Microscopic Nuclear Theory
RIA Summer School
August 1-6, 2005
Lawrence Berkeley National Laboratory
Lectures by
James P. Vary
ISU, LLNL, SLAC
“No-Core Shell Model (NCSM) Applications”
Overview of the previous 2 lectures
and plan for lecture 3
 Phenomenology - Extreme Single Particle Shell Model (ESPSM)
 Theory of Effective Operators
 Ab-initio No-Core Shell Model (NCSM)
 Applications with realistic NN interactions
 Applications with realistic NN + NNN interactions
 Reactions and Scattering with extensions of the NCSM
 Conclusions and Outlook
1
Cluster Approximation
Assume the model-space A-body effective Hamiltonian is a
superposition of a-body cluster effective Hamiltonians
H "H
(1)
= H (1)
+H
(a)
# HCM
"$ A %
'
# 2&
+
"$ A % "$ a%
' '
# a & # 2&
Sa
V˜123...a = e " S a H #
"
a,A e
A
(
V˜i1 i2 i3 ... ia ) H CM
i1 <i 2 <i 3 <... ia
a
$h
i
i
Ab-initio no-core shell model (NCSM) results
a = 2 for 2 < A < 16, 47, 48, 49
a = 3 for 3 < A < 16
Results in a harmonic oscillator basis space
PA characterized by:
Nm =
$ (2n
i
+ li ) % N min + N max
controls
momentum
scale limit
i"# P
h! controls length scale of basis functions (“box”)
Results compare well with VMC and GFMC results
Special thanks to Petr Navratil and Andrey Shirokov
for many results shown below
2
NN Interactions featured here with NCSM
 Traditional meson-exchange theory (Nijmegen X, CD Bonn X, AVX, etc.,)
 Effective field theory with roots in QCD (EFT X, Idaho X, NXLO, etc.,)
 Renormalization group reduced bare NN interactions (V-lowk X)
 Off-shell variations of bare NN interactions (INOY-X, etc.,)
 Inverse scattering theory (ISTP, JISPX, etc.,)
NNN Interactions featured here with NCSM
 Tucson-Melbourne prime (TM’)
 EFT at order N2LO (N2LO)
Development of high precision nuclear Hamiltonian
Nuclear Stand ard Model
Major role for RIA experiments and theory
Climb the walls of the valley of stability!
Test of convergence and comparison with other methods
All other methods
give same result
Egs = -6.91 MeV
Nmax
3
Nmax=2
Nmax=14
Nmax=0
Nmax=14
Egs=
-28.3 MeV
Nmax=2
Egs=
-28.3 MeV
Nmax=2
4
Theory
Experiment
5
NCSM calculations with the EFT N3LO NN interaction
Accurate NN potential at fourth order of chiral-perturbation theory (N3LO)
D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001(R ) (2003)
N3LO
7.85 MeV
4He 25.35(5) MeV
6Li
28.5(5) MeV
3H
Exp
8.48 MeV
28.30 MeV
31.99 MeV
Converged 6Li excitation energies
Correct level ordering, level spacing not right
NCSM calculations with the EFT N3LO NN interaction:
6Li binding energy convergence
6
NCSM calculations with the EFT N3LO NN interaction:
Convergence of 6Li excitation energies
Difficult convergence of the
binding energy
Good convergence of the
excitation energies
6Li
quadrupole moment
EFT N3LO NN potential
Exp
-0.08 e fm2
Theory extraploted
NCSM:
Good convergence
with Nmax
7
p-shell nuclei with realistic NN forces
• Correct level ordering for light p-shell nuclei
Old evaluation
NPA490,1(1988)
No 1/2-2
and
3/2-2, 7/2-2
reversed
New evaluation
NPA708,3(2002)
introduces 1/2-2
and orders the
states as in
calculation
Binding energy
35.5(5) MeV
Convergence of excitation energies
Realistic NN interactions provide reasonable description of nuclear structure
10B
•
•
using N3LO NN potential
Clearly, ground state is incorrectly predicted
In EFT, three-nucleon interaction appears
already at N 2LO
– Should be included in the Hamiltonian
– c1,c3,c4 parameters of the two-pion
term should be the same as those
used in the N3LO NN potential
• c1=-0.81, c3=-3.2, c4= 5.4
Binding energy:
56.3(2.0) MeV from Theory
