Nucleon correlations
Using nuclear reaction and structure data to proceed to the drip lines
• Scientific question I
• Green‘s function method / framework
• DOM: data ⇒ self-energy / propagator data-driven extrapolations ⇒ drip line
• Outlook
• Scientific question II (some other time)
• Dynamic pion exchange
• (p,2p) in inverse kinematics ⇔ (e,e’p)
Arctic FIDIPRO-
EFES Workshop 4-21-09
Wim DickhoffBob Charity
Jon MuellerBec Shane
Lee SobotkaMark BurnettJonathan MorrisSeth WaldeckerWashington University in St. LouisCarlo Barbieri, RIKENDimitri Van Neck, U GhentArnau Rios, MSU/SurreyArtur Polls, U Barcelona...
Nucleon correlations
Scientific Question I• How do the properties of protons and neutrons change as a
function of nucleon asymmetry?Properties: elastic p and n cross sections, bound single-particle properties, including removal energy, spectroscopic factors, strength distribution, one-body density matrix, E/A due to VNN.
Program• Employ framework of Green’s function method to link presently
available data to be cast in the form of the complex potentials (including their asymmetry dependence) that govern the behavior of nucleons in a large energy window: εF±200 MeV (or more) thereby linking (some) nuclear reactions with nuclear structure.
• Prediction of data for nuclei further along towards the dripline follows, allowing further improvements, etc.
Nucleon correlations
Description of the nuclear many-body problemIngredients: Nucleons interacting by “realistic interactions” Nonrelativistic many-body problemMethod: Green’s functions (Propagators) amplitudes instead of wave functions (efficient) keep track of all nucleons, including the high-momentum onesBook: Dimitri Van Neck & W.D. 2nd ed. 2008 Why: Physical insight and useful for all many-body systems Link between experiment and theory clear Can include all energy scales
Also available as a framework to analyze and interpret experimental data
Review: W.H.D. & C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004)
Nucleon correlations
Available applications:• Start from VNN (realistic) in infinite matter Rios et al.
– short-range correlations = SRC (ladders)
– fully self-consistent in nuclear and isospin asymmetric matter at finite temperature
• Start from VNN (realistic) in finite nuclei or Vee in atoms and molecules Barbieri et al.
– SRC ⇒ G-matrix (as effective interaction in model space)
– LRC : Faddeev-RPA (pp/hh and ph phonons)
– In progress: contribution of high-momenta to ground state and self-consistent Pauli principle (self-consistent model space)
• Start from DATA = Dispersive Optical Model St. Louis group
– Fit of complex potentials (self-energy) including nucleon asymmetry dependence ⇒predict more exotic systems
• Under construction: quasi-particle density functional theory StL-Ghent
(Effective)centralforce
Repulsivecore:500-600MeV
Attractivepocket:about30MeV
Bothofthesearesmallerthanweexpect.Thisisansweredbythequarkmassdependence.
The following diagram is contained.
This leads to the one pion exchangeat the large spatial separation.
N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,022001(’07).
Ishii talk at 3rd LACM-EFES-JUSTIPEN WorkshopOak Ridge 2009
Quarkmassdependenceofthecentralforce:(1) mπ=380MeV: Nconf=2034 [28 exceptional configurations have been removed](2) mπ=529MeV: Nconf=2000(3) mπ=731MeV: Nconf=1000
1S0
Strongquarkmassdependenceisfound.
Inthelightquarkmassregion,
therepulsivecoregrowsrapidly.
attractivepocketisenhancedmildly.
ThelatticeQCDcalculationatlightquarkmassregionisquiteimportant.
Ishii talk at 3rd LACM-EFES-JUSTIPEN WorkshopOak Ridge 2009
Nucleon correlations
Correlations for nuclei with N very different from Z?⇒ Radioactive beam facilities
• Short-Range Correlation ~ the same between pp, np, and nn
• Tensor force disappears for n when N >> Z but the opposite for p in matter (volume effect)
• Role of the surface?
