PHYS 3446 – Lecture #12Project: SubjectsHistory of Atomic Models cnt’dElastic ScatteringRutherford ScatteringTotal Cross SectionTotal X-Section of Rutherford ScatteringLab Frame and CM FrameRelationship of variables in Lab and CMQuantities in Special RelativityRelativistic VariablesSlide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 1PHYS 3446 – Lecture #12Monday, Oct. 18, 2010Dr. Andrew Brandt•Quiz•ReviewProject: Subjects•Quark-Gluon Plasma (RHIC)•Higgs Boson Theory•Higgs Boson Searches at LEP•Higgs Boson Searches at DZero•Higgs Boson Searches at CMS•Beyond the Standard Model Higgs rumors and CDF search•Supersymmetry or Blackhole Searches at ATLAS•Solar Neutrino Deficit•Long baseline neutrino experiments (neutrino mass)•G-2 experiments•HERA experiments: diffraction/large rapidity gapsMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 2Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 3History of Atomic Models cnt’d•Lec.2 •atomic models•elastic scattering•Rutherford scatteringMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 4Elastic Scattering•From momentum conservation•From kinetic energy conservation (Elastic only!)•From these, we obtainmtmmmtAfter Collisions0vrvartvr0v =r20v =21ttmvma� �- =� �� �ttm v m vmaaa+=r rttmv vmaa+r r2 2ttmv vmaa+2tv va�r rEq. 1.3Eq. 1.2look at limiting casesMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 5Rutherford Scattering•From the solution for b, we can learn the following 1. For fixed b, E and Z’–The scattering is larger for a larger value of Z.–Since Coulomb potential is stronger with larger Z.2. For fixed b, Z and Z’–The scattering angle is larger when E is smaller.–Since the speed of the low energy particle is smaller–The particle spends more time in the potential, suffering greater deflection3. For fixed Z, Z’, and E–The scattering angle is larger for smaller impact parameter b–Since the closer the incident particle is to the nucleus, the stronger Coulomb force it feels2'cot2 2ZZ ebEq=Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 6Total Cross Section•Total cross section is the integration of the differential cross section over the entire solid angle, : •Total cross section represents the effective size of the scattering center integrated over all possible impact parameters (and consequently all possible scattering angles)Totals =( )40,dddpsq f W=W�( )02 sindddpsp q q qW�lec. 3 diff +total xsec-24 21 barn = 10 cmMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 7Total X-Section of Rutherford Scattering•To obtain the total cross section of Rutherford scattering, one integrates the differential cross section over all :•What is the result of this integration?–Infinity!!•Does this make sense?–Yes•Why?–Since the Coulomb force’s range is infinite (particle with very large impact parameter still contributes to integral through very small scattering angle)•What would be the sensible thing to do?–Integrate to a cut-off angle since after certain distance the force is too weak to impact the scattering. (0>0); note this is sensible since alpha particles far away don’t even see charge of nucleus due to screening effects.Totals =( )02 sindddpsp q q q =W�22103' 18 sin4 2sin2ZZ edEqpq� �� �� �� �� �� ��Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 8Lab Frame and CM Frame•The CM is moving at a constant velocity in the lab frame independent of the form of the central potential•The motion is that of a fictitious particle with mass (the reduced mass) and coordinate r.•In the frame where the CM is stationary, the dynamics becomes equivalent to that of a single particle of mass scattering off of a fixed scattering center.•Frequently we define the Center of Mass frame as the frame where the sum of the momenta of all the interacting particles is 0.lec 4Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 9Relationship of variables in Lab and CM•The speed of CM is•Speeds of the particles in CM frame are•The momenta of the two particles are equal and opposite!!CM CMv R= =&1v =%CM2v =%and1 CMv v- =2 11 2m vm m+CMv =1 11 2m vm m+1 11 2m vm m+1v%1vCMVCMqLabqMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 10Quantities in Special Relativity•Fractional velocity •Lorentz factor•Relative momentum and the total energy of the particle moving at a velocity is•Square of four momentum P=(E,pc), rest mass EP =rE =v cb =rrv cb=rr211gb=-Mvg =rM cg br2 T +Re2stE =2 2 2 4P c M c+ =2Mcg2 2 2 2 2P Mc E p c= = -lec 5Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 11Relativistic Variables•The invariant scalar, s, is defined as:•So what is this in the CM frame?•Thus, represents the total available energy in the CM; At the LHC, eventually ( )21 2s P P= + =2 4 2 4 21 2 1 22m c m c E m c= + +( )( )2221 2 1 2CM CM CM CME E P P c= + - +r r( )21 2CM CME E= +( )2CMToTE=s0( )( )2221 2 1 2E E P P c+ - +r r14TeVs =( )21 2s P P= + =Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 12lec 6 (worked HW 2 probs)Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 13Monday, Sept. 15, 2008 PHYS 3446, Fall 2008Andrew Brandt14•The mass deficit is always negative and is proportional to the nuclear binding energy •What is the physical meaning of BE?–A minimum energy required to release all nucleons from a nucleusNuclear Properties: Binding Energy( ),M A ZD =( )2. ,B E M A Z c=D( ),M A Z( )p nZm A Z m- - -•Rapidly increase with A till A~60 at which point BE~9 MeV.•A>60, the B.E gradually decrease For most of the large A nucleus, BE~8 MeV.lec. 6Monday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 15•The size of a nucleus can be inferred from the diffraction pattern•All this phenomenological investigation resulted in a startlingly simple formula for the radius of the nucleus in terms of the number of nucleons or atomic number, A: Nuclear Properties: Sizes1 30R r A= �13 1 31.2 10 A cm-� =1 31.2 fmAMonday, Oct. 18, 2010 PHYS 3446, Fall 2010 Andrew Brandt 16•For electrons, e~B, where B is Bohr Magneton•For nucleons, magnetic dipole moment is measured in nuclear magneton, defined using proton mass•Measured magnetic moments of proton and
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