variational multiparticle-multihole configuration mixing approach using gogny force

[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay

Variational multiparticle-multihole

configuration mixing approach

using Gogny forceNathalie PilletCEA Bruyères-le-Châtel,

France

Collaborators: JF. Berger(1) , E. Caurier(2) , D. Gogny(3), H. Goutte(1)

(1) CEA, Bruyères-le- Châtel (2) IPHC, Strasbourg (3) LLNL, Livermore (4) IPN, Orsay

“Convergence of Particle-Hole expansions for the description of nuclear correlations”

Collaboration with N. Sandulescu(5), N. Van Giai(6) and JF Berger(5) NIPNE, Bucharest (6) IPN, Orsay


“Advances in Nuclear Physics”, vol.9, 1977

Computational methods for shell-model calculations

Introduction…


Brief history of … HTDA

approach

Variational mpmh configuration mixing approach

N. Pillet, P. Quentin and J. Libert, Nucl. Phys. A697 (2002) 141.

P. Quentin, H. Laftchiev, D. Samsoen et al., Nucl. Phys. A734 (2004) 477.

L. Bonneau, P. Quentin and K. Sieja, Phys. Rev. C76 (2007) 014304.

L. Bonneau Talk

L. Bonneau, J. Bartel and P. Quentin, arXiv: 0705.2587v1 [nucl-th].

High-K isomers in 178Hf

Isospin mixing

Rotating nuclei (kinetic and dynamical moments of inertia in 192Hg

and 194Pb)

Ground state properties of even-even N=Z nuclei for [56Ni;100Sn]

N. Pillet, JF Berger and E. Caurier, to be submitted to PRC.

Pairing correlations in Sn isotopes with D1S Gogny force

Introduction…


Plan

1. Formalism of the Variational mpmh Configuration Mixing

2. Applications to Pairing-type correlations

3. Summary and Outlook

MotivationsWave functions and symmetries Variational principle- Derivation of equations

Exactly solvable model- test of truncationsDescription of ground states of even-even Sn isotopes with variational mpmh configuration mixing using D1S Gogny force

Introduction…


Motivations

Symmetries Axial

symmetry:Eigen-solutions are specified by K quantum number

(K projection of total angular momentum J) Parity

Towards a unified description of long range correlations in the context of

beyond mean-field methods

Conservation of particle numbers + Pauli principle Description on the same footing of even-even, odd and odd-odd nuclei Description of both ground states and excited states

“Take advantage of both Mean-field and Shell-Model approaches”

Description of all correlations (Pairing, RPA, particle vibration coupling) All nucleons are considered for the description of states


Formalism

,K,K A

ia

Trial wave function(a priori for ground and yrast states)

+ + + …

0p0h 1p1h 2p2h mpmh

.

A.AA

.

and

Variational parameters:

Slater determinant

vacuum

- Mixing coefficients Aαπαν

- Orbitals a+i


Variational Principle applied to…

)(H)(

Present Prescription for one-body density:

0A

)(*

Determination of Mixing Coefficients

Determination of Optimized orbitals

0)(

*i

Functional:

ˆ




0A

)(*

A)(H)(HA

“Secular equation” equivalent to the diagonalization of H(ρ)+δH(ρ) in the multiconfiguration space

Highly non-linear equation because of δH(ρ)




Importance of consistencyJP. Blaizot and D. Gogny, Nucl. Phys. A284 (1977) 429-460.

D. Gogny and R. Padjen, Nucl. Phys. A293 (1977) 365-378.

''''

mnnm'mn''

mnnm'mn''

AVA

aa),(CHA

aa),(CHA

)r(B)r()r(rd),(C n

*m

3mn

kljiijkl

aaaal~

k)r(

Vij)r(B

with

Two-body correlation function

Residual interaction: two-body matrix elements + rearrangement terms

Example: for α≠α’

:V:'

mn

nm'mn :aa:),(C

jl,ik

kjliljkiklji aaaa


':V: npnh:

Pairing, RPA

Particle-vibration

RPA

Pairing

mpmh:'



Determination of Optimized orbitals

0)(

*i

VHH

rr

VrdV 3

P

Variation of the functional

Definition of projectors associated with the multiconfiguration space I

P1Q

Inside I Outside I

with




Thouless’ theorem 1Si

2 eC kl

lkkl aaSSwith

At first order:

SQi

0aa,rdrˆr

VH lk

3

)(G),,(h

with

G(σ) antisymmetric

mnpqnq,mp

ij

mnpqqnpm

ijmnnmij

q~pV

mn41

q~pV

mn21

n~jVimjKi),(h



Solution of the mpmh approach

)(G),,(h

A)(H)(HA

Solution of both equations

orbitals

Aαπαν

In present application: neglect of σ

0),(h



Test of Truncations in the mpmh wave function…

Richardson exact solution of Pairing Hamiltonian(*)

Pairing Hamiltonian

(*) R.W. Richardson and N. Sherman, Nucl. Phys. 52 (1964) 221.

f 'ff

'ffff bbgN2H

fffff aaaa

21

N

fff aab

f'ff'ff N21b,b


Test of the importance of the different terms in the mpmh wave function expansion (1 pair, 2pairs…)

Exact solution of Pairing Hamiltonian

Similarity between the many-fermion-pair system with pairing forces and the many-boson system with one-body forces

Exact wave function : mpmh wave function including all the configurations built as pair excitations

Exact solution obtained from a coupled system of algebraic equations deduced from variational principle

Richardson exact solution…

(*) R.W. Richardson, Phys.Rev. 141 (1966) 949.


