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The Illinois Extension to the Fujita-Miyazawa Three-Nucleon Force

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<strong>The</strong> <strong>Illinois</strong> <strong>Extension</strong> <strong>to</strong> <strong>the</strong><br />

<strong>Fujita</strong>-<strong>Miyazawa</strong> <strong>Three</strong>-<strong>Nucleon</strong> <strong>Force</strong><br />

Steven C. Pieper<br />

Physics Division, Argonne National Labora<strong>to</strong>ry<br />

WORK WITH<br />

Joe Carlson, Los Alamos<br />

Ken Nollett, Argonne<br />

Muslema Pervin, Argonne<br />

Rocco Schiavilla, Jefferson Lab. & Old Dominion U.<br />

Bob Wiringa, Argonne<br />

Work not possible without extensive computer resources:<br />

Argonne Labora<strong>to</strong>ry Computing Resource Center (Jazz)<br />

Argonne Math. & Comp. Science Division (BlueGene/L)<br />

NERSC IBM SP’s (Seaborg, Bassi)<br />

Physics Division<br />

Work supported by U.S. Department<br />

of Energy, Office of Nuclear Physics


GOAL OF ab-initio LIGHT-NUCLEI CALCULATIONS<br />

We want <strong>to</strong> understand nuclei as collections of nucleons interacting with <strong>the</strong> realistic (bare)<br />

potentials by reliably solving <strong>the</strong> many-nucleon Schrödinger equation.<br />

• Binding energies, excitation spectra, relative stability<br />

• Densities, transition amplitudes, cluster-cluster overlaps<br />

• Low-energy astrophysical reactions<br />

Only if such calculations fail <strong>to</strong> reproduce experiment can we claim <strong>to</strong> be seeing nuclear medium<br />

effects on sub-nucleonic (quarks, etc.) degrees of freedom in nucleons.<br />

At present <strong>the</strong> QMC methods used here are limited <strong>to</strong> light (A ≤ 12) nuclei


TWO PROBLEMS IN MICROSCOPIC FEW- & MANY-NUCLEON<br />

CALCULATIONS<br />

(I) What is <strong>the</strong> Hamil<strong>to</strong>nian?<br />

• NN force controlled by NN scattering<br />

– Argonne v ij<br />

• 3N force determined from properties of light nuclei<br />

– Recent <strong>Illinois</strong> models with 2π & 3π rings<br />

(II) Given H, solve <strong>the</strong> Schrödinger equation for A nucleons accurately.<br />

• Essential for comparisons of models <strong>to</strong> data<br />

• We have made much progress for A ≤ 12<br />

– VMC Ψ T can have correct asymp<strong>to</strong>tic behavior<br />

– GFMC refines Ψ T <strong>to</strong> give 1–2% accurate energies for given Hamil<strong>to</strong>nian.<br />

Our goal is a microscopic description of nuclear structure and reactions from bare NN & 3N<br />

forces and consistent currents.


PROGRESS IN ab-initio CALCULATIONS OF LIGHT NUCLEI<br />

Accurate Representations of Nuclear <strong>Force</strong>s<br />

• Meson-exchange <strong>the</strong>ory of Yukawa (1935)<br />

• <strong>Fujita</strong>-<strong>Miyazawa</strong> three-nucleon potential (1955)<br />

• First phase-shift analysis of NN scattering data (1957)<br />

• Gammel-Thaler, Hamda-Johns<strong>to</strong>n & Reid phenomenological potentials (1957–1968)<br />

• Bonn, Nijmegen & Paris field-<strong>the</strong>oretic models (1970s)<br />

• Nijmegen 1993 Partial Wave Analysis (PWA93) → χ 2 ∼ 1<br />

• Nijm I, Nijm II, Reid93, Argonne v 18 & CD-Bonn (1990s)<br />

• Effective field <strong>the</strong>ory at N 3 LO (2004)<br />

Accurate Solutions of Many-Body Schrödinger Equation<br />

• 2 H by Numerical Integration (1952)<br />

– “5 <strong>to</strong> 20 minutes for calculation and 10 minutes <strong>to</strong> print result”<br />

• 3 H by Faddeev (1975–1985)<br />

• 4 He by Green’s function Monte Carlo (GFMC) (1988)<br />

• A = 6 by GFMC & No-core shell model (NCSM) (1994-95)<br />

• A = 7 by GFMC & NCSM (1997-98)<br />

• A = 8 by GFMC & NCSM (2000)<br />

• 4 He benchmark by 7 methods <strong>to</strong> 0.1% (2001)<br />

• A = 9, 10 by GFMC & NCSM (2002)<br />

• 12 C by GFMC & NCSM (2004–)<br />

• 16 O by CC (2005–)


NUCLEAR HAMILTONIAN<br />

v ij + X<br />

V ijk<br />

H = X K i + X<br />

i i


NUCLEAR HAMILTONIAN – URBANA V ijk<br />

V UIX<br />

ijk<br />

= V 2π,P<br />

ijk<br />

+ V R<br />

ijk<br />

V 2π,P<br />

ijk<br />

: <strong>Fujita</strong>-<strong>Miyazawa</strong> ∼ p-wave πN scattering; in all realistic V ijk<br />

• Longest ranged V ijk<br />

• Attractive in all nuclei studied.<br />

V 2π,P<br />

ijk<br />

= A 2π,P<br />

X<br />

cyclic<br />

{X ij , X jk }{τ i · τ j , τ j · τ k } + 1 4 [X ij, X jk ][τ i · τ j , τ j · τ k ]<br />

X ij = Y (m π r ij ) σ i · σ j + T (m π r ij ) S ij<br />

Y (x) = e−x<br />

x<br />

ξ(r)<br />

„ 3<br />

T (x) =<br />

x + 3 « e<br />

−x<br />

2 x + 1 x<br />

ξ2 (r)<br />

π<br />

∆<br />

π<br />

ξ(r) = (1 − e −Cr2 )<br />

Vijk: R represents all else including relativistic effects – purely central and repulsive<br />

