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Understanding nuclear structure and reactions microscopically,
including the continuum. March 17-21, 2014, GANIL, France
Many-body methods for the description of bound
weakly bound and unbound nuclear states
George Papadimitriou
B. Barrett
N. Michel,
W. Nazarewicz,
J. Rotureau,
J. Vary, P. Maris.
• Nuclear Physics on the edge of stability
 Experimental and Theoretical endeavors
• The Gamow Shell Model (GSM)
 Applications on charge radii of Helium Halos
and neutron correlations
• Alternative method for extracting resonance parameters:
 The Complex Scaling Method in a Slater basis.
• Outlook, conclusions and Future Plans
Life on the edge of nuclear stability: Experimental highlights
New decay modes: 2n radioactivity?
Shell structure revisited: Magic
numbers disappear, other arise.
From: A.Gade
A.Spyrou et al
Marques et al (conflicting experiment)
New exotic resonant states: 7H, 13Li,
Nuclear Physics News 2013
PRC 87, 011304, PRL 110 152501, PRL 108 142503, PRL 109, 232501 recently)
Metastable states embedded in the continuum
are measured.
Very dilute matter distribution
Extreme clusterization close to particle thresholds.
Provide stringent constraints to theory
But also: Theory is in need for predictions and supporting certain experimental aspects
Dimension of the problem increases
One size does not fit all!
Fig: Bertsch, Dean, Nazarewicz, SciDAC review 2007
Life on the edge of nuclear stability: Theory
Dobaczewski et al Prog.Part.Nucl.Phys. 59, 432 (2007)
Dobaczewski, Nazarewicz Phil. Trans. R.Soc. A 356 (1998)
The very notion of the mean-field
and shell structure is under question
Nuclei are open quantum system and
the openness is governed by the Sn
• Weak binding and the proximity
of the continuum affects bulk properties
and spectra of nuclei.
Especially the clusterization of matter is a generic property of the coupling
to the continuum (or the impact of the open reaction channels).
Clusterization does not depend on the specific characteristics of the NN interaction
Okolowicz, Ploszajczak, Nazarewicz Fortschr. Phys. 61, 69 (2013)
Life on the edge of nuclear stability: Theory
Physics of nuclei
close to the drip-line
Coupling to
Additionally, complementary to the above: a new aspect is quality control
1) Cross check of codes/benchmarking
2) Statistical tools to estimate errors of calculations…
Recent Paradigms: DFT functionals, new chiral forces, new extrapolation techniques
Resonant and non-resonant states (how do they appear?)
 d2
l l  1 2 
  2  vr   2  k ul k , r   0
 dr
Solutions with
outgoing boundary
Solution of
the one-body Schrödinger
equation with outgoing
boundary conditions and
a finite depth potential
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
The Berggren basis (cont’d)
T.Berggren (1968)
NP A109, 265
The eigenstates of the 1b
Shrödinger equtaion form a complete basis, IF:
we also consider the L+ scattering states
are complex continuum states
along the L+ contour
(they satisfy scattering b.c)
The shape of the contour is arbitrary, and any state between
the contour and the real axis can be expanded in such as basis
(proof by T. Berggren)
In practice the continuum is discretized via a quadrature rule (e.g Gauss-Legendre):
Berggren’s Completeness relation and Gamow Shell Model
resonant states
(bound, resonances…)
N.Michel 2002
PRL 89 042502
along the contour
The GSM in 4 steps
Hermitian Hamiltonian
Hamiltonian matrix is built (complex symmetric):
SDi  ui1uiA
Hamiltonian diagonalized
Many body correlations and coupling
to continuum are taken into account simultaneously
 We assume an alpha core in our calculations..
 Hamiltonian free from spurious CM motion
“recoil” term coming from the
expression of H in relative
coordinates. No spurious states
 Appropriate treatment for proper description of the recoil of the core
and the removal of the spurious CoM motion.
Y.Suzuki and K.Ikeda
PRC 38,1 (1988)
Vij is a phenomelogical NN
interaction, fitted to spectra
of nuclei:
Minnesota force is used, unless
otherwise indicated.
