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ModernNuclear Structure Theory
From Realistic Interactions to the Nuclear Chart
Robert RothInstitut fur Kernphysik
Technische Universitat Darmstadt
1
Overview
Motivation
Modern Effective Interactions
• Correlations & Unitary Correlation Operator Method
Applications
• No Core Shell Model
• Hartree-Fock & Beyond
• Fermionic Molecular Dynamics
2
Nuclear Structure in the 21st Century
RISING, AGATA,REX-ISOLDE, ...
NUSTAR@ FAIR
NuclearAstrophysics
nuclei far-offstability
hyper-nuclei,...
reliablenuclear structure theory
for exotic nuclei
bridging betweenlow-energy QCD and
nuclear structure theory
3
Modern Nuclear Structure Theory
Nuclear Structure
Low-Energy QCD
RealisticNN-Potentials
ChiralInteractions
ab initioApproaches
EffectiveInteractions
Many-BodyMethods
DensityFunctional
Models
4
Realistic NN-Potentials
QCD motivated• symmetries, meson-exchange picture
• chiral effective field theory
short-range phenomenology• short-range parametrisation or contact terms
experimental two-body data• scattering phase-shifts & deuteron properties
reproduced with high precision
supplementary three-nucleon force• adjusted to spectra of light nuclei
Argonne V18
CD Bonn
Nijmegen I/II
Chiral N3LO
Argonne V18 +Illinois 2
Chiral N3LO +N2LO
5
Argonne V18 Potential
v(r) v(r) ~L2
v(r) S12 v(r) (~L · ~S) v(r) (~L · ~S)2
-100
0
100
.
[MeV
]
0 1 2r [fm]
-100
0
100
.
[MeV
]
0 1 2r [fm]
0 1 2r [fm]
(S, T )
(1, 0)(1, 1)(0, 0)(0, 1)
6
Ab initio Methods: GFMC
-100
-80
-60
-40
-20
Ene
rgy
(MeV
)
AV18
IL2 Exp
0+
4He
2+1+0+
0+2+
6He
1+
1+3+2+1+
6Li
5/2−5/2−
3/2−1/2−7/2−5/2−5/2−
7/2−
3/2−1/2−
7Li
1+0+2+
0+2+
8He1+
0+2+4+2+1+
3+4+
0+
2+
3+2+
8Be
9/2−
3/2−5/2−1/2−
7/2−
(3/2−)
(5/2−)
9Be
3+
4+
0+2+2+
(4+)
10Be 3+1+
2+
4+
1+
3+1+2+
10B
3+
1+
2+
4+
1+
3+
1+
2+
0+
12C
Argonne v18With Illinois-2
GFMC Calculations22 June 2004
12C results are preliminary.[S. Pieper, private comm.]
“exact” numericalsolution of interactingA-nucleon problem
75000CPU-hours
7
Modern Nuclear Structure Theory
Nuclear Structure
Low-Energy QCD
RealisticNN-Potentials
ChiralInteractions
ab initioApproaches
EffectiveInteractions
Many-BodyMethods
DensityFunctional
Models
8
Modern Nuclear Structure Theory
Nuclear Structure
Low-Energy QCD
RealisticNN-Potentials
ChiralInteractions
ab initioApproaches
EffectiveInteractions
Many-BodyMethods
DensityFunctional
Models
9
Why Effective Interactions?
Realistic Potentials
generate strong correlations inmany-body states
short-range central & tensor corre-lations most important
Many-Body Methods
rely on truncated many-nucleonHilbert spaces for A > 12
not capable of describing short-range correlations
extreme: Hartree-Fock based onsingle Slater determinant
Modern Effective Interactions
adapt realistic potential to the avail-able model space
conserve experimentally con-strained properties (phase shifts)
10
Unitary Correlation
Operator Method (UCOM)
11
Unitary Correlation Operator Method
Correlation Operatorintroduce short-range correlations by
means of a unitary transformation with re-spect to the relative coordinates of all pairs
C = exp[−i G] = exp[− i
∑
i<j
gij
]G† = GC†C = 1
Correlated States∣∣ψ⟩
= C∣∣ψ
⟩Correlated Operators
O = C† O C
⟨ψ
∣∣ O∣∣ψ′
⟩=
⟨ψ
∣∣ C† O C∣∣ψ′
⟩=
⟨ψ
∣∣ O∣∣ψ′
⟩
12
Central and Tensor Correlators
C = CΩCr
Central Correlator Cr
radial distance-dependent shiftin the relative coordinate of a nu-cleon pair
gr =1
2
[s(r) qr + qr s(r)
]
qr = 12
[~rr
· ~q + ~q · ~rr
]
Tensor Correlator CΩ
angular shift depending on the orienta-tion of spin and relative coordinate of anucleon pair
gΩ =3
2ϑ(r)
[(~σ1 · ~qΩ)(~σ2 ·~r) + (~r↔~qΩ)
]
~qΩ = ~q − ~rr
qr
s(r) and ϑ(r)for given potential determined
in the two-body system
13
Correlated States: The Deuteron
1 2 3 4 5r [fm]
0
0.05
0.1
0.15
.