X<br />

Vijk R = A R T 2 (r ij )T 2 (r jk )<br />

cyclic<br />

Two couplings fit <strong>to</strong> 3 H and nuclear matter saturation density


NUCLEAR HAMILTONIAN – ILLINOIS V ijk<br />

V ILx<br />

ijk<br />

= V 2π,P<br />

ijk<br />

+ V 2π,S<br />

ijk<br />

+ V 3π<br />

ijk + V R<br />

ijk<br />

V 2π,P<br />

ijk<br />

V 2π,S<br />

ijk<br />

& V R<br />

ijk: Same form as Urbana, new coupling constants<br />

: s-wave πN scattering term<br />

• From Tuscon-Melbourne V ijk<br />

• We use chiral-perturbation <strong>the</strong>ory coupling<br />

• Not different from V 2π,P<br />

ijk<br />

dependence on A, Z, J, T<br />

• but only 3%-4% of V 2π,P<br />

ijk<br />

π<br />

π<br />

Vijk: 3π 3π rings with ∆’s; new in <strong>Illinois</strong> V ijk<br />

• Extra p-shell, |N − Z| attraction<br />

• One ∆ in energy denomina<strong>to</strong>r<br />

• 2∆, 3∆ denomina<strong>to</strong>rs not yet considered<br />

• 〈Vijk〉 3π < ∼<br />

0.1〈Vijk〉<br />

2π<br />

V 3π<br />

ijk = A 3π<br />

ˆ 50<br />

3 S τ S σ + 26 3 A τ A σ˜<br />

∆<br />

π<br />

π<br />

π<br />

π<br />

π<br />

π<br />

∆<br />

∆<br />

S τ = 2 + 2 3 (τ i · τ j + τ j · τ k + τ k · τ i ) = 4P T =3/2<br />

A τ = − 1 6 [τ i · τ j , τ j · τ k ]


NUCLEAR HAMILTONIAN – ILLINOIS V ijk , CONTINUED<br />

IL2 had three couplings adjusted <strong>to</strong> fit 17 nuclear levels for A ≤ 8<br />

Model C A 2π,P A 2π,S A 3π A R<br />

fm −2 MeV MeV MeV MeV<br />

UIX 2.1 −0.0293 − − 0.00480<br />

IL2 2.1 −0.037 −1.0 0.0026 0.00705<br />

In light nuclei we find<br />

〈V ijk 〉 ∼ (0.02 <strong>to</strong> 0.09)〈v ij 〉 ∼ (0.15 <strong>to</strong> 0.6)〈H〉<br />

(Large cancellation of K and v ij )<br />

We expect<br />

〈V 4N 〉 ∼ 0.06〈V ijk 〉 ∼ (0.02 <strong>to</strong> 0.04)〈H〉 ∼ (0.5 <strong>to</strong> 2.) MeV<br />

But not possible <strong>to</strong> disentangle from V ijk uncertainties.


THE MANY-BODY PROBLEM<br />

Need <strong>to</strong> solve<br />

HΨ(⃗r 1 , ⃗r 2 , · · · , ⃗r A ; s 1 , s 2 , · · · , s A ; t 1 , t 2 , · · · , t A )<br />

= EΨ(⃗r 1 , ⃗r 2 , · · · , ⃗r A ; s 1 , s 2 , · · · , s A ; t 1 , t 2 , · · · , t A )<br />

s i are nucleon spins: ± 1 2<br />

t i are nucleon isospins (pro<strong>to</strong>n or neutron): ± 1<br />

“ ”<br />

2<br />

2 A AZ<br />

× complex coupled 2 nd order eqn in 3A − 3 variables<br />

(number of isospin states can be reduced)<br />

12 C: 270,336 coupled equations in 33 variables<br />

Coupling is strong:<br />

1<br />

1<br />

• 〈v Tensor 〉 is ∼ 60% of <strong>to</strong>tal 〈v ij 〉<br />

• 〈v Tensor 〉 = 0 if no tensor correlations<br />

fem<strong>to</strong>meter<br />

0<br />

0<br />

-1<br />

M J<br />

= 0<br />

M J<br />

= 1<br />

-1<br />

-1 0 1 -1 0 1<br />

fem<strong>to</strong>meter


Minimize expectation value of H<br />

VARIATIONAL MONTE CARLO<br />

E T = 〈Ψ T |H|Ψ T 〉<br />

〈Ψ T |Ψ T 〉<br />

≥ E 0<br />

Simplified trial wave function:<br />

|Ψ T 〉 = [1 + X<br />

U ijk are 3-body correlations from V ijk<br />

i


GREEN’S FUNCTION (DIFFUSION) MONTE CARLO<br />

VMC Ψ T propagated <strong>to</strong> imaginary time τ:<br />

Ψ(τ) = e −(H−Ẽ0)τ Ψ T<br />

Ψ T = Ψ 0 + X α i Ψ i ; Ψ(τ) = e −(E 0−Ẽ0)τ × [Ψ 0 + X α i e −(E i−E 0 )τ Ψ i ]<br />

Ψ 0 = lim<br />

τ→∞ Ψ(τ) ; HΨ 0 = E 0 Ψ 0<br />

Small time-step propaga<strong>to</strong>r:<br />

Ψ(τ) =<br />

Can be computed <strong>to</strong> order (△τ) 3<br />

h<br />

e −(H−Ẽ0)△τ i n<br />

ΨT ;<br />

τ = n△τ<br />

Ψ(R n , τ) =<br />

G βα (R ′ , R) = 〈R ′ , β|e −(H−Ẽ0)△τ |R, α〉<br />

Z<br />

G(R n , R n−1 ) · · · G(R 1 , R 0 )Ψ T (R 0 )dR n−1 · · · dR 0<br />

Done by Monte Carlo integration<br />

E(τ) = 〈Ψ T |H|Ψ(τ)〉<br />

〈Ψ T |Ψ(τ)〉<br />

≥ E 0


PROBLEMS WITH NUCLEAR GFMC<br />

(I) “Mixed-estimate” expectation values are used:<br />

exact for H, but must be extrapolated for o<strong>the</strong>r opera<strong>to</strong>rs<br />

〈Ψ(τ)|O|Ψ(τ)〉 ≈ 〈Ψ T |O|Ψ T 〉 + 2[〈Ψ(τ)|O|Ψ T 〉 − 〈Ψ T |O|Ψ T 〉]<br />

(II) Propaga<strong>to</strong>r cannot contain L 2 opera<strong>to</strong>rs:<br />

G βα (R ′ , R) has only v ′ 8<br />

〈v 18 − v ′ 8〉 computed perturbatively with extrapolation<br />

Reliable in Faddeev ( 3 H) and PHH & Yakubovsky ( 4 He) comparisons<br />

(III) Fermion sign problem limits maximum τ:<br />

G brings in lower-energy boson solution<br />

〈Ψ T |H|Ψ(τ)〉 projects back fermion solution<br />

Exponentially growing statistical errors<br />

Constrained-path propagation, removes steps that have<br />

Ψ † (τ, R)Ψ(R) = 0<br />

Any errors introduced are removed by 10 − 20 unconstrained steps before evaluating<br />

observables.