Applications of the Berggren basis –SpectraHelium isotopic chain (4He core plus valence neutrons in the p-shell)
Schematic NN force
Applications 
charge radii
L.B.Wang et al, PRL 93, 142501 (2004)
P.Mueller et al, PRL 99, 252501 (2007)
M. Brodeur et al, PRL 108, 052504 (2012)
RMS charge radii
L.B.Wang et al 1.67fm 2.054(18)fm
M.Brodeur et al 1.67fm 2.059(7)fm 1.959(16)fm
• 2n Very
data based
as a strong
pairon Isotopic Shifts measurements
are distributed
around thecalculations with
• 4n Extraction
of radii
Quantum Chemistry
charged core
a precision of up to 20 figures! (Hyllerraas basis calculations)
Z.-T.Lu, P.Mueller, G.Drake,W.Nörtershäuser,
S.C. Pieper, Z.-C.Yan
Rev.Mod.Phys. 2013, 85, (2013).
“Laser probing of neutron rich nuclei in light
• effects
independence of results
for the nuclear
Can• weStringent
calculate and
these correlations?
G. Papadimitriou et al PRC 84, 051304
s.o density and radii
also calculated by
S. Bacca et al PRC 86, 064316
AC +n
X = rpp2
( )
X +
( AC +n)
ri2 +
( AC +n)
å r ×r
i< j
Radii (and other operators different than Hamiltonian) are challenging
Courtesy of P. Maris
How to reliably extrapolate radial operators to the infinite basis?
Sid Coon et al, Furnstahl et al methods?
2) Renormalized operators?
3) Different basis?
Neutron correlations in 6He ground state
Halo tail
 Probability of finding the particles at distance r from the core with an angle θnn
See also I. Brida and F. Nunes NPA 847,1 (2010) and P. Navratil talk
Coupling to the continuum crucial for clusterization
G. Papadimitriou et al PRC 84, 051304
Only p3/2
• In the absence of continuum p1/2-sd states the neutrons show no preference
S=0 component (spin-antiparallel) dominant  Manifestation of the Pauli effect
Average opening angle calculated from the density: θnn = 68o
Neutron correlations in 6He 2+ excited state and spectroscopy
2+1 : [1.82, 0.1] MeV
2+2 : [4.13, 3.17] MeV
0+2 : [4.75, 8.6] MeV
1+1 : [4.4, 5.5 ] MeV
G.P et al PRC(R) 84, 051304, 2011
MN force fitted
just to the g.s. energy
of 6,8He.
 2+ neutrons almost uncorrelated…
Fig. from
Constructing an effective interaction in
GSM in the p and sd shell.
Effective interactions depend on the
position of thresholds…
Additional tools in our arsenal
The complex scaling
Belongs to the category of:
Bound state technique to calculate resonant parameters
and/or states in the continuum (see also talks by Lazauskas, Bacca, Orlandini)
Prog. Part. Nucl. Phys. 74, 55 (2014) and 68, 158 (2013) (reviews of bound state methods)
Nuclear Physics
Nuttal and Cohen PR 188, 1542 (1969)
Lazauskas and Carbonell PRC 72 034003 (2005)
Witala and Glöeckle PRC 60 024002 (1999)
Aoyama et al PTP 116, 1 (2006)
Horiuchi, Suzuki, Arai PRC 85, 054002 (2012)
Moiseyev Phys. Rep 302 212 (1998)
Y. K. Ho Phys. Rep. 99 1, (1983)
Additional tools in our arsenal
Complex Scaling Method in a Slater basis
A.T.Kruppa, G.Papadimitriou, W.Nazarewicz, N. Michel PRC 89 014330 (2014)
1) Basic idea is to rotate coordinates and momenta i.e. r  reiθ
Hamiltonian is transformed to H(θ) = U(θ)HoriginalU(θ)-1
H(θ)Ψ(θ) = ΕΨ(θ) complex eigenvalue problem
• The spectrum of H(θ) contains bound, resonances and continuum states.
2) Slater basis or Slater Type Orbitals (STOs):
Basically, exponential decaying functions
 Powerful method to obtain resonance parameters in Quantum Chemistry
 Involves L2 square integrable functions.
 Can (in general) be applied to available bound state methods techniques
(i.e. NCSM, Faddeev, CC etc)
Some results
Comparison between CS Slater and CSM
0+ g.s, 2+ 1st excited Force Minnesota, α-n interaction KKNN
Test the HO expansion of the NN force in
GSM for the unbound 2+ state.
In GSM the force is expanded in a HO basis:
Talmi-Moshinsky transformation
Numerical effort: Overlaps between HO and
Gamow states.
 Very weak dependence of results on b nnmax.
