Transcript FB19-Quaglionix

Lawrence Livermore National Laboratory
Scattering of light nuclei
Sofia Quaglioni
in collaboration with Petr Návratil
19th International IUPAP Conference
on Few-Body Problems in Physics
Bonn, September 4, 2009
Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551
This work performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
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Nuclear reactions

Nuclear physics underlying many key astrophysical processes
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•
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
Tools for studying exotic nuclei
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•

Formation of the chemical elements
Solar neutrino problem
Stellar evolution
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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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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 All nucleons are active
 Exact Pauli principle
 N-nucleus interactions
 (usually) inert core
Cluster few-body
Microscopic
Reaction approaches

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
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GFMC (Nollett et al.), NCSM/RGM (Navrátil, SQ), FMD (Neff et al.), …

Cluster techniques using semi-realistic NN interactions
• RGM, GCM (Descouvemont et al.), ...

Techniques using local/non-local
optical potentials
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
Faddeev, AGS (Deltuva et al.), …
Techniques using local optical potentials
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CDCC (Moro et al.), XCDCC (Summers et al.), DWBA,
adiabatic approaches (Baye et al.), …
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 Halo
effective-field
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PRC 79, 054007 (2009)
theories (Higa et al.), …
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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 > 5 fm), no coupling to continuum

Add microscopic description of nucleus-nucleus scattering
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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
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NCSM - single-particle degrees of freedom
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RGM - clusters and their relative motion
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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
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“realistic” (= reproduce NN data with high precision) NN potentials:
 coordinate space: Argonne …
 momentum space: CD-Bonn, EFT N3LO, …
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NNN interactions:
 Tucson-Melbourne TM’, EFT

N  N max  1
N2LO
Finite harmonic oscillator (HO) basis
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A-nucleon HO basis states
 Jacobi relative or Cartesian single-particle coordinates
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complete Nmaxħ model space
 translational invariance preserved even with Slater-determinant (SD) basis

Constructs effective interaction tailored to model-space truncation
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unitary transformation in a n-body cluster approximation (n=2,3)
Convergence
to exact solution with increasing Nmax
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Resonating-group method

Ansatz:

The many-body Schrodinger equation is mapped onto:
Norm
kernel
Hamiltonian
kernel
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Input:

Output (e.g., R-matrix method on Lagrange mesh):
,
 eigenstates of H(A-a), H(a) in the NCSM basis
, scattering matrix
NCSM/RGM: NCSM microscopic wave functions for the clusters involved,
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realistic
(bare
or derived NCSM effective) interactions among nucleons.
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(A-1)
Single-nucleon projectile: the norm kernel
(1,…,A-1)
(1)
(1,…,A-1)
(A)
(A)
SD
 (A-1) 
(A1) a a (A1)
1
1
SD

(A-1)
(1)
“Direct term”
treated
exactly.
“Exchange” term localized  d expanded in HO radial w.f.
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(A-1)
Single-nucleon projectile basis: the Hamiltonian kernel
(1,…,A-1)
(1,…,A-1)
(A)
(A)
+ terms containing NNN potential

(A1)
SD
1
a a  1

(1)
(A1)
SD
SD

(A-1) 
(A1) a a a a (A1)
1
1
SD
 (A-1)(A-2) 

“direct potential”
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“exchange potential”
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The RGM kernels in the single-nucleon projectile basis
(A-1)
(1)
 (A-1) 
“direct
potential”
+ (A-1) 
“exchange
potential”
 (A-1)(A-2) 

+ (2S ) is a Pauli-forbidden state, therefore g.s. in P wave
In
the A=5
system
the 1/2
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4He
NCSM/RGM ab initio calculation of n-4He phase shifts

NCSM/RGM calculation with n + 4He(g.s.)
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Low-momentum Vlowk NN potential:
convergence reached with bare interaction

EFT N3LO NN potential: convergence
reached with two-body effective interaction
n
No fit.
No free parameters.
Convergence in Nmax
under control.
Is everything else under
control? … need
verification against
independent ab initio
approach!
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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)
n
3H
p
3He
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!
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n-4He phase shifts with EFT N3LO NN interaction
4He


