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The (e,e‘p) reaction:

The (e,e‘p) reaction:. advantages access to nuclear structure probes the nucleon more information than inclusive clean definition of reaction. disadvantages: final state interaction proton-nucleus 2. detector needed smaller cross section due to acceptance.

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The (e,e‘p) reaction:

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  1. The (e,e‘p) reaction: • advantages • access to nuclear structure • probes the nucleon • more information than inclusive • clean definition of reaction • disadvantages: • final state interaction proton-nucleus • 2. detector needed • smaller cross section due to acceptance (e,e'p) advantages over (p,2p): • Electron interaction relatively weak: • OPEA reasonably accurate. • Nucleus is very transparent to electrons: • Can probe deeply bound orbits.

  2. Structure of the nucleus • nucleons are bound • energy (E) distribution • shell structure • nucleons are not static • momentum (k) distribution determined by N-N potential short-range in average: binding energy: ~ 8 MeV distance: ~ 2 fm repulsive core scan/test/nn_pot.agr r r long-range d attractive part

  3. 2 1 0 -1 -2 early hint on shell structure in the nucleus particular stable nuclei with Z,N = 2, 8, 20, 28, 50, 82, 126 (magic numbers) large separation energy ES 28 50 82 126 “average” (Weizsäcker Formula, no shell structure, bulk properties) ES - W.F 40 60 80 100 120 140 neutron number N shell closure

  4. Shell structure (Goeppert-Mayer, Jensen, 1949) expected: many collisions between nucleons, no independent-particle orbits But: experimental evidence for shell structure Pauli Exclusion Principle: nucleons can not scatter into occupied levels: Suppression of collisions between nucleons

  5. Independent Particle Shell model (IPSM) kF2 EF = 2M • single particle approximation: • nucleons move independently from each other • in an average potential created by the surrounded nucleons (mean field) • spectral function S(E, k): • probability of finding a proton with initial momentum k and • energy E in the nucleus • factorization into energy & momentum part nuclear matter: Z(E) Z(k) occupied empty occupied empty E EF kF k

  6. 6Li 12C 24Mg 40Ca 89Y 58Ni 118Sn 181Ta 208Pb Inclusive Quasielastic Electron Scattering: (e,e’)  getting the bulk features compared to Fermi model: fit paramter kF and e R.R. Whitney et al., Phys. Rev. C 9, 2230 (1974).

  7. Fit results: Moniz et al, PRL 26, 445 (1971) kFµwidth of q.e. peak, »250 MeV/c for medium-heavy nuclei

  8. The first (e,e'p) measurement: identification of different orbits  getting the details Frascati Synchrotron, Italy 12C(e,e'p) U. Amaldi, Jr. et al., Phys. Rev. Lett. 13, 341 (1964). 27Al(e,e'p) moderate resolution: FWHM: 20 MeV

  9. g.s. Jp 5.02 3/2- 4.45 5/2- 2.12 1/2- 3/2- 11B 0+ g.s. 12C NIKHEF: resolution 150 keV shape described by Lorentz function with central energy Ea + width Ga Steenhoven et al., PRC 32, 1787 (1985) incidental remark: 5/2- state can not be populated by 1-step process, because 1f5/2 state empty in 12C, but seen in (p,2p) electrons are a suitable probe to examine the nucleus

  10. shell model: describes basic properties like spin, parity, magic numbers ... NIKHEF results momentum distribution [(GeV/c)-3/sr] -100 0 100 200 -100 0 100 200 pm [MeV/c] Momentum distribution: - characteristic for shell (l, j) - Fourier transformation of Ylj(r)  info about radial shape

  11. NIKHEF 1.0 IPSM 65% Z/IPSM theory (solid line): Distorted wave impulse approximation (DWIA) solves the Schrödinger equation using an optical potential (fixed by p-12C) (Hartree-Fock, selfconsistent) real part: Wood-Saxon potential imaginary part: accounts for absorption in the nucleus Correction forCoulomb distortion well reproduced shape strength of the transition smaller! • definite number of nucleons • in each shell (IPSM): £ 2 j + 1 Spectroscopic factor Za kF ò Za= 4 p dE dk k2S(E, k) single particle state a = number of nucleons in shell