Some results
0+ g.s.
Valence neutrons radial density
Phenomenological NN
Minnesota interaction
Correct asymptotic behavior
Some results
2+ first excited state in 6He
The 2+ state is a many-body resonance (outgoing wave)
 GSM exhibits naturally this behavior
 but CS is decaying for large distances, even for a resonance state
This is OK. The solution Ψ(θ) is known to “die” off (L2 function)
 Perform a direct back-rotation. What is that?
In the case of the density this becomes:
The CS density has the correct asymptotic
behavior (outgoing wave)
• Back rotation is very unstable numerically. An Ill posed inverse problem.
Long standing problem in the CS community (in Quantum Chemistry as well)
• The problem lies in the analytical continuation of
a square integrable function in the complex plane.
• We are using the theory of Fourier transformations and a regularization process (Tikhonov)
to minimize the ultraviolet numerical noise of the inversion process.
2+ densities in 6He (real and imaginary part)
Conclusions/Future plans
 Berggren basis appropriate for calculations of weakly bound/unbound nuclei.
• GSM calculations provided insight behind the charge differences of
Helium halo nuclei.
 Construct effective interaction in the p and sd shell.
Use realistic effective interactions for GSM calculations that stem
from NCSM with a core, or Coupled Cluster or IM-SRG…
 GSM is the Shell Model technique to:
i) study 3N forces effects and continuum coupling for
the detailed spectroscopy of heavy drip line nuclei.
ii) exact treatment of many body correlations and coupling to continuum
 Complementary method to describe resonant states: Complex Scaling in
a Slater basis
 L2 integrable basis formulation.
 Slater basis  correct asymptotic behavior
 Back rotation inverse problem solved.
 Apart from complex arithmetics the computational expense is as “tough”
or as “easy” as for the solution of the bound state.
 Explore complex scaling in more depth
Back up
Back rotation is very unstable numerically.
Unsolved problem in the CS community (in QC as well)
The problem lies in the analytical continuation of
a square integrable function in the complex plane.
We are using the theory of Fourier transformations and
Tikhonov regularization process to obtain the original (GSM) density
To apply theory of F.T to the density, it should be defined in (-∞,+∞)
 Now defined from (-∞,+∞)
 F.T
Value of (1) for x+iy
(analytical continuation)
 Tikhonov regularization
x = -lnr , y = θ
Last slide before conclusions/future plans
 NN force: JISP16 (A. Shirokov et al PRC79, 014610) and
NNLOopt (A. Ekstrom et al PRL 110, 192502)
 Quality control: Verification/Validation, cross check of codes
MFDn: Vary, Maris
NCGSM: G.P, Rotureau, Michel…
 MFDn/NC-GSM + computer scientists at LBNL (Ng, Yang, Aktulga), collaboration
Goal: Scalable diagonalizations of complex symmetric matrices
Dimension comparison
 Lanczos: “brute” force diagon
of H.
 DMRG: Diagon of H in the space
where only the most important
degrees of “freedom” are considered
 Similar treatment by Caprio, Vary, Maris in Sturmian basis
Complex Scaling
 construction of a block in
Construct all many-body states associated with
the pole space P
Construct all many-body states associated with
the space of the discrete continua C.
Create many-body basis by coupling states in
P and C.
 construction of a superblock :
 truncation with the density matrix :
Nopt states that correspond to the largest
eigenvalues of the density matrix are kept
The process is reversed…
In each step (shell added) the Hamiltonian is diagonalized and Nopt states
are kept.
Iterative method to take into account all the degrees of freedom
in an effective manner.
In the end of the process the result is the same (within keV) with the one obtained by
“brute” force diagonalization of H.
Results: 4He against Fadeev-Yakubovsky
G.P., J.Rotureau, N. Michel, M.Ploszajczak, B. Barrett arXiv:1301.7140
 2 neutrons
 2 protons
 Pole space A:0s1/2 (p/n)
 Continuum space B:
p3/2,p1/2,s1/2 real
energy continua
H.O states
 156 s.p. states total
Dim for direct diagon: 119,864,088
Eab-initio = -29.15 MeV
EFY = -29.19 MeV
Neutron correlations in 8He ground state
G.Papadimitriou PhD thesis
Neutron correlations in 6He 2+ excited state
G.P et al PRC(R) 84, 051304, 2011
 2+ neutrons almost uncorrelated…