Lawrence Livermore National Laboratory

Scattering of light nuclei

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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

Sofia Quaglioni

in collaboration with Petr Návratil19th International IUPAP Conference

on Few-Body Problems in PhysicsBonn, September 4, 2009

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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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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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Mic

rosc

op

ic

All

nucl

eons

are

act

ive

E

xact

Pau

li pr

inci

ple

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 f

ew-b

od

y

N-n

ucle

us in

tera

ctio

ns

(us

ually

) in

ert c

ore

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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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

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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

• “realistic” (= reproduce NN data with high precision) NN potentials: coordinate space: Argonne …

momentum space: CD-Bonn, cEFT N3LO, …

• NNN interactions: Tucson-Melbourne TM’, cEFT 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

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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.

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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.

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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+aψν 1

(A−1)

SD

€

SDψ μ1

(A−1) a+a+ a a ψν 1

(A−1)

SD

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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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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

4He

n

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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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

3He

p

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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

4He

n

The resonances are sensitive to the inclusion of the first six excited states of 4He

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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

4He

n

Good agreement for energies beyond low-lying resonances

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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

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11Be bound states and n-10Be phase shifts

10Ben

NCSM/RGM NCSM

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-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−1

i< j=1

A−2

∑(1,…,A-2)

(A-1,A)

(1,…,A-2)

(A-1,A)

€

Nμl ' ,νl

(A−2,2) r',r( ) = δμν δl ' 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+ aa ψν 1

(A−2)

SD

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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

4He

d

Calculated two resonances: 2+0, 3+0 The 1+0 g.s. is still unbound: convergence moves towards bound state

6Li

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Toward the first ab initio calculation of theDeuterium-Tritium fusion

€

3

2

+ 1

2€

dr2

r

A1

∧

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

r

d

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

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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:• 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 h

h H

⎛

⎝ ⎜

⎞

⎠ ⎟c

ϕ

⎛

⎝ ⎜

⎞

⎠ ⎟= E

1 g

g N

⎛

⎝ ⎜

⎞

⎠ ⎟c

ϕ

⎛

⎝ ⎜

⎞

⎠ ⎟

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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!