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New Frontiers inNuclear Structure TheoryFrom Realistic Interactions to the Nuclear Chart
Robert RothInstitut fur Kernphysik
Technische Universitat Darmstadt
1
Overview
n Motivation
n Nucleon-Nucleon Interactions
n Solving the Many-Body Problem
n Correlations & Unitary Correlation Operator Method
n Applications
2
Nuclear Structure in the 21st Century
new frontiers innuclear structure physics
Experimentn fundamental astrophysical ques-
tions need nuclear input
n possibilities to investigate nucleifar off stability
n new nuclear structure facilities:FAIR@GSI, RIA,...
Theoryn improved understanding of funda-
mental degrees of freedom / QCD
n high-precision realistic nucleon-nucleon potentials
n ab initio treatment of the many-body problem
3
Astrophysical Challenges
4
Theoretical Contextbe
tterr
esol
utio
n/m
ore
fund
amen
tal n finite nuclei
n few-nucleon systemsn nucleon-nucleon interaction
n hadron structure
n quarks & gluonsn deconfinementQ
uant
umC
hrom
oD
ynam
ics
Nuc
lear
Str
uctu
re
5
Theoretical Contextbe
tterr
esol
utio
n/m
ore
fund
amen
tal
Qua
ntum
Chr
omo
Dyn
amic
s
Nuc
lear
Str
uctu
re“solve”
the interacting nuclearmany-body problem
“construct”a realistic nucleon-nucleon
interaction from QCD
6
RealisticNucleon-Nucleon Potentials
7
Nature of the NN-Interaction
∼ 1.6fm
ρ−1/30 = 1.8fm
n NN-interaction is not fundamental
n induced via mutual polarisation of virtualmeson cloud (simple picture)
n analogous to van der Waals interactionbetween neutral atoms
n short-ranged: acts only if the nucleonsoverlap
n genuine NNN-interaction is important
8
How to Construct the NN-Potential?
n QCD input
• symmetries
• meson-exchange picture
• chiral perturbation theory
n short-range phenomenology
• ansatz for short-range behaviour
n experimental two-body data
• scattering phase-shifts & deuteron properties
• reproduced with χ2/datum ≈ 1
ArgonneV18
CD Bonn
NijmegenI/II
Chiral...
9
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)
10
NuclearMany-Body Problem
11
Ab initio Calculations
solve the quantum many-bodyproblem for A nucleons interacting
via a realistic NN-potential
n exact numerical solution possible for small systems at an enormouscomputational cost
n Green’s Function Monte Carlo: Monte Carlo sampling of the A-body wave function in coordinate space; imaginary time cooling
n No-Core Shell Model: large-scale diagonalisation of the Hamilto-nian in a harmonic oscillator basis
12
Green’s Function Monte Carlo
-100
-80
-60
-40
-20E
nerg
y (M
eV)
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.]
75000CPU-hours
13
Our Goal
nuclear structure calculationsacross the whole nuclear chart
based on realistic NN-potentialsand as close as possible to
an ab initio treatment
bound to simpleHilbert spaces for large
particle numbers need to deal withstrong interaction-
induced correlations
14
What are Correlations?
correlations:
everything beyond theindependent particle picture
n the quantum state of A independent (non-interacting) fermionsis a Slater determinant
∣∣ψ⟩
= A (∣∣φ1
⟩⊗
∣∣φ2
⟩⊗ · · · ⊗
∣∣φA
⟩)
n Slater determinants cannot describe correlations by definition
15
Deuteron: Manifestation of Correlations
MS = 01√2(∣∣↑↓
⟩+
∣∣↓↑⟩)
MS = ±1∣∣↑↑⟩,
∣∣↓↓⟩
zn spin-projected two-body
density ρ(2)1,MS
(~r)
n exact deuteron solutionfor Argonne V18 potential
two-body density fullysuppressed at small particle
distances |~r|
central correlations
angular distributiondepends strongly on relative
spin orientation
tensor correlations
16
Central Correlations
-100
0
100
200
300
.
