Transcript Charge radii of 6,8He and 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

Charge radii of 6,8He
and 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, 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
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
 DMRG application to 8He
Conclusions and Future Plans
6,8He
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
Shell Model Theories that incorporate the continuum, selected references
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)
GSM HAMILTONIAN
We want a Hamiltonian free from spurious CM motion
Lawson method?
(G.Hagen et al PRL 103 062503 (2009))
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
 Appropriate treatment for proper description of the recoil of the core
and the removal of the spurious CoM motion.
EXPERIMENTAL RADII OF 6He, 8He, 11Li
RMS charge radii
4He
6He
8He
L.B.Wang et al 1.67fm 2.054(18)fm
Rcharge(6He) > Rcharge (8He)
P.Mueller et al 1.67fm 2.068(11)fm 1.929(26)fm
9Li
“Swelling” of the core is not negligible
~4%
~8%
Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)
charge radii determines
the correlations between
valence particles AND
reflects the radial extent
of the halo nucleus
R.Sanchez et al
11Li
2.217(35)fm
2.467(37)fm
10Be
11Be
W.Nortershauser et al 2.357(16)fm
2.460(16)fm
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)
Spin-orbit contribution to the charge radius
R p2  0.769 fm 2
 Usually point radii are converted to charge radii through:
Rn2   0.1161 fm 2
Darwin-Foldy term
3
 0.033 fm 2
2
4M p
finite size effects
 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))
Both terms appear explicitly in the expression of the single nucleon charge operator and they
enter a non-relativistic calculation to an order 1/m2 .
BUT …
 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)
Spin-orbit contribution to the charge radius
 The formulas to calculate the s.o correction are the following:
(J.Friar et al Adv.Nucl.Phys. 8, 219, (1975))
 As we shall see , the s.o can have a comparable contribution with the finite size effects!!
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 benchmark
test for nuclear structure theory!
Model space
GSM calculations for 6,8He nuclei
0p3/2 resonance only
i{p3/2} complex non-resonant part
i{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...
Modified Minnesota Interaction (MN) (NPA 286, 53)
3


V12  Vk0 Wk  M k P P  Bk P  H k P exp(  k r122 )
k 1
Im[k] (fm-1)
p-sd waves
(2.0,0.0)
Re[k] (fm-1)
B
0p3/2
A
(0.17,-0.15)
3.27
Lj
Parameterizations
• The two strength parameters of
the MN are adjusted to the g.s of
6,8He.
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


centerof mass correction
Generalization to n-valence particles is straightforward

2 AC  n
p
r

X  r

2 AC
p
n
1
1
nn  1 n  
2
X 
r 
ri rj

2  i
2
 AC  n i1
 AC  n 2 i j

 The ingredients of the calculation are the OBME and TBME
2
i ri j

ij ri rj kl
 Same formulas for heavier systems 11Li, 11Be
Results on charge radii of
6,8He
The s.o corrections are comparable to the D.W term
the finite size corrections in 8He.
with MMN interaction
6He
and comparable to
 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
We calculated also the correlation angle for 6He
Our result is:
nn  82.95o
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.
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.
Blue line: Core polarization + s.o effect
 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.
Comparison with other models and experiment
Density Matrix Renormalization Group (DMRG)
S.R White PRL 69 (1992) 2863
T.Papenbrock and D.Dean J.Phys.G 31 (2005) S1377
S.Pittel et al PRC 73 (2006) 014301
J.Rotureau et al PRC 79 (2009) 014304
 Truncation Method applied to lattice models, spin chains, atomic nuclei….
• Basic idea:
 
M

A, B 1
 Approximate


A B
where A and B are partitions of the system.
in terms of m < M basis states (truncation)
 These m states are eigenstates of the density matrix
 AA  AB A B
 The difference between the exact and the approximated
'

'
or
 BB  AB AB
'
B
'
A
, has the minimal norm.
 The partition of the system has to be decided by the practitioner.
In GSM+DMRG we
optimize the number
of non-resonant states
along the scattering
contours.
Density Matrix Renormalization Group application to
8He radius
 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…
 The radius operator has the same form (in second quantization) with H

2 AC  n
p
r

X  r

2 AC
p
n
1
1
nn  1 n  
2
X 
r 
ri rj

2  i
2
2
 AC  n i1
 AC  n
i j

 We calculate OBMEs and TBMEs of rpp
i ri2 j

ij ri rj kl
 In each DMRG step we calculate the expectation value the 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
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
Density Matrix Renormalization Group application to
8He proton radius
 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.
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 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)…
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   0.7424MeV
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 :
  0.7417MeV
Ac
 
p1  p2
He :
  0.1768,0.0824 MeV
Ac
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.
Density Matrix Renormalization Group application to
8He
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 result
of 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
ε = 10-8
dim DMRG = 3859
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
and u f r  ~ e
 ikr


0
0
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
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.