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3-1 Atoms and Radiation in Equilibrium3-2 Thermal Radiation Spectrum3-3 Quantization of Electromagnetic

Radiation3-4 Atomic Spectra and the Bohr Model

CHAPTER 3Planck’s ConstantPlanck’s Constant

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)

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.

Why is Black Body Radiation important?

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 (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.

58( )

1hc kT

hcu

eλ

π λλ

−

=−

E hf∆ =

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 Law

This data supports the Big

Bang Theory

Exercise 3-1: Derive Planck’s radiation law.

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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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 = ×

Planck’s Radiation LawExercise 3-2 : Show that Stefan’s Boltzmann law, Wein’s

displacement law and Rayleigh-Jean’s law can be derived from Planck’s law

Exercise 3-3: What is the average energy of an oscillator that has a

frequency given by hf=kT according to Planck’s calculations?

Exercise 3-4: 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?

Exercise 3-5: 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.

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).

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Properties of a 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

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∝

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 20

0 22 21 0

1

m hfE hf mc c m v c

cv c= = = ⇒ = − • =

−

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 StatisticsConsider 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.

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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 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.

Photo-electric Effect

Experimental Setup

Photo-electric effect

observationsThe 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

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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

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)

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:

21

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

= = −

Example – Photoelectric Effect

Exercise 3-6: 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.

Exercise 3-7: 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?

Exercise 3-8 Light of wavelength 400 nm and intensity 10-2 W/m2 is

incident on potassium. Estimate the time lag expected classically.

Newton discovers the

dispersion of light

Invention of Spectroscopy

Atomic Spectra

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Three Kinds of spectra

Solid, liquid or a dense gas excited to emit a continuous

spectrum

Light passing through low density

gas excites atoms to produce emission spectra

Light passing through cool low density gas results in absorption

spectra

Spectra

Chemical elements were observed to produce unique

wavelengths of light when burned or excited in an electrical discharge.

Line Spectra

Balmer Series

In 1885, Johann Balmer found an

empirical formula for the wavelength of the visible hydrogen line spectra in nm:

2

2364.6

4n

nnm

nλ =

−

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Rydberg-Ritz Formula

As more scientists

discovered emission lines at infrared and ultraviolet wavelengths, the Balmer

series equation was extended to the Rydberg equation:

2 2

7 1

H

7 1

1 1 1 for n>m, R=Rydberg constant

For Hydrogen R=R 1.096776 10

For Heavy atom R=R 1.097373 10

mn

Rm n

m

m

λ−

−

∞

= −

= ×

= ×

The Classical Atomic ModelConsider an atom as a planetary

system.

The Newton’s 2nd Law force of

attraction on the electron by the nucleus is:

2 2

2

kZe mvF

r r= =

22 2 21

22

kZev mv kze r

mr= ⇒ =

where v is the tangential velocity of the

electron:

The total energy is then:This is negative, so

the system is bound,

which is good.

2 2 2

2 2

kZe kZe kZeE K V

r r r= + = − = −

The Classical Atomic ModelExercise 7: Show that in the classical model the frequency of

radiation for an accelerating electron is

3/ 2

1f

r≈

and the Energy is

1E

r≈ −

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The Planetary Model is DoomedFrom classical E&M theory, an accelerated electric charge radiates

energy (electromagnetic radiation), which means the total energy must decrease. So the radius r must decrease!!

Physics had reached a turning point in 1900 with Planck’s

hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible.

Electron

crashes

into the

nucleus!?

Electron

does not

crash in

the Bohr

model

The Bohr Model of the Hydrogen Atom

Bohr’s general assumptions:

1. Stationary states, in which orbiting electrons do not radiate energy, exist in atoms and have well-defined energies, E.

Transitions can occur between them, yielding light of energy: Bohr frequency condition

E = Ei − Ef = hf

2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere.

3. The angular momentum of the nth state is: where n is called the Principal Quantum

Number.

hn

n = 1

n = 3

n = 2

Angular

momentum is

quantized!

Consequences of the Bohr Model

The angular momentum is:

1 22kZe

vmr

=

2 2 2

2 2

n kZe

m r mr=

h

hnrmL == v

mrn /v h=

So:

Solving for r:

So the velocity is:

a0 is called the Bohr radius.

a0

22

0

2n

n anr

mkZe Z= =

h

2 2

2From;

kZe mvF

r r= = ⇒

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Bohr Radius

The Bohr radius,

is the radius of the unexcited hydrogen atom.

The “ground” state Hydrogen atom diameter is:

2

0 20.0529a nm

mke≡ =

h

Energy of an electronExercise 3-9: Show that the energy of an electron in any atom at orbit n

is quantized and that it gives the ground state energy of Hydrogen atom to be -13.6eV.

2

0 2

2 4

0 2

1,2,3,...

where 2

n

ZE E n

n

mk eE

= − =

= −h

Rydberg-Ritz formula

Exercise 3-10: Derive the Ryderberg-Ritz formula

4

0

2 2 3

7 1

H

7 1

1 1 1 , =Rydberg constant

4

For Hydrogen R=R 1.096776 10

For Heavy atom R=R 1.097373 10

f i

E mkeR R

n n hc c

m

m

λ π

−

−

∞

= − = =

= ×

= ×

h

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Transitions in the Hydrogen AtomThe atom will remain in the excited state for a short time before

emitting a photon and returning to a lower stationary state. In equilibrium, all hydrogen atoms exist in n = 1.