⟨ r∣ ∣ φ
⟩
L = 0
1 2 3 4 5r [fm]
0
0.05
0.1
0.15
.
⟨ r∣ ∣ C
r
∣ ∣ φ⟩
1 2 3 4r [fm]
0
0.05
0.1
0.15
0.2
.
[fm
]
s(r)central
correlations
1 2 3 4 5r [fm]
0
0.05
0.1
0.15
.
⟨ r∣ ∣ C
ΩC
r
∣ ∣ φ⟩
L = 2
L = 0
1 2 3 4r [fm]
0
0.02
0.04
0.06
0.08
.
ϑ(r)
tensorcorrelations
constraint on rangeof tensor correlator
14
Correlated Interaction — VUCOM
H = T[1] + (T[2]+V[2]) + (T[3]+V[3]) + · · ·H = T + VUCOM + V[3]UCOM + · · ·
closed operator expression for the correlated interactionVUCOM in two-body approximation
correlated interaction and original NN-potential are phaseshift equivalent by construction
unitary transformation results in a pre-diagonalisation ofHamiltonian
momentum-space matrix elements of correlated interactionare similar to Vlow−k
15
Simplistic “Shell-Model” Calculation
expectation value of Hamiltonian (with AV18) for Slater de-terminant of harmonic oscillator states
-10
0
10
20
30
40
50
60
.
E/A
[MeV
]
4He 16O 48Ca 90Zr 132Sn 208Pb
-10
0
10
20
30
40
50
60
.
E/A
[MeV
]
4He 16O 48Ca 90Zr 132Sn 208Pb
Cr
-10
0
10
20
30
40
50
60
.
E/A
[MeV
]
4He 16O 48Ca 90Zr 132Sn 208Pb
Cr
CΩ
central & tensorcorrelations
essential to obtainbound nuclei
16
Application I
No-Core Shell Model
17
UCOM + NCSM
No-Core Shell Model+
Matrix Elements of CorrelatedRealistic NN-Interaction VUCOM
many-body state is expanded in Slater determinants of har-monic oscillator single-particle states
large scale diagonalisation of Hamiltonian within a truncatedmodel space (N~ω truncation)
assessment of short- and long-range correlations
NCSM code by Petr Navratil [PRC 61, 044001 (2000)]
18
4He: Convergence
VAV18
20 40 60 80~ω [MeV]
-20
0
20
40
60
.
E[M
eV]
Nmax
0 2 4
6
8
10121416EAV18
VUCOM
20 40 60 80~ω [MeV]
-20
0
20
40
60
.
E[M
eV]
residual state-dependentlong-range correlations
19
4He: Convergence
VAV18
20 40 60 80~ω [MeV]
-20
0
20
40
60
.
E[M
eV]
Nmax
0 2 4
6
8
10121416EAV18
VUCOM
20 40 60 80~ω [MeV]
-20
0
20
40
60
.
E[M
eV]
20 40 60 80
-30
-25
-20
-15
-10
omitted three- andfour-body contributions
20
Tjon-Line and Correlator Range
-8.6 -8.4 -8.2 -8 -7.8 -7.6E(3H) [MeV]
-30
-29
-28
-27
-26
-25
-24
.
E(4
He)
[MeV
]
AV18Nijm II
Nijm I
CD Bonn
XXX + 3-bodyExp.
Tjon-line : E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
21
Tjon-Line and Correlator Range
-8.6 -8.4 -8.2 -8 -7.8 -7.6E(3H) [MeV]
-30
-29
-28
-27
-26
-25
-24
.
E(4
He)
[MeV
]
AV18Nijm II
Nijm I
CD Bonn
VUCOM(AV18)Exp.
increasingCΩ-range
Tjon-line : E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
change of CΩ-correlator rangeresults in shift along Tjon-line
minimise netthree-body force
by choosing correlatorwith energies close to
experimental value
this VUCOM
is used in thefollowing
22
Tjon-Line and Correlator Range
-8.6 -8.4 -8.2 -8 -7.8 -7.6E(3H) [MeV]
-30
-29
-28
-27
-26
-25
-24
.
