Physics 3313 - Lecture 7

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Wednesday February 10, 2010Dr. Andrew Brandt

1. HW 2 (Ch 3 due today); HW3 (Ch 4) to be assigned today

2. Atomic Models3. Rutherford Scattering4. Bohr Atom

• 4.1 The Atomic Models of Thomson and Rutherford• 4.2 Rutherford Scattering• 4.3 The Classic Atomic Model• 4.4 The Bohr Model of the Hydrogen Atom• 4.5 Successes and Failures of the Bohr Model• 4.6 Characteristic X-Ray Spectra and Atomic Number

(skip)• 4.7 Atomic Excitation by Electrons (skip)

CHAPTER 4Structure of the AtomStructure of the Atom

In the present first part of the paper the mechanism of the binding of electrons by a positive nucleus is discussed in relation to Planck’s theory. It will be shown that it is possible from the point of view taken to account in a simple way for the law of the line spectrum of hydrogen.

- Niels Bohr, 19132/10/2010 23313 Andrew Brandt

Structure of the Atom

By 1900 scientists had evidence that indicated the atom was not a fundamental unit:

1) There seemed to be too many different kinds of atoms, each belonging to a distinct chemical element.

2) Electromagnetic properties and line spectra hinted at some underlying structure.

3) The problem of valence. Certain elements combined with some elements but not with others, a characteristic that hinted at an internal atomic structure.

4) The discoveries of radioactivity, of x rays, and of the electron.

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Evolution of Atomic Models• 1803: Dalton’s billiard ball

model• 1897: J.J. Thompson

Discovered electrons– Used cathode ray tubes– Called corpuscles– Made a bold claim that these

make up atoms– Measured charge/mass ratio

• 1904: J.J. Thompson Proposed a “plum pudding” model of atoms – Negatively charged electrons

embedded in a uniformly distributed positive charge

Cathode ray tube

Personally I prefer chocolate chip cookie model

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4.2 Rutherford Experiment• 1911: Geiger and Marsden with Rutherford performed a scattering

experiment firing alpha particles at a thin gold foil

Rutherford Scattering

• The actual result was very different—although most events had small angle scattering, many wide angle scatters were observed

• “It was almost as incredible as if you fired a 15 inch shell at a piece of tissue paper and it came back at you”

• Implied the existence of the nucleus. • We perform similar experiments at

Fermilab and CERN to look for fundamental structure

4 2

( )sin

2

KN

KE

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Scattering experiments help us study matter too small to be observed directly.

There is a relationship between the impact parameter b and the scattering angle θ.

Assume small particle+ thin target, small massive scatterer, dominated by Coulomb Force

When b is small,minimum r is small.Coulomb force gets large.θ can be large and the particle can be repelled backward.

Rutherford Scattering

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look at limiting cases for

Rutherford Example

• On blackboard demonstrate size of radius from distance of closest approach

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• In actual experiment a detector is positioned from θto θ+ dθ that corresponds to incident particles between b and b + db.

• The number of particles scattered per unit area is

Rutherford Scattering Equation

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The cross section σ = πb2 is related to the probability for a particle being scattered by a nucleus.

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Ruherford Atom• 1912: Rutherford’s planetary

model, an atomic model with a positively charged heavy core surrounded by circling electrons

• But many questions: a) Z=A/2, Z=atomic number

(number of electrons or protons) what is the other half of the atomic weight ?

b)what holds the nucleus together?

c)how do electrons move around the nucleus and does their motion explain observed atomic properties?

Hydrogen Atom: Electron Orbit• Consider a Hydrogen atom consisting of an electron and a proton

• Electron must be in motion or Coulomb Force would suck it into nucleus

• “Assume a spherical orbit” : this implies that the centripetal force must be balanced by the Coulomb force

• so

• Energy of electron is kinetic energy plus potential energy

(where potential energy is defined to be 0 at infinity and negative at closer radius since you have to input work to keep electron and proton apart)

• Can thus determine radius of Hydrogen atom given Binding Energy (-13.6 eV)

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2 2

20

1

4

mv e

r r

22

0

1

2 4

eE mv

r

2 2 2

0 0 08 4 8

e e eE

r r r

219 2 9

2 211

190

(1.6 10 ) 9 105.3 10

8 2( 13.6 1.6 10 / )

N mCe CR m

E eV J eV

04

ev

mr

This is knownas Bohr Radius

Quantum Effects• Classically an accelerating charge revolving with a

frequency would radiate at the same frequency. As it radiates, it loses energy, and radius decreases and frequency increases (death spiral)

• Law of physics in macro-world do not always apply in micro-world

• Quantum phenomena enter the picture• Evidence for quantum nature of atoms: discrete

line spectra emitted by low pressure gas when excited (by electric current)—only certain wavelengths emitted

• A gas absorbs light at some wavelengths of emission spectra, with the number intensity and wavelength of absorption lines depending on temperature, pressure, and motion of the source. This can be used to determine elements of a star and relative motion

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Spectral Lines: Balmer Series• In 1885, Johann Balmer found an empirical formula for wavelength of the

visible hydrogen line spectra in nm:

nm (where k = 3,4,5…)

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Rydberg Equation• As more scientists discovered emission lines at infrared and ultraviolet

wavelengths, the Balmer series equation was extended to the Rydberg equation:

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Bohr Atom Assumptions:1) The electron moves in circular orbits under

influence of Coulomb force

2) Only certain stable orbits at which electron does not radiate

3) Radiates when “jumps” from a more energetic initial state to a lower energy final state

4) The mean kinetic energy of the electron-nucleus system is K = nhforb/2, where forb is the frequency of rotation.

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i fE E hf

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20

1 02

hr a

me

2 2

02n

n hr

me

Bohr Atom Derivation

04

ev

mr

L r p mvr n

2

1nr n r 111 0 5.3 10r a m

/ 2KE nhf 21/ 2

2mv nhf 2mv nhf

/ 2f v r 2 / 2mv nhv r /mv n r

/v n mr

22 2

0

( / )4

ev n rm

mr

with

with

2 22 2 0

2

4 mrnr m

e

2 20

2

4n

nr

me

or

also gives /v n mr

Bohr Radius• The diameter of the hydrogen atom for stationary states is

Where the Bohr radius is given by

• The smallest diameter of the hydrogen atom is

• n = 1 gives its lowest energy state (called the “ground” state)

• recall: 219 2 9

2 211

190

(1.6 10 ) 9 105.3 10

8 2( 13.6 1.6 10 / )

N mCe CR m

E eV J eV

2 2 2 4

1 2 2 20 1 0 0 0

13.68 8 4 2(4 )

e e me meE eV

r