Folie 1

1
The (e,ep) 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
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)

• 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
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)
kF µ width 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
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
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 for Coulomb
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 k2 S(E, k)
single particle state a
number of nucleons in shell
12
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

• 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
spectroscopic factor strength of the
transitionYaA YbA-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 Dna ra Drcp
• n3S1/2 (206) - n3S1/2 (205) r(3S1/2)
  Drcp

Dn 0.64 0.06
18

• 2) input
• relative spectroscopic factors (from
  (e,ep)-experiment)
• R(205/208) Z3S1/2(205)/ Z3S1/2(208)
• n3S1/2(205)/ n3S1/2(208)

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,ep)
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 2j1 (Pick-up and
stripping reaction), assuming IPSM
But sumrule not fulfilled for Em lt 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
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 ltTgt ltVgt 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 lt kF single-particle contribution dominates k
kF SRC already dominates for E gt 50 MeV k gt
kF single-particle negligible
consequence search for SRC at large E, k method
(e,ep)-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)
• Greens function approach (2nd order)
• H. Müther, G. Knehr, A. Polls, Phys. Rev.
  C52, 2955 (1995)
• Self--consistent Greens 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 lt pm lt 300 MeV/c
k4que01_ipsmben250.eps
data on 12C IPSM0.85 CBF
Q2 1.5 (GeV/c)2
radiation tail
Spectral function containing SRC good agreement
with data
27
(e,ep)-reaction coincidence experiment measured
values momentum, angles
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
p
0.1-1.75 GeV/c
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 HMS
SOS
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

• 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
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
FWHM0.5ns
(Nij - Niju.g.) / e

Niju.g.
Niju.g.
L Pij
2ns
Efficiency, dead time ...
Luminosity
phase space from M.C.
31

• radiative corrections

redistribution of events in den (Em,pm) bins
S(Em,pm)ijderad S(Em,pm)ij Nijnorad
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,ep), D(e,ep),
  D(e,ep), (e,eK)
• also on nuclei spectral functions, cross
  sections
• internal external bremsstrahlung
• (e,ep) 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
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
non-PWIA contribution
distortion of the spectral function
1) FSI (e,ep) (p,pN)

• 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
1) Calculation of rescattering process in a
microscopic picture
(C. Barbieri)
d6sresc
ò

dT1
1/E2

dpf dk
rp(r1) K scc1 S(k,E) t(r1 r2) rN(r2)
t(r2 )
spN
propagation probability of p in medium
r1 - r22
(e,ep)
1
d6sresc
Sresc(Em,pm)
Rescattered strength
K scc1
dpf dk
Corrected exp. spectral function for each
(Em,Pm) bin

S(Em,Pm) Sexp(Em,Pm) - Sresc(Em,Pm)
36
2) Pion electroproduction
p contribution for large Em ³ mp
Resonances mainly transverse response,
qqk gt 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
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 ltEgtg.s via FHNC, Var. M.C.
CBF
corrections of 2. Ord.
spectral function S(E,k)
k (fm-1)
41
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 kgt kF
klt kF E ltEF
Shell structure of nucleus
SRC Tensor in basis states included
42
Greens 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
plane wave
oscillator wave fct.
SHF
S(2p1h)
S(2h1p)
S
Self energy
DS
Dyson Eq. g gHF gHF DS g g
1-particle Greens 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 lt Tcrit pair instabilities
due to transition to superfluid 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
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
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
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
exp. CBF theory G.F. 2.order
selfconsistent G.F.
12C
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
51
nuccomp_para
52
Other reaction mechanism?

• contribution of FSI in Au 10

• Meson exchange currents (MEC)

• Isobaric currents

im non-relativistic limit only transversal
components i.e. mainly perpendicular kinematics
C. Barbieri work in progress
53

