Ab initio many-body calculations of light-ion reactions

Lawrence Livermore National Laboratory

Ab initio many-body calculations of light-ion reactions

LLNL-PRES-425682

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

Petr NavratilCollaborators: Sofia Quaglioni (LLNL), R. Roth (TU Darmstadt), E. Jurgenson (LLNL)

6th ANL/MSU/JINA/INT FRIB Theory Workshop, Argonne National Laboratory, March 23 - 26, 2010

2LLNL-PRES--425682


Outline

Motivation

Ab initio no-core shell model (NCSM)

Extension of the no-core shell model by resonating group method (ab initio NCSM/RGM)• Nucleon-alpha scattering• n-3H, p-3He cross sections• 11Be parity-inverted ground state• n-7Li, N-12C, n-16O scattering

Calculations with wave functions from importance-truncated NCSM

• d-T fusion

Outlook

3LLNL-PRES--425682


Our goal is to develop an ab initio theory to understand nuclear structure and reactions of light nuclei

Nuclei are quantum many-body systems with bound

states, resonances, scattering states

• Bound-state techniques not sufficient

Our approach - combining the ab initio no-core shell

model (NCSM) with the resonating group method (RGM)

ab initio NCSM/RGM• NCSM - single-particle degrees of freedom

• RGM - clusters and their relative motion

PRL 99, 042501(2007)

NCSMNCSM

RGMRGM

The Hoyle state missing

Preserves Pauli principle and translational invariance

Important as nucleons are fermions and nuclei self-bound

Preserves Pauli principle and translational invariance

Important as nucleons are fermions and nuclei self-bound

4LLNL-PRES--425682


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

Realistic nuclear Hamiltonian

• High-precision nucleon-nucleon potentials

• Three-nucleon interactions

Finite harmonic oscillator (HO) basis

• A-nucleon HO basis states Jacobi relative coordinates

Cartesian single-particle coordinates

• complete Nmaxh model space Translational invariance preserved even with single-particle coordinate Slater-determinant (SD) basis

Effective interaction tailored to model-space truncation for NN(+NNN) potentials• Lee-Suzuki-Okamoto unitary transformation in n-body cluster approximation (n=2,3)

Or a sequence of unitary transformations in momentum space:• Similarity-Renormalization-Group evolved NN(+NNN) potential

• Soft: No further model-space dependent effective interaction needed Variational calculation

Convergence to exact solution with increasing Nmax for

bound states. No coupling to continuum.

N=0N=1

N=2

N=4N=3

N=5

5LLNL-PRES--425682


NCSM Convergence: 4He

Chiral N3LO NN plus N2LO NNN potential

• Bare interaction (black line) Variational calculation Strong short-range

correlations Large basis needed

• Similarity-renormalization group evolved effective interaction (red line) Unitary transformation Two- plus three-body

components, four-body omitted

Softens the interaction Variational calculation

Smaller basis sufficient

6LLNL-PRES--425682


The ab initio NCSM/RGM in a snapshot

Ansatz:

Non-local integro-differential coupled-channel equations:

Hamiltonian kernel Norm kernel

Many-body Schrödinger equation:

eigenstates of H(A-a) and H(a)

in the ab initio

NCSM basis

either bare interaction or NCSM effective interaction

NCSM/RGM: NCSM microscopic wave functions for the clusters involved,

and realistic (bare or derived NCSM effective) interactions among nucleons.

Proper boundary conditions for scattering and/or bound states

7LLNL-PRES--425682


Localized parts of kernels expanded in the HO basis

Single-nucleon projectile: the norm kernel(A-1)

(1)

(A-1)

(A-1) (1)

(1,…,A-1)

(A)

(1,…,A-1)

(A)

SD1

(A 1) aa 1

(A 1)

SD

8LLNL-PRES--425682


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 waveIn the A=5 system the 1/2+ (2S1/2) is a Pauli-forbidden state, therefore g.s. in P wave

9LLNL-PRES-405596


NCSM/RGM ab initio calculation of n-4He phase shifts

Fully ab initio. No fit. No free parameters. Good convergence with respect to Nmax

