Structure of even-even nuclei using a mapped collective hamiltonian

J.-P. Delaroche Pack Forest , June 2009

Structure of even-even nuclei using a mapped collective hamiltonian

and the D1S Gogny interaction

Structure of even-even nuclei using a mapped collective hamiltonian

and the D1S Gogny interaction

J.-P. Delaroche,M. Girod, H. Goutte, S. Hilaire, S. Péru, N.

Pillet(CEA Bruyères-le-Châtel, France)

J. Libert (IPN Orsay, France)

G. F. Bertsch (INT, Seattle, USA)


• Introduction• Reminder of formalism• Ground state properties • Yrast spectrum• Non yrast spectrum• Summary

OutlineOutline


• Motivations• Methodology• Calculations for ~1700 nuclei (dripline to dripline) (10<Z<110, N<200)

• Benchmarking• Predictions for future studies (SPIRAL 2, FAIR, RIA, ...)

Computing time : over 25 years of CPU time if calculation were performed on a single processor.

IntroductionIntroduction


1) Hartree-Fock-Bogoliubov equations with constraints

0 ZNQ Hii

qZNii

iq =Φλ−λ−λ−Φδ ∑

(Z) N )Z(Nii

qq =ΦΦ

iqiq q Qii

=ΦΦwith

Constraints on

FormalismFormalism

More details in: J. Libert et al., PRC60, 054301 (1999)

0Q and 2Q

2220 yxz2Q −−= 22

2 yxQ −=and

Self-consistent symmetries

2T π and parity


•Number of major shells: N0= 6 - 16

•Linear constraints used throughout

FormalismFormalism

•(q0, q2) Bohr coordinates (β, γ)

0 < β < 0.9 0 < γ < π/3

CHFB equations solved by expanding sp states onto triaxial harmonic oscillator basis

•CHFB equations solved on a grid

Δβ = 0.05 Δγ = 10°


CHFB -> GCM -> GOA -> 5DCH

No free parameters beyond those in theGogny D1S force.


FormalismFormalism

2) Collective Hamiltonian in 5 quadrupole collective coordinates

( ) ( ) ( )20202 and 0nm,

12/12/123

1

22

,,2

ˆ

2ˆ aaVaaV

aBD

aD

J

IH

nmn

mk k

kcoll Δ−+

∂

∂

∂

∂−= ∑∑

=

−−

=

hh

Jk(a0, a2): moment of inertia

)Bdet()a,a(J)a,a(D3,1k

20k20 ∏=

=

qq20 H)a,a(V ΦΦ=

.)vib.rot(ZPE)a,a(V 20 +=Δ ZPEpot neglected

γβ=γβ= sinacosa 20

Bmn(a0, a2): collective mass (vibration)

D(a0, a2): metric


FormalismFormalism

Approximations

Bmn(a0, a2): cranking (Inglis-Belyaev)

ValatinThoulessI

limJqkq

0k −⇔ω

ΦΦ=

ωω

→ω

IM)I(EIMHcoll =

∑=K

20IK IMK)a,a(gIM

∫=2

20IK20 )a,a(gdada)K(P

Notations

Correlation energy

DCH5minHFBcorr EEE −=

not fullfilled for ~80 nuclei at (near) double-closed-shells.0Ecorr >


FormalismFormalism

3) Limitations of present CHFB+5DCH theory

• Adiabatic approximation low spins only

•Quasiparticle degrees of freedom ignored

•No coupling to other collective modes


Shape coexistence

E.Clément et al. PRC75, 054313 (2007)


Transitional nucleus


GS static and dynamic deformations


Frequency distributions of β and γ deformations


Rigidity parameters


Charge radii


Charge radii for Sr isotopes


Correlation energy and residuals


2-nucleon separation energies


Energy weighted sum rules

)20;2E(B)2(ES 111+++ →=

∑ +++ →=i

i1i )20;2E(B)2(E)I(S

2

A

Z)I(S)II(S ⎟

⎠

⎞⎜⎝

⎛=

)X(Se

)20;2E(B)2(E)X(s

2111+++ →

=


First 2+ level collective properties

G.F. Bertsch et al., PRL 99, 032502 (2007)


Exp. Th.

Frequency distributions of the R42 ratio

R42= E(4+1) / E(2+

1)


R42 ratio versus deformation properties

δβ/<β><β>


R42: comparison Th. / Exp. R42 frequency distribution


R62 versus R42 : comparison Th. / Exp.


Probability distribution of K components for 22+ states

∫=2

20IK20 )a,a(gdada)K(P


P(K=2) frequency distribution for 22+ and 23

+ states


5DCH Systematics for 2+γ levels


Comparison Th. / Exp. for 22+ energies

γ vibration


Comparison Th. / Exp. for 02+ energies

Cranking masses too small !!!


Exp. and Th. for R02 versus R42

R02= E(0+2)/E(2+

1)


Energy distribution of 02+ levels


Model criteria for the occurrence of β-vibration

Crossover matrix elements

Relationship between quadrupole Transition operator for 21

+ 02+,

23+ 21

+, 23+ 01

+ transitions(Bohr and Mottelson, Eq. 4-219)

gEMJJJJgEMJ ggg 20020ˆ2 ββ ββ

Form the ratio of |M20|, |M02|, |M22| to their total.

Conditions for the existence of β vibration should be quite common.

222002 10

7MMM

10

7


Chart of the nuclei in the vicinity of the center of the triangle


E0 transition strengths versus neutron number


Ratio of transition strengths for excited K=0 over ground state bands versus

neutron number :indicator for shape coexistence


• ~1700 nuclei have been studied between drip-lines in the present microscopic model

• Yrast band properties: well described especially for well deformed nuclei

• 22+ levels: energy well described

most of these levels are 2γ + vibrations

• 02+ levels: energy high (cranking masses)

off-band E2 transition: β-vibration? E0 transition strength: high

→CHFB+5DCH questionable for the 02+ excitations.

Extension required to include coupling to quasiparticle and pairing vibration modes.

How with GOA? Better collective masses: Thouless-Valatin, QRPA

• Next to come: γ band properties

Summary and outlookSummary and outlook

Structure of even-even nuclei using a mapped collective hamiltonian

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