NCSM/RGM calculation with n + 4He(g.s., ex.)
EFT N3LO NN potential: convergence with 2-body effective interaction
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
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)
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n
0-0, 1-0 and 1-1 affect 2P1/2
2-0 and 2-1 affect 2P3/2
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The resonances
are sensitive
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Nucleon- phase-shifts with EFT N3LO NN interaction
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NCSM/RGM calculation with N+4He(g.s., 0+0
0-0
1-0
1-1
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2-1)
EFT N3LO NN potential: convergence with 2-body effective interaction

2S
1/2 in

Insufficient spin-orbit splitting between 2P1/2 and 2P3/2 (sensitive to interaction)
agreement with Expt. (dominated by N- repulsion - Pauli principle)
Fully abLivermore
initio, very
promising
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n+4He differential cross section and analyzing power
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NCSM/RGM calculations with
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
4He
n
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
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Polarized neutron experiment at Karlsruhe
Good
agreement
for energies beyond low-lying resonances
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7Li
NCSM/RGM ab initio calculation of n+7Li scattering



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:
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n
7Li
Calculated broad 1+ resonance
3+ resonance not seen when the 7/2- state of 7Li is not included
Expt: a01=0.87(7) fm
a02=-3.63(5) fm
Calc: a01=0.73 fm
a02=-1.42 fm
Predicted
narrow
0+ and
2+ resonances seen at recent p+7Be experiment at FSU
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11Be

Exotic nuclei: vanishing of magic numbers,
abnormal spin-parity of ground states, …
The g.s. of 11Be one of the best examples
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
Parity-inverted
g.s. of 11Be
understood!
2.5
Large-scale NCSM calculations, Forssen et al.,
PRC71, 044312 (2005)
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•

Observed spin-parity : 1/2+
p-shell expected: 1/2-
3.0
E [MeV]

bound states and n-10Be phase shifts
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
1/21/2+
Expt. NCSM/RGM NCSM
10Be
11Be
n
Several realistic NN potentials
Calculated g.s. spin-parity: 1/2-
NCSM/RGM calculation with CD-Bonn
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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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(A-2)
The deuteron-projectile formalism: norm kernel
(1,…,A-2)
A 2
(A-1,A)
'
N (Al '2,2)
r
,r d dl ' l

,l

1 
A 2
A
 Pˆij 
i1 k A 1
(1,…,A-2)
 Pˆi,A Pˆ j,A 1
i j1
(2)
(A-1,A)
d r'  r
r' r
(A2,2)JT
2(A  2) Rn ' l ' (r' ) (A2,2)JT
P

Rnl (r)
' '
A2,A1
nl
n l
n 'n


(A  2)(A  3)
(A2,2)JT
Rn ' l ' (r' ) (A2,2)JT
P
P

Rnl (r)
' '

A2,A1
A3,A
nl
n l
2
n 'n

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SD
(A2) a a (A2)
1
1
SD
SD
(A2) a a a a (A2)
1
1
SD
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4He
NCSM/RGM ab initio calculation of d-4He scattering


d
Nmax = 8 NCSM/RGM calculation with d(g.s.) + 4He(g.s.)
SRG-N3LO potential with Λ = 2.02 fm-1
6Li


Calculated two resonances: 2+0, 3+0
The 1+0 g.s. is still unbound: convergence moves towards bound state
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Toward the first ab initio calculation of the
Deuterium-Tritium fusion

3H
d
n
4H
e
✔
dr


 r’
r
A
H

E
A



1 
 1
n
n
2 
r 

r
 r’ 3H 
A
H

E
A


 d
2
1 
n

r  g1 (r) 
A1 H  E  A2
 


n
d   r 
 
✔


 3
r
g
(r)

r’ 3H A H  E A H
 2 d   2r 
2
d

r’


3H
0
3 1
2 2


Work in National
progress
on coupling between d + 3H and n + 4He bases
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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:
•
•

New results with SRG-N3LO:
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•
•

n-3H, n-4He, n-10Be and p-3,4He
S.Q. and P. Navrátil, PRL 101, 092501 (2008), PRC 79, 044606 (2009)
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
AJ  c AJ    dr  (r )Aˆ (Aa,a)
r
To do:
H h c  1
• Coupling of N+A and d+(A-1)


  E 
• Inclusion of NNN force
h
  g
• Heavier projectiles: 3H, 3He, 4He
• NCSM with continuum (NCSMC)
• Livermore
Three-cluster
and treatment of three-body continuum
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H
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
g c 
 
N  
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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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