  12. za/(2j+1) Massenzahl Long range correlations (LRC): 20 % Fragmentation of the 1-particle strength due to excitation of collective modes -- surface vibration -- configuration mixing -- Giant resonances  occupation of “empty” states close to the Fermi edge  broadening, fragmentation, excitation ~10 % Theoretical description: Random Phase Approximation (RPA) considerable mixing between hole states and 2h1p configurations

  13. SRC p2m Em ~ (+ 2EB) 2 M k -k • Short range and tensor correlations (SRC): ~ 15 % • repulsive core of NN force leads to scattering between • particles at short distances simple picture: max. of the s.f. expected at • Consequences: • scattering of nucleons to high lying states • depletion of IPSM shells occupation of states at large k and large E

  14. NIKHEF results: 208Pb(e,e¢p)207Tl SRC LRC Za/(2j+1) Bindungsenergie [MeV] spectroscopic factor: strength of the transitionYaAYbA-1 experimentally: identification via momentum distribution Comparison with CBF theory: modified for finite nuclei Im S fitted to the width of the state (PRC 41(1990) R24) + LRC for valence nucleons: binding energy » excitation energy for vibration deeper bound nucleons (inner shells): less easily excited, broadening close to occupation number

  15. LS splitting 2j +1 l - orbits W.H. Dickhoff, C. Barbieri,Prog.Part.Nucl.Phys. 52,(2004) 377

  16. Some definitions: spectroscopic factor Za probability to reach a final state a (n,l,j) when a nucleon is removed (added to) the target nucleus occupation number na number of nucleons in the state a, which might be fragmentated occupancy o: occupation number relative to completely filled shell

  17. Occupation numbers: • difficult to measure • fragmentation: strength is spread over a wide range of Em • deep-lying states: large width and start to overlap • absolute spectroscopic numbers are model dependent CERES: Clement et al, Phys.Lett B 183 (1987) 127 Combined Evaluation of Relative spectroscopic factors and Electron Scattering • 1) input: • difference in charge density from elastic electron scattering • Dr = r(206Pb) - r (205Tl) • assumption: • main contribution in charge difference Dr comes from 3S1/2 state • Dr = S Dnara+ Drcp • »[n3S1/2 (206) - n3S1/2 (205)] r(3S1/2) + Drcp Dn = 0.64 ±0.06

  18. 2) input: • relative spectroscopic factors (from (e,e¢p)-experiment) • R(205/208) = Z3S1/2(205)/ Z3S1/2(208) • » n3S1/2(205)/ n3S1/2(208) n(208Pb) = 1.51 ± 0.18 76 ±7% assumption: equal percentage depletion of all shells, nuclei are neighbours  similar shell structure in agreement with theory for nuclear matter n(206Pb) = 1.25 ± 0.1 n(205Tl) = 0.61 ± 0.1 63% • advantage: • only relative spectroscopic factors needed • avoids to sum up fragmented strength P.Grabmayr et al., PRC 49 (1994) 2971

  19. Spectroscopic factors from (d, 3He) Pick-up reaction should be equivalent to (e,e’p) reaction problem: very sensitive to the asymptotic tail of the wave function: DZ/Z » 7 Drrms/rrms reason: - strong interaction are absorptive - only sensitive to the surface rrms was chosen to fulfill the sumrule nparticles + nholes = 2j+1 (Pick-up and stripping reaction), assuming IPSM But: sumrule not fulfilled for Em < 10-20 MeV due to SRC

  20. NIKHEF Reanalysis (d,3He) Non-Local/Finite-range BSWF from (e,e'p) Kramer et al. NPA 679 (2001) 267 Original data (d,3He) Local/Zero-range BSWF chosen to fulfill sumrule 100

  21. Depletion in other systems: atoms (Ne,Ar) : 0.001 - 0.018: increases from the inner to outermost shells nuclei : » 0.2 - 0.3 L3He : » 0.4 - 0.5 Liquid 3He drops momentum distribution 0.4 quasi hole strength 0 0.5 1 1.5 2 1.0 momentum/kF 0.8 0.6 0.4 0.2 0.0 interaction increases IPSM large repulsive core in the interatomic potential: Lennard-Jones potential: strong repulsive core long-range attraction simple spin-independent function of the atomic distance

  22. Effect of correlations on the binding energy kinetic energy binding energy potential energy (Feldmeier, Neff) binding energy: <T> + <V> difference of 2 large values Cr, CW: Operator for SRC and tensor correlations Without correlations no bound nuclei!