V01(r
)[M
eV]
AV18 – central part(S, T ) = (0, 1)
0 0.5 1 1.5 2 2.5 3r [fm]
0
0.1
0.2
0.3
.
ρ(2)
01(r
)[f
m−
3]
4He
ρ(2)exact(r)
ρ(2)SD (r)
n strong repulsive core in central part ofrealistic interactions
n suppression of the probability den-sity for finding two nucleons within thecore Ô central correlations
“shift”the nucleons out of
the core region
17
Tensor Correlations
2.00 1.56
0.50
-0.56
-1.00
-0.56
0.50
1.562.00
-2.00-1.56
-0.50
0.56
1.00
0.56
-0.50
-1.56-2.00
Vtensor
anti-parallelspins
parallelspins
n analogy with dipole-dipole interaction
Vtensor ∼ −(3
(~σ1~r)(~σ2~r)
r2− ~σ1~σ2
)
n couples the relative spatial orientationof two nucleons with their spin orien-tation Ô tensor correlations
“rotate” nucleonstowards pole or equator
depending on spin
18
Unitary CorrelationOperator Method (UCOM)
19
Unitary Correlation Operator Method
Correlation Operatorintroduce correlations by means of an
unitary transformation with respectto the relative coordinates of all pairs
C = exp[−i G] = exp[− i
∑
i<j
gij
]
g = g(~r,~q; ~σ1, ~σ2, ~τ 1, ~τ 2)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′⟩
20
Central and Tensor Correlators
C = CΩCr
Central Correlator Cr
n 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
n 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)encapsulate the physics ofshort-range correlations
21
Correlated States
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
ρ(2)1,SM
(~r)
22
Correlated NN-Potential — VUCOM
H = T[1] + (T[2]+V[2]) + (T[3]+V[3]) + · · ·H = T + VUCOM + V[3]UCOM + · · ·
n closed operator expression for the correlated interactionVUCOM in two-body approximation
n correlated interaction and original NN-potential are phaseshift equivalent by construction
n unitary transformation results in a pre-diagonalisation ofHamiltonian
n momentum-space matrix elements of correlated interactionare similar to Vlow−k
23
Simplistic “Shell-Model” Calculation
n 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
24
Application I
No-Core Shell Model
25
UCOM + NCSM
No-Core Shell Model+
Matrix Elements of CorrelatedRealistic NN-Interaction VUCOM
n many-body state is expanded in Slater determinants of har-monic oscillator single-particle states
n large scale diagonalisation of Hamiltonian within a truncatedmodel space (N~ω truncation)
n assessment of short- and long-range correlations
n NCSM code by Petr Navratil [PRC 61, 044001 (2000)]
26
4He: Convergence
Vbare
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
27
4He: Convergence
Vbare
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 higher-ordercluster contributions
28
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 I
Nijm II
CD Bonn
XXX + 3-body
Exp. VUCOM(AV18)
increasingϑ-range
n Tjon-line: E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
n change in correlator range re-sults in shift along Tjon-line
choose correlatorwith energies close to
experimental value, i.e.,minimise net
three-body force
this VUCOMis used in the
following
29
Application II:
Hartree-Fock Calculations
30
UCOM-Hartree-Fock Approach
Standard Hartree-Fock+
Matrix Elements of CorrelatedRealistic NN-Interaction VUCOM
n many-body state is a Slater determinant of single-particlestates obtained by energy minimisation
n correlations cannot be described by Hartree-Fock states
n bare realistic NN-potential leads to unbound nuclei
31
Correlated Argonne V18
-8
-6
-4
-2
0
.
E/A
[MeV
]
4He16O
24O34Si
40Ca48Ca
48Ni56Ni
68Ni78Ni
88Sr90Zr
100Sn114Sn
132Sn146Gd
208Pb1
2
3
4
5
.
Rch
[fm
]
experiment l VUCOM
32
Missing Pieces
long-rangecorrelations
genuinethree-body forces
three-body clustercontributions
Beyond Hartree-Fock
n improve many-body states such that long-range correla-tions are included
n many-body perturbation theory (MBPT), configurationinteraction (CI), coupled-cluster (CC),...
