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Correlated nucleons in k- and r-space
Ingo Sick
historical development of nuclear physics
strongly influenced by mean-field ideas
existence of Quasi-Particle orbits
with fitted effective interactions explain many features of nuclei
can explain many features of nuclei
but: fitted parameters, limited to selected observables
more fundamental approach: start from N-N interaction
can do for nuclear matter, light nuclei
Faddeev, Hyperspherical, Variational MC for A<<
Bethe-Bruckner-Goldstone for NM
Correlated Basis Function (CBF) theory for NM
applicable to a priori all nuclei
decisive at higher densities as e.g. in stars, or for extreme N/Z
main difference
account for short-range N-N correlations
scattering of N to orbits E ≫ EF , k ≫ kFpartial depletion for E < EF
CBF ideal to expose correlations
appear explicitly as variational functions f(rij) in wave function
|N) = G|N ], G = S∏
j>i
F (i, j), F (i, j) =∑
nfn(rij)O
n(i, j)
Effect of correlations
components of potential f(rij)
correlation hole for some components, short-range enhancements for others
Consequences for NM momentum distribution and spectral function S(k,E):
• correlations give strength at both large k and large E
• strength very spread out, hard to identify experimentally
• correlated N have ∼20% probability (NM),
but give 37% of removal energy
47% of kinetic energy
(CBF calculation of Benhar, Fabrocini, Fantoni 1989)
→ mean-field approach cannot work
exception(?): differences of energies, spect. factors
insight for time being lost on shell-model community
calculations of ever increasing sophistication
ignoring 20% of nucleons
e.g. review Talmi, ”50 years of shell model”, 275pages
not one word on 20%, 37%, 47%
correlated N even more crucial for:
2N-dependent processes, MEC
enhancement of integrated transverse (e,e’) strength
NM study of A. Fabrocini
sum rule in 3He, 4He, J.Carlson et al.
effect of MEC with correlations 8 times larger
Analogous studies of liquid Helium
calculation easier, no spin-dependence
Lennard-Jones potential r−12 − r−6 even more repulsive at small r
→ even stronger correlations
But
find quasi-particle states for k < kFwith much reduced occupation
occupation of MF states ∼ 30%
= size of discontinuity at kFrest moved to k > kF mainly
”depopulation” of MF states
= main consequence for MF
Shown by Moroni et al.
via calculations for L3He and L4He
depopulation similar for bosonic/fermionic systems
mainly consequence of short-range VNN , not Pauli principle
Finite systems
more complicated
”occupation” requires concept of ”orbit”
not a priori obvious which one:
mean field, overlap, natural, ..?
Studies of L3He drops
Variational Monte Carlo
drops with A, A–1 atoms
deduce difference
find
for e.g. 3s-state
quasi-hole orbital close to MF orbitals +LDA ( )
ψQH = ψMF
√
z(ρ(r)) z=renormalization
= quantities observable in transfer, (e,e), (e,e’p)
main message
in nuclei and LHe find orbits ∼ quasi-particle states
R(r), R(k) ± as given by mean-field models
observed in transitions to E < 10 MeV
but
single particle states have partial occupation
rest of strength moved to very large k,E
experimental observation of depopulation
long in coming, no occupation numbers from transfer reactions
measure large-r properties of nucleon wave functions
nuclear interior not accessible
no integral information
large k’s, E’s not accessible
change: initiated by (e,e) and (e,e′p): radial sensitivity + quantitative interpretation
• magnetic scattering, measure R(r) of valence orbits
• density differences of isotones, measure R(r) in nuclear interior
• (e,e′p) reactions, measure R(k)
sensitivity of probes
Early evidence on partial occupation
inelastic (e,e), high multipolarity
little affected by configuration mixing
sensitive to interior
relative to MF reduced by factor ∼2
Systematic studies of (e,e’p)
more sensitive to nuclear interior
measure n(k) and absolute strength
experiments done mainly at NIKHEF
Today: have convincing data that QP strength z<1
208Pb(e,e′p)207Tl SRC
LRC
occupation of ∼ 0.7 (outer shells)
∼ 0.8 without surface + LRC effects
(note: peculiar behavior of 12C, see below)
striking example for MF R(r) despite depopulation:
3s-state in nuclei206Pb−205 T l
(e,e) sensitive to interior
radial distribution ∼ MF orbit
but: occupation = 0.7
unsatisfactory situation of experiment:
have identified missing strength
have fair theoretical understanding
have not seen correlated strength
measured: (1–correlated strength) → large uncertainty
blows up uncertainty by factor of 5
past attempts to identify strength at large k
• reactions of type (x,p)
low momentum transfer from x, observe high momentum p
e.g. (γ,p), (p,p) with high-momentum backward p
problem: Amado+Woloshyn, 1977
• in limit q→0 FSI cancels IA term from high-k component
no quantitative interpretation possible
(applies also to (p,2p), .. )
• (e,e) at large q, low ω
~k
~q~k + ~q
idea: small ω ∼ (~k + ~q)2/2m, large ~q → ~k ∼ −~q large
problem: FSI
• see Benhar, Fabrocini, Fantoni,...(1991)
provide first treatment of FSI using Glauber
but: not entirely satisfactory, still in the works
for either case: cannot address large E, where main strength is located
Best tool to measure strength at large k and E:
(e,e′p) at large q
complications:
• must look at large E
not large k and small E as done by ± all existing experiments
• strength spread out over 100-200MeV
small in given (k,E)-bin
very hard to observe
• reaction mechanism (see diagram)
p rescatters, reappears at lower kp′
simulates large missing energy E, large k
covers small genuine strength
• similar for (e, e′∆), with ∆ → p+ π(undetected)
study of all available data
compare experimental dσ/dΩdω to IA using realistic S(k,E)
n(k)MF + ScorrNM(ρ) in LDA
look if data ≃ theory, or >> theory
find
• most experiments give σexp ≫ σIA• standard perpendicular kinematics worst, // kinematics best
studies of kinematics of rescattering processes:
understand how (p, p′N) and (e, e′pπ) move strength
identify optimal kinematics: parallel (standard: perpendicular!)
