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Scattering of light nuclei. Sofia Quaglioni in collaboration with Petr Návratil 19 th International IUPAP Conference on Few-Body Problems in Physics Bonn, September 4, 2009. Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551. - PowerPoint PPT Presentation
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Lawrence Livermore National Laboratory
Scattering of light nuclei
LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
Sofia Quaglioniin collaboration with Petr Návratil
19th International IUPAP Conference on Few-Body Problems in Physics
Bonn, September 4, 2009
2LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Nuclear reactions
Nuclear physics underlying many key astrophysical processes• Formation of the chemical elements• Solar neutrino problem• Stellar evolution
Tools for studying exotic nuclei• Structure inferred from breakup reactions• Most low-lying states are unbound
A formidable challenge to nuclear theory …• Main difficulty: scattering states
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Lawrence Livermore National Laboratory
Disclaimer
As they deserve, nuclear reactions are attracting much attention There are many interesting new developments … … forgive me if I miss to mention some of them!
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Lawrence Livermore National Laboratory
Mic
rosc
opic
A
ll nu
cleo
ns a
re a
ctiv
e
Exa
ct P
auli
prin
cipl
e Few-nucleon techniques using realistic NN (+ NNN) interactions
• Faddeev, AGS (Deltuva et al.), FY (Lazauskas et al.), HH (Viviani et al.), LIT (Bacca et al.), RRGM (Hoffman et al.), …
Many-body techniques using realistic NN (+ NNN) interactions• GFMC (Nollett et al.), NCSM/RGM (Navrátil, SQ), FMD (Neff et al.), …
Cluster techniques using semi-realistic NN interactions• RGM, GCM (Descouvemont et al.), ...
Reaction approachesC
lust
er fe
w-b
ody
N
-nuc
leus
inte
ract
ions
(
usua
lly) i
nert
core
Techniques using local/non-local optical potentials
• Faddeev, AGS (Deltuva et al.), …
Techniques using local optical potentials• CDCC (Moro et al.), XCDCC (Summers et al.), DWBA,
adiabatic approaches (Baye et al.), …
Halo effective-field theories (Higa et al.), …
PRC 79, 054007 (2009)
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Lawrence Livermore National Laboratory
Our goal:ab initio approach to low-energy reactions of light nuclei
Start with the ab initio description of the structure of light nuclei• The ab initio no-core shell model (NCSM)
A successful ab initio approach to nuclear structure Capable of employing chiral effective field theory (EFT) NN + NNN potentials for
A>4 Covers nuclei beyond the s-shell Incorrect description of wave-function asymptotic (r >5fm), no coupling to
continuum
Add microscopic description of nucleus-nucleus scattering• The resonating-group method (RGM)
A successful microscopic cluster technique (also multi-cluster) Preserves Pauli principle, includes Coulomb force Describes reactions and clustering in light nuclei (also multichannel, transfer etc.) Usually simplified NN interactions and internal description of the clusters
Combine: NCSM/RGM ab initio bound & scattering states in light nuclei• NCSM - single-particle degrees of freedom• RGM - clusters and their relative motion
6LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
The ab initio no-core shell model (NCSM) in brief
The NCSM is a technique for the solution of the A-nucleon bound-state problem
Hamiltonian• “realistic” (= reproduce NN data with high precision) NN potentials:
coordinate space: Argonne … momentum space: CD-Bonn, EFT N3LO, …
• NNN interactions: Tucson-Melbourne TM’, EFT N2LO
Finite harmonic oscillator (HO) basis • A-nucleon HO basis states
Jacobi relative or Cartesian single-particle coordinates
• complete Nmaxħ model space translational invariance preserved even with Slater-determinant (SD) basis
Constructs effective interaction tailored to model-space truncation• unitary transformation in a n-body cluster approximation (n=2,3)
1max NN
Convergence to exact solution with increasing Nmax
7LLNL-PRES-XXXXXX
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Resonating-group method
Ansatz:
The many-body Schrodinger equation is mapped onto:
Input: ,
Output (e.g., R-matrix method on Lagrange mesh): , scattering matrix
Norm kerne
l
Hamiltoniankernel
eigenstates of H(A-a), H(a) in the NCSM basis
NCSM/RGM: NCSM microscopic wave functions for the clusters involved, and realistic (bare or derived NCSM effective) interactions among nucleons.
8LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Single-nucleon projectile: the norm kernel(A-1)
(1)
(A-1)
(A-1) (1)
(1,…,A-1)
(A)
(1,…,A-1)
(A)
€
SDψ μ1
(A−1) a+aψν 1
(A−1)
SD
“Direct term” treated exactly. “Exchange” term localized d expanded in HO radial w.f.
9LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Single-nucleon projectile basis: the Hamiltonian kernel
(A-1) (A-1)(A-2)
“direct potential”
“exchange potential”
(A-1)(1)
(1,…,A-1)
(A)
(1,…,A-1)
(A)
+ terms containing NNN potential
€
SDψ μ1
(A−1) a+aψ ν 1
(A−1)
SD
€
SDψ μ1
(A−1) a+a+ a aψν 1
(A−1)
SD
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Lawrence Livermore National Laboratory
The RGM kernels in the single-nucleon projectile basis
(A-1)(A-2)
(A-1)
(A-1)(1)
+ (A-1) “direct
potential”
“exchangepotential”
In the A=5 system the 1/2+ (2S1/2) is a Pauli-forbidden state, therefore g.s. in P wave
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Lawrence Livermore National Laboratory
NCSM/RGM ab initio calculation of n-4He phase shifts
NCSM/RGM calculation with n + 4He(g.s.)
Low-momentum Vlowk NN potential: convergence reached with bare interaction
EFT N3LO NN potential: convergence reached with two-body effective interaction
4Hen
Is everything else under control? … need verification against independent ab initio approach!
No fit. No free parameters. Convergence in Nmax under control.
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Lawrence Livermore National Laboratory
The A=4 system as a test ground for the NCSM/RGM approach within the single-nucleon-projectile basis
NCSM/RGM calculation with n + 3H(g.s.) and p + 3He(g.s.), respectively EFT N3LO NN potential: convergence with 2-body effective interaction Benchmark: AGS results (+), Deltuva & Fonseca, PRC75, 014005 (2007)
The omission of A = 3 partial waves with 1/2 < J ≤ 5/2 leads to effects of comparable magnitude on the AGS results. Need to include target excited (here breakup) states!
3Hn
3Hep
13LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
n-4He phase shifts with EFT N3LO NN interaction
Very mild effects of JpT = 0+0 on 2S1/2
The negative-parity states have larger effects on P phases (coupling to s-wave of relative motion)
• 0-0, 1-0 and 1-1 affect 2P1/2
• 2-0 and 2-1 affect 2P3/2
NCSM/RGM calculation with n + 4He(g.s., ex.) EFT N3LO NN potential: convergence with 2-body effective interaction
4Hen
The resonances are sensitive to the inclusion of the first six excited states of 4He
14LLNL-PRES-XXXXXX
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Nucleon- phase-shifts with EFT N3LO NN interaction
NCSM/RGM calculation with N+4He(g.s., 0+00-01-01-12-
02-1) EFT N3LO NN potential: convergence with 2-body effective interaction
2S1/2 in agreement with Expt. (dominated by N-repulsion - Pauli principle)
Insufficient spin-orbit splitting between 2P1/2 and 2P3/2 (sensitive to interaction)
Fully ab initio, very promising results. The resonances are sensitive to NNN force.
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n+4He differential cross section and analyzing power
NCSM/RGM calculations with• N + 4He(g.s., 0+0)• SRG-N3LO NN potential with Λ=2.02 fm-1
Differential cross section and analyzing power @17 MeV neutron energy
• Polarized neutron experiment at Karlsruhe
4Hen
Good agreement for energies beyond low-lying resonances
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Lawrence Livermore National Laboratory
NCSM/RGM ab initio calculation of n+7Li scattering7Li
n
Nmax = 8 NCSM/RGM calculation with n + 7Li(g.s.,1/2-, 7/2-) SRG-N3LO NN potential with Λ = 2.02 fm-1
Qualitative agreement with experiment:• Calculated broad 1+ resonance • 3+ resonance not seen when the 7/2- state of 7Li is not included
7Li
Predicted narrow 0+ and 2+ resonances seen at recent p+7Be experiment at FSU
Expt: a01=0.87(7) fm a02=-3.63(5) fmCalc: a01=0.73 fm a02=-1.42 fm
17LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
11Be bound states and n-10Be phase shifts
10Ben
NCSM/RGM NCSM
3.02.52.01.51.00.50.0-0.5-1.0
E [MeV]
Expt.
