1. PHYS 3446 – Lecture #2 Wednesday, Aug. 30, 2006 Dr. Jae Yu
Introduction
History on Atomic Models
Rutherford Scattering
Rutherford Scattering with Coulomb force
Scattering Cross Section
Measurement of Cross Sections
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

2. Announcements Mail distribution list is ready. Go ahead and subscribe!
5 points extra credit if done by next Wednesday, Sept. 6
3 points extra credit if done by next Friday, Sept. 8
Extra credit opportunities
Obtain a particle data booklet and a pocket diary
Subscribe to the Symmetry Magazine (free!)
Request for printed copies
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Send to Janice Voss at
3 points each if done by Wednesday, Sept. 6.
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

3. Why do Physics? Exp.{ Theory {
To understand nature through experimental observations and measurements (Research)
Establish limited number of fundamental laws, usually with mathematical expressions
Predict nature’s courses
Theory and Experiment work hand-in-hand
Theory works generally under restricted conditions
Discrepancies between experimental measurements and theory presents opportunities to quantum leap
Improves our everyday lives, though some laws can take a while till we see amongst us
Theory {
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

4. Condensed matter/Nano-Science/Chemistry
Structure of Matter
Matter
Molecule
Atom
Nucleus
High Energy Physics
Baryon
(Hadron) u
Quark
10-2m
10-9m
10-10m
10-14m
10-15m
<10-19m
Condensed matter/Nano-Science/Chemistry
protons, neutrons,
mesons, etc.
p,W,L...
top, bottom,
charm, strange,
up, down
Atomic Physics
Nuclear
Physics
Electron
(Lepton)
<10-18m
Give scale and relationship to the domain that high energy physicists work in.
Introduce the fundamental bits of matter loosely so that audience can point back to bulk matter.
Domain.
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

5. Theory for Microscopic Scale, QM
Difficult to describe Small scale phenom. w/ just CM and EM
The study of atomic structure led to quantum mechanics
Long range EM force is responsible for holding atoms together (why ?)
Yet sufficiently weak for QM to estimate properties of atoms reliably
In Nucleus regime, the simple Coulomb force does not work. Why?
The force in nucleus holds positively charged particles together
The known forces in nature
Strong ~ 1
Electro-magnetic ~ 10-2
Weak ~ 10-5
Gravitational ~ 10-38
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

6. Evolution of Atomic Models
1803: Dalton’s billiard ball model
1897: J.J. Thompson Discovered electrons
Built on all the work w/ cathode tubes
Called corpuscles
Made a bold claim that these make up atoms
Measured m/e 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
Thompson’s tubes
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

7. History of Atomic Models cnt’d
1911: Geiger and Marsden with Rutherford performed a scattering experiment with alpha particles shot on a thin gold foil
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

8. History of Atomic Models cnt’d
1912: Rutherford’s planetary model, an atomic model with a positively charged heavy core surrounded by circling electrons
Deficiency of instability. Why?
The electrons will eventually get pulled onto the nucleus, destroying the atom
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

9. History of Atomic Models cnt’d
1913: Neils Bohr proposed the Orbit Model, where electrons occupy well quantified orbits
Electrons can only transition to pre-defined orbits
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

10. History of Atomic Models cnt’d
1926: Schrödinger and de Broglie proposed the Electron Cloud Model based on quantum mechanics
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

11. Rutherford Scattering
A fixed target experiment with alpha particle as projectile shot on thin gold foil
Alpha particle’s energy is low  Speed is well below 0.1c (non-relativistic)
An elastic scattering of the particles
What are the conserved quantities in an elastic scattering?
Momentum
Kinetic Energy
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

12. Elastic Scattering From momentum conservationma q
After Collisions ma mt f mt
From momentum conservation
From kinetic energy conservation
From these two, we obtain
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

13. Analysis Case I If mt<<ma, left-hand side becomes positive
va and vt must be in the same direction
Using the actual masses
and
We obtain
If mt=me, then mt/ma~10-4. 
Thus, pe/pa0<10-4.
Change of momentum of alpha particle is negligible
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

14. Analysis Case II If mt>>ma,
left-hand side of the above becomes negative
va and vt is opposite direction
Using the actual masses
and
We obtain
If mt=mAu, then mt/ma~50. 
Thus, pe/pa0~2pa0.
Change of momentum of alpha particle is large
a particle can even recoil
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

15. Rutherford Scattering with EM Force 1
Let’s take into account only the EM force between the a and the atom
Coulomb force is a central force, so a conservative force
Coulomb potential between particles with Ze and Z’e electrical charge separated by distance r is
Since the total energy is conserved,
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

16. Rutherford Scattering with EM Force 2
The distance vector r is always the same direction as the force throughout the entire motion, so the net torque (rxF) is 0.
Since there is no net torque, the angular momentum (l=rxp) is conserved.  The magnitude of the angular momentum is l=mvb.
Impact parameter
From the energy relation, we obtain
From the definition of angular momentum, we obtain an equation of motion
From energy conservation, we obtain another equation of motion
Centrifugal barrier
Effective potential
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

17. Rutherford Scattering with EM Force 3
Rearranging the terms for approach, we obtain
and
Integrating this from r0 to infinity gives the angular distribution of the outgoing alpha particle
Distance of closest approach
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

18. Rutherford Scattering with EM Force 4
What happens at the DCA?
Kinetic energy reduces to 0.
The incident alpha could turn around and accelerate
We can obtain
This allows us to determine DCA for a given potential and c0.
Define scattering angle q as the changes in the asymptotic angles of the trajectory, we obtain
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

19. Rutherford Scattering with EM Force 5
For a Coulomb potential
DCA can be obtained for the given impact parameter b,
And the angular distribution becomes
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

20. Rutherford Scattering with EM Force 6
Replace the variable 1/r=x, and performing the integration, we obtain
This can be rewritten
Solving this for b, we obtain
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

21. Rutherford Scattering with EM Force 7
From the solution for b, we can learn the following
For fixed b, E and Z’
The scattering is larger for a larger value of Z.
Makes perfect sense since Coulomb potential is stronger with larger Z.
Results in larger deflection.
For fixed b, Z and Z’
The scattering angle is larger when E is smaller.
If particle has low energy, its velocity is smaller
Spends more time in the potential, suffering greater deflection
For fixed Z, Z’, and E
The scattering angle is larger for smaller impact parameter b
Makes perfect sense also, since as the incident particle is closer to the nucleus, it feels stronger Coulomb force.
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu

22. Assignments
Compute the masses of electron, proton and alpha particles in MeV/c2, using E=mc2.
Need to look up masses of electrons, protons and alpha particles in kg.
Compute the gravitational and the Coulomb forces between two protons separated by 10-10m and compare their strengths
Derive the following equations in your book:
Eq. # 1.3, 1.17, 1.32
Must show detailed work and accompany explanations
These assignments are due next Wednesday, Sept. 6.
Wednesday, Aug. 30, 2006
PHYS 3446, Fall 2006
Jae Yu