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Hampton University Graduate Studies 2003

(e,e ' p) and Nuclear Structure. Paul Ulmer Old Dominion University. Hampton University Graduate Studies 2003. Thanks to:. W. Boeglin T.W. Donnelly (Nuclear physics course at MIT) J. Gilfoyle R. Gilman R. Niyazov J. Kelly (Adv. Nucl. Phys. 23, 75 (1996)) B. Reitz Saha S. Strauch

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Hampton University Graduate Studies 2003

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  1. (e,e'p) and Nuclear Structure Paul Ulmer Old Dominion University Hampton University Graduate Studies 2003

  2. Thanks to: • W. Boeglin • T.W. Donnelly (Nuclear physics course at MIT) • J. Gilfoyle • R. Gilman • R. Niyazov • J. Kelly (Adv. Nucl. Phys. 23, 75 (1996)) • B. Reitz • Saha • S. Strauch • E. Voutier • L. Weinstein

  3. Outline • Introduction • Background • Experimental • Theoretical • Nuclear Structure • Medium-modified nucleons • Cross sections • Polarization transfer • Studies of the reaction mechanism • Few-body nuclei • The deuteron • 3,4He

  4. A(e,e'p)B e' B p q e A Known: e and ADetect: e' and p Infer: pm = q–p = pB

  5. p B B A–1 + N A–2 (e,e'p) - Schematically e' v + A e i.e. bound = Etc.

  6. Kinematics p e' scattering plane pq reaction plane (,q) e pA–1 x “out-of-plane” angle In ERLe: Q2 –qq  = q2 – 2 = 4ee' sin2/2 Missing momentum:pm = q–p = pA–1 Missing mass: m =  –Tp – TA–1

  7. Some (Very Few) Experimental Details …

  8. e' Detected e p e' “accidental” (uncorrelated) e e' “real” (correlated) p e

  9. # events a r relative time: te– tp

  10. Accidentals Rate = Re Rp  /DF  I 2 /DF Reals Rate = Reep I S:N = Reals/Accidentals DF /(I) Compromise: Optimize S:N and Reep

  11. Extracting the cross section NN (cm-2) e' Ne e (e, pe) (p, pp) p

  12. Some Theory …

  13. Cross Section for A(e,e'p)B in OPEA “A-1” where Current-Current Interaction

  14. Square of Matrix Element  W

  15. Cross Section in terms of Tensors Mott cross section Electron tensor Nuclear tensor

  16. 3 indep. momenta:Q , Pi , P (PA–1= Q + Pi – P) target nucleus ejectile 6 indep. scalars:Pi2, P2, Q2, Q•Pi ,Q•P , P•Pi = m2 = MA2 Consider Unpolarized Case Lorentz Vectors/Scalars

  17. Nuclear Response Tensor Xi are the response functions

  18. Impose Current Conservation Get 6 equations in 10 unknowns 4 independent response functions

  19. Putting it all together …

  20. Nuclear 4-current The Response Functions Use spherical basis with z-axis along q:

  21. Can choose: Q2,  , m , pm Q • Pi = MA P • Pi = E MA Q • P =  E – q p cos pq In lab: Response functions depend on scalar quantities Note: no x dependence in response functions

  22. Including electron and recoil proton polarizations

  23. Extracting Response Functions For instance: RLT and A (=A LT)

  24. A–1 e' p q p0 p0 e A Plane Wave Impulse Approximation (PWIA) spectator A-1 q – p = pA-1= pm= – p0

  25. The Spectral Function In nonrelativistic PWIA: e-p cross section nuclear spectral function For bound state of recoil system: proton momentum distribution

  26. The Spectral Function, cont’d. Note: S is not an observable!

  27. p p (+m, q) Elastic Scattering from a Proton at Rest (m,0) (,q) Before After Proton is on-shell  ( + m)2 q2 = m2 2 + 2m + m2 q2 = m2  = Q2 2m

  28. Vertex fcn p p p + p n 0 + p p p p point proton Scattering from a Proton , cont’d. + + + structure/anomalous moment

  29. Scattering from a Proton , cont’d. Vertex fcn: Dirac FF Pauli FF Sachs FF’s GE and GM are the Fourier transforms of the charge and magnetization densities in the Breit frame.

