Charge radii of 6,8Heand Halo nuclei

in Gamow Shell Model

G.Papadimitriou1

N.Michel6,7, 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.3 Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge4 Institute of Theoretical Physics, University of Warsaw, Warsaw.5 Grand Accélérateur National d'Ions Lourds (GANIL).6 CEA/DSM, Caen, France7 Department of Physics, Graduate School of Science, Kyoto University, Kyoto8 Department of Physics, University of Arizona, Tucson, Arizona

Outline

Drip line nuclei as Open Quantum Systems

Gamow Shell Model Formalism

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

Calculations on 6,8He nuclei

Results on charge radii of 6,8He

Comparison with other models

Conclusion 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)

Theories that incorporate the continuum

0,1 2

22

2

rkuk

r

llrv

dr

dl

The Gamow Shell Model (Open Quantum System)N.Michel et.al 2002PRL 89 042502

2

2

mE

k

statesscatteringrrkHCrkHCrku

resonancesstatesboundrrkHCrku

lll

ll

,),(),(~),(

,,),(~),(

Poles of the S-matrix

Berggren’s Completeness relation

n L

kknn dkuuuu 1~~

1~~ dkuuuu kn k

knn

T.Berggren (1968) NP A109, 265

resonant states(bound, resonances…)

Non-resonantContinuumalong the contour

Many-body discrete basis

Complex-Symmetric Hamiltonian matrix

Matrix elements calculated via complex scaling

GSM application for He chain

Borromean nature of 6,8He is manifested

GHF+SGI

PRC 70, 064313 (2004)N.Michel et al

Optimal basis for each nucleus via the GHF method

Helium anomaly is well reproduced

p model space

0p3/2 resonance0p1/2 resonance

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)

complex scaling does not apply to this particular integral…

We want a Hamiltonian free from spurious CM motion

Lawson method?

Jacobi coordinates?

pipj matrix elements

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

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 nucl-ex/0809.2607v1 (2008)

center of mass of the nucleus

6He

8He

EXPERIMENTAL RADII OF 6He, 8He, 11Li

Rcharge(6He) > Rcharge (8He)

“Swelling” of the core is not negligible

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

4He 6He 8He

L.B.Wang et al

P.Mueller et al

1.43fm 1.912fm

1.45fm 1.925fm 1.808fm

9Li 11Li

R.Sanchez et al 2.217fm 2.467fm

Point proton charge radii

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

10Be 11Be

W.Nortershauser et al 2.357fm 2.460fm

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

Charge radii provide a benchmarktest for nuclear structure theory!

GSM calculations for 6,8He nuclei

• WS or KKNN or HF basis

• WS is parameterized to the 0p3/2 s.p of 5He

• KKNN PTP 61, 1327 (1979)

3

2

2211

5

3

202

1

20

exp13.01exp

exp1exp

nn

LSn

lLSLS

kkk

l

kkkcentral

LScentral

rVrVrV

rVrVV

rVVrV

SL

Reproduces low-energy phase shifts of 4He-n system

• 4He + valence particles framework

Im[k

] (f

m-1)

Re[k] (fm-1)

3.270p3/2

jL

B

(0.17,-0.15)A

(2.0,0.0)

GSM calculations for 6,8He nuclei

jL

Model 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

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)

Schematic two-body interactions employed

1. Modified Minnesota Interaction (MN) (NPA 286, 53)2. Surface Delta Interaction (SDI) (PR 145, 830)3. Surface Gaussian Interaction (SGI) (PRC 70, 064313)

3

1

212

012 )exp(

kkkkkkk rPHPBPPMWVV

SGI

MN

0121012 RrrrVV SDI

021

2

21012 2exp, Rrr

rrTJVV

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

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.

Expression of charge radius in these coordinates

0

2 drrurru fi

correctionmassofcentercore

pp rrrrA

AZrAZr 2122

21

222 2

4

122,,

complex scaling cannot be applied!

Renormalization of the integral based on physical arguments (density)

Charge Radii calculations for 6,8He nuclei

In our calculations we carried out the radial integration until 20fm

Generalization to n-valence particles is straightforward

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

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

8He

rpp (6He) = 1.85 fm rpp (8He) = 1.801 fm

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 28.83 To be compared with

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

rpp(6He) = 1.924 fm rpp(8He) = 1.957 fm

GSM Jπ: 0+ = - 3.125 MeVExp Jπ: 0+ = - 3.11 MeV

p3/2 d5/2 p1/2 d3/2 s1/2

θnn = 81.80o

Results on charge radii of 6He with SDI interaction

rpp(6He) = 1.92 fm

θnn = 82.13o

Comparison with other models and experiment

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 modelling weakly bound nuclei with large radial extension.

For 6He we observed an overall weak sensitivity for both correlation angle and charge radius. However, when adding 2 more neutrons in the system he details of each interaction are revealed.

The next step: charge radii 8He, 11Li, 11Be assuming an 4He core. The rapid increase in the dimensionality of the space will be handled by the GSM+DMRG method. (J.Rotureau et al PRL 97 110603, 2006 and PRC 79 014304, 2009) Develop 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. The effective GSM interaction depends on the valence space, but also in the position of the thresholds and the position of the S-matrix poles

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

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…