E(4
He)
[MeV
]
AV18Nijm II
Nijm I
CD Bonn
VUCOM(AV18)VUCOM(Nijm II)VUCOM(Nijm I)
Exp.
Tjon-line : E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
change of CΩ-correlator rangeresults in shift along Tjon-line
minimise netthree-body force
by choosing correlatorwith energies close to
experimental value
23
10B: Benchmarking VUCOM
large-scale NCSM calculations troughout the p-shell in progress(with and w/o Lee-Suzuki transformation)
-2
-1
0
1
2
3
4
5
6
7
.
E−E
3+
[MeV
]
VUCOM Exp
3+
1+
0+
2+
4+
0~ω 2~ω 4~ω 6~ω 8~ω
~ω = 18MeV
-62.1 -64.7
calculations by Petr Navratil – preliminary
24
10B: Benchmarking VUCOM
large-scale NCSM calculations troughout the p-shell in progress(with and w/o Lee-Suzuki transformation)
-2
-1
0
1
2
3
4
5
6
7
.
E−E
3+
[MeV
]
VUCOM Exp CDBonn ChiralNN NN+NNN
3+
1+
0+
2+
4+
0~ω 2~ω 4~ω 6~ω 8~ω
~ω = 18MeV
8~ω 6~ω
-62.1 -64.7
-56.8-56.3
-64.0
calculations by Petr Navratil – preliminary
VUCOM gives correctlevel ordering withoutany NNN interaction
25
Application II:
Hartree-Fock & Beyond
26
UCOM + HF
Standard Hartree-Fock+
Matrix Elements of CorrelatedRealistic NN-Interaction VUCOM
many-body state is a Slater determinant of single-particlestates expanded in oscillator basis
correlations cannot be described by Hartree-Fock states
starting point for improved many-body calculations : MBPT,RPA, SM/CI, CC,...
27
Hartree-Fock with VUCOM
-8
-6
-4
-2
0
.
E/A
[MeV
]
4He16O
24O34Si
40Ca48Ca
48Ni56Ni
68Ni78Ni
88Sr90Zr
100Sn114Sn
132Sn146Gd
208Pb1
2
3
4
5
6
.
Rch
[fm
]
experiment HF
long-rangecorrelations are
missing
28
Perturbation Theory with VUCOM
-8
-6
-4
-2
0
.
E/A
[MeV
]
4He16O
24O34Si
40Ca48Ca
48Ni56Ni
68Ni78Ni
88Sr90Zr
100Sn114Sn
132Sn146Gd
208Pb1
2
3
4
5
6
.
Rch
[fm
]
experiment HF HF+PT2 HF+PT2+PT3
long-rangecorrelations are
perturbativeeasily tractable
within PT, SM/CI,CC, RPA,...
indications forpresence of residual
three-body force
29
Outlook: UCOM + RPA
0
20
40
60
R[f
m4 /M
eV]
0.07 fm3
0.08 fm3
0.09 fm3
0
100
200
300
0
100
200
300
400
R[f
m4 /M
eV]
0
500
1000
1500
0 10 20 30 40E[MeV]
0
1000
2000
3000
R[f
m4 /M
eV]
0 10 20 30 40E[MeV]
0
2500
5000
7500
10000
16O 40
Ca
48Ca
90Zr
132Sn
208Pb
0+
Iθ(S=1,T=0)
EXP.
EXP.
EXP.
Drozdz et al.Drozdz et al.
