new horizons in ab initio nuclear structure theory
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New Horizons in
Ab Initio Nuclear Structure Theory
Robert Roth
1
Ab Initio Nuclear Structure
Robert Roth – TU Darmstadt – 01/2013
Nuclear Structure Observables
NuclearLatticeSim
.
chiralEFTonlattice
ExactAb-Initio
Solutions
few-bodyetal. Exact Ab-Initio
Solutions
few-body, no-core
shell model, etc.
Approx. Many-Body Methods
controlled & im-
provable schemes
Energ
y-D
ensity-
FunctionalTheory
guidedbychiralEFT
Similarity Transformationsphysics-conserving transform. of observables
Chiral Interactionsconsistent & improvable NN, 3N,... interactions
Chiral Effective Field Theorysystematic low-energy effective theory of QCD
Low-Energy Quantum Chromodynamics
2
Ab Initio Nuclear Structure
Robert Roth – TU Darmstadt – 01/2013
Nuclear Structure Observables
NuclearLatticeSim
.
chiralEFTonlattice
ExactAb-Initio
Solutions
few-bodyetal.
Exact Ab-InitioSolutions
few-body, no-core
shell model, etc.
Approx. Many-Body Methods
controlled & im-
provable schemes
Energ
y-D
ensity-
FunctionalTheory
guidedbychiralEFT
Similarity Transformationsphysics-conserving transform. of observables
Chiral Interactionsconsistent & improvable NN, 3N,... interactions
Chiral Effective Field Theorysystematic low-energy effective theory of QCD
Low-Energy Quantum Chromodynamics
PREDICTIO
NVALID
ATIO
Nthere...
...andbackagain
3
Nuclear Interactions
from Chiral EFT
4
Nuclear Interactions from Chiral EFT
Robert Roth – TU Darmstadt – 01/2013
Weinberg, van Kolck, Machleidt, Entem, Meissner, Epelbaum, Krebs, Bernard,...
low-energy effective field theoryfor relevant degrees of freedom (π,N)based on symmetries of QCD
long-range pion dynamics explicitly
short-range physics absorbed in con-tact terms, low-energy constants fit-ted to experiment (NN, πN,...)
hierarchy of consistent NN, 3N,...interactions (plus currents)
NN 3N 4N
LO
NLO
N2LO
N3LO
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many ongoing developments
• 3N interaction at N3LO, N4LO,...
• explicit inclusion of Δ-resonance
• YN- & YY-interactions
• formal issues: power counting,renormalization, cutoff choice,...
5
Chiral NN+3N Hamiltonians
Robert Roth – TU Darmstadt – 01/2013
standard Hamiltonian:
• NN at N3LO: Entem / Machleidt, 500 MeV cutoff
• 3N at N2LO: Navrátil, local, 500 MeV cutoff, fit to T1/2(3H) and E(3H, 3He)
standard Hamiltonian with modified 3N:
• NN at N3LO: Entem / Machleidt, 500 MeV cutoff
• 3N at N2LO: Navrátil, local, with modified LECs and cutoffs, refit to E(4He)
consistent N2LO Hamiltonian:
• NN at N2LO: Epelbaum et al., 450,...,600 MeV cutoff
• 3N at N2LO: Epelbaum et al., nonlocal, 450,...,600 MeV cutoff
consistent N3LO Hamiltonian:
• coming soon...
6
Similarity
Renormalization Group
Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)
Roth, Neff, Feldmeier — Prog. Part. Nucl. Phys. 65, 50 (2010)
Roth, Reinhardt, Hergert — Phys. Rev. C 77, 064033 (2008)
Hergert, Roth — Phys. Rev. C 75, 051001(R) (2007)
7
Similarity Renormalization Group
Robert Roth – TU Darmstadt – 01/2013
Wegner, Glazek, Wilson, Perry, Bogner, Furnstahl, Hergert, Roth, Jurgenson, Navratil,...