33
Long-Range Correlations: MBPT
n many-body perturbation theory: second-order energy shift gives esti-mate for influence of long-range correlations
∆E(2) = −1
4
occu.∑
i,j
unoccu.∑
a,b
|⟨φaφb
∣∣ Tint + VUCOM∣∣φiφj
⟩|2
εa + εb − εi − εj
4He16O
24O34Si
40Ca48Ca
48Ni56Ni
68Ni78Ni
88Sr90Zr
100Sn114Sn
132Sn146Gd
208Pb
-8
-6
-4
-2
0
.
E/A
[MeV
]
Nmax = 12
experiment l HF HF+PT2 HF+PT2+PT3
34
Long-Range Correlations: MBPT
13 14 15 16 17 18 19 20 21 22 23 24-10
-8
-6
-4
-2
0
.E/A
[MeV
]
AO
48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78-10
-8
-6
-4
-2
0
.
E/A
[MeV
]
ANi
100 104 108 112 116 120 124 128 132-10-8-6-4-20
.
E/A
[MeV
]
ASn
l HF HF+PT2
experiment
35
Missing Pieces
long-rangecorrelations
genuinethree-body forces
three-body clustercontributions
Beyond Hartree-Fockn residual long-range correla-
tions are perturbative
n mostly long-range tensor cor-relations
n easily tractable within MBPT,CI, CC,...
Net Three-Body Forcen small effect on binding ener-
gies for all masses
n cancellation does not work forall observables
n construct simple effectivethree-body force
36
Application III
Fermionic MolecularDynamics (FMD)
37
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
Diagonalisationin sub-space spannedby several non-ortho-
gonal Slater deter-minants
∣∣Qi
⟩
38
Variation: Chart of Nuclei
H2 H3
He3 He4 He5 He6 He7 He8
Li5 Li6 Li7 Li8 Li9 Li10 Li11
Be7 Be8 Be9 Be10Be11Be12Be13Be14
B8 B9 B10 B11 B12 B13 B14 B15 B16 B17
C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20
N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22
O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24
F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26
Ne17Ne18Ne19Ne20Ne21Ne22Ne23Ne24Ne25Ne26Ne27Ne28
Na20Na21Na22Na23Na24Na25Na26Na27Na28Na29Na30Na31
Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34
Al24 Al25 Al26 Al27 Al28 Al29 Al30 Al31 Al32 Al33 Al34 Al35 Al36
Si22 Si24 Si25 Si26 Si27 Si28 Si29 Si30 Si31 Si32 Si33 Si34 Si35 Si36 Si37 Si38 Si39
P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41
S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 S41 S42 S43 S44
Cl31 Cl32 Cl33 Cl34 Cl35 Cl36 Cl37 Cl38 Cl39 Cl40 Cl41 Cl42 Cl43 Cl44 Cl45
Ar32 Ar33 Ar34 Ar35 Ar36 Ar37 Ar38 Ar39 Ar40 Ar41 Ar42 Ar43 Ar44 Ar45 Ar46 Ar47
K35 K36 K37 K38 K39 K40 K41 K42 K43 K44 K45 K46 K47 K48 K49 K50
Ca36Ca37Ca38Ca39Ca40Ca41Ca42Ca43Ca44Ca45Ca46Ca47Ca48Ca49Ca50Ca51Ca52Ca53Ca54
Sc39Sc40Sc41Sc42Sc43Sc44Sc45Sc46Sc47Sc48Sc49Sc50Sc51Sc52Sc53Sc54
Ti41 Ti42 Ti43 Ti44 Ti45 Ti46 Ti47 Ti48 Ti49 Ti50 Ti51 Ti52 Ti53 Ti54 Ti55
V45 V46 V47 V48 V49 V50 V51 V52 V53 V54 V55 V56
Cr46 Cr47 Cr48 Cr49 Cr50 Cr51 Cr52 Cr53 Cr54 Cr55 Cr56 Cr57
Mn49Mn50Mn51Mn52Mn53Mn54Mn55Mn56Mn57Mn58
Fe50Fe51Fe52Fe53Fe54Fe55Fe56Fe57Fe58Fe59
Co53Co54Co55Co56Co57Co58Co59Co60
Ni48 Ni54 Ni55 Ni56 Ni57 Ni58 Ni59 Ni60 Ni61
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34N
2
4
6
8
10
12
14
16
18
20
22
24
26
28
.