AV18
ERPA/SRPA :long-rangecorrelations
HFB: pairingwith realisticinteractions
effect of simplethree-nucleon
forces
30
Application III
Fermionic MolecularDynamics (FMD)
31
UCOM-FMD Approach
Gaussian Single-Particle States
∣∣q⟩
=n∑
ν=1
cν
∣∣aν,~bν
⟩⊗
∣∣χν
⟩⊗
∣∣mt
⟩
⟨~x∣∣aν,~bν
⟩= exp
[−
(~x−~bν)2
2 aν
]
aν : complex width χν : spin orientation~bν : mean position & momentum
Slater Determinant∣∣Q
⟩= A
( ∣∣q1
⟩⊗
∣∣q2
⟩⊗ · · · ⊗
∣∣qA
⟩)
Correlated Hamiltonian
H = T + VUCOM + δVc+p+ls
Variation⟨Q
∣∣ H−Tcm
∣∣Q⟩
⟨Q
∣∣Q⟩ → min
Projectionrestoration of rotationaland inversion symmetry
PAV / VAP
Multi-Configurationmixing of several
intrinsic configurationsGCM
32
Intrinsic One-Body Density Distributions
ρ(1
)(~x
)[ρ
0]4He 16O 40Ca
9Be 20Ne 24Mg
capable of describing sphericalshell-model as well as intrinsically
deformed and α-cluster states
33
Structure of 12C
-95
-90
-85
-80
-75
-70
-65
E@M
eVD
0+
0+
2+
4+
3-
0+
2+
4+
0+
2+
1-2-
3-
0+
2+
H0+ L
0+
H2+ L1+4+
3-1-2-H2- L
12C
Variation PAVΠ Multi-Config Experiment
E [ MeV] Rch [ fm] B(E2) [e2 fm4]
V/PAV 81.4 2.36 -VAP α-cluster 79.1 2.70 76.9PAVπ 88.5 2.51 36.3VAP 89.2 2.42 26.8Multi-Config 92.2 2.52 42.8
Experiment 92.2 2.47 39.7 ± 3.3
V/PAV VAPα
-6 -4 -2 0 2 4 6y [fm]
-6
-4
-2
0
2
4
6
z [fm
]
Vπ/PAVπ VAP
-6 -4 -2 0 2 4 6x [fm]
-6
-4
-2
0
2
4
6
y [fm
]
Multi-Config
-6 -4 -2 0 2 4 6y [fm]
-6
-4
-2
0
2
4
6
z [fm
]
-6 -4 -2 0 2 4 6x [fm]
-6
-4
-2
0
2
4
6
y [fm
]
-6 -4 -2 0 2 4 6y [fm]
-6
-4
-2
0
2
4
6
z [fm
]
-6 -4 -2 0 2 4 6y [fm]
-6
-4
-2
0
2
4
6
z [fm
]
34
Structure of 12C — Hoyle State
-95
-90
-85
-80
-75
-70
E@M
eVD
0+
2+
4+
0+
2+
1-2-
3-
0+
2+
4+
0+
2+0+2+
1-2-
3-
2-
0+
2+
H0+ L
0+
H2+ L
1+4+
3-1-2-H2- L
12C
Multi-ConfigH4L Multi-ConfigH14L Experiment
Multi-Config Experiment
E [ MeV] 92.4 92.2Rch[ fm] 2.52 2.47
B(E2, 0+1 → 2+
1 ) [e2 fm4] 42.9 39.7 ± 3.3
M(E0, 0+1 → 0+
2 ) [ fm2] 5.67 5.5 ± 0.2
-5 0 5y [fm]
-5
0
5
z [fm
]
-5 0 5y [fm]
-5
0
5
z [fm
]
-5 0 5y [fm]
-5
0
5
z [fm
]
∣∣⟨ ∣∣0+2
⟩∣∣ = 0.76
∣∣⟨ ∣∣0+2
⟩∣∣ = 0.71
∣∣⟨ ∣∣0+2
⟩∣∣ = 0.50
35
Outlook: Resonances & Scattering in FMD
-38
-36
-34
-32
-30
-28
-26
B @MeVD
FMD Exp. Frozen Frozen+PAVΠ Exp.
3 He+α 3 He+α 7 Be7 Be 7 Be
32-12-
72-
52-
E[M
eV]
0 2 4 6 8 10E [MeV]
-120
-100
-80
-60
-40
-20
0
δ(E
) [d
eg]
Frozen+PAVπ
3He-α
1/2+
3/2+
5/2+
collective coordinate rep-resentation as tool for thedescription of continuumstates in FMD
first steps towardsfully microscopic andconsistent description
of structure andreactions
36
Conclusions
Unitary Correlation Operator Method (UCOM)
• explicit description of short-range central and tensor correlations
• universal phase-shift equivalent correlated interaction VUCOM
Innovative Many-Body Methods
• No-Core Shell Model
• Hartree-Fock, MBPT, SM/CI, CC, RPA, ERPA, SRPA,...
• Fermionic Molecular Dynamics
unified description of nuclearstructure across the whole
nuclear chart is within reach
37
Epilogue
thanks to my group & my collaborators
• H. Hergert, N. Paar, P. Papakonstantinou, A. ZappInstitut fur Kernphysik, TU Darmstadt
• T. NeffNSCL, Michigan State University
• H. Feldmeier, A. Cribeiro, K. LangankeGesellschaft fur Schwerionenforschung (GSI)
supported by the DFG through SFB 634“Nuclear Structure, Nuclear Astrophysics andFundamental Experiments...”
38