george papadimitriou georgios@iastate
DESCRIPTION
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 [email protected]. B. Barrett N . Michel , - PowerPoint PPT PresentationTRANSCRIPT
George Papadimitriou
Many-body methods for the description of bound
weakly bound and unbound nuclear states
Understanding nuclear structure and reactions microscopically,
including the continuum. March 17-21, 2014, GANIL, France
B. Barrett
N. Michel,
W. Nazarewicz,
M.Ploszajczak,
J. Rotureau,
J. Vary, P. Maris.
Outline
• 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
• New exotic resonant states: 7H, 13Li, 10He,26O…
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.
• New decay modes: 2n radioactivity?
Life on the edge of nuclear stability: Experimental highlights
• Shell structure revisited: Magic
numbers disappear, other arise.
Provide stringent constraints to theory
But also: Theory is in need for predictions and supporting certain experimental aspects
From: A.Gade
Nuclear Physics News 2013A.Spyrou et al
Marques et al (conflicting experiment)
Fig: Bertsch, Dean, Nazarewicz, SciDAC review 2007
Dimension of the problem increases
One size does not fit all!
Life on the edge of nuclear stability: Theory
• Weak binding and the proximity
of the continuum affects bulk properties
and spectra of nuclei.
• 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
Dobaczewski et al Prog.Part.Nucl.Phys. 59, 432 (2007)
Dobaczewski, Nazarewicz Phil. Trans. R.Soc. A 356 (1998)
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)
Input
Forces
Many-body
Methods
techniques
Open
Channels
Coupling to
continuum
Physics of nuclei
close to the drip-line
Life on the edge of nuclear stability: Theory
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
0,1 2
22
2
rkuk
r
llrv
dr
dl
Resonant and non-resonant states (how do they appear?)
2
2
mE
k
statesscatteringrrkHCrkHCrku
resonancesstatesboundrrkHCrku
lll
ll
,),(),(~),(
,,),(~),(
Solution of
the one-body Schrödinger
equation with outgoing
boundary conditions and
a finite depth potential
Solutions with
outgoing boundary
conditions
The Berggren basis (cont’d)
The eigenstates of the 1b
Shrödinger equtaion form a complete basis, IF:
T.Berggren (1968)
NP A109, 265
are complex continuum states
along the L+ contour
(they satisfy scattering b.c)
In practice the continuum is discretized via a quadrature rule (e.g Gauss-Legendre):
with
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)
we also consider the L+
scattering states
Berggren’s Completeness relation and Gamow Shell Model
resonant states
(bound, resonances…)
Non-resonant
Continuum
along the contour
Many-body basis
Hermitian Hamiltonian
The GSM in 4 steps
iSD
iAii uuSD 1
N.Michel et.al 2002
PRL 89 042502
Hamiltonian diagonalized
Hamiltonian matrix is built (complex symmetric):
Many body correlations and coupling
to continuum are taken into account simultaneously
GSM HAMILTONIAN
“recoil” term coming from the
expression of H in relative
coordinates. No spurious states
Y.Suzuki and K.Ikeda
PRC 38,1 (1988)
Hamiltonian free from spurious CM motion
Appropriate treatment for proper description of the recoil of the core
and the removal of the spurious CoM motion.
We assume an alpha core in our calculations..
Vij is a phenomelogical NN
interaction, fitted to spectra
of nuclei:
Minnesota force is used, unless
otherwise indicated.
Applications of the Berggren basis –Spectra-
Helium isotopic chain (4He core plus valence neutrons in the p-shell)
Schematic NN force
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)
6,8He charge radiiApplications
M.Brodeur et al
4He 6He 8He
L.B.Wang et al 1.67fm 2.054(18)fm
1.67fm
RMS charge radii
2.059(7)fm 1.959(16)fm
• Very precise data based on Isotopic Shifts measurements
• Extraction of radii via Quantum Chemistry calculations with
a precision of up to 20 figures! (Hyllerraas basis calculations)
• Model independence of resultsZ.-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 atoms”
6He: 2n as a strong correlated pair
8He: 4 n are distributed more symmetrically around the charged core
Other effects also…
Can we calculate and quantify these correlations?
• Stringent test for the nuclear Hamiltonian
6He
8He
G. Papadimitriou et al PRC 84, 051304
s.o density and radii
also calculated by
S. Bacca et al PRC 86, 064316
Radii (and other operators different than Hamiltonian) are challenging
Example:
Courtesy of P. Maris
1) 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
Probability of finding the particles at distance r from the core with an angle θnn
Halo tail
See also I. Brida and F. Nunes NPA 847,1 (2010) and P. Navratil talk
Coupling to the continuum crucial for clusterization
• 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
G. Papadimitriou et al PRC 84, 051304
• Average opening angle calculated from the density: θnn
= 68o
Full continuumOnly p3/2
Neutron correlations in 6He 2+ excited state and spectroscopy
2+ neutrons almost uncorrelated…
G.P et al PRC(R) 84, 051304, 2011
Constructing an effective interaction in
GSM in the p and sd shell.
Effective interactions depend on the
position of thresholds…
2+2
: [4.13, 3.17] MeV
0+2
: [4.75, 8.6] MeV
1+1
: [4.4, 5.5 ] MeV
2+1
: [1.82, 0.1] MeV
GSM
MN force fitted
just to the g.s. energy
of 6,8He.
21
+
02
+
22
+
1+
Fig. from http://www.tunl.duke.edu/nucldata/
Additional tools in our arsenal
• 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)
The complex scaling
Belongs to the category of:
• 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)
Nuclear Physics
Chemistry
• 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)
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)
1) Basic idea is to rotate coordinates and momenta i.e. r reiθ
Hamiltonian is transformed to H(θ) = U(θ)Horiginal
U(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
Some results
• Comparison between CS Slater and CSM
0+ g.s, 2+ 1st excited Force Minnesota, α-n interaction KKNN
0+
2+
• 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
6He 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)
Solution
Perform a direct back-rotation. What is that?
In the case of the density this becomes:
Back-rotation
The CS density has the correct asymptotic
behavior (outgoing wave)
2+ densities in 6He (real and imaginary part)
• 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.
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
Solution
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/NC-GSM + computer scientists at LBNL (Ng, Yang, Aktulga), collaboration
Goal: Scalable diagonalizations of complex symmetric matrices
MFDn: Vary, Maris
NCGSM: G.P, Rotureau, Michel…
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 :
construction of a superblock :
superblock
block
• 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.
truncation with the density matrix :
Nopt
states that correspond to the largest
eigenvalues of the density matrix are kept
truncation
“up”
truncation
“down”
• 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.
Sweep-downSweep-up
Results: 4He against Fadeev-Yakubovsky
2 neutrons
2 protons
Pole space A:0s1/2 (p/n)
Continuum space B:
p3/2,p1/2,s1/2 real
energy continua
d5/2-d3/2
f5/2-f7/2 H.O states
g7/2-g9/2
156 s.p. states total
Dim for direct diagon: 119,864,088
Eab-initio
= -29.15 MeV
EFY
= -29.19 MeV
G.P., J.Rotureau, N. Michel, M.Ploszajczak, B. Barrett arXiv:1301.7140
Neutron correlations in 8He ground state
G.Papadimitriou PhD thesis
Neutron correlations in 6He 2+ excited state
2+ neutrons almost uncorrelated…
G.P et al PRC(R) 84, 051304, 2011