comprehensive treatment of correlations at different energy scales in nuclei...

Green‘s functions IV 1

Comprehensive treatment of correlations at differentenergy scales in nuclei using Green’s functions

Wim DickhoffWashington University in St. Louis

CISS07 8/30/2007

Lecture 1: 8/28/07 Propagator description of single-particle motion and the link with experimental dataLecture 2: 8/29/07 From Hartree-Fock to spectroscopic factors < 1:

inclusion of long-range correlationsLecture 3: 8/29/07 Role of short-range and tensor correlations

associated with realistic interactionsLecture 4: 8/30/07 Dispersive optical model and predictions for nuclei towards

the dripline

Adv. Lecture 1: 8/30/07 Saturation problem of nuclear matter& pairing in nuclear and neutron matter

Adv. Lecture 2: 8/31/07 Quasi-particle density functional theory


Outline

• Dispersion relation for self-energy• Self-energy and nucleon optical potential• Description of elastic nucleon scattering• Empirical information on optical potentials• Subtracted dispersion relation• Discussion of time and space nonlocality• Dispersive optical model fits for 40Ca and 48Ca• Extrapolation to the dripline• Data-driven extrapolations & missing data• Inclusion of nonlocal potentials


Theory & Framework

Calculations: Include SRC and LRC as good as possible

Compare with experiment; add more physics

Determine propagator from data!?!!

Illustrated with (e,e’p) reaction below the Fermi energy

Today also elastic scattering data

Answers:What do nucleons do in the nucleusand how does their behavior change as a function of asymmetry

{Theoryhard…

{Framework“easy”


Correlations for nuclei with N very different from Z?⇒ Radioactive beam facilities

• SRC about the same between pp, np, and nn• Tensor force disappears for n when N >> Z but …• Empirically p more bound with increasing asymmetry (N-Z)/A

• Any surprises?• Ideally: quantitative predictions based on solid foundation

Some pointers: one from theory and one from experiment

Nuclei are TWO-component Fermi liquids


SCGF for isospin-polarized nuclear matterincluding SRC ⇒ momentum distribution

0.16 fm-3

0.32 fm-3neutrons

protons

asymmetry = (N-Z)/A

Frick et al.PRC71,014313(2005)

CDBonnArV18Reid

n(k=0)

SRCcan be handled


A. Gade et al., Phys. Rev. Lett. 93, 042501 (2004)

Z=18N=14

Z=8N=14

neutrons more correlated with increasing proton numberand accompanying increasing separation energy.

RS ≠ not spectroscopic factor

Reduction w.r.t. shell model

Program at MSU initiated by Gregers Hansen P. G. Hansen and J. A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003)


Dyson Equation and “experiment”Equivalent to …

!!2"2

2m#

n

N!1a"

r m#0

N+ d

" r $

m '

% '&'* (" r m," r 'm';E

n

!) #

n

N!1a"

r 'm'#0

N= E

n

! #n

N!1a"

r m#0

N

Self-energy: non-local, energy-dependent potentialWith energy dependence: spectroscopic factors < 1⇒ as observed in (e,e’p)

En

!= E

0

N! E

n

N!1Schrödinger-like equation with:

S = !n

N"1a#qh

!0

N2

=1

1"$%'

*#qh ,#qh;E( )$E

En"

αqh solution of DE at En-

DE yields

!n

N"1a!

r m!0

N=#

n

N"1 ! r m( )

!0

Na!

r m!

k

N +1=#

k

N +1 ! r m( )

!E

c,N"1a!

r m!0

N= $

c

N"1 ! r m;E( )

!0

Na!

r m!

E

c,N +1= $

c

N +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)


Does the nucleon self-energy also have animaginary part above the Fermi energy?

Loss of flux in the elastic channel

Answer: YES!


