Charge radii of 6,8Heand Halo nuclei

in Gamow Shell Model

G.Papadimitriou1

N.Michel7, W.Nazarewicz1,2,4, M.Ploszajczak5, J.Rotureau8

1 Department of Physics and Astronomy, University of Tennessee,Knoxville.2 Physics Division, Oak Ridge National Laboratory, Oak Ridge, USA3 Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge, USA4 Institute of Theoretical Physics, University of Warsaw, Warsaw.5 Grand Accélérateur National d'Ions Lourds (GANIL) CEA/DSM Caen Cedex, France7 Department of Physics, Post Office Box 35 (YFL), FI-40014 University of Jyväskylä, Finland8 Department of Physics, University of Arizona, Tucson, Arizona, USA

Outline

Drip line nuclei as Open Quantum Systems

Gamow Shell Model Formalism

Experimental Radii of 6,8He (11Li and 11Be)

Spin-orbit density effect on charge radii

Calculations on charge radii of 6,8He

DMRG application to 8He

Conclusions and Future Plans

I.Tanihata et alPRL 55, 2676 (1985)

0

2

964

1797

1867

6He

4He +2n

5He +n

Proximity of the continuum

It is a major challenge of nuclear theory to developtheories and algorithms that would allows us to understandthe properties of these exotic systems.

Continuum Shell Model (CSM) • H.W.Bartz et al, NP A275 (1977) 111• A.Volya and V.Zelevinsky PRC 74, 064314 (2006)

Shell Model Embedded in Continuum (SMEC)• J. Okolowicz.,et al, PR 374, 271 (2003) • J. Rotureau et al, PRL 95 042503 (2005)

Gamow Shell Model (GSM)• N. Michel et al, PRL 89 042502 (2002)• R. Id Betan et.al PRL 89, 042501 (2002)• N. Michel et al., Phys. Rev. C67, 054311 (2003)• N. Michel et al., Phys. Rev. C70, 064311 (2004)• G. Hagen et al, Phys. Rev. C71, 044314 (2005)• N. Michel et al, J.Phys. G: Nucl.Part.Phys 36, 013101

(2009)

Shell Model Theories that incorporate the continuum, selected references

GSM HAMILTONIAN

“recoil” term coming from the expression of H in the COSMcoordinates. No spurious states

Y.Suzuki and K.Ikeda PRC 38,1 (1988)

We want a Hamiltonian free from spurious CM motion

Lawson method?

Jacobi coordinates?

Appropriate treatment for proper description of the recoil of the core and the removal of the spurious CoM motion.

(G.Hagen et al PRL 103 062503 (2009))

L.B.Wang et al, PRL 93, 142501 (2004) P.Mueller et al, PRL 99, 252501 (2007) R.Sanchez et al PRL 96, 033002 (2006) W.Nortershauser et al PRL 102, 062503 (2009)

EXPERIMENTAL RADII OF 6He, 8He, 11Li

Rcharge(6He) > Rcharge (8He)

“Swelling” of the core is not negligible

4He 6He 8He

L.B.Wang et al

P.Mueller et al

1.67fm 2.054(18)fm

1.67fm 2.068(11)fm1.929(26)fm

9Li 11Li

R.Sanchez et al 2.217(35)fm 2.467(37)fm

RMS charge radii

charge radii determinesthe correlations betweenvalence particles ANDreflects the radial extentof the halo nucleus

10Be 11Be

W.Nortershauser et al 2.357(16)fm 2.460(16)fm

~4% ~8%

Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)

Spin-orbit contribution to the charge radius

Usually point radii are converted to charge radii through:

finite size effects

Darwin-Foldy term

22 769.0 fmRp 22 1161.0 fmRn

22

033.04

3fm

M p

It was proposed that the D.F term should be treated as a part of the charge radius because it appears in the charge density of the proton (J.L. Friar et al PRA 56, 6 (1997))

Additionally the spin-orbit density could have a non-negligible effect on the charge radius.

Contributes on a noticeable change to the charge radius between 40Ca and 48Ca(W. Bertozzi et al , PLB 41, 408 (1972))

Finite size effects and relativistic D.F term are consistently considered in theoretical calculations

The s.o effect is almost never considered… (except maybe)H.Esbensen et al PRC 76, 024302 (2007)

A.Ong et al PRC 82, 014320 (2010)

Both terms appear explicitly in the expression of the single nucleon charge operator and theyenter a non-relativistic calculation to an order 1/m2 .