continuous transformation drivingHamiltonian to band-diagonal form
with respect to a chosen basis
unitary transformation of Hamiltonian (and other observables)
eHα = Uα†HUα
evolution equations for eHα and Uα depending on generator ηαd
dαeHα =ηα, eHα
d
dαUα = −Uαηα
dynamic generator: commutator with the operator in whoseeigenbasis H shall be diagonalized
ηα = (2μ)2Tint, eHα
simplicity and flexibilityare great advantages of
the SRG approach
solve SRG evolutionequations using two- &
three-body matrixrepresentation
8
SRG Evolution in Three-Body Space
Robert Roth – TU Darmstadt – 01/2013
perform SRG evolution for three-body Jacobi-HO matrix elements
0 . . . 18 20 22 24 26 28
(E, )
28
26
24
22
20
18
..
.
0
.
(E′,′)
α = 0 fm4
0 . . . 18 20 22 24 26 28
(E, )
α = 0.04 fm4
0 . . . 18 20 22 24 26 28
(E, )
α = 0.16 fm4
Jπ=
1 2
+,T=
1 2,ℏΩ=28MeV
0 2 4 6 8 10 12 14 16 18 20Nmx
-8
-6
-4
-2
.
E[MeV]
exp.
0 2 4 6 8 10 12 14 16 18 20Nmx
0 2 4 6 8 10 12 14 16 18 20Nmx
NCSM calculation3H ground state
suppress off-diagonal couplingimprove convergence behavior
9
Hamiltonian in A-Body Space
Robert Roth – TU Darmstadt – 01/2013
evolution induces n-body contributions eH[n]α
to Hamiltonian
eHα = eH[1]α+ eH[2]
α+ eH[3]
α+ eH[4]
α+ . . .
truncation of cluster series inevitable — formally destroys unitarityand invariance of energy eigenvalues (independence of α)
SRG-Evolved Hamiltonians
NN only: start with NN initial Hamiltonian and keep two-bodyterms only
NN+3N-induced: start with NN initial Hamiltonian and keep two-and induced three-body terms
NN+3N-full: start with NN+3N initial Hamiltonian and keep two-and all three-body terms
α-variation provides adiagnostic tool to assessthe contributions of omittedmany-body interactions
10
Importance Truncated
No-Core Shell Model
Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)
Navrátil, Roth, Quaglioni — Phys. Rev. C 82, 034609 (2010)
Roth — Phys. Rev. C 79, 064324 (2009)
Roth, Gour & Piecuch — Phys. Lett. B 679, 334 (2009)
Roth, Gour & Piecuch — Phys. Rev. C 79, 054325 (2009)
Roth, Navrátil — Phys. Rev. Lett. 99, 092501 (2007)
11
No-Core Shell Model
Robert Roth – TU Darmstadt – 01/2013
Barrett, Vary, Navratil, Maris, Nogga, Roth,...
NCSM is one of the most powerful anduniversal exact ab-initio methods
construct matrix representation of Hamiltonian using a basis of HOSlater determinants truncated w.r.t. HO excitation energy NmxℏΩ
solve large-scale eigenvalue problem for a few extremal eigenvalues
all relevant observables can be computed from the eigenstates
range of applicability limited by factorial growth of basis with Nmx & A
adaptive importance truncation extends the range of NCSM by reduc-ing the model space to physically relevant states
we have developed a parallelized IT-NCSM/NCSM code capable ofhandling 3N matrix elements up to E3mx = 16
12
Importance Truncated NCSM
Robert Roth – TU Darmstadt – 01/2013
Roth, PRC 79, 064324 (2009); PRL 99, 092501 (2007)
converged NCSM calcula-tions essentially restrictedto lower/mid p-shell
full 10ℏΩ calculation for16O getting very difficult(basis dimension > 1010)
0 2 4 6 8 10 12 14 16 18 20Nmx
-150
-140
-130
-120
-110
.
E[MeV]
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
ImportanceTruncation
reduce model spaceto the relevant basis
states using an a prioriimportance measurederived from MBPT
0 2 4 6 8 10 12 14 16 18 20Nmx
-150
-140
-130
-120
-110
.
