1. Physics 3313 - Lecture 7 Wednesday February 10, 2010 Dr. Andrew Brandt
HW 2 (Ch 3 due today); HW3 (Ch 4) to be assigned today
Atomic Models
Rutherford Scattering
Bohr Atom
2/10/2010
3313 Andrew Brandt

2. CHAPTER 4 Structure of the 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)
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, 1913
2/10/2010
3313 Andrew Brandt

3. Structure of the Atom
By 1900 scientists had evidence that indicated the atom was not a fundamental unit:
There seemed to be too many different kinds of atoms, each belonging to a distinct chemical element.
Electromagnetic properties and line spectra hinted at some underlying structure.
The problem of valence. Certain elements combined with some elements but not with others, a characteristic that hinted at an internal atomic structure.
The discoveries of radioactivity, of x rays, and of the electron.
2/10/2010
3313 Andrew Brandt

4. Evolution of Atomic Models
Cathode ray tube
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
Personally I prefer chocolate chip
cookie model
2/10/2010
3313 Andrew Brandt

5. 4.2 Rutherford Experiment
1911: Geiger and Marsden with Rutherford performed a scattering experiment firing alpha particles at a thin gold foil
2/10/2010
3313 Andrew Brandt

6. 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
2/10/2010
3313 Andrew Brandt

7. Rutherford Scattering
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.
look at limiting
cases for 
2/10/2010
3313 Andrew Brandt

8. Rutherford Example
On blackboard demonstrate size of radius from distance of closest approach
2/10/2010
3313 Andrew Brandt

9. Rutherford Scattering Equation
In actual experiment a detector is positioned from θ to θ + dθ that corresponds to incident particles between b and b + db.
The number of particles scattered per unit area is
The cross section σ = πb2 is related to the probability for a particle being scattered by a nucleus.
2/10/2010
3313 Andrew Brandt

10. 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?
2/10/2010
3313 Andrew Brandt

11. 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)
This is known
as Bohr Radius
2/10/2010
3313 Andrew Brandt

12. 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
2/10/2010
3313 Andrew Brandt

13. 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…)
2/10/2010
3313 Andrew Brandt

14. Rydberg Equation
As more scientists discovered emission lines at infrared and ultraviolet wavelengths, the Balmer series equation was extended to the Rydberg equation:
2/10/2010
3313 Andrew Brandt

15. Bohr Atom Assumptions:
The electron moves in circular orbits under influence of Coulomb force
Only certain stable orbits at which electron does not radiate
Radiates when “jumps” from a more energetic initial state to a lower energy final state
The mean kinetic energy of the electron-nucleus system is K = nhforb/2, where forb is the frequency of rotation.
2/10/2010
3313 Andrew Brandt

16. Bohr Atom Derivation with with or also gives 2/10/2010
3313 Andrew Brandt

17. 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: