PHYS 3446 – Lecture #13History of Atomic Models cnt’dElastic ScatteringRutherford ScatteringTotal Cross SectionTotal X-Section of Rutherford ScatteringLab Frame and CM FrameRelationship of variables in Lab and CMSome Quantities in Special RelativityRelativistic VariablesWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt1PHYS 3446 – Lecture #13Wednesday, October 8, 2008Dr. Andrew Brandt• Review• Test ch 1-4 Monday Oct. 13• HW solutions posted on HW pageWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt2History of Atomic Models cnt’d•Lec.2 • atomic models• elastic scattering• Rutherford scatteringWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt3Elastic Scattering• From momentum conservation• From kinetic energy conservation (Elastic only!)• From these, we obtainmtmαmαθmtφAfter Collisions0vrvαrtvr0v =r20v =21ttmvmα⎛⎞−=⎜⎟⎝⎠ttmv mvmααα+=rrttmvvmαα+rr22ttmvvmαα+2tvvα⋅rrEq. 1.3Eq. 1.2look at limiting casesWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt4Rutherford 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'cot22ZZ ebEθ=Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt5Total Cross Section• Total cross section is the integration of the differential cross section over the entire solid angle, Ω: • Total cross section represents the effective size of the scattering center integrated over all possible impact parameters (and consequently all possible scattering angles)Totalσ=()40,dddπσθφΩ=Ω∫()02sindddπσπθθ θΩ∫lec. 3 diff +total xsec-24 21 barn = 10 cmWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt6Total X-Section of Rutherford Scattering• To obtain the total cross section of Rutherford scattering, one integrates the differential cross section over all θ:• 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. (θ=θ0>0); note this is sensible since alpha particles far away don’t even see charge of nucleus due to screening effects.Totalσ=()02sindddπσπθθθ=Ω∫22103'18sin42sin2ZZ edEθπθ⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠∫Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt7Lab 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 μ(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 μ 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 4Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt8Relationship 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 CMvR==&1v =%CM2v=%and1 CMvv−=2112mvmm+CMv=1112mvmm+1112mvmm+1v%1vCMVCMθLabθWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt9Some Quantities in Special Relativity• Fractional velocity •Lorentzγ 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=vcβ=rrvcβ=rr211γβ=−Mvγ=rMcγβr2 T +Re2stE=22 24Pc Mc+=2Mcγ22222PMcEpc==−Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt10Relativistic 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 ()212sPP=+ =24 24 212 122mc mc Emc=++()()22212 12CM CM CM CMEE PPc=+ −+rr()212CM CMEE=+()2CMToTE=s0()()22212 12EE PPc+−+rr14TeVs =lec 6()212sPP=+ =Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt11• Binding Energy per nucleon (BE) is total binding energy (B) divided by number of nucleons (A) Nuclear Properties: Binding Energy()2,MAZ cA−Δ=()()()2,pnZmAZmMAZcA+− −=• Rapidly increase with A till A~60 at which point BE~9 MeV.• A>60, the B.E gradually decrease Î For most of the large A nucleus, BE~8 MeV.BE =BAlec 71stlecof nucpropsWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt12• 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: Sizes130RrA=≈Does this formula make sense? why 1/3 power?13 1 31.2 10 Acm−×=131.2 fmAWednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt13• For electrons, μe~μB, where μBis Bohr Magneton• For nucleons, magnetic dipole moment is measured in nuclear magneton, defined using proton mass• Measured magnetic moments of proton and neutron:Nuclear Properties: Magnetic Dipole MomentsBμ=2Npemcμ=h2.79pNμμ≈1.91nNμμ≈−2eemc=h115.79 10 MeV/T−×lec. 8Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt14• The number of protons and neutrons inside stable nuclei are– A<40: Equal (N=Z)– A>40: N~1.7Z– Neutrons outnumber protons– Most are even-p + even–n• See table 2.1– Supports strong pairing Nuclear Properties: StabilityN~1.7ZN=ZN Z NnuclEven Even156Even Odd 48Odd Even 50Odd Odd 5Wednesday, Oct. 8, 2008 PHYS 3446, Fall 2008Andrew Brandt15• A square well nuclear potential Î provides the basis of quantum theory with discrete energy levels and corresponding bound state just like in atoms– Presence of nuclear quantum states have been confirmed through •
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