Some pointers: from theory and experiment
Nuclei are TWO-component Fermi liquids
Nucleon correlations
Self-Consistent-Green’s-Function calculation for isospin-polarized nuclear matterincluding SRC ⇒ momentum distribution ⇒ n(k=0)
0.16 fm-3
T=10 MeVneutrons
protons
asymmetry = (N-Z)/A
Frick, Rios et al.PRC71,014313(2005)
CDBonnArV18Reid
n(k=
0)
SRCcan be handled!
{
small uncertainty
Nucleon correlations
Temperature vs. correlations Rios, Polls, WD
arXiv:0904.2183
Nucleon correlations
Fixed by phase shifts?!
arXiv:0904.2183
Nucleon correlations
Gade et al. Phys Rev C77, 044396 (2008)
⇒ Spectroscopic factors become very small; way too small?
RS ≠ not spectroscopic factor
Reduction w.r.t. shell model
neutrons more correlated with increasing proton numberand accompanying increasing separation energy & vice versa
Nucleon correlations
Faddeev-RPA ⇒ Barbieri, WD 2009
Trend similar but magnitude ...
arXiv:0901.1920
Nucleon correlations
History for atoms
Nucleon correlations
F-RPA application to Neon atomC. Barbieri, D. Van Neck, and WD, Phys. Rev. A76, 052503 (2007)
Spectroscopic factors in brackets
*
* Consistent with results from Heidelberg group
One of the developers of the no-core shell model:“F-RPA/Green’s function method maybe the only way to do heavy nuclei”
Nucleon correlations
Periodic table: Ne ionization energy
HF: 23.12 eV
Σ(2): 20.32 eVF-RPA: 21.73 eVExp: 21.57 eV
Nucleon correlations
Green’s functions as a framework:DOM = Dispersive Optical Model
Vehicle for data-driven extrapolations / predictions to the dripline
Green’s function formulation ⇒ “Mahaux analysis” Goal: extract “propagator”/”self-energy” from data
C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991)
Nucleon correlations
FRAMEWORK FOR EXTRAPOLATIONS BASED ON EXPERIMENTAL DATA
Combined analysis of protons/neutrons in 40Ca and 48CaCharity, Sobotka, & WD, Phys. Rev. Lett. 97, 162503 (2006)Charity, Mueller, Sobotka, & WD, Phys. Rev. C76, 044314 (2007).
Goal: Extract asymmetry dependence ⇒ δ/β = (N - Z)/A ⇒ Predict proton properties at large asymmetry ⇒ 60Ca ⇒ Predict(?) neutron properties … the dripline based on data!
Large energy window (> 200 MeV)
Recent extension of DOM
Nucleon correlations
Dyson Equation and “experiment”Equivalent to …
�
− 2∇2
2mΨn
N−1 a r m Ψ0N + d
r ∫
m '∑ 'Σ'* ( r m, r 'm';En
−) ΨnN−1 a r 'm' Ψ0
N = En− Ψn
N−1 a r m Ψ0N
Self-energy: non-local, energy-dependent potentialWith energy dependence: spectroscopic factors < 1 ⇒ as observed in (e,e’p) �
En− = E0
N − EnN−1
Schrödinger-like equation with:
�
S = ΨnN−1 aαqh
Ψ0N 2
= 1
1−∂Σ'* αqh ,αqh;E( )
∂EEn
−
αqh solution of DE at En-
DE yields
�
ΨnN−1 a r m Ψ0
N =ψnN−1 r m( )
Ψ0N a r m Ψk
N +1 =ψkN +1 r m( )
ΨEc,N−1 a r m Ψ0
N = χcN−1 r m;E( )
Ψ0N a r m ΨE
c,N +1 = χcN +1 r m;E( )
Bound states in N-1 Bound states in N+1 Scattering states in N-1
Elastic scattering in N+1
Elastic scattering wave function for (p,p) or (n,n)
Nucleon correlations
Dispersion relations (nuclear matter or nuclei)
Imaginary part both above and below εF
Above εF:required by elastic nucleon scattering data
Below εF:required by e.g. (e,e’p) and (p,2p) data
Consequences:Fermi sea is depletedand there is occupation of“particle-like” states (high-momenta e.g.)