64.75 MeV from Experiment
8
16O
ground and excited 0+ 0 and 3- 0 states
•Ground state changes structure
•0hΩ less than 50%, large 2hΩ and 4hΩ components
•Energy consistent with the UM OA result
•Excited 3- 0 state dominated by 1hΩ; follows the ground state
•Excited 0+ 0 state 2hΩ dominated; stable
•The 4hΩ dominated state still higher in the 8hΩ model space
9
Binding Energy of 16O (MeV)
Experiment
127.619
+/- 0.001
CD-Bonn 2000
117.4
+/- 3
AV8’
115.5
+/- 5
N3LO
110.9
+/- 5
INOY-3
138.3
+/- 3
Again - see role for NNN interactions
Convergence for 3H with a real three-body interaction
Needed to reproduce
experimental binding
energy
Tucson-Melbourne force
E [MeV]
]
V
e
M
[
E
-7.0
-7.2
-7.4
-7.6
-7.8
-8.0
-8.2
-8.4
-8.6
-8.8
-9.0
3
H
AV18
AV18 +TM'(81)
AV18 exact
NCSM
Faddeev
calculation
AV18+TM'(81) exact
4 6
8 10 12 14 16 18 20 22 24 26 28 30
Nmax
Paves the way for including the V3b in the NCSM p-shell calculations
10
Neutrino scattering on 12C
•
•
Exclusive 0+ 0 → 1+ 1 cross section & transistions
Extremely sensitive to the spin-orbit interaction strength
–
B(GT) (B(M1)) - στ,
• No spin-orbit 0+ 0 and 1+ 1 in different SU(4) irreps
–
•
12C
–
Transition underestimated by a factor of six
N.B.
V2b+V3b up to 4hΩ
–
Different processes dominated by different Q
•
•
2
Underestimates by a factor of 2-3
–
Exp
AV8'+TM'(99)
V2b up to 6hΩ - saturation
•
+
2.5
NCSM - no fit, no free parameters
–
+
C B(M1; 0 0 -> 1 1)
3
B(M1)
•
12
no transition
ground state 8 nucleons in p 3/2
CD-Bonn
AV8'
1.5
1
Significant improvement
0.5
0
Correlation with M1 transverse form factor
-2
0
2
4
6
8
10
Nmax
AV8' AV8'+TM’(99) Exp
B(GT) 0.26
0.67
0.88
CD-Bonn AV8'+TM’(99) Exp
(<νe,e-)
3.69
6.8
8.9±0.3±0.9
(<ν!,!-)
0.312
0.537 0.56±0.08±0.1
!-capture 2.38
4.43
6.0±0.4
V3b increases the strength of the spin-orbit force
EFT N2LO three-nucleon interaction
•
Two-pion exchange term
– Used in standard Three Nucleon Interaction
(TNI) models
•
•
•
•
Fujita-Miyazawa
Tucson-Melbourne
Urbana
Illinois
– Low-energy constants c1, c3, c4
• Determined by the corresponding EFT NN
interaction
–
•
Consistent NN & TNI
One-pion exchange plus contact term
– Low-energy constant cD
• Must be determined from experiment
•
Contact term
– Low-energy constant cE
• Must be determined from experiment
•
A regulator appears in all terms
– Depends on cutoff parameter Λ
exp["(Q2 /#2 ) 2 ]
• Taken consistently from that used in the
corresponding EFT NN interaction
!
11
Determination of the cD and cE low-energy constants
•
•
Fit the 3H and 4He binding energies
– Suggested and done by A. Nogga
– Two solutions
• 3NF-A
– cD=-1.11
– cE=-0.66
• 3NF-B
– cD=8.14
!
– cE=-2.02
– Regulator depending on Jacobi coordinates
Present work: Two-pion term local in coordinate space
– Change regulator: depending on momentum transfer
– Need to re-fit cD and cE
• A=3 done
• 4He under way
• Presented results
– “3NFA”: cD=-1.11, cE=-0.25
– “3NFB”: cD=8.14, cE=-1.15
3NFA and 3NFB dominated by different terms
– 3NFA two-pion term dominant
– 3NFB one-pion term dominant
– Contact term repulsive in both cases
Important for saturation properties
10B
•
exp["(( p 2 + q 2 ) /#2 ) 2 ]
exp["(Q2 /#2 ) 2 ]
!
cE
•
cD
= "1.09
= "0.07
= 0.25
= "0.69
= "0.88
= 0.37
!
!
!
!
!
!
using N3LO NN plus consistent N2LO TNI
N2LO TNI 3NF-A dominated by twopion exchange term
– Results close to the TM’
– Smaller radius
– Larger binding energy
• EB=68.36 MeV
•
N2LO TNI 3NF-B dominated by onepion exchange plus contact term
– Visible difference in particular for
higher-lying states
– Reasonable radius
– No overbinding
Both cure the gs spin problem!