  23. CBF IPSM » kF c12_spectheocomp_nice.agr k < kF: single-particle contribution dominates k » kF: SRC already dominates for E > 50 MeV k > kF: single-particle negligible consequence: search for SRC at large E, k method: (e,e’p)-experiment

  24. spectral/c12_momtheocomp_nice1.agr signature of SRC: additional strength at high momentum • Modern many-body theories: • Correlated Basis Function theory (CBF) • O. Benhar, A. Fabrocini, S. Fantoni, Nucl. Phys. A505, 267 (1989) • Green’s function approach (2nd order) • H. Müther, G. Knehr, A. Polls, Phys. Rev. C52, 2955 (1995) • Self--consistent Green’s function (T = 2 MeV) • T. Frick, H. Müther, Phys. Rev. C68 (2003) 034310

  25. in the IPSM region: c12_benh_mom1s_nicesh

  26. Missing strength already at moderate pm compared to IPSM 200 MeV/c < pm < 300 MeV/c k4que01_ipsmben250.eps data on 12C IPSM*0.85 CBF Q2 = 1.5 (GeV/c)2 radiation tail Spectral function containing SRC: good agreement with data

  27. (e,e¢p)-reaction: coincidence experiment measured values: momentum, angles MA e¢ e initial momentum k pp ¢=k+q |E| ==Emk == - pm q pm = q - pp¢ electron energy:Ee proton:pp’ electron: ke’ Ee’ = |ke’| reconstructed quantites: missing energy: Em = Ee- Ee¢- Tp¢- TA-1 missing momentum: in PWIA: direct relation between measured quantities and theory:

  28. Setup in HallC: SOS detector system 0.1-1.75 GeV/c p¢ iron magnets D HMS D 0.5-7.4 GeV/c e¢ target detector system e Q Q Q D collimator 0.8-6 GeV superconducting magnets, quadrupole: focussing/defocussing performance HMSSOS momentum range (GeV): 0.5-7.4 0.1-1.75 acceptance d (%): ± 10 ± 15 p = po (1 + d) solid angle (msr): 6.7 7.5 target acceptance (cm): ± 7 ± 1.5

  29. k q k^ q parallel kinematics perpendicular kinematics kinp1 kin3,qqpi<45 o kin3,qqpi >45 o kin4 kinp2 kin5 • Data at high pm, Em measured in Hall C at Jlab: • targets: C, Al, Fe, Au • kinematics: 3 parallel • 2 perpendicular Covered Em-pm range: EmPm_allkins.eps high Em- region: dominated by D resonance

  30. ds ds = K sepS(Em,pm)TA = K sep S(Em,Pm)ijTp dWe dWp dE¢e dE¢p dWe dWp dE¢e dE¢p ( ) e-p c.s.. ij Extraction of the spectral function: only in PWIA possible, care for corrections later exp. c.s.: • Binning of the data (Em,pm)ij: e & p: DEm= 10-50 MeV, Dpm= 40 MeV/c Nij FWHM:0.5ns (Nij- Niju.g.) / e = Niju.g. Niju.g. L Pij 2ns Efficiency, dead time ... Luminosity phase space from M.C.

  31. pm Em • radiative corrections: redistribution of events in den (Em,pm) bins [S(Em,pm)ij]derad= S(Em,pm)ijNijnorad spectral function for M.C. needed Nijrad Iteration- process simulated events Fit to exp. spectral function pm = 490 MeV/c controlling/adjusting the spectral fct. via histograms M.C. data