4He

n

n-4He phase shifts: SRG-N3LO, =2.02 fm-1

Similarity-renormalization-group (SRG) evolved chiral N3LO NN interaction (R. Roth)

Low-momentum Vlowk NN potential

convergence reached with bare interaction

VlowkVlowk

10LLNL-PRES-405596


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

NNN missing: Good agreement only for energies beyond low-lying 3/2- resonanceNNN missing: Good agreement only for energies beyond low-lying 3/2- resonance

11LLNL-PRES-405596


p+4He differential cross section and analyzing power

12LLNL-PRES--425682


Neutron-triton elastic scattering at 14 MeV

Important for the NIF physics• deuteron-triton fusion generates 14 MeV neutrons

Experimental situation confusing Good data for p+3He elastic scattering

Use NCSM/RGM calculation to relate the two reactions and predict n+3H cross section

13LLNL-PRES--425682


B(E1;1/2+->1/2-)=0.02 e2 fm2

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

What happens? Substantial drop of the relative kinetic energy due to the rescaling of the relative wave function when the Whittaker tail is recovered

14LLNL-PRES--425682


NCSM/RGM with Importance-Truncated (IT-NCSM)

IT-NCSM, Roth & Navratil, PRL99, 092501 (2007) makes possible:

• large Nmax fortarget g.s. + excited states

• good convergence for integration kernels

7Li

• NCSM up to Nmax=10 (12 possible)

• IT-NCSM up to Nmax=18

12C, 16O • NCSM up to Nmax= 8

• IT-NCSM up to Nmax= 18(!)

Benchmark with NCSM in smaller model spaces: perfect agreement

Combining the NCSM/RGM with the IT-NCSM highly promising. Access to medium mass nuclei.

15LLNL-PRES--425682


NCSM/RGM ab initio calculation of n+7Li scattering7Li

n

Nmax = 12 NCSM/RGM calculation with n + 7Li(g.s.,1/2-, 7/2-)

SRG-N3LO NN potential with Λ = 2.02 fm-1

• 8Li bound states: 2+ and 1+

• 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 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= 1.24 fm a02= -0.61 fm

Expt: a01= 0.87(7) fm a02= -3.63(5) fmCalc: a01= 1.24 fm a02= -0.61 fm

0+

2+

16LLNL-PRES--425682


13C bound states and n-12C scattering12C

n

Nmax = 16 NCSM/RGM calculation with n + 12C(g.s.,2+1)


• Three 13C bound states: 1/2-, 3/2-, 1/2+ ( 5/2+ still unbound )• 5/2+ narrow resonance

Excitation energy of the 1/2+ state drops by 4 MeV when n-12C long-range correlations included Excitation energy of the 1/2+ state drops by 4 MeV when n-12C long-range correlations included

17LLNL-PRES--425682


13N ground state and p-12C scattering12C

p

Experiments with a polarized proton target under way Nmax = 16 NCSM/RGM calculation with n + 12C(g.s.,2+

1)


• 13N 1/2- ground state (bound by 2.9 MeV), other states unbound • 1/2+ and 5/2+ narrow resonance

18LLNL-PRES--425682


Nmax=12

16O

n17O bound states and n-16O scattering

Nmax = 12 NCSM/RGM calc. with n+16O(g.s., 3-,1-,2-)


• 17O bound states: 5/2+, 1/2+ ( 1/2-, 5/2- unbound )• Narrow resonances only when 16O excited states

included• Impact of incomplete 16O description• 13C+alpha not taken into account yet

( )

( )

Nmax=18

19LLNL-PRES--425682


Deuterium-Tritium fusion: a future energy source

The d+3Hn+4He reaction• The most promising for the production of fusion energy

in the near future• Will be used to achieve inertial-confinement (laser-

induced) fusion at NIF, and magnetic-confinement fusion at ITER

NIFNIF

ITERITER

Resonance at Ecm =48 keV (Ed=105 keV) in the J=3/2+ channelCross section at the peak: 4.88 b