EXAMPLES OF GFMC PROPAGATION<br />

E (MeV)<br />

−26.8<br />

−27.0<br />

−27.2<br />

−27.4<br />

−27.6<br />

−27.8<br />

4 He<br />

E(τ) (MeV)<br />

-24<br />

-26<br />

-28<br />

6 Li<br />

2 +<br />

3 +<br />

gs<br />

−28.0<br />

−28.2<br />

-30<br />

α+d<br />

➙<br />

−28.4<br />

0.000 0.002 0.004 0.006 0.008 0.010<br />

τ (MeV −1 )<br />

-32<br />

0 0.5 1 1.5 2 2.5<br />

τ (MeV -1 )<br />

Curve has exp(−E i τ) with<br />

E i = 1480, 340 & 20.2 MeV<br />

(20.2 MeV is first 4 He 0 + excitation)<br />

Ψ T has small amounts of 1.5 GeV<br />

contamination<br />

g.s. (1 + ) & 3 + stable after τ = 0.2 MeV −1<br />

2 + (a broad resonance) never stable –<br />

decaying <strong>to</strong> separated α & d<br />

E(τ=0.2) is best estimate of resonance energy<br />

Should use scattering boundary conditions<br />

(see later)


-20<br />

Energy (MeV)<br />

-25<br />

-30<br />

-35<br />

-40<br />

-45<br />

-50<br />

-55<br />

-60<br />

4 He<br />

AV18<br />

0 +<br />

UIX<br />

1 +<br />

2 +<br />

α+2n<br />

0 +<br />

IL2<br />

Exp<br />

1 +<br />

2 +<br />

3 +<br />

α+d<br />

6 He 1 +<br />

6 Li<br />

7 Li<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

3/2 −<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

α+t<br />

1/2 −<br />

3/2 −<br />

8 He<br />

1 +<br />

2 +<br />

6 He+2n<br />

0 +<br />

Argonne v 18<br />

8 Li<br />

4 +<br />

0 +<br />

1 +<br />

3 +<br />

7 Li+n<br />

1 +<br />

2 +<br />

without & with various V ijk<br />

GFMC Calculations<br />

8 Be<br />

3 +<br />

1 +<br />

2 +<br />

4 +<br />

2 +<br />

0 +<br />


-30<br />

-35<br />

-40<br />

7/2 −<br />

?<br />

5/2 −<br />

3/2 −<br />

8 Li+n<br />

1/2 −<br />

Argonne v 18<br />

without & with V ijk<br />

GFMC Calculations<br />

AV18 UIX IL2 Exp<br />

Energy (MeV)<br />

-45<br />

-50<br />

9 Li<br />

3/2 −<br />

3 +<br />

1 +<br />

-55<br />

-60<br />

9 Be<br />

1/2 −<br />

5/2 −<br />

2α+n<br />

3/2 −<br />

3,2 +<br />

9 Be+n<br />

0 +<br />

2 +<br />

6 Li+α<br />

-65<br />

10 Be<br />

0 +<br />

10 B<br />

1 +<br />

3 +


Energy (MeV)<br />

0<br />

-10<br />

-20<br />

-30<br />

-40<br />

-50<br />

-60<br />

-70<br />

-80<br />

1 +<br />

2 H 1/2+<br />

3 H<br />

0 +<br />

4 He<br />

1 +<br />

AV18<br />

2 +<br />

3 +<br />

5/2 −<br />

7/2 −<br />

6 1/2<br />

Li −<br />

3/2 −<br />

7 Li<br />

8 Li<br />

4 +<br />

0 +<br />

3 +<br />

1 +<br />

2 +<br />

8 Be<br />

3 +<br />

1 +<br />

4 +<br />

2 +<br />

0 +<br />

9 Li<br />

Argonne v 18<br />

With <strong>Illinois</strong>-2<br />

GFMC Calculations<br />

7/2 −<br />

5/2 −<br />

1/2 −<br />

3/2 −<br />

9 Be<br />

9/2 −<br />

7/2 +<br />

1/2 +<br />

7/2 −<br />

5/2 +<br />

3/2 +<br />

1/2 −<br />

5/2 −<br />

3/2 −<br />

4 + 2 +<br />

1 + 3 +<br />

3 + 1 +<br />

2 + ,3 +<br />

0 +<br />

2 +<br />

0 +<br />

10 Be<br />

10 B<br />

4 +<br />

2 +<br />

1 +<br />

3 +<br />

4 + 0 +<br />

-90<br />

-100<br />

IL2<br />

Exp<br />

12 C


Excitation energy (MeV)<br />

25<br />

20<br />

15<br />

10<br />

5<br />

0<br />

Argonne v 18<br />

With <strong>Illinois</strong>-2<br />

GFMC Calculations<br />

6 He<br />

0 +<br />

2 +<br />

1 +<br />

2 +<br />

0 +<br />

6 Li<br />

1 +<br />

1 +<br />

2 +<br />

3 +<br />

1 +<br />

7 Li<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

1/2 −<br />

3/2 −<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

1/2 −<br />

3/2 −<br />

8 Li<br />

4 +<br />

3 +<br />

2 +<br />

1 +<br />

0 +<br />

2 +<br />

1 +<br />

3 +<br />

1 +<br />

2 +<br />

8 Be<br />

2 +<br />

3 +<br />

0 +<br />

2 +<br />

4 +<br />

1 +<br />

3 +<br />

1 +<br />

2 +<br />

4 +<br />

2 +<br />

0 +<br />

9 Li<br />

7/2 −<br />

?<br />

3/2 −<br />

5/2 −<br />

1/2 −<br />

3/2 −<br />

9 Be<br />

3/2 +<br />

7/2 +<br />

5/2 +<br />

3/2 −<br />

9/2 −<br />

7/2 +<br />

5/2 −<br />

1/2 +<br />

9/2 +<br />

5/2 +<br />

