Post on 20-Aug-2018
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
Green‘s functions IV 2
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
Green‘s functions IV 3
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”
Green‘s functions IV 4
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
Green‘s functions IV 5
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
Green‘s functions IV 6
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)
Green‘s functions IV 7
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)
Green‘s functions IV 8
Does the nucleon self-energy also have animaginary part above the Fermi energy?
Loss of flux in the elastic channel
Answer: YES!
Green‘s functions IV 9
“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
Green‘s functions IV 10
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
Green‘s functions IV 11
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!
Green‘s functions IV 12
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
Green‘s functions IV 13
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
Green‘s functions IV 14
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
Green‘s functions IV 15
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
Green‘s functions IV 16
Fit and predictions of n & p elastic scattering cross sections
Green‘s functions IV 17
Present fit and predictions of polarization data
Green‘s functions IV 18
Spin rotation parameter (not fitted)
Green‘s functions IV 19
Fit and predictionsOf reaction cross sections
Green‘s functions IV 20
Present fit to (e,e’p) data
radii ofbound state wave functions
spectroscopicfactors
widths of strengthdistribution
Green‘s functions IV 21
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!
Green‘s functions IV 23
What’s the physics? GT resonance?
NPA369,258(1981)PRC31,1161(1985)
IAS
Green‘s functions IV 24
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)
Green‘s functions IV 25
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
Green‘s functions IV 26
Extrapolationfor large Nof sp levels
Old 48Ca(p,pn) dataJ.W.Watson et al.Phys. Rev. C26,961 (1982)~ consistent with DOM
Green‘s functions IV 27
Spectroscopic factors as a function of δ
Occupation numbers
Protons more correlated with δ
Neutrons not much change
neutrons
Green‘s functions IV 28
Driplines
Proton dripline wrong by 1
Neutron dripline more complicated: 60Ca and 70Ca particle boundIntermediate isotopes unboundReef?
Green‘s functions IV 29
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!!
Green‘s functions IV 30
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)
Green‘s functions IV 31
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
Green‘s functions IV 32
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)
Green‘s functions IV 33
Charge density & High-momentum components
Only 2% high-momentum strength⇒ Modify self-energy to include more high-momentum strengthConsistent with theoretical experienceand Jlab data!
Green‘s functions IV 34
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)
Green‘s functions IV 35
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
Green‘s functions IV 36
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”
Green‘s functions IV 37
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!
Green‘s functions IV 38
Isospin analysis