“Mahaux analysis” ⇒ Dispersive Optical Model (DOM)

C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991)

There is empirical information about the nucleon self-energy!!⇒ Optical potential to analyze elastic nucleon scattering data⇒ Extend analysis from A+1 to include structure information in A-1 ⇒ (e,e’p) data⇒ Employ dispersion relation between real and imaginary part of self-energy

FRAMEWORK FOR EXTRAPOLATIONS BASED ON EXPERIMENTAL DATA

Combined analysis of protons in 40Ca and 48CaCharity, Sobotka, & WD nucl-ex/0605026, Phys. Rev. Lett. 97, 162503 (2006)Charity, Mueller, Sobotka, & WD, Phys. Rev. C (2007) submitted

Goal: Extract asymmetry dependence ⇒ δ = (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


General dispersion relation

Re! ",#;E( ) = !"HF" ",#( ) $1

%P dE '

Im! ",#;E '( )

E $ E '+1

%P dE '

Im! ",#;E '( )

E $ E '$&

ET

$

'ET

+

&

'

Re! ",#;E0( ) = !"HF" ",#( ) $

1

%P dE '

Im! ",#;E '( )

E0$ E '

+1

%P dE '

Im! ",#;E '( )

E0$ E '$&

ET

$

'ET

+

&

'

At E0 for example the Fermi energy

Subtract

Re! ",#;E( ) = Re! ",#;E0( )

$1

%E0$ E( )P dE '

Im! ",#;E '( )

E $ E '( ) E0$ E '( )

+1

%E0$ E( )P dE '

Im! ",#;E '( )

E $ E '( ) E0$ E '( )$&

ET

$

'ET

+

&

'

“HF” from Mahaux

Note here: ImΣ < 0 for “2p1h” energies but >0 for “2h1p” energies


Diagonal approximation

Further simplification: assume no mixing between major shells

!(2) ";E( ) =1

2

"h3V p

1p2

2

E # $p1 + $p2 #$h3( ) + i%+

"p3V h

1h2

2

E # $h1 + $h2 #$p3( ) # i%h1h2p3

&p1p2h3

&'

( )

* )

+

, )

- )

Corresponding Dyson equation

G !;E( ) = GHF

!;E( ) + G !;E( )"(2) !;E( )GHF!;E( ) =

1

E #$!# "

(2)!;E( )

Assume discrete poles in Σ, then discrete solution (poles of G) for

En!

= "!

+ #(2)

!;En!( )

With residue (spectroscopic factor)

Rn!

=1

1"#$

(2)!;E( )

#EEn!


f r,ri,ai( ) = 1+ e

r!ri A1/3

ai

"

#

$ $

%

&

' '

!1

Σ(rm,r´m´;E) ⇒ U(r,E) = -V(r,E) + Vso(r) + VC(r) - iWv(E) f(r,rv,av) + 4iasWs(E) f´(r,rs,as)

V(r,E) = VHF(E) f(r,rHF,aHF) + ΔV(r,E)

ΔV(r,E) = ΔVv(E) f(r,rv,av) - 4as ΔVs(E) f´(r,rs,as)

Employed equations

!ViE( ) =

P

"W

iE '( )

1

E '#E#

1

E '#EF

$

% &

'

( ) dE '

#*

*+

Woods-Saxon form factor

“HF” includes maineffect of nonlocality⇒ k-mass

Subtracted dispersion relationequivalent to earlier slide

“Time”nonlocality⇒ E-mass


Features of simultaneous fit to 40Ca and 48Ca data

• Surface contribution assumed symmetric around EF• Represents coupling to low-lying collective states (GR)

• Volume term asymmetric w.r.t. EF taken from nuclear matter• Geometric parameters ri and ai fit but the same for both nuclei• Decay (in energy) of surface term identical also• Possible to keep volume term the same (consistent with exp) and

independent of asymmetry• HF and surface parameters different and can be extrapolated to

larger asymmetry• Surface potential stronger and narrower around EF for 48Ca


Locality and other approximations

Mahaux

VHF (! r m,! r 'm') = Re!(

! r m,! r 'm';EF )"VHF (r;E) = UHF (E) f (X HF )

with

f (XHF ) = 1+ exp XHF( )[ ]!1

XHF =r ! RHF

aHF

RHF = rHFA1/ 3

UHF (E) =UHF (EF ) + 1!mHF

*

m

"