E[MeV]
IT-NCSM
+ full NCSM
13
4He: Ground-State Energies
Robert Roth – TU Darmstadt – 01/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 16 ∞Nmx
-29
-28
-27
-26
-25
-24
-23
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 14 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 14 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
14
12C: Ground-State Energies
Robert Roth – TU Darmstadt – 01/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 ∞Nmx
-110
-100
-90
-80
-70
-60
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
15
16O: Ground-State Energies
Robert Roth – TU Darmstadt – 01/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 ∞Nmx
-180
-160
-140
-120
-100
-80
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
in progress:
explicit inclusion of
induced 4Nclear signature of
induced 4N originatingfrom initial 3N
16
16O: Lowering the Initial 3N Cutoff
Robert Roth – TU Darmstadt – 01/2013
standard
500 MeV
2 4 6 8 1012141618Nmx
-150
-145
-140
-135
-130
-125
-120
-115
-110
.
E[MeV]
cD= −0.2cE = −0.205
reduced 3N cutoff(cE refit to 4He binding energy)
450 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = −0.016
400 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = 0.098
350 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = 0.205
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1NN+3N-full
ℏΩ = 20MeV
lowering theinitial 3N cutoff
suppresses induced4N terms
17
Spectroscopy of 12C
Robert Roth – TU Darmstadt – 01/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
0
2
4
6
8
10
12
14
16
18
.
E[MeV]
NN+3N-induced
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
NN+3N-full
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
12CℏΩ = 16MeV
α = 0.04 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 1.88 fm−1
18
Spectroscopy of 12C
Robert Roth – TU Darmstadt – 01/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
0
2
4
6
8
10
12
14
16
18
.
E[MeV]
NN+3N-induced
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
NN+3N-full
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
12CℏΩ = 16MeV
α = 0.04 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 1.88 fm−1
spectra largelyinsensitive toinduced 4N
19
The Bottom Line...
Robert Roth – TU Darmstadt – 01/2013
beyond the lightest nuclei, SRG-induced 4N contributions af-fect the absolute energies (but not the excitation energies)
with the inclusion of the leading 3N interaction we already obtain agood description of spectra (and ground states)
breakthrough in computation, transformation and managementof 3N matrix-elements
applications: spectroscopy of p- and sd-shell nuclei and groundstates with reduced initial 3N cutoff
next-generation SRG: include induced 4N contributions or sup-press many-body terms with modified SRG-generators
next-generation chiral 3N: use consistent chiral Hamiltoniansand propagate uncertainties to many-body observables
20
Ab Initio Calculations
for p- and sd-Shell Nuclei
21
Spectroscopy of Carbon Isotopes
Robert Roth – TU Darmstadt – 01/2013
PRELI
MINARY
0+ 1
2+ 1
2+ 1
2 4 6 8 Exp.0
2
4
6
8
.
E[MeV]
10C
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 02+ 1
4+ 0
2 4 6 8 Exp.0
2
4
6
8
10
12
14
16
12C
0+ 1
0+ 1
0+ 1
1+ 1
2+ 1
2+ 1
2+ 14+ 1
2 4 6 8 Exp.0
2
4
6
8
10
14C
0+ 2
0+ 2
2+ 2
2+ 23? 24+ 2
2 4 6 Exp.Nmx
0
1
2
3
4
.
E[MeV]
16C
0+ 3
2+ 3
2+ 3
2 4 6 Exp.Nmx
0
1
2
3
4
18C
0+ 4
2+ 4
2 4 6 Exp.Nmx
0
1
2
3
4
20C
NN+3N-full
Λ3N = 400MeV
α = 0.08 fm4
ℏΩ = 16MeV
22
Spectroscopy of Carbon Isotopes
Robert Roth – TU Darmstadt – 01/2013
PRELI
MINARY
2 4 6 80
1
2
3
4
5
6
7
8
.
B(E2,2+→
0+)[e2fm
4] 10C
2 4 6 80
1
2
3
4
5
6
7
8
12C
2 4 6 80
1
2
3
4
5
6
7
8
14C
2 4 6Nmx
0
0.5
1
1.5
2
2.5
3
.