Z
(E − Eexp)/A[MeV]
-0.5
0
0.5
1
1.5
VUCOM+ δVc+p+ls
H2 H3
He3 He4 He5 He6 He7 He8
Li5 Li6 Li7 Li8 Li9 Li10 Li11
Be7 Be8 Be9 Be10 Be11 Be12 Be13 Be14
B8 B9 B10 B11 B12 B13 B14 B15 B16 B17
C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20
N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22
O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24
F16 F17 F18 F19 F20 F21 F22 F23
Ne17 Ne18 Ne19 Ne20 Ne21 Ne22 Ne23 Ne24 Ne25 Ne26 Ne27 Ne28
Na20 Na21 Na22 Na23 Na24 Na25 Na26 Na27 Na31
Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34
2 4 6 8 10 12 14 16 18 20 22
2
4
6
8
10
12
oneGaussian
per nucleon
twoGaussiansper nucleon
39
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
40
Beyond Simple Variation
n Projection after Variation (PAV)• restore inversion and rotational symmetry by
angular momentum projection
n Variation after Projection (VAP)• find energy minimum within parameter space of
parity and angular momentum projected states
• implementation via generator coordinate method(constraints on multipole moments)
n Multi-Configuration• diagonalisation within a set of different Slater determinants
16O
41
Intrinsic Shapes of 12C
⟨H
⟩
⟨T
⟩
⟨Vls
⟩√⟨
r2⟩
intrinsic projected-81.4 -81.5
212.1 212.1
-39.8 -40.2
2.22 2.22
intrinsic projected-77.0 -88.5
189.2 186.1
-12.0 -17.1
2.40 2.37
intrinsic projected-74.1 -85.5
182.8 179.0
-5.8 -8.0
2.45 2.42
intrinsic projected-57.0 -75.9
213.9 201.4
0.0 0.0
2.44 2.42
42
Structure of 12C
-95
-90
-85
-80
-75
-70
-65
E@MeVD 0+
0+
2+
4+
3-
0+
2+
4+
0+
2+
1-2-
3-
0+
2+
H0+ L0+
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 42.8Experiment 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
]
43
Structure of 12C — Hoyle State
-95
-90
-85
-80
-75
-70
E@MeVD
0+
2+
4+
0+
2+
1-
2-
3-
0+
2+
4+
0+
2+
0+2+
1-
2-
3-
2-
0+
2+
H0+ L0+
H2+ L1+
4+
3-1-2-
H2- L
12C
Multi-ConfigH4L Multi-ConfigH14L Experiment
Multi-Config ExperimentE [ MeV] 92.4 92.2Rch[ fm] 2.52 2.47B(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
44
Conclusions
n exciting times for nuclear structure physics!
n realistic NN-potentials & ab initio calculations
n systematic schemes to derive effective interactions
n innovative ways to treat the many-body problem
unified description of nuclearstructure across the whole
nuclear chart is within reach
45
Epilogue
n thanks to my group & my collaborators
• H. Hergert, N. Paar, P. Papakonstantinou, A. ZappInstitut fur Kernphysik, TU Darmstadt
• T. NeffNSCL, Michigan State University
• H. FeldmeierGesellschaft fur Schwerionenforschung (GSI)
supported by the DFG through SFB 634“Nuclear Structure, Nuclear Astrophysics andFundamental Experiments...”