7/2 −<br />

3/2 +<br />

3/2 −<br />

3/2 +<br />

5/2 +<br />

1/2 −<br />

5/2 −<br />

1/2 +<br />

3/2 −<br />

AV18 +IL2<br />

10 Be<br />

4 +<br />

2 +<br />

0 +<br />

1 +<br />

3 +<br />

2 + ,3 +<br />

2 −<br />

0 +<br />

2 +<br />

1 − 2 −<br />

2 + 4 +<br />

1 +<br />

2 +<br />

3 +<br />

0 + 2 +<br />

3 +<br />

1 +<br />

1 +<br />

Exp<br />

3 +<br />

4 +<br />

2 +<br />

1 +<br />

2 −<br />

3 +<br />

2 +<br />

1 +<br />

1 +<br />

3 +<br />

-5<br />

10 B


ORDERING OF STATES IN 10 BE AND 10 B<br />

• NN potentials with no NNN predict 1 + ground state for 10 B<br />

– <strong>Illinois</strong>-2 NNN potential fixes this and gives correct 3 + ground state<br />

• First two excited states in 10 Be are both 2 +<br />

– VMC and GFMC calculations predict large positive and negative quadrupole moments<br />

– VMC also predicts large B(E2) <strong>to</strong> <strong>the</strong> g.s. for only one of <strong>the</strong>m<br />

– GFMC calculations predict that UIX or <strong>Illinois</strong>-2 changes <strong>the</strong> ordering of <strong>the</strong> states<br />

20<br />

Excitation energy (MeV)<br />

15<br />

10<br />

5<br />

0<br />

-5<br />

Argonne v 18<br />

Without and with UIX or <strong>Illinois</strong>-2<br />

10 Be; T=1<br />

AV18 UIX IL2 Exp<br />

4 +<br />

2 +<br />

0 +<br />

1 +<br />

3 +<br />

3,2 +<br />

0 +<br />

2 +<br />

2 +<br />

0 +<br />

4 +<br />

1 +<br />

2 +<br />

3 +<br />

2 +<br />

3 +<br />

1 +<br />

1 +<br />

10 B; T=0<br />

3 +<br />

4 +<br />

2 +<br />

1 +<br />

3 +<br />

2 +<br />

1 +<br />

1 +<br />

3 +<br />

-10<br />

• An ATLAS expt for B(E2) and quadrupole moments of <strong>the</strong>se states will be made


RMS RADII OF HELIUM ISOTOPES<br />

Recent measurements of 6 He charge radius at Argonne and 8 He by Argonne group at GANIL<br />

• Single 4 He or 6,8 He a<strong>to</strong>ms trapped<br />

• Iso<strong>to</strong>pe shift of an a<strong>to</strong>mic transition measured<br />

• Small 〈r 2 〉 1/2 dependence of shift extracted using precise a<strong>to</strong>mic calculations<br />

• 6 He half life only 0.807 sec.; 8 He only 0.119 sec.<br />

GFMC radius strongly dependent on propagated separation energy<br />

8 He charge radius smaller than for 6 He – both calc. and expt.<br />

rms r p<br />

(fm)<br />

2.06<br />

2.02<br />

1.98<br />

1.94<br />

AV8’ modified<br />

SSCC V8’ modified<br />

IL2, o<strong>the</strong>r<br />

IL2, V R<br />

’=1.254<br />

IL6, V R<br />

’, E S<br />

=1.84<br />

IL6, V R<br />

’, E S<br />

=1.48<br />

IL6, 3 trm, R WS<br />

=4.0<br />

IL6, 3 trm, E S<br />

=1.4<br />

rms r p<br />

(fm)<br />

1.88<br />

1.86<br />

1.84<br />

1.82<br />

8 He<br />

AV8’ modified<br />

SSCC V8’ modified<br />

IL6, VR’ varied<br />

IL6, VR’ varied<br />

IL6, 3=trm, l3bc=6<br />

IL6, 3=trm, l3bc=5<br />

Experiment<br />

1.90<br />

6 He<br />

1.80<br />

1.86<br />

0.8 0.9 1 1.1 1.2 1.3<br />

E sep<br />

′ = H′( 4 He) - H′( 6 He) (MeV)<br />

1.78<br />

2 2.2 2.4 2.6 2.8 3 3.2<br />

E sep<br />

′ = H′( 6 He) - H′( 8 He) (MeV)


GFMC FOR SCATTERING STATES<br />

GFMC treats nucleus as particle-stable system<br />

– Should be good for narrow resonances<br />

Many cases should be done as scattering states<br />

– Wide resonances: 5,7 He, 6 Li(2 + ), 8 Be(2 + ,4 + ), ...<br />

– Will get widths of resonances<br />

– Capture reactions: 4 He(d,γ) 6 Li, 7 Be(p,γ) 8 B, ...<br />

1987 – early 1990’s:<br />

– Carlson et al. do 5 He states by VMC scattering<br />

– Carlson also does preliminary 5 He GFMC scattering<br />

Present:<br />

– Joe Carlson doing 5 He for parity violation studies<br />

– Ken Nollett has modified Argonne GFMC program for scattering and done 5 He


GFMC FOR SCATTERING STATES – METHOD<br />

• Pick a logarithmic derivative, χ, at some large boundary radius (R ≥ 7 fm)<br />

• GFMC propagation, using method of images <strong>to</strong> preserve χ at R, finds E(R, χ)<br />

