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PHYS 3446 – Lecture #2

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. Announcements.

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PHYS 3446 – Lecture #2

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  1. PHYS 3446 – Lecture #2 Wednesday, Aug. 30, 2006 Dr. JaeYu • Introduction • History on Atomic Models • Rutherford Scattering • Rutherford Scattering with Coulomb force • Scattering Cross Section • Measurement of Cross Sections 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 • http://pdg.lbl.gov/pdgmail/ • Subscribe to the Symmetry Magazine (free!) • http://www.symmetrymagazine.org • Request for printed copies • Subscribe to the printed copies of CERN Courier Magazine • http://www.cerncourier.com/ • Send e-mail to Janice Voss at vosses@aol.com • 3 points each if done by Wednesday, Sept. 6. 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 PHYS 3446, Fall 2006 Jae Yu

  4. Matter Molecule Atom Nucleus Baryon Quark (Hadron) u Electron (Lepton) High Energy Physics Structure of Matter 10-14m 10-9m 10-10m 10-15m <10-19m 10-2m Condensed matter/Nano-Science/Chemistry protons, neutrons, mesons, etc. p,W,L... top, bottom, charm, strange, up, down Atomic Physics Nuclear Physics <10-18m 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 toquantum 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 PHYS 3446, Fall 2006 Jae Yu

  6. Evolution of Atomic Models Cathode ray tube • 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 Thompson’s tubes 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 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 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 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 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 PHYS 3446, Fall 2006 Jae Yu

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

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