Ch 27 1

Chapter 27

Early Quantum Theory and Models of the Atom

© 2006, B.J. LiebSome figures electronically reproduced by permission of Pearson Education, Inc.,

Upper Saddle River, New Jersey Giancoli, PHYSICS,6/E © 2004.

Ch 27 2

Properties of the Electron

•Called “cathode ray” because appeared to come from the cathode

•J. J. Thomson did the first experiments to discover its properties and received Nobel Prize for work.

•First discovered in experiments where electricity is discharged through rarefied gases.

•First experiments measured charge /mass.

•Another experiment measured e and thus mass could be determined

Ch 27 3

Electron Charge to Mass Ratio

Magnetic Force = Centripetal Force

r

vmBve

2

rB

v

m

e

The electric field is adjusted until it balances the magnetic field and thus eE = evB . This gives

B

Ev

This is a velocity selector since all electrons that pass through have the same velocity. The final equation is

rB

E

m

e2

Ch 27 4

Blackbody Radiation

•Intensity vs. wavelength is shown above

•Radiation emitted by any “hot” object

•In Ch 14, we learned Intensity T4

•The wavelength of the peak of the spectrum P depends on the Kelvin temperature by

KmTp 3109.2

Ch 27 5

Planck’s Quantum Hypothesis

•Attempts to explain the shape of the blackbody curve were unsuccessful

•In 1900 Max Planck proposed that radiation was emitted in discrete steps called quanta instead of continuously

•He did not see this as revolutionary

•Introduced a new constant-now called Planck’s constant

h =6.626 x 10-34 Js = 4.14 x 10-15 eV·s

Ch 27 6

Planck’s Quantum Hypothesis•We can understand quantized energy by considering the energy of a box on stairs vs. box on a ramp.

E = m g y

•If the height of each step is Δ y, can you derive an equation for the quantized potential energy of the box on the steps.

E = m g ( n Δ y )

where n is an integer.

Ch 27 7

Photoelectric Effect

•Light shines on a metal and electrons (called photoelectrons ) are given off

•Easy to measure kinetic energy of electrons

•There was a threshold frequency below which no electrons were emitted

Ch 27 8

Photoelectric Effect

•Wave theory predicts that:

•Number of electrons Intensity

•Maximum electron kinetic energy intensity

•Frequency of light should not affect kinetic energy

•No threshold frequency

•This can not explain the photoelectric effect

Ch 27 9

Photon Theory of Light

•In 1905 Einstein proposed that in some experiments light behaved like particles instead of waves

•Light consisted as stream of photons, each with energy:

fhE

Where h is Planck’s constant

•Each photon had wavelike properties

Ch 27 10

Explanation of Photoelectric Effect•Work function W0 is the energy necessary to free the least tightly bound electron

•A single photon with energy hf gives all of this energy to a single electron

•A photon with frequency below the threshold lacks sufficient energy to free the electron, so hf0 = W0

•This electron thus escapes from the metal with kinetic energy

0WhfKE max

Ph0

5

0 2

Fr

Ch 27 11

Photons

fhE •where h is Planck’s Constant

•The rest mass of a photon must be zero because it travels at the speed of light

•The photon has momentum (p) because if we substitute m0 = 0 in 42

0222 cmcpE

We get the following equation

h

c

ch

c

hf

c

Ep

and thus

h

p

The energy of a photon is given by

Ch 27 12

Example 27-1 (20) In a photoelectric effect experiment it is observed that no current flows unless the wavelength is less than 570 nm. What is the work function of the material?

)(0max thresholdKE

000

chfhW

ms

mseVW

9

815

0 10570

)1000.3)(1014.4(

eVW 18.20

What is the stopping voltage if light of wavelength 400 nm is used?

eVseVm

smseV

KEMAX 93.018.210400

)1000.3)(1014.4(9

815

So the stopping voltage is 0.93 volts.

0max WfhEK

Ch 27 13

Evidence for Photon Nature of Electromagnetic Radiation

Compton Effect: photon scatters off of electron, photon looses energy and electron gains energy. This effect shows that momentum and energy is conserved.

Ch 27 14

Further Evidence for Photon Nature of Electromagnetic Radiation

Pair Production: a photon passing a nucleus is converted into an electron-positron pair. (A positron is a positive particle with all the other properties of an electron.)

Since the mass of the electron is 0.511MeV/c2 and two electrons must be produced, the kinetic energy shared by the two electrons is

MeVEKE 022.1

Ch 27 15

Electron Microscopes

•Electrons accelerated through 100,000 V have 0.004nm and can achieve a resolution of 0.2 nm which is a factor of 1000x better than optical microscopes.

•Use magnetic lenses to focus electron beam.

•Scanning Tunneling Microscope has a probe that moves up and down to maintain a constant tunneling current.

Ch 27 16

Thomson Model of the Atom

•J.J. Thomson Model: had negatively charged electrons inside a sphere of positive charge.

•Assumed that the electrons would oscillate due to electric forces.

•An oscillating charge produces electromagnetic radiation which should match agree with atomic spectra.

Ch 27 17

Rutherford Experiment

•Ernest Rutherford performed an experiment to probe the structure of the atom.

•He aimed a beam of alpha particles at a thin gold foil and measured how they were scattered.

•Alpha particle is the nucleus of a He atom (two protons and two neutrons) and thus was positively charged.

•Alpha particles are emitted by a radioactive nucleus.

Ch 27 18

Rutherford Experiment Results

•Most alpha particles passed through foil without scattering

•A few were scattered through large angles

Ch 27 19

Rutherford Experiment Results

•Concluded that foil was mainly empty space with some small but massive concentrations of positive charge.

•An alpha particle that happened to pass near a nucleus was repelled without ever touching the nucleus.

