Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model G.Papadimitriou 1 N.Michel 7 , W.Nazarewicz 1,2,4 , M.Ploszajczak 5 , J.Rotureau 8 1 Department of Physics and Astronomy, University of Tennessee,Knoxville. 2 Physics Division, Oak Ridge National Laboratory, Oak Ridge, USA 3 Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge, USA 4 Institute of Theoretical Physics, University of Warsaw, Warsaw. 5 Grand Accélérateur National d'Ions Lourds (GANIL) CEA/DSM Caen Cedex, France 7 Department of Physics, Post Office Box 35 (YFL), FI-40014 University of Jyväskylä, Finland 8 Department of Physics, University of Arizona, Tucson, Arizona, USA
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Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model
Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model. G.Papadimitriou 1 N.Michel 7 , W.Nazarewicz 1,2,4 , M.Ploszajczak 5 , J.Rotureau 8. 1 Department of Physics and Astronomy, University of Tennessee,Knoxville. 2 Physics Division, Oak Ridge National Laboratory, Oak Ridge, USA - PowerPoint PPT Presentation
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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)
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
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