B(E2,2+→
0+)[e2fm
4] 16C
2 4 6Nmx
0
0.5
1
1.5
2
2.5
3
18C
2 4 6Nmx
0
1
2
3
4
5
20C
NN+3N-full
Λ3N = 400MeV
α = 0.08 fm4
ℏΩ = 16MeV
B(E2,2+1→ 0+
gs)
B(E2,2+2 → 0+gs)
NEW
EXPE
RIME
NT
ANL,Mc
Cutch
anetal.
NEW
EXPE
RIME
NT
NSCL, Petri e
t al.
NEW
EXPE
RIME
NT
NSCL, Voss e
t al.
NEW
EXPE
RIME
NT
NSCL, Petri e
t al.
23
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 01/2013
Hergert, Binder, Calci, Langhammer, Roth; in prep.
PRELI
MINARY
NN+3N-induced(chiral NN)
2 4 6 8 10 12 14 16 18Nmx
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
12O
14O
16O18O20O22O24O26O
NN+3N-full(chiral NN+3N)
2 4 6 8 10 12 14 16 18Nmx
12O
14O
16O18O20O22O26O24O
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
24
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 01/2013
Hergert, Binder, Calci, Langhammer, Roth; in prep.
PRELI
MINARY
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
parameter-freeab initio calculations with
full 3N interactionshighlights predictive power
of chiral NN+3N Hamiltonians
25
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 01/2013
Hergert, Binder, Calci, Langhammer, Roth; in prep.
PRELI
MINARY
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM IM-SRG (prel.)
(H. Hergert)
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
26
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 01/2013
Hergert, Binder, Calci, Langhammer, Roth; in prep.
PRELI
MINARY
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM IM-SRG (prel.)
(H. Hergert)
È CCSD
Î Λ-CCSD(T)
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
different many-bodyapproaches using sameNN+3N Hamiltonian give
consistent results minor differences are
understood (NO2B, E3mx,...)
27
Ab Initio Calculations
for Heavy Nuclei
Roth, Binder, Vobig et al. — Phys. Rev. Lett. 109, 052501 (2012)
Binder, Langhammer, Calci et al. — arXiv:1211.4748
Hergert, Bogner, Binder et al. — arXiv:1212.1190
28
Coupled-Cluster Method
Robert Roth – TU Darmstadt – 01/2013
Coester, Kuemmel, Bischop, Dean, Piecuch, Walet, Papenbrock, Hagen, Binder,...
CC is one of the most efficientmethods for the description of ground statesof medium-mass or heavy closed-shell nuclei
many-body state parametrized as exponential wave operator appliedto single-determinant reference state
ref
ΨCC= Ωref= expT1 + T2 + T3 + · · ·+ TA
ref
truncation with respect to n-particle-n-hole excitation operators Tn
solve non-linear system of equations for the amplitudes in T1, T2, T3,...
extensions to near-closed-shell nuclei and excited states throughequations-of-motion methods
we have developed a parallelized CC code for CCSD and Λ-CCSD(T)
29
Inclusion of 3N Interactions
Robert Roth – TU Darmstadt – 01/2013
premium option: explicit 3N
• extend coupled-cluster equations for explicit 3N interactions
• CCSD-3B, Λ-CCSD(T)-3B are feasible, but much more expensive
low-cost option: normal-ordered two-body approximation
• write 3N interaction in normal-ordered form with respect to the actualA-body reference determinant (HF state)
V3N =∑
V3N
†††
=W0B +∑
W1B
†+∑
W2B
††
+∑
W3B
†††
• discard normal-ordered three-body term and use two-body coupled-cluster formalism
30
CCSD with Explicit 3N Interactions
Robert Roth – TU Darmstadt – 01/2013
Binder et al.; arXiv:1211.4748
NN+3N-induced
-130
-120
-110
-100
-90
.
E[MeV]
exp.
4 6 8 10 12emx
-170
-160
-150
-140
-130
-120
-110
.
E[MeV]
exp.
