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Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model G.Papadimitriou 1 N.Michel 6,7,...
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Transcript of Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model G.Papadimitriou 1 N.Michel 6,7,...
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…