confirmed by MC calculations of Barbieri (see below)
Perpendicular kinematics
reasons for choice
experimentally convenient
need to change only angle of p-spectrometer
can keep q and spectrometer-momenta constant
convenient for DWIA-analysis
± same optical potential for all kinematics
However
maximizes multi-step contribution
maximizes MEC
Optical potential anyway wrong approach
must describe protons that have inelastically interacted
and not been ”swallowed up” by Im(V)
need Glauber-type approach to describe dominating (p,p’N)
Detector stack:
2 drift chambers 4 scintillator planes 1 Cerenkov
HMS
SOS
Q Q Q D
D
D
e
e′
p′
3.2 GeV
0.85-1.7 GeV/c
2.05-2.75 GeV/c
Target spectrometer
(e,e’p) experiment
JLab hall C:
Daniela Rohe et al.
Can achieve with JLab:
large q → low FSI, treatable with Glauber
acceptable true/accidental ratio
despite unfavorable (//) kinematics
Tests
in single-particle region, kinematics with same Ep′ as production runs
use: T=0.6... (Benhar+Pieper), integrate over E <80MeV
find: occupation agrees with CBF S(k,E)
Results for correlated region
0.1 0.2 0.3 0.4E
m (GeV)
1e-13
1e-12
1e-11
1e-10
S(E
m,p
m)
[MeV
-4 s
r-1]
0.1700.2100.2500.2900.3300.3700.4100.4500.4900.5300.1700.2100.5700.6100.650
Spectral function for C using ccparallel: kin3, kin4, kin5
pm
(GeV/c)Em
= p2
m2 M
p
main observation on E-dependence
maximum of S(k,E) of theories at too large E
understood by recent calculation of Muther+Polls?
selfconsistent GF, ladder approximation, finite T
momentum dependence
0.2 0.4 0.6p
m [GeV/c]
10-3
10-2
10-1
n(p m
) [f
m3 s
r-1]
parallel
CBF theoryGreens function approachexp. using cc1(a)exp. using cc
→ theory and experiment ± agree
how about standard perpendicular kinematics?
used for overwhelming majority of experiments
find
(distorted) S(k,E) >> S from parallel kinematics
ckp1p2_e01trec16n_cc1on_nice.agr
confirmed by calculation of Barbieri
includes multistep (p,p′N) via Glauber
0 0.1 0.2 0.3 0.4 0.5E
m (GeV)
1e-13
1e-12
1e-11
1e-10
S(E
m,p
m)
[MeV
-4 s
r-1]
input S(k,E) distorted S(E
m,p
m) in parallel kin.
distorted S(Em
,pm
) in perpendicular kin.
Parallel vs. perpendicular kinematics for 12
C
330 MeV/c
410 MeV/c
490 MeV/c
570 MeV/c
C. Barbieri,preliminary
lesson
in parallel kinematics multi-step = manageable correction
perpendicular kinematics useful only to check multistep calculations
How much correlated strength??
cannot integrate over entire correlated region
∆-excitation and QP strength cover part of correlated strength
integrate over ’clean’ region, both data and theory
80%
250
80 300
1.5%
5%
700
0
4.5%
8% 1%
Em(MeV)
pm(MeV/c)used region
correlated protons in 12C used total
integral over S from experiment 0.59
integral over S from CBF 0.64 1.32
integral over S from SGGF 0.61 1.27
→ good agreement
→ can believe total from theory
→ 20%, integrated over k,E
heavier nuclei
experiment performed for C, Al, Fe, Au
interest in A>>
→ nuclear matter
ratio to C of correlated strength
find ratio ∼ 1 as expected
0 50 100 150 200mass number A
11.21.41.61.8
22.22.4
spec
tral
rat
io
parallel kinematicsperpendicular
Ratio Al, Fe, Au to C spectral functionintegrated over correlated region
∆-resonance!
enhancement for Au
not yet understood
consequence of n-p correlations as N > Z??, rescattering ??