Nucleon correlations
Fit and predictions of n & p elastic scattering cross sections
Above εF
Nucleon correlations
Mougey et al., Nucl. Phys. A335, 35 (1980) 16O(e,e’p)
Energy
Moment
um
∝ Cross section
Nucleon correlations
40Ca spectral functionRecent theoretical development: nonlocal self-energy below the Fermi energyVan Neck, Charity, Sobotka,WD 2009
Old (p,2p) data from Liverpool
Below εF
Nucleon correlations
HIMAC data: Kobayashi et al. Nucl. Phys. A805, 431c (2008)
Nucleon correlations
What about neutrons?• Almost no elastic scattering data on 48Ca until
recently!• Most pressing information• Also very relevant for future applications of
transfer reactions involving deuterons
Protons:Surface absorption increaseswith increasing asymmetry
Nucleon correlations
Extrapolation in δNaïve: p/n ⇒ D1 ⇒ ± (N-Z)/A
Cannot be extrapolated for n
Less naïve:
D2⇒ p ⇒ +(N-Z)/A
D2⇒ n ⇒ 0Emphasizes coupling to GT resonanceConsistent with n+AMo data
Need n+48Ca elastic scattering data!!!Performed recently at TUNL (Sobotka & Charity)
D1D2
Nucleon correlations
WashU exp: Sobotka, Shane, Mueller, Charity at TUNL
Nucleon correlations
Prelim
inary$$$
Nucleon correlations
Total neutron cross section 48Ca vs. 40Ca
Prelim
inary
WashU exp: Sobotka, Shane, Mueller, Charity at LANL
Nucleon correlations
Preliminary fit including n data
Correlations at large asymmetry
Fragmentation of strength E. Quint, Ph.D.thesis NIKHEF, 1988
Spectroscopic factor S(2s1/2) =0.65I. Sick & P de Witt Huberts,
Comm. Nucl. Part. Phys.,20, 177 (1991)
Intermediatefragmentation
Strong fragmentation of deeply-bound strength ⇒ self-energy has significant imaginary part!
Towards a “global” DOM
208Pb
DOM spectroscopic factors
Correlations at large asymmetry
M. van Batenburg & L. Lapikás from 208Pb (e,e´p) 207Tl NIKHEF in preparation
Up to 100 MeV missing energy and270 MeV/c missing momentum
Covers the whole mean-field domainfor the FIRST time!!
Occupation of deeply-bound proton levels from EXPERIMENT
Confirms predictions for depletion
SRCLRC
n(0) ⇒ 0.85 Reid 0.87 Argonne V18 0.89 CDBonn
Nuclear matter
Nucleon correlations
Project DOM-DRIP How to do it:Experiment
✦ Elastic scattering n+48Ca, total neutron cross sections (& other N ≠ Z nuclei)
✦ Elastic scattering of radioactive beams off p
✦ Heavy-on knock-out reactions (DOM + ...)
✦ (p,2p & pn) inverse kinematics (DOM + NN T-matrix) to match (e,e’p)
✦ Transfer reactions (DOM + Johnson, Tostevin, Nunes approach)
Extend DOM (⇒ more nuclei)
๏ Nonlocality (Van Neck)
๏ Isospin (Waldecker)
๏ Include charge density data
๏ Include (e,e’p) data including high-momentum JLab results
๏ Include higher energy elastic scattering data ⇒ relativity (Piekarewicz)
Theory
✴ Calculate self-energy microscopically; include tensor force explicitly (Barbieri)
✴ Input for quasiparticle DFT (Burnett, Van Neck)
Nucleon correlations
ConclusionProject DOM-DRIP
• suggests many new experiments (some in the pipeline)
• allows data-driven extrapolation to the drip line
• many reactions can be included in DOM framework including (n,γ)
• room for lots of theoretical input
• room for lots of improvements and collaborations