• asymmetric nuclear matter

number of neutrons gt number of protons
for Au (118 79) a 0.2
20-30 more correlated strength for p-spectral
function
reason larger contribution of the tensor force
in n-p interaction
occupancy
Frick et al., PRC 71, 014313 (2005)
54
Inclusive electron scattering (e,e) xB gt 1
quasielastic scattering on the nucleons ?
probing its momentum distribution
large xB ?? large pm
Bjorken variable
Q2
xB
2Mw
55
(No Transcript)
56
momentum distribution at high pm is similar
? Ratios of cross section from two nuclei should
scale at corresponding (Q2, xB) combination
57
Compare with simple nucleus with known momentum
distribution
3He (pm gt 250 MeV/c) 0.0800.016 (2N-SRC)
0.00180.0006
(3N-SRC)
M. Sargsian et al., PRC 71, 044614 (2005)
In xB gt 2 region mainly 2 - Body configuration
(M.Sargsian et al., PRC 71, 044615 (2005))
58
K.Egiyan et al., PRC68, 014313 (2003)
2N-SRC
Scaling onset xB 1.5, Q2gt1.5 GeV2
?? pm _at_ 250 MeV/c
3N-SRC
taking the calculated 2N and 3N SRC strength in
3He, gives absolute numbers, occupancies
Note (23 SRC) Fe
1.2
(23 SRC) C
Single particle () 2N SRC () 3N SRC ()
56Fe 76 0.2 4.7 23.0 0.2 4.7 0.79 0.03 0.25
12C 80 02 4.1 19.3 0.2 4.1 0.55 0.03 0.18
4He 86 0.2 3.3 15.4 0.2 3.3 0.42 0.02 0.14
3He 92 1.6 8.0 1.6 0.18 0.06
2H 96 0.7 4.0 0.7 -----
Fractions
Nucleus
59
Information from inclusive electron scattering
nuclear matter (NM)
at D, heavy nuclei
CBF theory
Compare d, NM momentum distribution
(NM/D)CBF 5.0
Benhar, Fabrocini, Fantoni, Sick, PLB 343 (1995)
47
197 MeV/c
Correlated Glauber Theorie
large sensitivity to 2-N correlations
(NM/D)exp 5.5 0.8
a2(NM)
only half of the correlated strength at k large
Pandharipande,Sick, deWitts Huberts, RMP 69
(1997) 981
xB
60
(e,ep)-experiment access to the interior of the
nucleus
charge density distribution in 12C
measured
IPSM- part
correlated part
exp.
theory
Müther, Sick PRC 70, 041301(R) (2004)
61
Settings 1-5 parallel kinematics (pm lt 400
MeV/c) Settings 6,7 perpendicular kinematics (pm
gt 400 MeV/c)
K.I. Blomqvist et al., Phys. Lett. B 344, 85
(1995).
62
16O(e,ep) at Q2 0.8 (GeV/c)2 in Hall A
parallel kinematics also LT separation, even at
moderate pm150MeV/c data exceed the DWIA
calculation of Kelly
calculation Ryckebusch describes only half of
the yield at high Em
N. Liyanage et al., PRL 86, 5670 (2001)
63
Other methods to look for SRC
1-particle region
only central SRC
dominated by tensor correlation

• SRC ltlt 1-particle contribution
• better selection needed
• detection of the correlated pair(.,.pp),(.,.np)
• suppression of D, MEC, FSI in superparallel
  kinematics

• smaller cross section
• additional detector

• choice of one particular shell (L0), requires
  good energy resolution

• LT-separation?
• sensitive to non-PWIA contributions sL klein,
  sT groß
• might help to get info about reaction mechansm

not only (e,e...) g only transverse
response Watts et
al., PRC 62 (2000) 014616
p strong interaction, less
radiation corrections
Tang et al., PRL 90 (2003) 042301
64
12C(e,epp), Em lt 70 MeV
0.6fm HC
MAMI
MAMI
G-Matrix
D
VMC
FHNC
Blomqvist et al., PLB 421 (1998) 71
3He(e,epp)n
superparallel
NIKHEF
MEC
Faddeev calculation Bochum-Krakow group
Groep et al., PRC 63 (2001) 14005
p3
65
Experiment E97-111 in Hall A (Jlab)
4He(e,ep) in parallel and perpendicular
kinematics 0 ground state
only central SRC
Theoretical prediction
Experimental result
Location of the minimums sensitive to SRC
parallel kinematics

• no minimum !
• ? FSI
• MECIC
• contribution small
• good
• agreement with
• full theory

measurement
n(k) (fm3)
full
FSI
PWIA
0 2 4 6 8
k(fm-1)
calculation J.-M. Laget, Analysis B. Reitz
S. Tadokoro et al., Prog. Theor, Phys. 79
(1987)732
66
Photoabsorption 12C(g,pn) wide angular range
compared to momentum distribution using harmonic
oscillator wave functions
calculation Orlandini, Sarra
Watts et al., PRC 62, 14616 (2003)
67
12C(p,ppn)
BNL
c.m. momentum of the p-n pair is zero
Tang et al., PRL 90 (2003) 042301
68
7Li(e,e'p) Spectroscopic Strength
7Li
spectroscopic strength 0 2
0 2 --------------------------------------
------- Exp 0.42(4) 0.16(2) 0.58(5) -----------
---------------------------------- VMC 0.41
0.19 0.60 MFT 0.59 0.40 0.99 -----------------
----------------------------
69
Picture of SRC and LRC
everything understood?
W.H. Dickhoff, C. Barbieri, Prog.Part.Nucl.Phys.
52 (2004) 377
70
Summary spectroscopic factors
71
q dependence of spectroscopic factors for
16O(e,ep) ?
q ? 0.45 GeV/c (NIKHEF) q ? 1.0 GeV/c (JLab)
Z62
Z70
1p1/2 Z0.71,0.72
1p3/2 Z0.71,0.67
Difference of 5 10 (relativistic
description?) or problem with DWIA ?
H. Gao et al., PRL 84, 3265 (2000)