would like to get S(k,E) for N6=Z
overall
• have now experiment with optimized kinematics
to minimize multi-step contributions
• have identified strength at large k,E
• theory produces S(k,E) with ± correct strength
SGFT, CBF+LDA
E-dependence does not entirely agree
strength at too low E
enhancement for large A not understood
• would want kinematics more strictly parallel
rather restrictive kinematics
unfavorable true/accidental ratio
but it’s worth it!
for details:
see habilitation work of Daniela Rohe
Orthogonal look:
where correlated strength in r-space?
motivation: difficulties with QP-R(r)
• QP radial wave function fitted to ρ(r)
poorly explain F(q) of QP-dominated transitions
• QP wave functions poorly explain ρ(r) at small r
reason: ρ(r) contains correlated contribution
correlated radial distribution presumably 6= QP distribution
→ question: radial distribution of correlated strength = ?
approach:
1. study via selfconsistent Green’s function theory SGFT
H. Muther
2. determine from (e,e) and (e,e′p)
ρ(r) = sum of QP and correlated density
(e,e′p) gives QP piece in k-space
transform to r-space
difference in r-space yields ρcorr(r) = ρ(r) − ρQP (r)
1. S(k,E) from Green’s function method (Muther, Polls, ..)
split S into QP plus correlated piece
ρ(r) =∑
lj
SQPlj (r, r) +∑
lj
∫ ∞
ε2h1p
dE Scontlj (r, r;E)
= ρQP (r) + ρcorr(r) ,
CD-Bonn NN interaction → 1.0 correlated protons (low?)
observations
ρcorr concentrated much more toward small r
does not contribute at large r
there tail of QP dominates completely
ρcorr at small r despite contributions of large l
31% l=0, 37% l=1, rest large l
large E of states pulls R(r) to small r
at small r ρcorr contributes ∼30% of ρ(r)
explains failure of QP wave functions
2. ρcorr from (e,e)+(e,e′p) data
ρcorr(r) = ρ(r)point − FT (ρQP (k))
point density of C
have very precise (e,e) data up to large q
have µ-X-ray data
do modelindependent analysis (SOG)
→ charge density with small δρ
unfold nucleon size to get point density
QP wave functions from (e,e′p)
extensive set of (e,e′p) data
• low-q from NIKHEF, Saclay
analyzed with DWBA
optical potentials from (p,p)
• high-q data from SLAC, JLAB
analyzed with theoretical transparencies
confirmed by data
comprehensive analysis: L. Lapikas et al, PRC61, 64325
problem
analyses apparently not consistent
low q: 3.4 uncorrelated protons
low as compared to other nuclei
high-q: 5 - 5.6 uncorrelated protons
discrepancy
embarrassing for practitioners of (e,e′p)!
deathblow to (e,e′p) as quantitative tool
question: which is true occupation? is q-dependent??
issues: quality optical potentials, role MEC’s,
coupled-channel effects, value of T
one reason for difference obvious: sloppy interpretation of data
low-q data:
E ≤ 50MeV , k ≤ 180MeV/c
high-q data:
E ≤ 80MeV , k ≤ 300MeV/c
integrate over part of correlated strength, must remove before comparing!
correction of high-q result
use analysis of QP region of Rohe et al.
agrees with previous SLAC/JLAB data, uses most reliable T
calculate correlated contribution using theory, remove
result:
high-q: 4.5 uncorrelated protons
low-q: 3.4 uncorrelated protons
→ discrepancy much reduced
but: still larger than desirable, larger than believed ±10%
my choice: (corrected) high-q result
low-q have coupled-channel effects
Steenhoven: 20% (12C=soft nucleus)
low-q have significant MEC effects
Boffi: up to 20%
low-q would imply unrealistic correlated strength
would disagree with Rohe et al.
use high-q result
choice confirmed by consistency check, see below
further adjustment needed
1s fitted to data E < 50MeV
this region contains already correlated strength
n(k)corr falls less quickly than n(k)QPmust correct shape of Lapikas 1s n(k)
correction of shape:
• can do using QP and correlated S(k,E) from theory
• can analyze Rohe data with QP+correlated parts
find same result:
n(k) compressed by 11%
ρcorr from (e,e)+(e,e′p)
start with point density
subtract QP contribution, Fourier-transformed
shape+occupation from high-q (corrected)
result
observations
ρcorr concentrated toward small r
as was seen in theory
ρcorr gives ∼30% contribution at small r
explains failure of QP models
reasonable agreement with theory
(uncertainty of ρcorr ∼20%)
in exp. density perhaps more l > 1 strength
important consistency check: large r
perfect agreement ρQP ... ρpointshould occur as ρcorr cannot contribute
confirms choice of (high-q) QP strength
large-r = the region where MF ± OK
conclusions of r-space study
shape of ρcorr differs strongly from shape of ρQP
ρcorr gives 30% contribution in nuclear interior
explains failure of QP models, cannot be ’compensated’ using eeff , etc.
reasonable agreement with Green’s function theory
Conclusions
have finally data on correlated strength
... some 15 years after CBF calculation
± agrees with modern many-body theories
... which were amazingly good!
for quantitative understanding of nuclei:
must go beyond mean-field, include correlated nucleons
Collaborators:
JLab hall-C, D. Rohe
C. Barbieri, O. Benhar, H. Muther