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Charge radii of 6,8He
and 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 Ridge
4 Institute of Theoretical Physics, University of Warsaw, Warsaw.
5 Grand Accélérateur National d'Ions Lourds (GANIL).
6 CEA/DSM, Caen, France
7 Department of Physics, Graduate School of Science, Kyoto University, Kyoto
8 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 al
PRL 55, 2676 (1985)
Proximity of the
continuum
5
It is a major challenge of nuclear theory to develop
theories and algorithms that would allows us to understand 
the properties of these exotic systems.
1867
He +n
2  1797
4
He +2n



6
He
964

0
Theories that incorporate the continuum
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)
The Gamow Shell Model (Open Quantum System)
N.Michel et.al 2002
PRL 89 042502
 d2
l l  1 2 
  2  vr   2  k ul k , r   0
r
 dr

k
2mE
2
Poles of the
S-matrix
ul (k , r ) ~ C H l (k , r ) , r   bound states, resonances
ul (k , r ) ~ C H l (k , r )  C H l (k , r ) , r   scattering states
Berggren’s Completeness relation
T.Berggren (1968)
NP A109, 265
u
n
n
u~n   uk u~k dk  1
L
resonant states
(bound, resonances…)
u
n
n
Non-resonant
Continuum
along the contour
u~n   uk u~k dk  1
k
Many-body discrete basis
Complex-Symmetric Hamiltonian matrix
Matrix elements calculated via complex scaling
GSM application for He chain
GHF+SGI
p model space
0p3/2 resonance
0p1/2 resonance
 Optimal basis for each nucleus via the GHF method
 Borromean nature of
6,8He
is manifested
 Helium anomaly is well reproduced
PRC 70, 064313 (2004)
N.Michel et al
GSM HAMILTONIAN
We want a Hamiltonian free from spurious CM motion
Lawson method?
Jacobi coordinates?
Y.Suzuki and K.Ikeda
PRC 38,1 (1988)
“recoil” term coming from the
expression of H in the COSM
coordinates. No spurious states
 pipj matrix elements
complex scaling does not apply to this particular integral…
Recoil term treatment
PRC 73 (2006) 064307
Two methods which are equivalent from a numerical point of view
i) Transformation to momentum space
ii) Expand
pi
in HO basis
α,γ are oscillator shells
a,c are Gamow states
pi  ki
 No complex scaling is involved for
the recoil matrix elements
Fourier transformation to return back
to r-space
 
k k
0 6 He : 1 2   0.7424 MeV
Ac
 
k k
2 6 He : 1 2   0.1771,0.0824 MeV
Ac
No complex scaling is involved
Gaussian fall-off of HO states provides
convergence
Convergence is achieved with a truncation
of about Nmax ~ 10 HO quanta
0
2
 6
 6
 
p1  p2
He :
  0.7417 MeV
Ac
 
p1  p2
He :
  0.1768,0.0824 MeV
Ac
EXPERIMENTAL RADII OF 6He, 8He, 11Li
Point proton charge radii
6He
charge radii determines
the correlations between
valence particles AND
reflects the radial extent
of the halo nucleus
center of mass of the
nucleus
8He
4He
1.43fm 1.912fm
P.Mueller et al
1.45fm 1.925fm 1.808fm
9Li
(6He)
> Rcharge
8He
L.B.Wang et al
R.Sanchez et al
Rcharge
6He
(8He)
2.217fm
10Be
W.Nortershauser et al
2.357fm
11Li
2.467fm
11Be
2.460fm
“Swelling” of the core is not negligible
Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)
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)
Comparison of
6,8He
radius data with nuclear theory models
Charge radii provide a benchmark
test for nuclear structure theory!
GSM calculations for 6,8He nuclei
Im[k] (fm-1)
• 4He + valence particles framework
• WS or KKNN or HF basis
• WS is parameterized to the 0p3/2 s.p of 5He
• KKNN
(2.0,0.0)
Re[k] (fm-1)
B
0p3/2
PTP 61, 1327 (1979)
A
(0.17,-0.15)
V r   Vcentral  VLS r 
2


Vcentral   V exp   k r   1
k 1
0
k
2
5
l
V
k 3
0
k

exp   k r 2
3.27
Lj

3
 LS


2
l
VLS r   L  S V1 exp  1 r  1  0.3  1  VnLS exp   n r 2 
n2





 Reproduces low-energy phase shifts of 4He-n system


Model space
p-sd waves
0p3/2 resonance only
j
i{p3/2} complex non-resonant part L
i{s1/2}, i{p1/2}, i{d3/2}, i{d5/2} real continua (red line)
Im[k] (fm-1)
GSM calculations for 6,8He nuclei
(2.0,0.0)
Re[k] (fm-1)
B
0p3/2
A
(0.17,-0.15)
with i=1,…Nsh
3.27
Lj
We limit ourselves to 2 particles occupying continuum
orbits.
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