Successes and Failures of the Bohr Model

Why should the nucleus of the atom be

kept fixed?

The electron and hydrogen nucleus

actually revolve about their mutual

center of mass.

Conservation of momentum require that

the momenta of nucleus and electron

equal in magnitude. The total kinetic

energy is then2 2 2

2

2 2 2 2

where 1

k

p p M m pE p

M m mM

mM m

M m m M

µ

µ

+= + = =

= =+ +

Success:

The electron mass is replaced

by its reduced mass:

Limitations of the Bohr Model

Works only for single-electron (“hydrogenic”) atoms.

Could not account for the intensities or the fine structure of the spectral lines (for example, in magnetic fields).

Could not explain the binding of atoms into molecules.

Failures:

The Bohr model was a great

step in the new quantum theory, but it had its limitations.

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Rydberg-Ritz Formula -Example

Exercise 3-11 Use the Rydberg-Ritz formula to calculate the first

line of Balmer, Lyman and Paschen series for the Hydrogen atom.

The Correspondence Principle

In the limits where classical and

quantum theories should agree,

the quantum theory must reduce

the classical result.

Bohr’s correspondence

principle is rather obvious:

When energy levels are

very close quantization

should have little effect

The Correspondence PrincipleExercise 3-12: Show that in the limit of large quantum number the Bohr

frequency is the same as the classical frequency.

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Fine Structure Constant

� In Bohr’s theory we know that transitions can

occur for ∆n≥1, for small n values. If we allow this

for large n and calculate the classical and Bohr

frequencies as in previous exercise, we will find

that they do not agree.

� To avoid this disagreement, A. Sommerfeld

introduced special relativity and elliptical orbits.

� From Bohr orbit in hydrogen for n=1, we have

( )

2

2 21

21.44 . 1

197.3 . 137

mvr n

kev

mr m mke

v ke eV nm

c c eV nmα

=

= = =

∴ = = ≈ =

h

h h

hh

h

α is called the fine structure constant

We will skip the

mathematical treatment of A. Sommerfeld work

Fine Structure Constant

� The fine structure constant can be understood in the following way.

For each circular orbit rn and energy En a set of n elliptical orbits exist whose major axis are the same but they have different

eccentricities and thus different velocities and Energies.

Electron transitions depend on the eccentricities of the initial and

final orbits and on the major axes, thus resulting in splitting of energy levels of n called fine-structure splitting.

Fine structure constant leads to the notion of electron spin

ExampleExercise 3-13: Show that the energy levels of oscillators in simple

harmonic motion are quantized.

Do it yourself exercise: solve the differential equation

22

20

dxx

dtω+ =

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ExampleExercise 3-14: Derive the Bohr quantum condition from Wilson-

Sommerfeld quantization rule

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= −

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

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.

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

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

Shells have letter names:

K shell for n = 1

L shell for n = 2

The atom is most stable in its

ground state.

When it occurs in a heavy atom, the radiation emitted is an x-ray.

It has the energy E (x ray) = Eu − Eℓ.

X-Ray Spectra

An electron from higher

shells will fill the inner-shell vacancy at lower energy.

Bohr-Rutherford picture of the atom

can also be applied to heavy elements

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Atomic Number and Moseley

The x-rays have names:

L shell to K shell: Kα x-ray

M shell to K shell: Kβ x-ray

etc.

G.J. Moseley studied x-ray

emission in 1913.

Atomic number Z = number

of protons in the nucleus.

Moseley found a

relationship between the

frequencies of the

characteristic x-ray and Z.

1 2 ( )n

f A Z b= −

Moseley found this relation

holds for the Kα x-ray with b=1 and different An values (from quantum mechanics):

Moseley’s Empirical ResultsThe Kα x-ray is produced from the n = 2 to n = 1 transition.

In general, the K series of x-ray frequencies are: Form the Bohr model with Z=Z-1

Moseley’s research clarified the importance of Z and the electron

shells for all the elements, not just for hydrogen.

We use Z-1 instead of Z because one electron is already

present in the K-shell and so shields the other's from the

nucleus’ charge.

2 42 2

3 2 2 2

1 1 1( 1) ( 1) 1

4 1

mk ef Z cR Z

n nπ∞

= − − = − −

h

2

2

11nA cR

n∞

= −

where

Moseley’s Empirical ResultsFor the L series of x-ray wavelength the

frequencies are

2

2 2

1 1( 7.4)

2f cR Z

n∞

= − −

a) Neodymium Z=60 and

Samarium Z=62

b) Promethium Z=61

c) All three elements together

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Example X-Ray SpectraExercise 3-15: Calculate the wavelength of Kα line of molybdenum, Z=42,

and compare with the value λ=0.0721nm measured by Moseley.

Aguer (oh-zhay) EffectIn 1923 Pierre Auger discovered an alternative to X-ray emission.

The atom may eject a third electron from a higher-energy outer shell via radiationless process called Auger effect

3 2 1

3

E E E E

KE E E

< ∆ = −

= ∆ −

EdN(e)/dE plot:

a) Auger spectrum of Cu

b) Al and Al2O3 togethernote the energy shift in the larges peak due to

adjustment in Al electron shell energies

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)

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

Examples

Exercise 3-16: What is the shortest wavelength present in a radiation if

the electrons are accelerated to 50,000 volts?

Exercise 3-16: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?