BUT …

Spin-orbit contribution to the charge radius

As we shall see , the s.o can have a comparable contribution with the finite size effects!!

The formulas to calculate the s.o correction are the following:(J.Friar et al Adv.Nucl.Phys. 8, 219, (1975))

The charge distribution in Helium halos is consistently described by:

The orbital motion of the core around the center of mass of the nucleus The polarization of the core by the valence neutrons

The s.o contribution caused by the anomalous magnetic moment of the neutron

Comparison of 6,8He radius data with nuclear theory models

Charge radii provide a benchmarktest for nuclear structure theory!

GSM calculations for 6,8He nucleiModel space

p-sd waves

0p3/2 resonance onlyi{p3/2} complex non-resonant parti{s1/2}, i{p1/2}, i{d3/2}, i{d5/2} real continua (red line) with i=1,…Nsh

Nsh = 30 for p3/2 contour and Nsh

= 20 for each real cont.

Total 111 single particle states.

We limit ourselves to 2 particles occupying continuum orbits...

Im[k

] (f

m-1)

Re[k] (fm-1)

3.270p3/2

jL

B

(0.17,-0.15)A

(2.0,0.0)

Modified Minnesota Interaction (MN) (NPA 286, 53)

3

1

212

012 )exp(

kkkkkkk rPHPBPPMWVV

Parameterizations

• The two strength parameters of the MN are adjusted to the g.s of 6,8He.

Expression of charge radius in these coordinates

correctionmassofcentercore

pp rrrrA

AZrAZr 2122

21

222 2

4

122,,

Charge Radii calculations for 6,8He nuclei

Generalization to n-valence particles is straightforward

n

i

n

jiji

C

i

C

Ap

nAp rr

nn

nAr

nAXrXr CC

12

22

22

2

111

The ingredients of the calculation are the OBME and TBME

jri i2 klrrij ji

Same formulas for heavier systems 11Li, 11Be

Results on charge radii of 6,8He with MMN interaction

The s.o corrections are comparable to the D.W term 6He and comparable to the finite size corrections in 8He.

The s.o as compared to a maximal estimate they are not very different.

Good overall agreement of the radii with experiment. Experimental trend is satisfied.

Results on charge radii of 6,8He with MMN interaction (configuration mixing and correlation angle (for 6He) )

8He

201083onn131878onn

PRC 76, 051602

201083 o

nn

131878 o

nn

Angles estimatedfrom the availableB(E1) data and the average distances betweenneutrons.

We calculated also the correlation angle for 6He

Our result is:

onn 95.82 To be compared with

The charge radius of 6He as a function of the S2n

Black line: Core polarization was not included.

Red line: Core polarization is taken into account.We use a 4% increase of the α-core pp radius as it was estimated by GFMC calculations.

The narrow experimental error bars suggest that the S2n should be calculated with a high precision if one aims in a detailed description of the 6He radial extent.

When this condition is met the p3/2 state had a dominant occupation of about 90% in the 6He g.s.

For this p3/2 percentage and the correct S2n, the geometry of the neutrons (correlations) and the radial extent is such, so as the calculated radius is in a satisfactory agreement with the experiment.

Blue line: Core polarization + s.o effect

Comparison with other models and experiment

Density Matrix Renormalization Group (DMRG)S.R White PRL 69 (1992) 2863T.Papenbrock and D.Dean J.Phys.G 31 (2005) S1377S.Pittel et al PRC 73 (2006) 014301J.Rotureau et al PRC 79 (2009) 014304

Truncation Method applied to lattice models, spin chains, atomic nuclei….

BAM

BA

1,

• Basic idea: where A and B are partitions of the system.

Approximate in terms of m < M basis states (truncation)

These m states are eigenstates of the density matrix A

ABABBBB

BAABAA '''' or

The difference between the exact and the approximated , has the minimal norm.

The partition of the system has to be decided by the practitioner.

In GSM+DMRG we optimize the numberof non-resonant statesalong the scattering contours.

Key point: In DMRG the wave function is not stored. But the second quantized operators that define the Hamiltonian are calculated and stored in each step…

n

i

n

jiji

C

i

C

Ap

nAp rr

nn

nAr

nAXrXr CC

12

22

22

2

111

The radius operator has the same form (in second quantization) with H

We calculate OBMEs and TBMEs of rpp jri i2 klrrij ji

In each DMRG step we calculate the expectation value the radius

Density Matrix Renormalization Group application to 8He radius

Density Matrix Renormalization Group application to 8He radius

In the following we slightly renormalized the strengths of the MN interaction so as to reproduce the g.s 0+ energy of 8He.