• Phase shift, δ(E), is function of R, χ, E<br />

• Repeat for a number of χ until δ(E) is mapped out<br />

Example for 5 He( 1 − )<br />

2<br />

• “Bound-state” boundary<br />

condition does not<br />

give stable energy;<br />

Decaying <strong>to</strong> n+ 4 He<br />

threshold<br />

• Scattering boundary<br />

condition produces<br />

stable energy.<br />

E(τ) (MeV)<br />

-23<br />

-24<br />

-25<br />

-26<br />

"bound state"<br />

Log-deriv = -0.168 fm -1<br />

-27<br />

0 0.1 0.2 0.3 0.4<br />

τ (MeV -1 )


δ LJ<br />

(degrees)<br />

GFMC FOR 5 HE AS n+ 4 HE SCATTERING STATES<br />

• Black curves: Hale phase shifts from R-matrix analysis up <strong>to</strong> J = 9 of data<br />

2<br />

• AV18 with no V ijk underbinds 5 He( 3 − ); overbinds 5 He( 1 − )<br />

2<br />

2<br />

• AV18+IL2 was not fit <strong>to</strong> 5 He, reproduces locations and widths of both P -wave resonances<br />

– Spin-orbit splitting well reproduced by AV18+IL2<br />

180<br />

150<br />

120<br />

90<br />

60<br />

1+<br />

2<br />

3-<br />

2<br />

AV18<br />

AV18+UIX<br />

AV18+IL2<br />

R-Matrix<br />

σ LJ<br />

(b)<br />

7<br />

6<br />

5<br />

4<br />

3<br />

2<br />

+<br />

1<br />

2<br />

1<br />

2<br />

3<br />

2<br />

-<br />

-<br />

R-Matrix<br />

Pole location<br />

30<br />

1-<br />

2<br />

0<br />

0 1 2 3 4 5<br />

E c.m.<br />

(MeV)<br />

1<br />

0<br />

0 1 2 3 4 5<br />

E c.m.<br />

(MeV)<br />

K.M. Nollett, S.C. Pieper, R.B. Wiringa, J. Carlson, G. M. Hale, Phys. Rev. Lett. 99, 022502 (2007)


NEW ILLINOIS POTENTIALS – PROGRESS REPORT I<br />

• <strong>Illinois</strong> 1–5 parameters determined in 2000.<br />

– Fits made <strong>to</strong> A ≤ 8 only<br />

– Preliminary nuclear matter calculations at Urbana (Morales, Pandharipande, Ravenhall)<br />

suggested at most IL2 is viable<br />

– Improved GFMC results in worse 8 He agreement<br />

• Started new fitting up <strong>to</strong> A = 10<br />

• Michele Viviani (Pisa) finds sign error in one piece of A σ in V 3π<br />

ijk<br />

– Formula was published correctly, but incorrectly programmed<br />

– Increased attraction for all nuclei<br />

• New fit made with corrected A σ : IL7<br />

– parameters weaker than for IL2 because of increased attraction<br />

– better quality reproduction of energies than IL2<br />

– so far have not found any significant difference in o<strong>the</strong>r observables


Energy (MeV)<br />

-20<br />

-25<br />

-30<br />

-35<br />

-40<br />

-45<br />

-50<br />

-55<br />

-60<br />

-65<br />

-70<br />

4 He<br />

0 +<br />

6 He<br />

0 +<br />

2 +<br />

1 +<br />

2 +<br />

0 +<br />

6 Li<br />

1 +<br />

1 +<br />

2 +<br />

3 +<br />

1 +<br />

IL2<br />

7 He<br />

3/2 −<br />

5/2 +<br />

3/2 −<br />

1/2 +<br />

5/2 −<br />

(5/2) −<br />

1/2 −<br />

3/2 −<br />

IL7<br />

7 Li<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

1/2 −<br />

3/2 −<br />

5/2 −<br />

5/2 −<br />

7/2 −<br />

1/2 −<br />

3/2 −<br />

Argonne v 18<br />

With <strong>Illinois</strong> V ijk<br />

GFMC Calculations<br />

22 Oc<strong>to</strong>ber 2007<br />

Exp<br />

2 +<br />

0 +<br />

1 +<br />

2 +<br />

0 +<br />

4 +<br />

3 +<br />

1 +<br />

8 0<br />

He +<br />

2 +<br />

8 Li<br />

2 +<br />

1 +<br />

3 +<br />

1 +<br />

2 +<br />

8 Be<br />

2 +<br />

3 +<br />

0 +<br />

2 +<br />

4 +<br />

3 +<br />

1 +<br />

1 +<br />

2 +<br />

4 +<br />

2 +<br />

0 +<br />

9 He<br />

9 Li<br />

1/2 −<br />

7/2 −<br />

?<br />

5/2 −<br />

3/2 −<br />

1/2 −<br />

3/2 −<br />

9 Be<br />

0 +<br />

10 He<br />

3/2 +<br />

7/2 +<br />

5/2 +<br />

3/2 −<br />

9/2 −<br />

7/2 +<br />

5/2 −<br />

1/2 +<br />

9/2 +<br />

7/2 −<br />

5/2 +<br />

3/2 −<br />

3/2 +<br />

3/2 +<br />

5/2 +<br />

1/2 −<br />

5/2 −<br />

1/2 +<br />

3/2 −<br />

10 Li<br />

0 +<br />

2 +<br />

1 +<br />

(1 - ,2 - )<br />

2 +<br />

4 +<br />

0 +<br />

1 +<br />

3 +<br />

3,2 +<br />

2 −<br />

0 +<br />

2 +<br />

1 −<br />

2 +<br />

0 +<br />

3 +<br />

4 +<br />

2 +<br />

1 +<br />

2 −<br />

3 +<br />

2 +<br />

1 +<br />

1 +<br />

10 3<br />

Be +<br />

10 B


NEW ILLINOIS POTENTIALS – PROGRESS REPORT II<br />

Stefano Gandolfi (Tren<strong>to</strong>) has started Auxiliary Field Diffusion Monte Carlo (AFDMC)<br />

