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Cross Sections

Cross Sections. Definition of Cross Section Why its useful. Breit-Wigner Resonances Rutherford Scattering. Cross-Sections. Why concept is important Learn about dynamics of interaction and/or constituents (cf Feynman’s watches). Needed for practical calculations. Experimental Definition

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Cross Sections

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  1. Cross Sections • Definition of Cross Section • Why its useful. • Breit-Wigner Resonances • Rutherford Scattering Nuclear Physics Lectures

  2. Cross-Sections • Why concept is important • Learn about dynamics of interaction and/or constituents (cf Feynman’s watches). • Needed for practical calculations. • Experimental Definition • How to calculate s • Fermi Golden Rule • Breit-Wigner Resonances • QM calculation of Rutherford Scattering Nuclear Physics Lectures

  3. Definition of s • a+bx • Effective area for reaction to occur is s Na(0) particles type a/unit time hit target b Nb atoms b/unit volume Number /unit area= Nb dx Probability interaction = s Nbdx dNa=-Na Nb dx s Na(x)=Na(0) exp(-x/l) ; l=1/(Nbs) Beam a Na dx Nuclear Physics Lectures

  4. Reaction Rates • Na beam particles/unit volume, speed v • Flux F= Na v • Rate/target b atom R=Fs • Thin target x<<l: R=(NbT) F sTotal • This is total cross section. Can also define differential cross sections, as a function of reaction product, energy, transverse momentum, angle etc. • dR(a+bc+d)/dE=(NbT) F ds(a+bc+d)/dE Nuclear Physics Lectures

  5. Breit-Wigner Line Shape • Start with NR Schrödinger equation: X by f*n and integrate Start in state m  exponential decay Nuclear Physics Lectures

  6. Breit-WignerLine Shape - 2 For Nuclear Physics Lectures

  7. Breit-Wigner Line Shape -3 Normalised Breit-Wigner line shape Q: where have you seen this shape before? We will see this many times in NP and PP. Nuclear Physics Lectures

  8. Breit-Wigner Resonance • Important in atomic, nuclear and particle physics. • Uncertainty relationship • Determine lifetimes of states from width. • , G=FWHM; Nuclear Physics Lectures

  9. Fermi Golden Rule • Want to be able to calculate reaction rates in terms of matrix elements of H. • Warning: We will use this many times to calculate s but derivation not required for exams, given here for completeness. Nuclear Physics Lectures

  10. Discrete  Continuum • Decays to a channel i (range of states n). Density of states ni(E). Assume narrow resonance Nuclear Physics Lectures

  11. Cross Section • Breit Wigner cross section. • Definition of s and flux F: Nuclear Physics Lectures

  12. Breit-Wigner Cross Section • Combine rate, flux & density states  Nuclear Physics Lectures

  13. Breit-Wigner Cross Section n + 16O 17O Nuclear Physics Lectures

  14. Low Energy Resonances • n + Cd total cross section. • Cross section scales s ~ 1/E1/2 at low E. • B-W: 1/k2 and G~n(E)~k Nuclear Physics Lectures

  15. Rutherford Scattering 1 Nuclear Physics Lectures

  16. Rutherford Scattering 2 Nuclear Physics Lectures

  17. Rutherford Scattering 3 • Fermi Golden Rule: Nuclear Physics Lectures

  18. Rutherford Scattering 4 pf pi Compare with experimental data at low energy Q: what changes at high energy ? Nuclear Physics Lectures

  19. Low Energy Experiment • Scattering of a on Au & Ag  agree with calculation assuming point nucleus dN/dcosq Sin4(q/2) Nuclear Physics Lectures

  20. Higher Energy Electron - Gold • Deviation from Rutherford scattering at higher energy  determine charge distribution in the nucleus. • Form factors is F.T. of charge distribution. Nuclear Physics Lectures

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