinge on atoms

� Photon StatisticsConsider the radiation as a gas of photon. Photons move randomlylike molecules in a gas and have wide range of energies but samevelocity

Statistics of photons is described by Bose-Einstein. We can talk about intensity and temperature in the same way as density and temperature.

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3-3 Photoelectric Effect

Methods of electron emission:

Thermionic emission: Applying heat allows electrons to gainenough energy to escape.

Secondary emission: The electron gains enough energy by transferfrom 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

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Photo-electric Effect Experimental Setup

Image adapted from http://www.chem1.com/acad/webtut/atomic/qprimer/#Q8

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Photo-electric effect observations

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.

Electron

kinetic

energy

Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png

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Photo-electric effect observations

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.

Electron

kinetic

energy

Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png

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Photo-electric effect observations

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.

(number of

electrons)

Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png

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Einstein’s Theory: Photons

Einstein suggested that the electro-magnetic radiation field is quantized into particles called photons. Each photon has the energy quantum:

where f is the frequency of the light and h is Planck’s constant.

Alternatively,

E hf=

where:

E hω= 2h h π=

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Einstein’s TheoryConservation of energy yields:

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2hf mvφ= +

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):

where φ is the work function of the metal (potential energy to be overcome before an electron could escape).

2

0

max

1

2eV mv hf φ

= = −

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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.

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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?

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Example – Photoelectric Effect

Light of wavelength 400 nm and intensity 10-2 W/m2 is incident on potassium. Estimate the time lag expected classically.

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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:

fE

iE

hν

f iE E hf= −

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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?

“The Inverse photoelectric effect”

This was discovered before the work of Planck and Einstein

fE

iE

hν

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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/

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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.

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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.

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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

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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

max

min

o

hceV hf

λ= =

2

0

max

1

2eV mv hf φ

= = −

From photoelectric effect

Duane-Hunt rule

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X-Ray ScatteringMax 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.

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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)

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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.

The Bragg Spectrometer

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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 X-

rays?

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Compton EffectIf photons behave like particles than collisions between a photon and a particle can be studied just like billiard ball collisions.

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:

E h hp

c c

ν

λ= = =

This yields the change in wavelength of the scattered photon, known as the Compton effect:

eE

/hc λ

/hc λ′

The electron energy is:

2 1 (1 cos )h

mcλ λ λ θ∆ = − = −

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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 havean effective mass equal t that of the atom. The separation in wavelength of the peaks is given by the Compton scattering equation