NN+3N-full
16OℏΩ = 20 MeV
4 6 8 10 12emx
24OℏΩ = 20 MeV
CCSD-3B( Î)
CCSD-NO2B( ◊ Í)
α = 0.02 fm4
◊ α = 0.04 fm4
Î Í α = 0.08 fm4
HF basis
E3mx = 12
Λ3N = 400 MeV
31
CCSD with Explicit 3N Interactions
Robert Roth – TU Darmstadt – 01/2013
Binder et al.; arXiv:1211.4748
NN+3N-induced
-380
-360
-340
-320
-300
-280
-260
.
E[MeV]
exp.
4 6 8 10 12emx
-500
-450
-400
-350
-300
-250
.
E[MeV]
exp.
NN+3N-full
40CaℏΩ = 24 MeV
4 6 8 10 12emx
48CaℏΩ = 28 MeV
CCSD-3B( Î)
CCSD-NO2B( ◊ Í)
α = 0.02 fm4
◊ α = 0.04 fm4
Î Í α = 0.08 fm4
HF basis
E3mx = 12
Λ3N = 400 MeVNO2B approximationreproduces explicit-3Nresults with better than
1% accuracy
32
CCSD with Explicit 3N Interactions
Robert Roth – TU Darmstadt – 01/2013
Binder et al.; arXiv:1211.4748
NN+3N-induced
4 6 8 10 12emx
-550
-500
-450
-400
-350
.
E[MeV]
exp.
NN+3N-full
4 6 8 10 12emx
56NiℏΩ = 28 MeV
CCSD-3B( Î)
CCSD-NO2B( ◊ Í)
α = 0.02 fm4
◊ α = 0.04 fm4
Î Í α = 0.08 fm4
HF basis
E3mx = 12
Λ3N = 400 MeV
E3mx truncation of 3N matrix elements has sig-nificant effects for A ¦ 60
many-body framework is ready to go to heaviernuclei... still cheap with NO2B approximation
33
Λ-CCSD(T) with NO2B Approximation
Robert Roth – TU Darmstadt – 01/2013
Binder et al.; arXiv:1211.4748
NN+3N-full
-130
-120
-110
-100
.
E[MeV]
16OℏΩ = 20 MeV
4 6 8 10 12emx
-380
-360
-340
-320
-300
-280
-260
.
E[MeV]
40CaℏΩ = 24 MeV
NN+3N-full
-170
-160
-150
-140
-130
-120
-11024OℏΩ = 20 MeV
4 6 8 10 12emx
-500
-450
-400
-350
-300
-25048CaℏΩ = 28 MeV
Λ-CCSD(T)( Î)
CCSD( ◊ Í)
α = 0.02 fm4
◊ α = 0.04 fm4
Î Í α = 0.08 fm4
HF basis
E3mx = 14
Λ3N = 400 MeV
all NO2B
34
Conclusions
35
Conclusions
Robert Roth – TU Darmstadt – 01/2013
new era of ab-initio nuclear structure and reaction theoryconnected to QCD via chiral EFT
• chiral EFT as universal starting point... propagate uncertainties &provide feedback
consistent inclusion of 3N interactions in similarity transfor-mations & many-body calculations
• breakthrough in computation & handling of 3N matrix elements
innovations in many-body theory: extended reach of exactmethods & improved control over approximations
• versatile toolbox for different observables & mass ranges
many exciting applications ahead...
36
Epilogue
Robert Roth – TU Darmstadt – 01/2013
thanks to my group & my collaborators
• S. Binder, A. Calci, B. Erler, E. Gebrerufael,H. Krutsch, J. Langhammer, S. Reinhardt, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, K. Vobig, R. WirthInstitut für Kernphysik, TU Darmstadt
• P. NavrátilTRIUMF Vancouver, Canada
• J. Vary, P. MarisIowa State University, USA
• S. Quaglioni, G. HupinLLNL Livermore, USA
• P. PiecuchMichigan State University, USA
• H. Hergert, K. HebelerOhio State University, USA
• P. PapakonstantinouIPN Orsay, F
• C. ForssénChalmers University, Sweden
• H. Feldmeier, T. NeffGSI Helmholtzzentrum
JUROPA LOEWE-CSC HOPPER
COMPUTING TIME
37
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