/ee/ectq.tex

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

explains light nuclei amazingly well

.... for light nuclei where calculations can be done

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