46
Supplements
47
Correlated Operators
Cluster Expansion
O = C† O C = O[1] + O[2] + O[3] + · · ·
Two-Body Approx.OC2 = O[1] + O[2]
ClusterDecomposition Principle
if the correlation range is smallcompared to the mean particle
distance, then higher orders aresmall
operators of allobservables can be and have to be
correlated consistently
48
Correlated NN-Potential — VUCOM
VUCOM =∑
p
12
[vp(r) Op + Op vp(r)
]
O = 1, (~σ1 ·~σ2), ~q2, ~q2(~σ1 ·~σ2), ~L
2, ~L
2(~σ1 ·~σ2),
(~L · ~S), S12(~r,~r), S12(~L, ~L),
S12(~qΩ,~qΩ), qr S12(~r,~qΩ), ~L2(~L · ~S),
~L2S12(~qΩ,~qΩ), . . . ⊗ 1, (~τ1 ·~τ2)
n Cr-transformation evaluated directly
n CΩ-transformation through Baker-Campell-Hausdorff expansion
n vp(r) uniquely determined by bare potential and correlation functions
49
Optimal Correlation Functions
n s(r) and ϑ(r) determined by two-body energy minimisation
n constraint on range of the tensor correlators ϑ(r) to isolate state in-dependent short-range correlations
s(r)
0
0.05
0.1
0.15
0.2
.
[fm
]
(S, T ) = (0, 0)
0 0.5 1 1.5 2 2.5r [fm]
0
0.05
0.1
0.15
0.2
.
[fm
]
(S, T ) = (0, 1)
(S, T ) = (1, 0)
0 0.5 1 1.5 2 2.5r [fm]
(S, T ) = (1, 1)
ϑ(r)
0
0.02
0.04
0.06
0.08
.
(S, T ) = (1, 0)
0 0.5 1 1.5 2 2.5r [fm]
0
0.02
0.04
0.06
0.08
.
(S, T ) = (1, 1)
50
Correlated NN-Potential — VUCOM
HC2 = T[1] + T[2] + V[2] = T + VUCOM
n closed operator expression for the correlated interactionVUCOM in two-body approximation
n correlated interaction and original NN-potential are phaseshift equivalent by construction
n unitary transformation results in a pre-diagonalisation ofHamiltonian
n momentum-space matrix elements of correlated interactionare similar to Vlow−k
51
Momentum-Space Matrix Elements3S1
02
46
0
2
4
6-40
-20
0
20
02
46
02
46
0
2
4
6-40
-20
0
20
02
46
q [fm−1]
q′
[fm−1]
3S1−3D1
02
46
0
2
4
6
-30
-20
-10
0
02
46
02
46
0
2
4
6
-30
-20
-10
0
02
46
q [fm−1]
q′
[fm−1]
Vbare
VUCOM
⟨q(LS)JT
∣∣ ∣∣q′(L′S)JT
⟩
AV18
pre-diagonalisationof Hamiltonian
52
Comparison with Vlowk
-40
-20
0
20
.
[MeV
]
1S0
0 0.5 1 1.5 2 2.5q [fm−1]
-40
-20
0
20
.
[MeV
]
3S1
-10
-8
-6
-4
-2
0
.
[MeV
]
3S1 − 3D1
0 0.5 1 1.5 2 2.5q [fm−1]
0
1
2
3
4
5
.
[MeV
]
3D1
AV18bare
VUCOM(Iϑ = 0.09 fm3)
Vlowk
(Λ = 2.1 fm−1)
53
UCOM / Lee-Suzuki / Vlowk
Lee-Suzukin decoupling of P and Q space by
similarity transformation
n same representation as used inmany-body method
n (state dependent)
Vlowk
n decimation to low-momentum Pspace; Q space discarded
n uses momentum representation
n state independent
n phase-shift equivalent
UCOMn pre-diagonalisation with respect to short-
range correlations
n no specific model-space or representation
n state independent
n phase-shift equivalent
54
Correlated Oscillator Matrix Elements
⟨n(LS)JT
∣∣ C†rC
†Ω H CΩCr
∣∣n′(L′S)JT⟩
=⟨n(LS)JT
∣∣ T + VUCOM∣∣n′(L′S)JT
⟩
calculate usinguncorrelated states andoperator form of VUCOM
map correlator onto statesand use bare interaction(avoids BCH expansion)
n Talmi-Moshinsky transformation & recoupling to obtain jj-coupledmatrix elements
n input for all kinds of many-body methods (HF, NCSM, CC,...)