  32. Monte Carlo simulation of Hall C: tracks all particles from the reaction vertex to last scintillator in the spectrometer and back • diff. reactions: (e,e’), (e,e’p), [D](e,e’p), [D](e,e’p), (e,e’K) • also on nuclei: spectral functions, cross sections • internal + external bremsstrahlung: • (e,e’p): contains interference term (Ent et al., Phys. Rev. C64 (2001) 054610) • Coulomb corrections • Rastering of the electron beam + correction • multiple scattering, energy loss • forward and backward transfer matrix, offsets • reaction kinematics  Focal plane variables  target variables • spectrometer resolution • different collimators (also slits) • different kind of targets (solid targets, cryo targets) • normalization for a given luminosity  spectra (PAW ntuples) to compare with (corrected) data

  33. Extraction of the spectral function: ds = K sepS(Em,pm)TA dWe dWp dE¢e dE¢p ò nexp(pm) = dEmS(Em, pm) ò ò ZC = 4p dpm pm2 dEmS(Em,pm) >kF only valid in PWIA: exp. W.Q.: K : kinematical factor sep : cross section for moving bound nucleon (off-shell) TA : nuclear transparency • momentum distribution: • strength = „spectroscopic factor“ in the correlated region

  34. p¢ e¢ N¢ p N k e non-PWIA contribution: distortion of the spectral function 1) FSI: (e,e¢p) + (p,p¢N) • FSI (rescattering) rekonstructed Em,pm ¹ E, k nucleus reducing FSI due to choice of kinematics: • q || k (parallel) • high q • different nuclei (C, Al, Fe, Au) corrections small (C. Barbieri) perpendicular kinematics: large correction • contribution of the D(+ higher) -resonance onset of the resonance region clearly visible solution: cut

  35. Considering 2-step process:(e,e¢p¢) + (p¢ ,p¢ N¢) 1 1 f 2 N2 p¢ N¢ p1 p 2 f 1 e- r2 r1 V (p¢ ,p¢ N¢) 1 f 2 e¢- ò ò dr1 dr2 V V 1) Calculation of rescattering process in a microscopic picture (C. Barbieri) d6sresc ò ¢ dT1 1/E¢2 = dpf dk¢ rp(r1)Kscc1 S(k,E)t(r1 r2) rN(r2)t(r2 ¥) spN propagation probability of p in medium |r1 - r2|2 (e,e¢p¢) 1 d6sresc Rescattered strength: Sresc(Em,pm) = K scc1 dpf dk¢ Corrected exp. spectral function: for each (Em,Pm) bin ~ S(Em,Pm) = Sexp(Em,Pm) - Sresc(Em,Pm)

  36. e¢ p¢ D p k e 2) Pion electroproduction p : contribution for largeEm³ mp Resonances: mainlytransverse response, qqk > 45 o, non-parallel kinematics in the acceptance of the detector setup • Simulation with MAID • Calculation taking the response functions of MAID reactions: e + n = e’ + p’ + p- e + p = e’ + p’ + po MAID: unitary isobar model resonant + non-resonant Terme Maid2000: 8 Resonanzen Maid2003: 13 with W up to 2 GeV fit to available data + input from theoy

  37. Calculation: C. Barbieri

  38. ck5k4k3_e01trec5n_cc1on_nice.agr

  39. Theoretical treatment of SRC: main problem: Divergence of realistic NN-Potentials at small distances usual Hartree-Fock approach: unbound states  use of effective potentials, usually fitted to data in IPSM region contains no SRC! • Modern Many-Body theories: • approaches to avoid divergences • Correlated Basis Function (CBF) • O. Benhar, A. Fabrocini, S. Fantoni, Nucl. Phys. A505, 267 (1989) • Greens function (2. Ord.) • H. Müther, G. Knehr, A. Polls, Phys. Rev. C52, 2955 (1995) • Self consistent Greens function (T = 2 MeV) • T. Frick, phD thesis, Uni Tübingen, 2004 • T. Frick, H. Müther, Phys. Rev. C68 (2003) 034310 • T. Frick et al., Phys. Rev. C 70, 024309 (2004)

  40. P S yV = ( fp(rij) Opij) yp.w. i>j p=1..8 Correlated Basis Function Theorie (CBF) O.Benhar et. al, Nucl.Phys.A579,493,1994 O. Benhar, A. Fabrocini, S. Fantoni, Nucl. Phys. A505, 267 (1989) Ansatz for nuclear matter occupancy 0.9 variational method correlations already contained in the ground state (0. Ord.) CBF without tensor correlations Minimizing of <E>g.s via FHNC, Var. M.C. CBF + corrections of 2. Ord. spectral function S(E,k) k (fm-1)