V12   Vk0 Wk  M k P P  Bk P  H k P exp(   k r122 )
MN
  r  r  2 
 
V12 V0 J , T   exp   1 2      r1  r2  2  R0 
    
SGI
k 1
 
V12 V0  r1  r2  r1  R0 
SDI
GSM calculations for 6,8He nuclei
Forces
• 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.
with Nmax = 10 and b = 2fm
• 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
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.
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 continua
i{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
Basis set 2
Jπ : 0+ = - 0.9801 MeV
Jπ : 0+ = - 0.9779 MeV
Differences of the order of ~ 0.22 keV…
GSM calculations for 6,8He nuclei
Radial density of the
6He g.s.
red and green curves
correspond to the
two different basis sets.
 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)…
Charge Radii calculations for
6,8He
nuclei
Expression of charge radius in these coordinates
r Z , A  r Z , A  2



2
p
2
p
core
2
 
 2  1 2 2
 
r1  r2  2r1  r2
4
A


center of mass correction
Generalization to n-valence particles is straightforward

2


u
r
r
u f r dr
 i
complex scaling cannot be applied!
0
Renormalization of the integral based on physical arguments (density)
In our calculations we carried out the radial integration until 20fm
Radial density of valence neutrons for the 6He
cut
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
Results on charge radii of
6,8He
rpp (6He) = 1.85 fm
with MN interaction
rpp (8He) = 1.801 fm
8He
We calculated also the correlation angle for 6He
Our result is:
 nn  83.28o
To be compared with
PRC 76, 051602
20
 nn  83o 10
13
 nn  78o 18
Angles estimated
from the available
B(E1) data and the
average distances between
neutrons.
Results on charge radii of
GSM Jπ: 0+ = - 3.125 MeV
Exp Jπ: 0+ = - 3.11 MeV
θnn = 81.80o
d5/2
with SGI interaction
rpp(8He) = 1.957 fm
rpp(6He) = 1.924 fm
p3/2
6,8He
p1/2
d3/2
s1/2
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
Results and discussion
 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.
Complex Scaling
•Diagonalization of Hamiltonian matrix
•Large Complex Symmetric Matrix
•Two step procedure
“pole approximation”
non-resonant
continua
resonance
bound state
Full
space
resonance
bound state
•Identification of physical state by maximization of
0 
Integral regularization problem between scattering states
For ui r  ~ e  e
ikr
the
 ikr
and


0
0
u f r  ~ e
ik ' r
e
 ik ' r
2
2


u
r
r
u
(
r
)
dr
~
Const

r
dr
f
 i

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.
From radii...to stellar nucleosynthesis!
Experiment
◊ 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 models
Very precise measurements on charge radii
Provide critical test of nuclear models
Charge radii of Halo nuclei is a very
important observable that needs
theoretical justification
Figures are taken from
PRL 96, 033002 (2006) and
PRL 93, 142501 (2004)
SHELL MODEL (as usually applied to closed quantum systems)
H1,... A  E1,...A
pi2
H 
i 1 2m
A

A
V
i  j 1
ij
A

 pi2
  A
 H   
 U i r     Vij   U i r 
i 1  2m
i 1
 i  j 1

A
single particle Harmonic
Oscillator (HO) basis
nice mathematical properties:
Lawson method applicable…
Largest tractable M-scheme
dimension ~ 109
HEAVIER SYSTEMS
Explosion of dimension
Hamiltonian Matrix is dense+non-hermitian
Lanczos converges slowly
B
nonresonant
continua
A
bound-states
resonances
Density Matrix Renormalization Group
S.R.White., 1992 PRL 69, 2863; PRB 48, 10345
T.Papenbrock.,D.Dean 2005., J.Phys.G31 S1377
J.Rotureau 2006., PRL 97, 110603
J.Rotureau et al. (2008), to be submitted
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
V (r1 , r2 )  V ( J , T )  exp( 
r12  r22

2
) Y (1)Y (2)
GI
EXPERIMENTAL RADII OF Be ISOTOPES
11Be
1-neutron halo
W.Norteshauser et all
nucl-ex/0809.2607v1
interaction cross section measurements
GFMC
PRC 66, 044310, (2002)
and
Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)
 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
NCSM
PRC 73 065801 (2006)
and
PRC 71 044312 (2005)
FMD
The message is that changes in charge distributions
provides information about the interactions in the
different subsystems of the strongly clustered nucleus!
Closed Quantum System
(nuclei near the valley of stability)
infinite well
Open quantum system
(nuclei far from stability)
scattering continuum
resonance
discrete states
(HO) basis
nice mathematical properties:
Exact treatment of the c.m,
analytical solution…
bound states