for neutron matter using AV8 ′ +ILx.<br />

• IL2 and IL7 much <strong>to</strong>o soft<br />

need much stronger short-ranged<br />

repulsion in V R<br />

ijk<br />

• Investigating two changes<br />

1) Change cu<strong>to</strong>ff in T 2 (r ij ) <strong>to</strong><br />

„ 3<br />

x 2 + 3 x + 1 « e<br />

−x<br />

x (1 − e−Cr2 ) 3/2<br />

2) Use<br />

Vijk R = (A R + A R,3/2 P T =3/2 )<br />

X<br />

T 2 (r ij )T 2 (r jk )<br />

cyclic<br />

<strong>to</strong> get repulsion <strong>to</strong> counter T = 3/2<br />

attraction in V 3π<br />

ijk<br />

We are still working on this.<br />

V (MeV)<br />

V (MeV)<br />

600<br />

400<br />

200<br />

0<br />

V R in equilateral triangles<br />

× 10<br />

-200<br />

6000 0.5 1 1.5 2<br />

IL2: P=2, C=2.1, A=.0075<br />

P=3/2, C=2.1, A=.0075<br />

400<br />

P=3/2, C=2.6, A=.004<br />

AV18 v c<br />

200<br />

× 500<br />

0<br />

V R in L, 2L, 2L isosceles triangles<br />

-200<br />

0 0.5 1 1.5 2<br />

L (fm)


TWO-NUCLEON KNOCKOUT – (e, e ′ pN)<br />

• Recent (still being analyzed) JLAB expt. for 12 C(e, e ′ pN)<br />

• Measured back <strong>to</strong> back pp and np pairs<br />

• Pairs with relative momentum 2–3 fm −1 show 10–20 × np enhancement (preliminary).<br />

ρ NN<br />

(q,Q=0) (fm 6 )<br />

10 5<br />

10 4<br />

10 3<br />

10 2<br />

10 1<br />

10 0<br />

pp<br />

np<br />

AV18+UIX 4 He<br />

AV4’ 4 He<br />

ρ NN<br />

(q,Q=0) (fm 6 )<br />

10 6<br />

10 5<br />

10 4<br />

10 3<br />

10 2<br />

10 1<br />

10 0<br />

3<br />

He<br />

4<br />

He<br />

6<br />

Li<br />

8<br />

Be<br />

10 -1<br />

10 -1<br />

0 1 2 3 4 5<br />

q (fm -1 )<br />

0 1 2 3 4 5<br />

q (fm -1 )<br />

• VMC calculations for 3 He, 4 He, and 8 Be show this effect<br />

• Effect disappears when tensor correlations are turned off<br />

• Shows importance of tensor correlations <strong>to</strong> > 3 fm −1 .


CAN MODERN NUCLEAR HAMILTONIANS TOLERATE A BOUND<br />

TETRANEUTRON?<br />

AV18 + IL2 does not bind 4 n; E ∼ +2. MeV<br />

Attempt minimal modifications <strong>to</strong> AV18+IL2 <strong>to</strong> give E( 4 n) ∼ −0.5 MeV<br />

Check effects for o<strong>the</strong>r nuclei<br />

Modify 1 S 0 or 3 P J part of AV18<br />

• 1 S 0 binds 2 n<br />

• 3 P J must be insanely strong<br />

Add V ijk (T = 3 2 ) attraction<br />

• No effect on NN scattering<br />

• No effect on 3 H, 3 He, 4 He<br />

• Huge effects elsewheres<br />

Add V ijkl (T = 2) attraction<br />

(not shown)<br />

• No effect on 4 H, 5 He, 6 Li<br />

• Extreme (GeV scale)<br />

binding of 5,6,8 n, 6 He, etc.<br />

Energy (MeV)<br />

A bound 4 n is incompatible with<br />

our understanding of nuclear forces.<br />

S.C. Pieper, Phys. Rev. Lett. 90, 252501 (2003)<br />

0<br />

-20<br />

-40<br />

-60<br />

-80<br />

-100<br />

-120<br />

0 +<br />

2 1/2<br />

n + 4 n<br />

3 H<br />

0 +<br />

4 He<br />

Argonne v 18<br />

With <strong>Illinois</strong>-2<br />

+ modifications<br />

1 S 0<br />

V ijk<br />

Exp<br />

0 +<br />

5 H<br />

1/2 +<br />

3/2 −<br />

5 He<br />

6 n<br />

0 +<br />

0 +<br />

6 Li<br />

6 He<br />

1 +


WHY ARE THE 1 S 0 NN & T = 3 2<br />

NNN MODIFICATIONS SO<br />

DIFFERENT?<br />

v NN acts pairwise; pairs typically very far apart – rms radius = 8.9 fm<br />

4 n looks like two well separated 2 n<br />

Change of v NN ( 1 S 0 ) not that big<br />

V NNN requires triples <strong>to</strong> be close <strong>to</strong>ge<strong>the</strong>r – rms radius = 1.9 fm<br />

Hence V NNN (T = 3 ) must be large<br />

2<br />

10 -1<br />

10 -2<br />

1 S 0<br />

modified<br />

V ijk<br />

modified<br />

10 -2<br />

ρ nn<br />

(S 12<br />

= 0,1)<br />

1 S 0<br />

; S 12<br />

=0<br />

1 S 0<br />

; S 12<br />

=1<br />

V ijk<br />

; S 12<br />

=0<br />

10 0 r (fm)<br />

10 -3<br />

10 -4<br />

10 -5<br />

10 -3<br />

10 -1 r 12<br />

(fm)<br />

4 n − ρ n<br />

0 2 4 6 8 10 12 14<br />

V ijk<br />

; S 12<br />

=1<br />

2 × 2 n(S 12<br />

=0)<br />

10 -6<br />

10 -7<br />

10 -8<br />

10 -9<br />

0 2 4 6 8 10 12 14 16 18 20 22<br />

10 -4<br />

10 -5


CONCLUSIONS<br />

We have made much progress in calculating light nuclei<br />

• 1 − 2% calculations of A = 6 – 12 nuclear energies are possible<br />

• <strong>Illinois</strong> V ijk give average binding-energy errors < 0.7 MeV for A = 3 − 12<br />