• EB=63.14 MeV
•
•
6hΩ needed to check convergence of
spectra
Calculation to be re-done after proper
fitting to 4He
Both 3NF-A and 3NF-B resolve the 10B ground state spin problem
Similarly like TM’
TM’ , Illinois 3NF, but unlike Urbana IX
12
ab initio NCSM wave functions
NN only at a=2 level + corrections for long range tails
currently being used to investigate problems relevant to
nucleosynthesis by Livermore group:
7Be(p,γ
Be(p,γ)8 B
S-factor
3He(α,γ
He(α,γ))7 Be
S-factor
S-factor
n + 4He elastic scattering phase shifts
10Be(n,γ
Be(n,γ)11Be
cross section
Light ion reactions are important for astrophysics
Understanding
our Sun
p-p chain
Solar neutrinos
Eν < 15 MeV
Observed at SNO, Super K
- neutrino oscillations
13
7Be(p,γ
Be(p,γ)8B
S-factor from ab initio wave functions
• Sensitivity to NN interaction
– CD-Bonn 2000 vs. INOY
Conclusions and outlook from lecture 3
•
Ab initio no-core shell model
– Method for solving the nuclear structure problem for light nuclei
– Apart from the GFMC the only working method for A>4 at present
– Advantages
• applicable for any NN potential
• Presently the only method capable to apply the QCD χPT NN+NNN interactions to p-shell nuclei
• Easily extendable to heavier nuclei
• Calculation of complete spectra at the same time
– Success - importance of three-nucleon forces for nuclear structure
Work in progress - examples shown
•
•
•
Better NN interactions whose off-shell properties tuned to nuclei
Calculations with realistic three-body forces in the p-shell
– Better determination of the three-body force itself
Coupling of the NCSM to nuclear reactions theories
– Direct reactions
• Density from NCSM plus folding approaches
– Low-energy resonant and nonresonant reactions
• RGM-like approach
–
–
-161 MeV -147 MeV
Exotic nuclei: RIA
T hermonuclear reaction rates: Astrophysics
14
Overall Conclusions
 Phenomenological shell models, such as the Extreme Single Particle Shell
Model (ESPM of Lecture 1) provide physical insight and guide our choices of
theoretical paths
 Ab-initio NCSM is founded on this established physical intuition (Lecture 2)
 High precision comparisons between theory and experiment are emerging
 Applications up through A = 16 tell us that NNN interactions are needed
with realistic local NN interactions
 New era of interplay between theory and experiment has emerged
Outstanding Problems/Challenges (Sample)
 Role of NNN interactions with non-local NN interactions
 Converged saturation properties in better agreement with experiment
 RMS radius of 6-He not understood (discrepancy ~ 9%)
 Applications with NN + NNN potentials to reactions and scattering
 Applications to unstable light nuclei and to heavier
(stable and unstable) nuclei
Acknowledgment of material from many collaborations
With thanks to many collaborators
K. Joseph Abraham, Oleksiy Atramentov, Peter Peroncik, Bassam Shehadeh,
Richard Lloyd, John R. Spence, James P. Vary, Thomas A. Weber, Iowa State University
Petr Navratil, W. Erich Ormand, Lawrence Livermore National Laboratory
Bruce R. Barrett, U. van Kolck, Hu Zhan, Ionel Stetcu, University of Arizona
Andreas Nogga, Institute of Physics, Juelich, Germany
E. Caurier, Institute Reserche Subatomique, Strasbourg, France
Anna Hayes, Los Alamos National Laboratory
M. Slim Fayache, S. Aroua, University of Tunis, Tunisia
Cesar Viazminsky, University of Aleppo, Syria
Mahmoud A. Hasan, University of Jordan, Jordan
Andrey Shirokov, Moscow State University, Russia
Alexander Mazur, Sergei Zaytsev, Khabarovsk State Technical University, Russia
Alina Negoita, Sorina Popescu, Sabin Stoica, Institute of Atomic Physics, Romania
Avaroth Harindranath, Dipankar Chakrabarty, Saha Institute of Nuclear Physics, India
Grigorii Pivovarov, Victor Matveev, Institute for Nuclear Research, Moscow, Russia
Lubo Martinovic, Institute of Physics Institute, Bratislava, Slovakia
Kris Heyde, N. Smirnova, University of Gent, Belgium
Larry Zamick, Rutgers University
15