# $

%

& ' E ! EF( )

Dispersive part: - assumed large E contribution and m*HF correlated

⇒ can use nuclear matter model and introduces asymmetry in Im part - nonlocality of Im Σ smooth ⇒ replace by local form identified with the imaginary part of the optical-model potential with volume and surface contributions


Infinite matter self-energy

Real and imaginary part of the

(retarded) self-energy

• kF = 1.35 fm-1

• T = 5 MeV

• k = 1.14 fm-1

Note differences due to

NN interaction

Asymmetry w.r.t. the Fermi energy related to phase space for p and h


Fit and predictions of n & p elastic scattering cross sections


Present fit and predictions of polarization data


Spin rotation parameter (not fitted)


Fit and predictionsOf reaction cross sections


Present fit to (e,e’p) data

radii ofbound state wave functions

spectroscopicfactors

widths of strengthdistribution


PotentialsSurface potential strengthenswith increasing asymmetryfor protons

Volume integrals

rms radii

Pairing of protons dueto pn correlations?!

Proton single-particle structure and asymmetry

Increased correlationswith increasing asymmetry!


What’s the physics? GT resonance?

NPA369,258(1981)PRC31,1161(1985)

IAS


Influence of Gamow-Teller Giant Resonanceor σ1.σ2 τ1.τ2 (& tensor force) ph interaction

Sum rule for strength: S(β+)- S(β-)=3(N-Z)

For N>Z only p affected

p

p

GT+ resonancep

p n

nn

For Z>N only n affected

n

n

n

n

pp

pGT- resonance

Related issue:Change in magic numbers with increasing asymmetry e.g. Otsuka et al., Phys. Rev. Lett. 95, 232502 (2005)


Extrapolation in δ

Naïve: p/n ⇒ D1 ⇒ ± (N-Z)/A

Cannot be extrapolated for n

Less naïve:

D2⇒ p ⇒ +(N-Z)/AD2⇒ n ⇒ 0

Emphasizes coupling to GT resonanceConsistent with n+AMo data

Need n+48Ca elastic scattering data!!!

D1D2


Extrapolationfor large Nof sp levels

Old 48Ca(p,pn) dataJ.W.Watson et al.Phys. Rev. C26,961 (1982)~ consistent with DOM


Spectroscopic factors as a function of δ

Occupation numbers

Protons more correlated with δ

Neutrons not much change

neutrons


Driplines

Proton dripline wrong by 1

Neutron dripline more complicated: 60Ca and 70Ca particle boundIntermediate isotopes unboundReef?


Outlook

• Explore the Gamow-Teller connection • link with excited states

• More experimental information from elastic nucleon scattering is important!• lots of informative experiments to be done with radioactive beams

• Neutron experiments on 48Ca and 48Ca(p,d) in the 47Ca continuum• Data-driven extrapolations to the neutron dripline

• More DOM analysis• Exact solution of the Dyson equation with nonlocal potentials (in progress)• Employ information of nucleon self-energy to generate functionals for QP-DFT = Quasi-Particle Density Functional Theory (Van Neck et al. ⇒ PRA)

DFT that includes a correct description of QP properties!!


Inclusion of VNN (or parts of it)

Self-energy

Requires one-body density matrixAlready “determined” from experimentCan take explicit realistic tensor force VTRefit to dataUseful for asymmetry dependence!