Picket fence model (*)

gεi

εi+1

d

System of 2N particles in 2N equispaced and doubly-degenerated levels

System of identical fermions

Constant pairing interaction strength

Prototype of axially deformed nuclei


(*) R.W. Richardson, Phys.Rev. 141 (1966) 949.


N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , 044306 (2005).

Ground state Correlation energy

Ecorr=E(g≠0)-

E(g=0) ΔEcorr = Ecorr (exact) – Ecorr (mpmh)

Truncation in excitation energy

Truncation in mpmh order of excitation

g (Pairing interaction strength)



N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , 044306 (2005).

Ground state occupation probabilities



Variational mpmh configuration mixing applied to Pairing-type

correlations in Sn isotopes using D1S Gogny force (*)

mpmh wave function

Usual pairing-type correlations (pp and nn) No residual proton-neutron interaction

A.AA

Configurations with:

Ground states of even-even spherical nuclei

000 .

Kp=Jp=0+

Kp=0+One excited pair of nucleons

. …Several excited Pairs of nucleons Kp=0

+

Kp=0+

Kp=0+

(*) N. Pillet, JF Berger and E. Caurier, to be submitted to PRC.


Link between mpmh and PBCS wave functions

BCS wave function

BeNBCS

0j

jcosN

0jjj btgB jj ucos

0N

N

!NB

BCS

Projected BCS (PBCS) wave function

Nj...2j1j0Nj1jNj1jN2 b...btg...tgN

Component of |BCS> with 2N nucleons

N

1hhbHF

HF-type reference state

fff aab

Variational mpmh configuration mixing applied to pairing…


Link between mpmh and PBCS wave functions

mpmh wave function similar to PBCS one with more general mixing coefficients

PBCS wave function

n

1kkhkp

0nnh...1h0np...1p0 nh1h

np1p'N2 HFbb

tg...tg

tg...tgN

p0p0h

h' cossinNN

wit

h

mpmh wave function:

n

1kkhkp

HFbbA



Contributions in Spin-Isospin ST channels for D1S

Parameterization

D1S Gogny force

Residual interaction


central

Spin-orbite


Three pairing regimes

Weak pairing

Medium pairing

Strong pairing

100Sn

106Sn

116Snhttp://www-phynu.cea.fr/HFB-Gogny.htm

S. Hilaire and M. Girod, EPJ A33 (2007) 237.



Results without Self-Consistency

Convergence properties

Some Dimensions

11 shell harmonic oscillator basis (286 neutron +286 proton states)Number of configurations


Shell-model “Standard” dimensions

(E. Caurier)



Correlation energy (MeV)

HFHHFHEcorr

Configurations with 1 and 2 excited pairs are required Configurations with 3 excited pairs are negligible




Correlation energy (MeV)

Coulomb ~ 700 keVS=0 T=1 (Central+ s.o.) ~ 99% of Ecorr without Coulomb

100Sn

116Sn

Contributions associated with:

1 pair : 4.474 MeV

2 pairs : 0.967 MeV

1 pair : 3.397 MeV

2 pairs : 0.275 MeV


Protons ~ 1.7 MeV



Structure of correlated wave functions

106Sn

116Sn

100Sn


65% ~ (92%)π x (71%)ν

ji2

A)j,i(T

3s1/2→ 1d3/2

3s1/2→ 1h11/2

No specific configurations

2d5/2→ 1g7/2



Effect of a truncated space

Nucleus |Ecorr| |Ecorr|

116Sn 5.44 3.45106Sn 4.62 3.54100Sn 3.67 2.79

Total

Truncated

Total

Truncated

~141 neutron states

~ 98 proton states



Effect of Approximate Self-Consistency

First step: neglecting of the two-body correlation matrix σ

Correlation energy

0),(h

Use of the truncated space for: the number of valence orbitals

the order of excitation



Effect of Approximate Self-Consistency

Correlated wave function

PBCS after variation



Polarization of single particle states

7/2

7/2



Effects of Pairing correlations on proton and neutron single particle spectrum



Occupation

Probabilities

ii

2i aav



Neutron Skin

Charge Radii

Exp.


2p

2nnp rrr

ZN

1161.0)bB(23

rr 22pc


mpmh configuration mixing: Summary and outlook

Formalism for the description of ground states and yrast states

First applications to nuclear superfluidity quite encouraging

Specify the fundamental nature of correlations induced in our study

Study the effect of the two body correlation function σ

Still a lot of work to do !

Study of the effect of pn pairing-type correlations on Sn ground states

Study of different prescriptions for ρ


More general correlations…

Unique interaction for both mean-field and residual part

pairing, RPA and particle-vibration correlations

Interaction with good properties in T=1 and T=0 residual channels

Shell-Model interactions: a guide for effective interactions?

Challenge for Gogny density-dependent interaction

Different valence spaces

Shell-model matrix elements: fitted to reproduced excited states

Different truncations in excitation order of the wave function?


Matrix elements of Gogny force in sd shell

J J



JJ

J J

D2: Gogny force with a finite range density-dependent term, PhD thesis of F. Chappert.



J

J JJ JJ

variational multiparticle-multihole configuration mixing approach using gogny force

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