– V ijk required for overall P -shell energies<br />

– Also required for S.O. splittings and several level orderings<br />

• Charge radii are generally in good agreement with expt.<br />

• GFMC values of A=6,7 electromagnetic and weak transitions improve on VMC<br />

• GFMC for scattering states is being done!<br />

and <strong>the</strong>re is still much <strong>to</strong> do<br />

• Lots of scattering states and reactions <strong>to</strong> be done<br />

– n+ 3 H, p+α, n+ 6 He, n+ 8 He, n+ 9 Li, α+α, etc.<br />

– astrophysical reactions:<br />

3 He+α → 7 Be, p+ 7 Be → 8 B, n+(α+α) → 9 Be, etc.<br />

– All big-bang nucleosyn<strong>the</strong>sis, solar neutrino, & some r-process seeding reactions<br />

are accessible.<br />

• GFMC calculations of<br />

– overlaps, spectroscopic amplitudes, and asymp<strong>to</strong>tic normalization coefs<br />

– electromagnetic and weak transitions in A ≥ 8 nuclei<br />

– unnatural-parity states<br />

• 12 C including 2 nd 0 + (Hoyle) state


MIMICKING OXYGEN ISOTOPES AS NEUTRON DROPS<br />

Neutron-rich nuclei are of great experimental interest – major thrust of RIA<br />

Generally A is <strong>to</strong>o big for GFMC calculations<br />

We are exploring <strong>the</strong> use of neutron drops, collections of neutrons in an artificial external well<br />

which represents <strong>the</strong> pro<strong>to</strong>ns, <strong>to</strong> study oxygen iso<strong>to</strong>pes.<br />

Also provides “experimental” input <strong>to</strong> Skyrme models for isospin spin-orbit terms<br />

Preliminary central well was adjusted<br />

so that 8,9,10 n energy differences<br />

match 16,17,18 O<br />

• Well binds only 0S and 0P<br />

– SD-shell binding results<br />

from AV18+IL2 in H<br />

• 8 n ρ(r) similar <strong>to</strong> 16 O ρ(r)<br />

To be done:<br />

Energy (MeV)<br />

-100<br />

-110<br />

-120<br />

-130<br />

-140<br />

-150<br />

• Spin-orbit splitting <strong>to</strong>o small -160<br />

– need V SO term in well<br />

• Calculate <strong>to</strong> drip line and o<strong>the</strong>r states (2 + )<br />

6 n<br />

0 +<br />

7 n<br />

3/2 −<br />

1/2 −<br />

8 n<br />

IL2<br />

-19<br />

0 +<br />

N+8 O<br />

Exp.<br />

External Well<br />

with<br />

Argonne v 18<br />

+ IL2<br />

3/2 +<br />

1/2 +<br />

5/2 +<br />

9 n 0 +<br />

1/2 +<br />

10 n<br />

5/2 +<br />

11 n<br />

0 +<br />

12 n<br />

1/2 +<br />

5/2 +<br />

13 n 0 +<br />

14 n<br />

B.S. Pudliner, A. Smerzi, J. Carlson, V.R. Pandharipande, S.C. Pieper, and D.G. Ravenhall, Phys. Rev. Lett. 76, (1996)<br />

S.C. Pieper, Nucl. Phys. 751, 516c (2005)


CHARGE SYMMETRY BREAKING<br />

GFMC isovec<strong>to</strong>r and isotensor energy coefficients a (n)<br />

A,T<br />

E A,T (T z ) = X<br />

n≤2T<br />

a (n)<br />

A,T Q n(T, T z )<br />

Q 0 = 1 ; Q 1 = T z ; and Q 2 = 1 2 (3T 2 z − T 2 )<br />

Argonne v 18 + IL2<br />

(keV)<br />

3 H– 3 He<br />

7 Li– 7 Be<br />

7 He, 7 Li, 7 Be, 7 B<br />

T n v Coul v o<strong>the</strong>rE&M v CSB+CD K CSB Total Expt.<br />

1<br />

2<br />

1 649 29 64 14 757 764<br />

1<br />

2<br />

1 1458 40 83 23 1605 1644<br />

3<br />

2<br />

1 1286 14 49 17 1366 1373<br />

3<br />

2<br />

2 132 7 34 174 175<br />

8 Li, 8 Be, 8 B 1 1 1692 24 78 24 1818 1770<br />

1 2 140 5 −5 140 145<br />

8 He, 8 Li, 8 Be, 8 B, 8 C 2 1 1719 13 83 26 1840 1659<br />

2 2 153 7 42 203 153


ISOSPIN MIXING IN 8 BE<br />

• 8 Be has pairs of T =0,1 states around 17 MeV excitation energy<br />

• Isospin mixing of <strong>the</strong>se has been expt. known for decades<br />

• GFMC gives reasonable values for <strong>the</strong> mixing matrix elements<br />

20<br />

4 +<br />

3 +<br />

3 + ;1<br />

1 +<br />

1 + ;1<br />

2 +<br />

2 + ;1<br />

– Charge-symmetry-breaking term in AV18 is important<br />

15<br />

Isospin mixing matrix elements in keV<br />

J P GFMC Expt<br />

Coulomb Strong CSB O<strong>the</strong>r Total<br />

2 nd 2 + 78 21 16 115 144<br />

1 + 80 18 4 102 120<br />

3 + 61 15 14 90 63<br />

1 st 2 + 4 0.4 1 6 –<br />

Excitation energy (MeV)<br />

10<br />

5<br />

4 +<br />

Possible V −A in 8 Li(β − ) 8 Be(1 st 2 + ) at ATLAS and <strong>the</strong> CPT<br />

