george papadimitriou [email protected]

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George Papadimitriou [email protected] 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.

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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 [email protected]. B. Barrett N . Michel , - PowerPoint PPT Presentation

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Page 1: George Papadimitriou georgios@iastate.edu

George Papadimitriou

[email protected]

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.

Page 2: George Papadimitriou georgios@iastate.edu

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

Page 3: George Papadimitriou georgios@iastate.edu

• 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 theoryBut also: Theory is in need for predictions and supporting certain experimental aspects

From: A.Gade

Nuclear Physics News 2013A.Spyrou et alMarques et al (conflicting experiment)

Page 4: George Papadimitriou georgios@iastate.edu

Fig: Bertsch, Dean, Nazarewicz, SciDAC review 2007

Dimension of the problem increases

One size does not fit all!

Page 5: George Papadimitriou georgios@iastate.edu

Life on the edge of nuclear stability: Theory

• Weak binding and the proximityof the continuum affects bulk propertiesand 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 S

n

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 couplingto 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)

Page 6: George Papadimitriou georgios@iastate.edu

InputForces

Many-body

Methods

techniques

Open

Channels

Coupling to

continuum

Physics of nucleiclose 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

Page 7: George Papadimitriou georgios@iastate.edu

0,1 222

2

rkuk

rllrv

drd

l

Resonant and non-resonant states (how do they appear?)

2

2mEk

statesscatteringrrkHCrkHCrku

resonancesstatesboundrrkHCrku

lll

ll

,),(),(~),(

,,),(~),(

Solution ofthe one-body Schrödingerequation with outgoingboundary conditions anda finite depth potential

Solutions with outgoing boundaryconditions

Page 8: George Papadimitriou georgios@iastate.edu

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

Page 9: George Papadimitriou georgios@iastate.edu

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 2002PRL 89 042502

Hamiltonian diagonalized

Hamiltonian matrix is built (complex symmetric):

Many body correlations and couplingto continuum are taken into account simultaneously

Page 10: George Papadimitriou georgios@iastate.edu

GSM HAMILTONIAN

“recoil” term coming from the expression of H in relativecoordinates. 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, unlessotherwise indicated.

Page 11: George Papadimitriou georgios@iastate.edu

Applications of the Berggren basis –Spectra-

Helium isotopic chain (4He core plus valence neutrons in the p-shell)

Schematic NN force

Page 12: George Papadimitriou georgios@iastate.edu

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: 4n are distributed more symmetrically around the charged core

Other effects also…

Can we calculate and quantify these correlations?

• Stringent test for the nuclear Hamiltonian

Page 13: George Papadimitriou georgios@iastate.edu

6He

8He

G. Papadimitriou et al PRC 84, 051304

s.o density and radii

also calculated by

S. Bacca et al PRC 86, 064316

Page 14: George Papadimitriou georgios@iastate.edu

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?

Page 15: George Papadimitriou georgios@iastate.edu

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

Page 16: George Papadimitriou georgios@iastate.edu

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

Page 17: George Papadimitriou georgios@iastate.edu

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/

Page 18: George Papadimitriou georgios@iastate.edu

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)

Page 19: George Papadimitriou georgios@iastate.edu

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)H

originalU(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

Page 20: George Papadimitriou georgios@iastate.edu

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.

Page 21: George Papadimitriou georgios@iastate.edu

Some results

6He 0+ g.s. Valence neutrons radial density

Phenomenological NN Minnesota interaction

Correct asymptotic behavior

Page 22: George Papadimitriou georgios@iastate.edu

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)

Page 23: George Papadimitriou georgios@iastate.edu

Solution

Perform a direct back-rotation. What is that?

In the case of the density this becomes:

Back-rotation

Page 24: George Papadimitriou georgios@iastate.edu

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 ofa 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.

Page 25: George Papadimitriou georgios@iastate.edu

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

Page 26: George Papadimitriou georgios@iastate.edu

Back up

Page 27: George Papadimitriou georgios@iastate.edu

Solution

Back rotation is very unstable numerically. Unsolved problem in the CS community (in QC as well)

The problem lies in the analytical continuation ofa square integrable function in the complex plane.

We are using the theory of Fourier transformations andTikhonov 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

Page 28: George Papadimitriou georgios@iastate.edu

Last slide before conclusions/future plans

NN force: JISP16 (A. Shirokov et al PRC79, 014610) and NNLO

opt (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…

Page 29: George Papadimitriou georgios@iastate.edu

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

Page 30: George Papadimitriou georgios@iastate.edu

Similar treatment by Caprio, Vary, Maris in Sturmian basis

Page 31: George Papadimitriou georgios@iastate.edu
Page 32: George Papadimitriou georgios@iastate.edu

Complex Scaling

Page 33: George Papadimitriou georgios@iastate.edu

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.

Page 34: George Papadimitriou georgios@iastate.edu

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-down Sweep-up

Page 35: George Papadimitriou georgios@iastate.edu

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

Page 36: George Papadimitriou georgios@iastate.edu

Neutron correlations in 8He ground state

G.Papadimitriou PhD thesis

Page 37: George Papadimitriou georgios@iastate.edu

Neutron correlations in 6He 2+ excited state

2+ neutrons almost uncorrelated…

G.P et al PRC(R) 84, 051304, 2011