  41. R ò dr S(E, k)[r] r(r) 0 for finite nuclei: Local density approximation (LDA) S(E,k)[r] for different densities theory does not contain the shell structure, i.e. no LRC S(E,k) = S1p(E,k) + Scorr(E,k) dominant for k> kF k< kF E <EF Shell structure of nucleus SRC + Tensor in basis states included

  42. p1 p2 h1 h1 h2 p1 Green’s function-approach (2. Ord.) H. Müther, A. Polls, W.H. Dickhoff: Phys.Rev.C 51, 3040 (1995) Kh. Gad, H. Müther: Phys. Rev. C66, 044301 (2002) Bethe Goldstone Equ. G-Matrix (nuclear matter) N-N potential oscillator wave fct. plane wave S = + + SHF S(2p1h) S(2h1p) Self energy DS Dyson Eq.: g = gHF + gHFDSg g: 1-particle Green’s fct. S(E,k) = Im g(E,k) /p contains SRC + LRC for finite nuclei

  43. ck4k3_e01trec7nd_cc1on_ovbenhmueth_nice.agr

  44. Self consistent Greens function approach at finite temperature T. Frick, phd thesis, Uni Tübingen, 2004 T. Frick, H. Müther, Phys. Rev. C68 (2003) 034310 T. Frick et al., nucl-th/0406010 for nuclear matter: Translation invariance, no shell structure (k2/2M), no LRC  complictated calculation possible Self consistent:nucleon distribution  potential due to interacting nuclei propagator is dressed (complete spectral function, no approximation) ladder diagrams and self energy inserts in all orders solution for finite temperature only: T = 2 MeV experiment: T = 0 T < Tcrit: pair instabilities due to transition tosuperfluid phase (Superposition of NN-pairs with opposite spin + momentum) comparison with 12C: r =1/2 rNM

  45. T. Frick et al., Phys. Rev. C 70, 024309 (2004) ck4k3_e01trec7nd_cc1on_ovfrick_nice2.agr

  46. SRC Em = pm2 2 M k -k Em dependence of the spectral function: Frick et al., Phys. Rev. C 70, 024309 (2004) • missing strength at • low Em LRC ck4k3_e01trec7nd_cc1on_ ovfrickcomp410_nice2 • shift of the max. • of exp. S.F. to smaller Em simple picture: max. of the s.f. expected at

  47. < D (pm) ò nexp(pm) = dEmS(Em, pm) 40MeV in the correlated region: eep/auswert/ana_prg/output/c12_kin3/plots momdistrpara_e01theofrickup_nice_cc integration limits chosen such that 1-particle + resonance region eliminated

  48. 670 ò 230MeV/c Integrated strength in the covered Em-pm region: ò ZC = 4p dpm pm2 dEm S(Em,pm) Strength distribution in % (from CBF) 800 region used for integral 7.5 1.5 contains » half of the total strength 1.7 pm (MeV/c) 300 6.5 2 1.3 240 76 3.5 Em(MeV) 0 80 350

  49. “correlated strength” in the chosen Em-pm region: 12C exp. CBF theory G.F. 2.order selfconsistent G.F. experimental area 0.61 0.64 » 10 % 0.46 0.61 in total (correlated part) 22 % 12% »20% contribution from FSI: -4 % • » 10% of the protons in 12C at high pm, Em found • first time directly measured comparing to theory leads to conclusion that » 20 % of the protons are beyond the IPSM region Rohe et al., Phys. Rev. Lett. 93, 182501 (2004)

  50. Comparison of the results C to Al, Fe, Au • shape of the s.f.: • C, Al, Fe: quite similar • Au : contribution of (broad) resonances to the correlated region • max. of the s.f.: • » similar to C • Au : shift to larger Em (particulary at large pm) • (consequence of the resonance contribution) • correlated strength: (normalized to Z = 1) • C, Al, Fe  Au : increasing increase of the strength from 12C to 197Au: 1.7 nuc_ratiocomp_up2

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