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Transcript of [email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay Variational...
[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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
“Advances in Nuclear Physics”, vol.9, 1977
Computational methods for shell-model calculations
Introduction…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Variational Principle applied to…
)(H)(
Present Prescription for one-body density:
0A
)(*
Determination of Mixing Coefficients
Determination of Optimized orbitals
0)(
*i
Functional:
ˆ
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Variational Principle applied to…
Determination of Mixing Coefficients
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)
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Variational Principle applied to…
Determination of Mixing Coefficients
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
':V: npnh:
Pairing, RPA
Particle-vibration
RPA
Pairing
mpmh:'
Variational Principle applied to…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
Variational Principle applied to…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Variational Principle applied to…
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
Variational Principle applied to…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Solution of the mpmh approach
)(G),,(h
A)(H)(HA
Solution of both equations
orbitals
Aαπαν
In present application: neglect of σ
0),(h
Variational Principle applied to…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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.
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
Richardson exact solution…
(*) R.W. Richardson, Phys.Rev. 141 (1966) 949.
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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)
Richardson exact solution…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , 044306 (2005).
Ground state occupation probabilities
Richardson exact solution…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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.
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Contributions in Spin-Isospin ST channels for D1S
Parameterization
D1S Gogny force
Residual interaction
Variational mpmh configuration mixing applied to pairing…
central
Spin-orbite
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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.
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Results without Self-Consistency
Convergence properties
Some Dimensions
11 shell harmonic oscillator basis (286 neutron +286 proton states)Number of configurations
Variational mpmh configuration mixing applied to pairing…
Shell-model “Standard” dimensions
(E. Caurier)
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Results without Self-Consistency
Correlation energy (MeV)
HFHHFHEcorr
Configurations with 1 and 2 excited pairs are required Configurations with 3 excited pairs are negligible
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Results without Self-Consistency
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
Variational mpmh configuration mixing applied to pairing…
Protons ~ 1.7 MeV
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Results without Self-Consistency
Structure of correlated wave functions
106Sn
116Sn
100Sn
Variational mpmh configuration mixing applied to pairing…
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Results without Self-Consistency
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
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Effect of Approximate Self-Consistency
Correlated wave function
PBCS after variation
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Polarization of single particle states
7/2
7/2
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Effects of Pairing correlations on proton and neutron single particle spectrum
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Occupation
Probabilities
ii
2i aav
Variational mpmh configuration mixing applied to pairing…
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Neutron Skin
Charge Radii
Exp.
Variational mpmh configuration mixing applied to pairing…
2p
2nnp rrr
ZN
1161.0)bB(23
rr 22pc
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
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?
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Matrix elements of Gogny force in sd shell
J J
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Matrix elements of Gogny force in sd shell
JJ
J J
D2: Gogny force with a finite range density-dependent term, PhD thesis of F. Chappert.
[email protected] CEA-Bruyères-le-Châtel ESNT, 7-10 April 2008, Saclay
Matrix elements of Gogny force in sd shell
J
J JJ JJ