•Rutherford proposed a positive heavy nucleus with radius of 10-15 m with electrons in orbit

•Problem was electrons should radiate energy away.

Ch 27 20

Hydrogen Spectra

•The visible spectra from hydrogen gas has a very distinctive pattern that can be represented by the Balmer formula

....5,4,3,1

2

1122

n

nR

where R = 1.0974x10-7 m-1.

Ch 27 21

Hydrogen Spectra

....4,3,2,1

1

1122

n

nR

....6,5,4,1

3

1122

n

nR

•The Balmer formula could also be modified to fit the Lyman series that was discovered in the ultraviolet

•And the Paschen series in the infrared

Ch 27 22

Bohr Model of the Atom•Niels Bohr was a Danish physicist who studied at the Rutherford lab. He decided to try to add the quantum effects of Planck and Einstein to the Rutherford planetary model of the atom

•He knew that the answer had to be the Balmer formula but the task was to develop a set of assumptions that would lead to it.

•Problem was that a charged particle in orbit is like an antenna--it should emit radiation and gradually loose energy until it fell into the nucleus

•The discrete wavelengths emitted by hydrogen suggested a quantum effect as in the stair example

Ch 27 23

Bohr’s Assumptions•Bohr was like a student who looked up the answer in the back of the book and needed to find a way to get that answer

•He said electron could remain in possible orbit called a stationary state without emitting any radiation

•Each stationary state is characterized by a definite energy En

•When electron changes from the upper to the lower stationary state (or orbit) it emits a photon of energy equal to the difference in the states:

lu EEhf

Ch 27 24

Bohr’s Assumptions

•Radiation is only emitted when an electron changes from one stationary state to another

•He found he could derive the Balmer formula if he assumed that the electrons moved in circular orbits with angular momentum (L) satisfied the following quantum condition:

...3,2,1,2

nh

nmvrL n

nmr

nhv

2and thus

Ch 27 25

Bohr Radius•Z is the number of protons so Qnucleus = Ze

•The electrical force equals the centripetal force

nn r

mv

r

eZek

2

2

))((

22

222

2

2 4

hn

mrZek

mv

kZer n

n

1

2

rZ

nrn

mmke

hr 10

22

2

1 10529.04

)10529.0( 102

mZ

nrn

nmr

nhv

2

Ch 27 26

Bohr Energy Levels

The electric potential of the nucleus is

r

kZe

r

kQV

2

With potential energy r

ZekeVPE

2

nn r

kZemvE

22

2

1

3,2,16.132

2

nn

ZeVEn

Substituting Bohr radius equation and values gives

Ch 27 27

Summary of Bohr Model

3,2,16.132

2

nn

ZeVEn

•Electrons obit in stationary states that are characterized by a quantum number n and a discrete energy En. Sometimes this is called a energy level.

•En is negative indicating a bound electron

•At room temperature, most H atoms have their electron in the n=1 energy level

•When electron changes to a lower n it emits a photon of energy equal to the energy difference.

•Electron must be given energy to move to a higher n

•This formula can be used for any single electron atom or ion such as a singly-ionized He ion in which case Z=2.

•The radius of the orbit is given by )10529.0( 102

mZ

nrn

Ch 27 28

Energy Level Diagram

•Red arrows indicate transitions where electron emits a photon and moves to a lower state

•Vertical scale is energy.

Ch 27 29

Example 27-2A. A hydrogen atom initially in its ground state (n=1) absorbs a photon and ends up in the n=3 state. Calculate the energy and wavelength of the absorbed photon.

)1(6.136.13

22

2

zn

V

n

zeVEn

eVeV

E 6.131

6.1321

eVeVEE 1.12)6.13(51.113

ch

fhE

eVeV

E 4.32

6.1322

eVeV

E 51.13

6.1323

E

hc nm

eVJeV

smJs102

)/106.1)(1.12(

)/1000.3)(10626.6(19

834

First calculate the energy of the first three states.

Ch 27 30

Example 27-2B. A hydrogen atom initially in its ground state (n=1) absorbs a photon and ends up in the n=3 state. Calculate the energy and wavelength of the absorbed photon.

When the atom returns to the ground state, what possible energy photons could be emitted?

eVeVeVE 89.1)4.3(51.123

eVeVeVE 2.10)6.13(4.312

eVEn 51.13 eVEn 40.32

eVEn 6.131

eVeVE 1.12)6.13(51.113

Ch 27 31

Example 27-3. Singly ionized 4He consists of a nucleus with two protons and two neutrons with a single electron in orbit around this nucleus. Use the Bohr model to calculate the energy of a photon that is emitted when the electron goes from the first excited state to the ground state of singly ionized 4He.

2

2

2

2 26.136.13

n

eV

n

zeVEn

eVeVEEEE 8.40)4.54(6.1312

eVeV

E 4.541

4.5421

eVeV

E 6.132

4.5422

Calculate the ionization energy of singly ionized 4He.

Ionization energy = eVeV 4.54)4.54(0

Could you use the Bohr model for atomic 4He?

Ch 27 32

Wave Nature of Particles

Louis de Broglie proposed that if light had a wave-particle duality, then perhaps particles, such as electrons, also had a wave nature. He assumed that the following equation for photons

h

p

mv

h

also applied to electrons

Ch 27 33

Diffraction of Electrons

Later experiments showed that a beam of electrons was diffracted just like light.

mv

h

Ch 27 34

De Broglie Hypothesis and Hydrogen

De Broglie was able to give a reason for the Bohr quantum hypothesis by assuming that allowed electron orbits had to be in standing waves around orbit.

Circumference =2 rn = n where n = 1, 2, 3...

mv

h

and we get the following equation which was Bohr’s original quantum assumption.

2

nhmvrn

Combine with