17.64 MeV energy released:14.1 MeV neutron and 3.5 MeV alpha

20LLNL-PRES--425682


Toward the first ab initio calculation of theDeuterium-Tritium fusion

3H

d 4He

n

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’r’

n

r

n

r’r’

d

rr

d

r’r’

n

r

n

r’r’

d

rr

d

✔

• d+3H d+3H norm kern• Direct and exchange part• S-wave channel: J=3/2+,J=1/2+

• d, 3H spins parallel, anti-parallel

21LLNL-PRES--425682


Toward the first ab initio calculation of theDeuterium-Tritium fusion

3H

d 4He

n

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’r’

n

r

n

r’r’

d

rr

d

r’r’

n

r

n

r’r’

d

rr

d

✔

• d+3H n+4He norm kernel• S-wave channel: J=1/2+

• d, 3H spins anti-parallel• d+3H S-wave to n+4He D-wave transition: J=3/2+

• Important for fusion

✔

2 x -3 x

22LLNL-PRES--425682


Toward the first ab initio calculation of theDeuterium-Tritium fusion: Phase shifts

3H

d 4He

n

D-T fusion happens through the S-wave d+3H to D-wave n+4He transitionD-T fusion happens through the S-wave d+3H to D-wave n+4He transition

Ab initio phase shift calculations of the d+3H elastic scattering show

resonance in the 4S3/2 channel

No resonance in the 2S1/2

channel: Pauli principle

Ab initio phase shift calculations of the d+3H elastic scattering show

resonance in the 4S3/2 channel

No resonance in the 2S1/2

channel: Pauli principle

Phase shift of the n+4He elastic scattering show slight impact of the d+3H channels

on P-waves

Effect of resonance in the 3/2+ D-wave just above the d-3H threshold

Phase shift of the n+4He elastic scattering show slight impact of the d+3H channels

on P-waves

Effect of resonance in the 3/2+ D-wave just above the d-3H threshold

23LLNL-PRES--425682


aaAaaA veZZE

EEEES

,2 /)(

)](2exp[)()(

Toward the first ab initio calculation of theDeuterium-Tritium fusion: Cross section

3H

d 4He

n

First ab initio results of d-T and d-3He fusion: promising, correct physics, more work needs to be done…First ab initio results of d-T and d-3He fusion: promising, correct physics, more work needs to be done…

Correct features:

Resonance just above threshold, lower for d-T

S-factor of d+3He flat as E0: Experimental rise due to electron screening

Correct features:

Resonance just above threshold, lower for d-T

S-factor of d+3He flat as E0: Experimental rise due to electron screening

Incorrect features:

Resonances higher than in experiment: 150 keV vs. 50 keV (d-T)

250 keV vs. 200 keV (d-3He)

Cross sections way too low, it gets increased by including 4He resonances (2- 0 in particular)

Incorrect features:

Resonances higher than in experiment: 150 keV vs. 50 keV (d-T)

250 keV vs. 200 keV (d-3He)

Cross sections way too low, it gets increased by including 4He resonances (2- 0 in particular)

Still preliminary, incomplete: Nmax=13, SRG-N3LO NN

(Λ=2.02 fm-1), no NNN, ground states of d, 3H, 4He only.

Still preliminary, incomplete: Nmax=13, SRG-N3LO NN

(Λ=2.02 fm-1), no NNN, ground states of d, 3H, 4He only.

24LLNL-PRES--425682


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. Quaglioni and P. N., PRL 101, 092501 (2008), PRC 79, 044606 (2009)

New results with SRG-N3LO: • N-4He, n-7Li, N-12C and n-16O

Breakthrough due to the importance-

truncated NCSM approach

• First results for 3H(d,n)4He• Development for 3H, 3He projectiles

To do:• Heavier projectiles: 4He• NCSM with continuum (NCSMC)• Inclusion of NNN force • Three-cluster NCSM/RGM and treatment of three-body continuum

AJ c AJ d

r (

r ) ˆ A

r

(A a,a )

H h

h H

c

E

1 g

g N

c

7Li

n

Ab initio many-body calculations of light-ion reactions

Documents

Transcript of Ab initio many-body calculations of light-ion reactions