55
Nuclear Matter: Equation of State
0.5 1 1.5 2kF [fm−1]
-20
-15
-10
-5
0
5
.
E/A
[MeV
]
AV18αAV18α + δVc+p+ls
n symmetric nuclear matter
n Slater determinant of plane-wave states |~k| ≤ kF
n correlated momentum spacematrix elements
n saturation point:
(E/A)0 ≈ −16.0 MeV
ρ0 ≈ 0.14 fm−3
K0 ≈ 280 MeV
n HvH theorem fulfilled
56
Parity and Angular Momentum Projection
7Li
0
2
4
6
8
10
12
@MeVD
12-
32-
52-
72-
32-12-
72-
52-
32-12-
72-
52-52-
PAV VAπP Exp.
-32.0 -34.4 -39.2
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
0.00
1
0.00
1
0.001
0.01
0.01
0.1
0.1
0.5
1.0
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
7Li
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
0.001
0.01
0.10.2
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
7Li
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
0.001
0.001
0.00
1
0.01
0.01
0.1
0.1
0.5
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
7Li
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
0.001
0.01
0.1
0.2
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
7Li
20Ne
0
1
2
3
4
5
6
@MeVD
0+
2+
2+
4+
0+
0+
2+
2+
2+
3+
4+
4+
4+
1-
2-
3-
3-
0+
2+
4+
PAV VAπP Exp.
-153.9 -157.2 -160.6
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
0.001
0.001
0.01
0.01
0.01
0.1
0.1
0.1
0.5
0.5
1.01.
0
1.0
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
20Ne
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
0.001
0.01
0.1
0.2
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
20Ne
-4 -2 0 2 4x [fm]
-4
-2
0
2
4y
[fm]
0.001
0.001
0.01
0.01
0.1
0.1
0.5
0.5
1.0
1.0 1.0
-4 -2 0 2 4x [fm]
-4
-2
0
2
4y
[fm]
20Ne
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
0.001
0.01
0.1
0.2
-3 -2 -1 0 1 2 3kx [fm
-1]
-3
-2
-1
0
1
2
3
k y [f
m-1]
20Ne
57
Helium Isotopes: Energies & Radii
He4 He5 He6 He7 He8
-32
-30
-28
-26
-24
-22
@MeVD 0+32-
0+32-
0+
0+
32-
0+
32-
0+0+
32-
0+32-
0+
PAVΠ
Multi-ConfigExperiment
Binding energies
He4 He5 He6 He7 He8
1.5
2.0
2.5
3.0
3.5
@fmD
0+
ch
0+
ch
0+
0+
ch
32-0+
ch
32-0+
ch
0+
ch
32-
0+
ch
32-0+
ch
Matter & charge radii
58
Helium Isotopes: Density Profiles
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
8He
totalneutronproton
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
6He
totalneutronproton
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
0 2 4 6 8 10r [fm]
10-4
10-3
10-2
10-1
1
ρ(r)
[ρ0]
4He
totalneutronproton
neutronhalo
59
Summary
n Unitary Correlation Operator Method (UCOM)
• short-range central and tensor correlations treated explicitly
• long-range correlations have to be accounted for by model space
n Correlated Realistic NN-Potential VUCOM
• low-momentum / phase-shift equivalent / operator representation
• robust starting point for all kinds of many-body calculations
60
Summary
n UCOM + No-Core Shell Model• dramatically improved convergence
• tool to assess long-range correlations & higher-order contributions
n UCOM + Hartree-Fock• closed shell nuclei across the whole nuclear chart
• basis for improved many-body calculations
n UCOM + Fermionic Molecular Dynamics• clustering and intrinsic deformations in p- and sd-shell
• projection / multi-config provide detailed structure information
61
Outlook
n residual “long-range” correlations• many-body perturbation theory
• configuration interaction & coupled-cluster calculations
• pairing correlations, Hartree-Fock-Bogoliubov
n collective excitations• random phase approximation (RPA, SRPA, QRPA)
n make contact to experiment!!!
62