1/2-
1/2+
Parity-inverted g.s. of 11Be understood!
11Be
Exotic nuclei: vanishing of magic numbers, abnormal spin-parity of ground states, …
The g.s. of 11Be one of the best examples• Observed spin-parity : 1/2+• p-shell expected: 1/2-
Large-scale NCSM calculations, Forssen et al., PRC71, 044312 (2005)
• Several realistic NN potentials• Calculated g.s. spin-parity: 1/2-
NCSM/RGM calculation with CD-Bonn• n + 10Be(g.s.,21
+,22+,11
+)• Calculated g.s. spin-parity : 1/2+
What happens? Substantial drop of the relative kinetic energy due to the rescaling of the relative wave function when the Whittaker tail is recovered
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The deuteron-projectile formalism: norm kernel(A-2)
(2)
€
1− ˆ P ijk= A−1
A
∑i=1
A−2
∑ + ˆ P i,A ˆ P j,A−1i< j=1
A−2
∑(1,…,A-2)
(A-1,A)
(1,…,A-2)
(A-1,A)
€
Nμl ' ,νl(A−2,2) r',r( ) = δμν δ
l 'l
δ r' − r( )r'r
€
−2(A − 2) Rn ' l ' (r')
n 'n
∑ Φμn ' l '(A−2,2)JT PA−2,A−1 Φνnl
(A−2,2)JT Rnl (r)
€
(A − 2)(A − 3)
2R
n ' l ' (r')n 'n
∑ Φμn ' l '(A−2,2)JT PA−2,A−1PA−3,A Φνnl
(A−2,2)JT Rnl (r)
€
SDψ μ1
(A−2) a+aψν 1
(A−2)
SD
€
SDψ μ1
(A−2) a+a+ a aψν 1
(A−2)
SD
19LLNL-PRES-XXXXXX
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NCSM/RGM ab initio calculation of d-4He scattering
Nmax = 8 NCSM/RGM calculation with d(g.s.) + 4He(g.s.) SRG-N3LO potential with Λ = 2.02 fm-1
4Hed
Calculated two resonances: 2+0, 3+0 The 1+0 g.s. is still unbound: convergence moves towards bound state
6Li
20LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Toward the first ab initio calculation of theDeuterium-Tritium fusion
€
32
+ 12
€
dr 2rA1
∧H − E( ) A1
∧A1
∧H − E( ) A2
∧
A2
∧H − E( ) A1
∧A2
∧H − E( ) A2
∧
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
g1(r)r
g2(r)r
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟= 0
€
∫r’
n
r n
r’ 3H
d
rd
3H
r’
n
r n
r’ 3H
d
rd
3H
3H
d 4He
n
✔
✔
Work in progress on coupling between d + 3H and n + 4He bases
21LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Conclusions and Outlook
With the NCSM/RGM approach we are extending the ab initio effort to describe low-energy reactions and weakly-bound systems
Recent results for nucleon-nucleus scattering with NN realistic potentials:• n-3H, n-4He, n-10Be and p-3,4He • S.Q. and P. Navrátil, PRL 101, 092501 (2008), PRC 79, 044606 (2009)
New results with SRG-N3LO: • N-4He, n-7Li, (also N-12C and
N-16O, not presented here)• Initial results for d-4He scattering• First steps towards 3H(d,n)4He
To do:• Coupling of N+A and d+(A-1)• Inclusion of NNN force• Heavier projectiles: 3H, 3He, 4He• NCSM with continuum (NCSMC) • Three-cluster NCSM/RGM and treatment of three-body continuum
€
ΨAJ = cλ AλJ∑ + d
r r ϕ ν∫ (
r r ) ˆ A Φν
r r
(A−a,a )∑
€
H hh H ⎛ ⎝ ⎜
⎞ ⎠ ⎟cϕ ⎛ ⎝ ⎜
⎞ ⎠ ⎟= E
1 gg N ⎛ ⎝ ⎜
⎞ ⎠ ⎟cϕ ⎛ ⎝ ⎜
⎞ ⎠ ⎟
22LLNL-PRES-XXXXXX
Lawrence Livermore National Laboratory
Thanks
Petr Navrátil, without whom much of this work would not have been possible
Our collaborators:• R. Roth, GSI, on the Importance-truncation NCSM• S. Bacca, TRIUMF, on the NCSMC
Thank you for your attention!
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