George Papadimitriou

georgios@iastate.edu

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