Nuclear Structure from ScratchNuclear Structure from Scratch

Erich Ormand Petr NavratilChristian ForssenVesselin GueorguievLawrence Livermore National Laboratory

Erich Ormand Petr NavratilChristian ForssenVesselin GueorguievLawrence Livermore National Laboratory

Feb 6, 2004

Collaborators:Bruce Barrett, U of ArizonaIonel Stecu, U of ArizonaAndreas Nogga, U of WashingtonJames Vary, Iowa State UniversityE. Caurier, CNRS, Strasbourg Calvin Johnson, San Diego State University

Collaborators:Bruce Barrett, U of ArizonaIonel Stecu, U of ArizonaAndreas Nogga, U of WashingtonJames Vary, Iowa State UniversityE. Caurier, CNRS, Strasbourg Calvin Johnson, San Diego State University

UCRL-PRES-205602

This work was carried out under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. This work was supported in part from LDRD contract 00-ERD-028

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Nuclear structure at the dawn of the 21st centuryNuclear structure at the dawn of the 21st century

• Nuclear structure has been studied for a long time

• But there are still many mysteries– Especially for light p-shell nuclei– But they are a rich laboratory to test a

basic understand of nuclear physics– Vast array of physics: deformation,

clustering, super-allowed beta decay, three-body forces, parity inversion, halos, etc.

• Newer methods and faster computers are making a comprehensive study from first principles possible

– GFMC– No-core shell model– Coupled cluster methods

Exactly how nuclei are put together is one of the most fundamental questions in nuclear physicsExactly how nuclei are put together is one of the most fundamental questions in nuclear physics

• Two nucleons A-nucleons• Properties: binding energies, decay mechanisms, etc.

– Do we need three-body forces? (Yes)– How about four-body?

Goal: Exact calculation of Nuclear Structure for light nucleiGoal: Exact calculation of Nuclear Structure for light nuclei

• Hasn’t the Shell Model already solved this problem?– Yes and NO!

• The Shell Model as practiced is an empirical tool that often requires arbitrary and unsatisfying approximations to make it work

The goal of the No-core Shell Model is to implement effective interaction theory to apply the shell model correctly

We want to take the model out of the shell model

The many-body HamiltonianThe many-body Hamiltonian

€

H =r p i

2

2mi

∑ + VNN

r r i −

r r j( )

i< j

∑

– The Bad:• Radial behavior is not right for large r

• Provides a confining potential, so all states are effectively bound• Some dependence on oscillator parameter (goes away with

larger model space)Ωh

– The Good:• Provides a convenient basis to build the many-body Slater

determinants• Does not affect the intrinsic motion• Exact separation between intrinsic and center-of-mass motion

• Add the center-of-mass oscillator potential

€

HCM =1

2AmΩ2

r R 2 =

1

2mΩ

r r i

2

i

∑ −mΩ2

2A

r r i −

r r j( )

2

i< j

∑

Many-body solutionsMany-body solutions

• Typical eigenvalue problem

0s N=0

0d1s N=2

0f1p N=4

1p N=1

€

φ= 1

A!

φi r1( ) φi r2( ) K φi rA( )

φ j r1( ) φ j r2( ) φ j rA( )

M O M

φl r1( ) φl r2( ) K φl rA( )

= al+K a j

+ai+ 0⎟⎟

⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

NNN

N

HH

HH

HHH

K

OM

L

1

2221

11211

• Calculate Hamiltonian matrix Hij=j|H|i

– Diagonalize to obtain eigenvalues€

Ψi = Cinφn

n

∑

• Construct many-body states |i so that

€

HΨi = E iΨi

Problem: Short-range repulsion requires an infinite space

Strong repulsion at Strong repulsion at ~ 0.5 fm~ 0.5 fmjj|H||H|ii large large

Can we get around this problem?Effective interactionsCan we get around this problem?Effective interactions

• Choose subspace of for a calculation (P-space)– Include most of the relevant physics

– Q -space (excluded - infinite)

• Effective interaction:

nφ

€

Heffˆ P Ψi = E i

ˆ P Ψi

QP

QQXXHHXX-1-1PP=0=0

HHeffeff==PPXXHHXX-1-1PP

• Lee-Suzuki:

QQ

PP

QQPP

P-space defined by ΩhmaxN

The general idea behind effective interactionsThe general idea behind effective interactions

H

HeffQXHX-1P=0

Exact reproduction of N eigenvalues

Heff has one-, two, three-, … A-body terms

€

Qe−ωHeωP = 0; ω = QωP

α Q ω α P = α Q k ˜ k α P

k

d P

∑ ; ˜ k α P α P ′ k α P

∑ = δk ′ k

β P Heff α P = β P 1+ ω+ω( )−1 2

′ α P ′ α P ˜ k E k˜ k ′ ′ α P ′ ′ α P 1+ ω+ω( )

−1 2α P

′ ′ α P

∑′ α P

∑k

dP

∑

β P 1+ ω+ω( ) α P = β P˜ k ˜ k α P

k

d P

∑Hermitian effective interactionHermitian effective interaction

The matrix P|Heff|P exactly reproduces dp solutions of the full problem

• Choose P-space for A-body calculation, with dimension dp

– P-space basis states: and Q-space basis states:

– Need dP solutions, |k, in the “infinite” space, i.e.,

– Write X=e-

kEkH k=P Q

Effective interactions with the Lee-Suzuki method

The NCSM uses a variety of realistic interactionsThe NCSM uses a variety of realistic interactions

• Bonn potentials • Based on meson exchange

Non-local, off shell behavior

Charge-dependence, CD-Bonn

• Effective field theory

EFT

• Guided by QCD

Idaho-A, N3LO, etc

Softer, faster convergence

• INOY • Two-body

Inside non-local, outside Yukawa

Fit to NN data and 3H and 3He

• Three-body • n=3 clusters in Heff

Tucson-Melbourne

Urbana and EFT-based

• Potential model, v18, v8’

Local

Fit to Nijmegan scattering data

• Argonne potentials

Application to A=3 and convergenceApplication to A=3 and convergence

• 3H with CD-Bonn and N3LO

Potential Egs (MeV)

AV18 -7.62

N3LO -7.86

CD-Bonn -8.00

INOY -8.48

Exp -8.48

Results with two-body - frequency dependenceResults with two-body - frequency dependence

• 9Be with CD-Bonn

•NCSM result taken in region with least dependence on frequency •Note NCSM is not variational

Binding energies with Av8’Binding energies with Av8’

Results with three-body effective interactionsResults with three-body effective interactions

400 keV

1.8 MeV, Clustering?

Oscillator parameterOscillator parameter Model spaceModel space

Three-body effective interactions represent a significant improvement Results are within 400 keV of the GFMC

Excitation spectra with NN-interactionsExcitation spectra with NN-interactions

So far, things are looking pretty good!

The NN-interaction clearly has problemsThe NN-interaction clearly has problems

Note inversion of spins!

•The NN-interaction by itself does not describe nuclear structure•Also true for A=11

Three-body interaction in a nucleusThree-body interaction in a nucleus

The three-nucleon interaction plays a critical role in determining the structure of nuclei

Future directions for the No-core Shell ModelFuture directions for the No-core Shell Model

• We want to get better convergence– Extend Nmax

• For three-body interactions up to Nmax=6

– Include higher clusters in the effective interaction• Four-body

• Shell-model program REDSTICK– Two-body

• Competitive with ANTOINE

– Three-body• Now finished and production runs expected this Fall!

Some computational aspects of the NCSMSome computational aspects of the NCSM

• Model-space limitation ~ 108 states

A=3, Nmax = 36, Jacobi coordinates

A=4, Nmax = 20, Jacobi coordinates

A=6, Nmax = 14, M-scheme

7 A 11, Nmax ~ 10, M-scheme

A 12, Nmax = 6, M-scheme

• Two-body interactions

• Limited by number of three-body matrix elements

For Nmax=4, 39,523,066

For Nmax=6 over 600,000,000

Practical limit is Nmax = 6 for all the p-shell

Four-body - Nmax = 4

• Three-body interactions

REDSTICK3B – Timing for three-bodyREDSTICK3B – Timing for three-body

Four-body interactionG.S. 10 iterations ~454 hr

12C: N ~ 32.5x106

G.S. 10 iterations ~ 1305 hr

A new formalism for nuclear reactionsA new formalism for nuclear reactions

€

3He × 4 He[ ]s

×YLˆ r ( )gJ r( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

JM

Entrance-channel wave function

Relative wave functionOrbital angular momentum L

€

−h2

2M

d2

dr2+

L(L +1)

r2

⎛

⎝ ⎜

⎞

⎠ ⎟+ VFold + E A−a + E A − E

⎡

⎣ ⎢

⎤

⎦ ⎥uΓ r( ) + drRnL r( )HΓn, ′ Γ ′ n R ′ n ′ L ′ r ( )u ′ Γ ′ r ( )

0

∞

∫′ Γ n ′ n

∑ = 0

Sum over all channels

Radial-cluster overlap enters in these matrix elements of H

Harmonic oscillator functions

• Nuclear reactions from scratch - Petr Navratil

• Ingredients:– Exact intrinsic wave functions for

entrance and exit channels - ab initio shell model

– Wave function for composite– Radial-cluster form factor– Solve couple integro-differential

equation for relative wave function– Extract cross section from

asymptotic normalization

3He 4He 7Be+

Example: 4He + t 7Li and 6Li + n 7Li 4He + t Example: 4He + t 7Li and 6Li + n 7Li 4He + t

Example: 7Be(p,)8BExample: 7Be(p,)8B

• NCSM cluster overlap

• Fit tail to binding energy

• Potential model for capture

PreliminaryP. Navratil and C. Bertulani

SummarySummary

• Extensions and improvements:– Determine the form of the NNN-interaction– Implementation of effective operators for transitions– Four-body effective interactions– Effective field-theory potentials; are they any good for structure?– Integrate the structure into some reaction models (underway)

• Questions and open problems to be addressed:– Is it possible to improve the mean field?– Can we improve the convergence of the higher states?– Unbound states. Can we use a continuum shell model?– How high in A can we go?– Can we use this method to derive effective interactions for

conventional nuclear structure studies?

Ωh

• Significant progress towards an exact understanding of nuclear structure is being made!

• These are exciting times!!