Conclusion and Future Plans

The very precise measurements of 6,8He halos charge radii provide a valuable test of the configuration mixing and the effective interaction in nuclei close to the drip-lines.

The GSM description is appropriate for modeling weakly bound nuclei with large radial extension.

Using a finite range force (MN) and adjusting the strengths to the g.s energies of 6,8He we reproduced the experimental trend of Helium halo charge radii.

Charge radii are primarily sensitive on the p3/2 occupation and the S2n.

The core polarization by the valence neutrons is a small but NOT negligible effect.

Our calculations showed that the s.o contribution in the conversion of the point-proton radii can be comparable to the D.F term and the finite size effects

The next step: charge radii and properties of 11Li, 11Be assuming an 4He core in a GSM+DMRG framework. Develop the effective interaction for GSM applications in the p and p-sd shells that will open a window for a detailed description of weakly bound systems.

Radial density of valence neutrons for the 6He

With an adequate number of points along the contour the fluctuations become minimal

We “cut” when for a given number of discretization points the fluctuations are smeared out

cut

Convergence properties of the DMRG are met for both radius and energy. DMRG converges on the right value. We compare a 2p-2h calculation with a full (4p-4h). The differences depict the model space effect on the observables (energy/radius).

The energy in DMRG is more attractive and the radius is smaller compared to the 2p-2h.

Density Matrix Renormalization Group application to 8He proton radius

ε = 10-8

GSM calculations for 6,8He nuclei

Example: 6He g.s with MN interaction.

Basis set 1: p-sd waves with 0p3/2 resonant and ALL the rest continuai{p3/2}, i{p1/2}, i{s1/2}, i{d3/2}, i{d5/2}30 points along the complex p3/2 contour and 25 points for each real continuum Total dimension: dim(M) = 12552

Basis set 2: p-sd waves with 0p3/2 resonant and i{p3/2}, i{p1/2}, i{s1/2}non-resonant continua BUT d5/2 and d3/2 HO states.nmax = 5 and b = 2fm (We have for example 0d5/2, 1d5/2, 2d5/2, 3d5/2, 4d5/2 for nmax = 5)Total dimension: dim(M) = 5303

g.s energies for 6He

Basis set 1

Jπ : 0+ = - 0.9801 MeVBasis set 2

Jπ : 0+ = - 0.9779 MeV

Differences of the order of ~ 0.22 keV…

GSM calculations for 6,8He nuclei

Energies and radial properties are equivalent in both representations.

The combination of Gamow states for low values of angular momentum and HO for higher, captures all the relevant physics while keeping the basis in a manageable size.

Applicable only with fully finite range forces (MN)…

Radial density of the6He g.s. red and green curves correspond to thetwo different basis sets.

Recoil term treatment

i) Transformation to momentum space

ii kp

Fourier transformation to return back to r-space

ii) Expand ip in HO basis

α,γ are oscillator shellsa,c are Gamow states

No complex scaling is involvedGaussian fall-off of HO states provides convergenceConvergence is achieved with a truncation of about Nmax ~ 10 HO quanta

Two methods which are equivalent from a numerical point of view

PRC 73 (2006) 064307

No complex scaling is involved for the recoil matrix elements

MeVA

kkHe

c

7424.0:0 216

MeV

A

ppHe

c

7417.0:0 216

MeVA

kkHe

c

0824.0,1771.0:2 216

MeVA

ppHe

c

0824.0,1768.0:2 216

Different interactions lead to different configuration mixing.

6He charge radius (Rch) is primarily related to the p3/2

occupation of the 2-body wavefunction.

The recent measurements put a constraint in our GSM Hamiltonian

which is related to the p3/2 occupation.

We observe an overall weak sensitivity for both radii and the correlation angle.