2 +<br />

• Must be pure T = 1 decay <strong>to</strong> exclude Fermi decay<br />

0<br />

0 +<br />

• Our calculated mixing of T = 1 in 8 Be(1 st 2 + ) is small<br />

enough <strong>to</strong> make experiment feasible.<br />

8 Be; T=0


GFMC CALCULATIONS OF ELECTROWEAK MATRIX ELEMENTS<br />

Electromagnetic Transitions of A = 6, 7 Nuclei – Widths in MeV<br />

Ji P → Jf P Transition VMC GFMC Expt<br />

6 Li(3 + ; 0) → 6 Li(1 + ; 0) BE2 (10 −10 ) 2.68 4.73 ± 0.05 4.4 ± 0.34<br />

6 Li(0 + ; 1) → 6 Li(1 + ; 0) BM1 (10 −6 ) 7.09 6.85 ± 0.02 8.19 ± 0.17<br />

7 Li( 1 2<br />

7 Li( 1 2<br />

7 Li( 7 2<br />

7 Be( 1 2<br />

7 Be( 1 2<br />

− ) → 7 Li( 3 − ) BE2 (10 −13 ) 2.68 3.33 ± 0.12 3.3 ± 0.2<br />

2<br />

−<br />

) → 7 Li( 3 − ) BM1 (10 −9 ) 4.81 4.67 ± 0.15 6.3 ± 0.31<br />

2<br />

−<br />

) → 7 Li( 3 − ) BE2 (10 −8 ) 1.26 1.7 ± 0.02 0.6<br />

2<br />

−<br />

) → 7 Be( 3 − ) BE2 (10 −13 ) 4.30 6.07 ± 0.19 –<br />

2<br />

−<br />

) → 7 Be( 3 − ) BM1 (10 −9 ) 2.67 2.66 ± 0.03 3.43 ± 0.45<br />

2<br />

Weak Transitions of A = 6, 7 Nuclei – log(ft)<br />

J P i → J P f Weak Current VMC GFMC Expt Half life(s)<br />

6 He(0 + ) → 6 Li(1 + ) GT 2.90 2.92 2.91 0.81<br />

7 Be( 3 − ) → 7 Li( 3 − ) GT and F 3.28 3.24 3.22 4.6×10 6<br />

2<br />

2<br />

7 Be( 3 − ) → 7 Li( 1 − ) GT 3.52 3.54 3.45<br />

2<br />

2


4,6,8 HE DENSITIES<br />

• 4 He central density twice that of nuclear matter!<br />

• Neutrons drag 4 He center of mass around – spread out density<br />

– results in charge radius of 6 He > 4 He (2.08 fm vs 1.66 fm)<br />

• 6 He & 8 He have large neutron halos due <strong>to</strong> weak binding of neutrons<br />

• Neutron halo of 6 He more diffuse than that of 8 He – smaller E sep<br />

0.15<br />

4 He - p or n<br />

6 He - Pro<strong>to</strong>n<br />

4 He - p or n<br />

6 He - Pro<strong>to</strong>n<br />

6 He - Neutron<br />

6 He - Neutron<br />

Density (fm -3 )<br />

0.10<br />

0.05<br />

8 He - Pro<strong>to</strong>n<br />

8 He - Neutron<br />

Density × r 2 (fm -1 )<br />

10 -1 r (fm)<br />

10 -2<br />

10 -3<br />

10 -4<br />

8 He - Pro<strong>to</strong>n<br />

8 He - Neutron<br />

0.00<br />

0 0.5 1 1.5 2 2.5 3<br />

r (fm)<br />

10 -5<br />

0 2 4 6 8 10 12


IS AN ALPHA PARTICLE IN A SEA OF NEUTRONS<br />

STILL AN ALPHA PARTICLE?<br />

<strong>The</strong> α core of 6,8 He is pushed around by <strong>the</strong> neutrons<br />

− rms charge radius changed just by C.M. effects<br />

ρ pp (r 12 ) is probability of two pro<strong>to</strong>ns separated by r 12<br />

− not effected by C.M. motion of <strong>the</strong> α.<br />

Only pp pair in 4,6,8 He is “in” <strong>the</strong> α core<br />

–ρ pp (r 12 ) measures size of α core<br />

ρ pp less peaked in 6,8,10 He<br />

− Core polarization<br />

− Charge-exchange correlations<br />

with valence neutrons<br />

− Implies 80−350 keV excitation of α or<br />

0.4−2% admixture of 20 MeV excited state<br />

ρ pp<br />

(fm -3 )<br />

0.020<br />

0.015<br />

0.010<br />

0.005<br />

4 He<br />

6 He - Pro<strong>to</strong>n<br />

8 He - Pro<strong>to</strong>n<br />

0.000<br />

0 0.5 1 1.5 2 2.5 3<br />

r 12<br />

(fm)


INTRINSIC DENSITY OF 8 BE<br />

8 Be w.f.: 4 He core + 4 p-shell nucleons + pair corr.<br />

M. C. ρ(r): random walk in |Ψ(r 1 , r 2 , · · · , r 8 )| 2 & periodically for each set (r 1 , r 2 , · · · , r 8 )<br />

Lab ρ(r): bin r 1 , r 2 , · · · , r 8<br />

Intrinsic ρ(r): find eigenvec<strong>to</strong>rs of moment of inertia matrix:<br />

0<br />

1<br />

x 2 i x i y i x i z i<br />

M = X i<br />

B<br />

@<br />

y i x i y 2 i y i z i<br />

C<br />

A ,<br />

rotate <strong>to</strong> <strong>the</strong>m, and bin r ′ 1, r ′ 2, · · · , r ′ 8.<br />

6<br />

z i x i z i y i z 2 i<br />

6<br />

4<br />

2<br />

.005<br />

.02<br />

.1<br />

8 Be(0 + )<br />

.005<br />

.02<br />

.1<br />

.3<br />

4<br />

2<br />

.005<br />

.02<br />

8 Be(4 + )<br />

.005<br />

.02<br />

.1<br />

.3<br />

z (fm)<br />

0<br />

z (fm)<br />

0<br />

.1<br />

-2<br />

-4<br />

-6<br />

.05<br />

.01<br />

.2<br />

.002<br />

.05<br />

.01<br />

.002<br />

Labora<strong>to</strong>ry<br />

Intrinsic<br />

-4 -2 0 2 4 -2 0 2<br />

r = (x 2 +y 2 ) 1/2 (fm)<br />

-2<br />

-4<br />

-6<br />

.05<br />

.01<br />

.002<br />

.2<br />

.05<br />

Labora<strong>to</strong>ry<br />

Intrinsic<br />

.01<br />

.002<br />

-4 -2 0 2 4 -2 0 2<br />

r = (x 2 +y 2 ) 1/2 (fm)

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