See alsoOtsuka, Matsuo, and Abe, PRL97, 162501 (2006)


Improvements in progress Replace treatment of nonlocality in terms of local equivalent but energy-dependent potential by explicitly nonlocal potential ⇒ Necessary for exact solution of Dyson equation

• Yields complete spectral density as a function of energy OK• Yields one-body density OK• Yields natural orbits OK• Yields charge density OK• Yields neutron density OK• Data for charge density can be included in fit• Data for (e,e’p) cross sections near EF can be included in fit• High-momentum components can be included (Jlab data)• E/A can be calculated/ used as constraint ⇒ TNI• NN Tensor force can be included explicitly• Generate functionals for QP-DFT


Exact solution of Dyson equation Coordinate space technique employed for atoms can be employedto solve Dyson equation including any true nonlocality (Van Neck) Yields

Sh(!,";E) = #

0

Na"† #

n

N$1 #n

N$1

n

% a! #0

N & E $ E0

N $ En

N$1( )( )spectral density (spectral function for α = β) and therefore

n(!,") = dE Sh",!;E( )

#$

% F#

& = '0

Na!† '

n

N#1 'n

N#1

n

( a" '0

N= '

0

Na!†a" '

0

N

the one-body density matrix including occupation numbers (α = β), charge density, etc. and last but not least

E0

N=1

2! T " n !,"( ) + dE E Sh !;E( )

#$

% F#

&!

'! ,"

'(

)

* *

+

,

- -

=1

2dk k

2(2 j +1)

!2k2

2mn"j k( )

0

$

& + (2 j +1) dkk2

dE E S"j k;E( )

#$

% F#

&0

$

&"j

'"j

'(

)

* *

+

,

- -

the ground state energy ⇒ useful constraints (includes also Z & N)


Charge density & High-momentum components

Only 2% high-momentum strength⇒ Modify self-energy to include more high-momentum strengthConsistent with theoretical experienceand Jlab data!


Summary•• ProtonProton sp properties sp properties inin stable closed stable closed-shell-shell nuclei understood nuclei understood ((mostlymostly))

Study of N≠Z nuclei based on DOM framework and experimental data

• Description of huge amounts of data• Sensible extrapolations to systems with large asymmetry• More data necessary to improve/pin down extrapolation• More theory

Predictions

• N≠Z p more correlated while n similar (for N>Z) and vice versa• Proton closed-shells with N>>Z ⇒ may favor pp pairing• Neutron dripline may be more complicated (reef)


Deep-inelastic neutron scattering off quantum liquids

Liquid 3He Response at 19.4 Å-1

Probe: neutronsR.T. Azuah et al., J. Low Temp. Phys. 101, 951 (1995)

Theory: Monte Carlo n(k) & FSE (ρ2) beyond IAF. Mazzanti et al., Phys. Rev. Lett. 92, 085301 (2004)

J(Y ) =1

2! 2"dk k n

Y

#

$ (k) IA result

Momentum distribution liquid 3He

S. Moroni et al., Phys. Rev. B55, 1040 (1997)Comparison of DMC, GFMC, and VMC & HNC

Y =m!

q"q

2scaling variable


protons in Ca

electrons in NeData from (e,2e)

protons in stableclosed-shell nuclei

(e,e’p) DOM

Neutron-proton asymmetry

ZFZF

weak correlations

very strong correlationsData from (n,n’)

asymmetry (BE) “knob”


New framework to do self-consistent sp theoryQuasiparticle density functional theory ⇒ QP-DFT

D. Van Neck et al., Phys. Rev. A74, 042501 (2006)

Ground-state energy and one-body density matrix fromself-consistent sp equations that extend the Kohn-Sham scheme.

Based on separating the propagator into a quasiparticle part and abackground, expressing only the latter as a functional of the density matrix.⇒ in addition yields qp energies and overlap functions

Reminder: DFT does not yield removal energies of atomsRelative deviation [%] DFT HF

He atom 1s 37.4 1.5Ne atom 2p 38.7 6.8Ar atom 3p 36.1 2.0

While ground-state energies are closer to exp in DFT than in HF

Can be developed for nuclei from DOM input!


Isospin analysis

comprehensive treatment of correlations at different energy scales in nuclei...

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