Results and discussion

p-sd shells (5 partial-waves) , 47 shells total.Edmrg = -3.284 MeV, EGSM = -3.112 MeV the system gained energy in DMRG as a resultof the larger model space (4p-4h). The difference is ~150keV.Remember that EGSM (2p-2h) is the experimental value (MN was fitted in this way.)Truncation in the DMRG sector is governed by the trace of the density matrix

dim GSM = 9384683

dim DMRG = 3859ε = 10-8

Density Matrix Renormalization Group application to 8He

Complex Scaling

•Diagonalization of Hamiltonian matrix

•Large Complex Symmetric Matrix

•Two step procedure

“pole approximation”

•Identification of physical state by maximization of 0

resonance

bound state

resonance

bound state

Fullspace

non-resonantcontinua

0 0

22 ~)(

~~''

drrConstdrrurruthe

eeruandeeruFor

fi

rikrikf

ikrikri

Integral regularization problem between scattering states

For this integral it cannot be found an angle in the r-complex plane to regularize it…

Density Matrix Renormalization Group

From the main configuration space all the |k>A are built (in J-coupled scheme)

Succesivelly we add states from the non resonant continuum state and construct states |i>B

In the {|k>A|i>B}J the H is diagonalized

ΨJ=ΣCki {|k>A|i>B}J is picked by the overlap method

From the Cki we built the density matrix and the N_opt states are corresponding to the maximum eigenvalues of ρ.

◊ NCSM P.Navratil and W.E Ormand PRC 68 034305

∆ GFMC S.C.Pieper and R.B.Wiringa Annu.Rev.Nucl.Part.Sci. 51, 53 Collective attempt to calculate the charge radius by all modern structure modelsVery precise measurements on charge radiiProvide critical test of nuclear models

Charge radii of Halo nuclei is a very important observable that needs theoretical justification

Experiment

From radii...to stellar nucleosynthesis!

Figures are taken fromPRL 96, 033002 (2006) andPRL 93, 142501 (2004)

A

ji

A

iiij

A

ii

iA

jiij

A

i

i rUVrUm

pHV

m

pH

1 11

2

11

2

22

AEAH ,...1,...1

SHELL MODEL (as usually applied to closed quantum systems)

Largest tractable M-scheme dimension ~ 109

single particle HarmonicOscillator (HO) basis

nice mathematical properties:Lawson method applicable…

HEAVIER SYSTEMS

Explosion of dimension

Hamiltonian Matrix is dense+non-hermitian

Lanczos converges slowly

Density Matrix Renormalization Group

S.R.White., 1992 PRL 69, 2863; PRB 48, 10345T.Papenbrock.,D.Dean 2005., J.Phys.G31 S1377 J.Rotureau 2006., PRL 97, 110603J.Rotureau et al. (2008), to be submitted

bound-statesresonances

non-resonantcontinua

A

B

Separation of configuration space in A and B

Truncation on B by choosing the most important configurations

Criterion is the largest eigenvalue of the density matrix

Form of forces that are used

SGI

SDI

Minnesota

)2()1()exp(),(),(

2

22

21

21 YYrr

TJVrrV

GI

EXPERIMENTAL RADII OF Be ISOTOPES

W.Norteshauser et allnucl-ex/0809.2607v1

interaction cross section measurements

GFMC

NCSMPRC 73 065801 (2006) andPRC 71 044312 (2005)

PRC 66, 044310, (2002) andAnnu.Rev.Nucl.Part.Sci. 51, 53 (2001)

FMD

7Be charge radius provides constraints for the S17 determination

Charge radius decreases from 7Be to 10Be and then increases for 11Be

11Be increase can be attributed to the c.m motion of the 10Be core

The message is that changes in charge distributionsprovides information about the interactions in thedifferent subsystems of the strongly clustered nucleus!

11Be 1-neutron halo

Closed Quantum System Open quantum system

scattering continuum

resonance

bound states

discrete states

(nuclei near the valley of stability)

infinite well

(nuclei far from stability)

(HO) basis

nice mathematical properties:

Exact treatment of the c.m,analytical solution…

GSM calculations for 6,8He nuclei

Forces

with Nmax = 10 and b = 2fm

States with high angular momentum (d5/2, d3/2)

• The large centrifugal barrier results in an enhanced localization of the d-waves

• ONLY for d5/2, d3/2 or higher orbits we may use HO basis states for our calculation

• s-p waves are always generated by a complex WS or KKNN basis.

• SGI/SDI parameters are adjusted to the g.s of 6He

• The 2+ state of 6He is used to adjust the V(J=2,T=1) strength of the SGI

• The two strength parameters of the MN are adjusted to the g.s of 6,8He

• The matrix elements of the MN were calculated with the HO expansion method, like in the recoil case.

• In all the following we employed SGI+WS, SDI+WS and KKNN+MN for 6He

• For 8He we used the spherical HF potential obtained from each interaction