  30. Phase difference: Form Factor r k k' Amplitude at q:

  31. Cross section for ep elastic However, (e,e'p) on a nucleus involves scattering from moving protons, i.e. Fermi motion.

  32. (E,p) (,q) p p After (+E, q+p) Elastic Scattering from a Moving Proton Before ( + E)2 – (q+p)2 = m2 2 + 2E + E2 q2 2p•q  p2 = m2 Q2 = 2E 2p•q  (E/m) = (Q2 2m) + p•q m

  33. Cross section for ep elastic scattering off moving protons Follow same procedure as for unpolarized (e,e'p) from nucleus We get same form for cross section, with 4 response functions …

  34. Response functions for ep elastic scattering off moving protons

  35. Quasielastic Scattering For E  m:   (Q2 2m) + p•q m If we “quasielastically” scatter from nucleons within nucleus: Expect peak at:   (Q2 2m) Broadened by Fermi motion: p•q m

  36. Electron Scattering at Fixed Q 2 Elastic Nucleus Deep Inelastic  Quasielastic N*  Proton Elastic Deep Inelastic  N* 

  37. 6Li 12C 24Mg 40Ca 89Y 58Ni 118Sn 181Ta 208Pb Quasielastic Electron Scattering R.R. Whitney et al., Phys. Rev. C 9, 2230 (1974).

  38. Data: P. Barreau et al., Nucl. Phys. A402, 515 (1983). y-scaling analysis: J.M. Finn, R.W. Lourie and B.H. Cottman, Phys. Rev. C 29, 2230 (1984).

  39. Nuclear Structure

  40. First, a bit of history: The first (e,e'p) measurement Frascati Synchrotron, Italy 12C(e,e'p) 27Al(e,e'p) U. Amaldi, Jr. et al., Phys. Rev. Lett. 13, 341 (1964).

  41. (e,e'p) advantages over (p,2p) • Electron interaction relatively weak: OPEA is reasonably accurate. • Nucleus is very transparent to electrons: Can probe deeply bound orbits. However: ejected proton is strongly interacting. The “cleanness” of the electron probe is somewhat sacrificed. FSI must be taken into account.

  42. Recall, in nonrelativistic PWIA: where q – p = pm= – p0 FSI destroys simple connection between the measured pm and the proton initial momentum (not an observable).

  43. Final State Interactions (FSI) p A–1 FSI e' p0' q e p0 A

  44. Distorted Wave Impulse Approximation (DWIA) Treat outgoing proton distorted waves in presence of potential produced by residual nucleus (optical potential). “Distorted” spectral function

  45. Optical potential is constrained by proton elastic scattering data. • Problems with this approach: • Residual nucleus contains hole state, unlike the target in p+A scattering. • Proton scattering data is surface dominated, whereas ejected protons in (e,e'p) are produced within entire nuclear volume.

  46. 100 MeV data is significantly overestimated by DWIA near 2nd maximum. NIKHEF-K Amsterdam J.W.A. den Herder, et al., Phys. Lett. B 184, 11 (1987).

  47. At pm160 MeV/c, wf is probed in nuclear interior. J.W.A. den Herder, et al., Phys. Lett. B 184, 11 (1987).

  48. Adjusting optical potential renders good agreement while maintaining agreement with p+A elastic. J.W.A. den Herder, et al., Phys. Lett. B 184, 11 (1987).

  49. Saclay Linac, France 12C(e,e'p)11B J. Mougey et al., Nucl. Phys. A262, 461 (1976).

  50. 12C(e,e'p)11B p-shell l=1 Saclay Linac, France s-shell l=0 J. Mougey et al., Nucl. Phys. A262, 461 (1976).

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