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Correlation energy contribution to nuclear masses

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A study for the influence of collective motion on nuclear ground state energy

- To describe many physical observables of many-body systems (e.g. (screened) coulomb interaction in metals, level density at the Fermi energy in atomic nuclei) one is forced to go beyond mean field
- Effects due to the coupling of particles and collective vibrations have to be taken into account
- The aim of this work is to study the influence zero point fluctuations have on the nuclear masses

Correlation energy contribution

to nuclear masses

Simone Baroni

Erice, September 2006

University of Milan, Italy

Nuclear masses: the state of the art… ground state energy

The state of the art…rms errors

- Liquid drop model

2.970 MeV

Weizsacker formula (1935)……………….………..

Finite-range droplet method. (Möller et al.)….….…

0.689 MeV

(1654 nuclei fitted)

- Microscopic theories

(mean field approximation)

0.738 MeV

0.674 MeV

HFBCS (Goriely et al.)…………………………………

(with Skyrme interaction)

HFB (Goriely et al.)……………………………………

(with Skyrme interaction)

(1888 nuclei fitted)

(2135 nuclei fitted)

0.709 MeV

(1719 nuclei fitted)

ETFSI (Myers, Swiatecki)…….………………………..

Extended Thomas-Fermi plus Strutinsky integral

1 P.Möller et al., At. Data Nucl. Data Tables 59 (1995) 185

2 S.Goriely et al., Phys. Rev. C 66 (2002) 024326-1

…and the need of a better mass formula ground state energy

…the need of a better formulaSn 7.40 MeV

208Pb

S2n 300 keV

11Li

An accuracy of about 700 keV is remarkable

We need a formula at least a factor of two more

accurate than present microscopic ones!!

How to improve the prediction?

Beyond mean field approximation ground state energy

Beyond MF- vibrations of the surface
- change angular momentum
- in two (three) units

- pairing vibrations
- change particle number
- in two units

Correlation energy associated to zero point fluctuations

affects the binding energy of the nuclear system

Collective modes of the nucleus:

Calculation details ground state energy

Calculation details- HF approximation
- box calculations, configuration space
- Skyrme interaction MSk7 (rmsd = 0.754 MeV)
- Wigner term for correlations

- BCS approximation
- Vpair(r,r’)=V0 (r,r’)
- energy cutoff at h = 41·A-1/3 MeV

Pairing channel:

121 spherical even-even nuclei have been studied

Starting point

Mean field (MF)

Deviation of ground state energycomputed mean field (MF) ground state energies

from experimental values for the set of 121 nuclei we studied

Calculation details ground state energy

Surface vibrations

- QRPA in configuration space
- residual interaction derived from MF two-body interaction
- 2+ and 3- multipolarities
- only most collective low-lying QRPA phonons

Pairing vibrations

- PV included only in nuclei with N or Z magic number
- RPA, based on Woods-Saxon s.p. levels
- separable interaction with constant matrix elements G
- = 0+ (2+) contribution of the lowest (n=1) PV phonon
- G calculated in double closed shell nuclei

Results ground state energy

SV (2+,3-)

PV (0+)

PV (2+)

Correlation energies in Ca and Pb isotopic chains

Ca

Pb

Summary and conclusions ground state energy

Summary and conclusions- SV: 2+, 3- (QRPA)
- PV: 0+ (2+) (RPA)
- linear refit

HF+BCS

MF+SV+PV

rmsd = 0.724 MeV

rmsd = 0.667 MeV

( - 8 % )

- A systematic analysis of the contribution of medium polarization effects associated with collective surface and pairing vibrations on the binding energies of even-even spherical nuclei.

- The contributions of ZPF associated with collective nuclear vibrations (surface and pairing modes) to the nuclear binding energy are important.
- Future work: a more systematic refit which is able to make a wider survey
- of the parameters space.

People working on this project:

S.Baroni, F.Barranco, P.F.Bortignon, R.A.Broglia, G.Colò, E.Vigezzi

(Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353)

(Baroni et al., Phys. Rev. C 74 (2006) 024305)

Other existing work on the subject: ground state energy

(Bender, Bertsch, Heenen PRC 73 (2006) 034322)

605 even-even (spherical and deformed) nuclei have been studied.

In the GCM framework, by projection on good angular momentum:

- quadrupole Surface Vibration correlations
- correlations due to static deformation

My work:

(Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353)

(Baroni et al., Phys. Rev. C 74 (2006) 024305)

121 spherical even-even nuclei in the QRPA framework:

- quadrupole and octupole Surface Vibration correlations
- Pairing Vibrations correlations

Results ground state energy

Influence of the various contributions

full correlations

added

fluctuations

added

MF: Mean Field

SV: Surface Vibrations

PV: Pairing Vibrations

- quadrupole PV can give an important contribution.
- (however they have to be included only where experimentally observed)

Alcuni dati ground state energy

- Variazione % dei coeff. di Skyrme:
- MSk7
- MSk7 modified
- X0 0,58 0,58 -0,2420%
- X1 -0,50 -0,50 0,0000%
- X2 -0,50 -0,50 0,0000%
- X3 0,79 0,79 -0,0211%
- T0 -1828,23 -1828,76 -0,0291%
- T1 259,40 255,31 1,5761%
- T2 -292,84 -286,71 2,0926%
- T3 13421,70 13412,76 0,0666%
- W0 118,81 116,14 2,2476%
- Alpha 0,33 0,33 0,0000%

A brief summary of pairing vibrations ground state energy

PV summarySpatial (quadrupole) deformations

Pairing deformations

deformation of the surface

deformation of the Fermi surface

rotational symmetry (3D space)

gauge symmetry (gauge space)

I

N

Nilsson wavefunction

BCS wavefunction

and and Euler angles

and gauge angle

Spherical nuclei

Normal nuclei

Excited by coulomb excitations or inelastic scattering

Excited by two-particle transfer reactions

surface vibrations

pairing vibrations

Where corrections… ground state energy

Where are corrections expected to be important?

2+(one phonon state)

vibrational spectrum

strong B(E2)

0+ (g.s.)

2+ (vibrational)

additional rotational structure

6+

4+

weak B(E2)

2+

0+

- no pairing distortion
- high collectivity of pairing vibrational modes

by analogy

- permanent pairing distortion (eq 0)
- most of collectivity is found in pairing
- rotational band

In a spherical nucleus

In a deformed nucleus

In a closed shell nucleus

In an open shell nucleus

Refit procedure ground state energy

HF+BCS

MF

binding energies

interaction

interaction

corrected

MF b.e.

+

(Q)RPA

SV, PV

correlations

Refit procedure

Results ground state energy

HF+BCS

MF

binding energies

interaction

interaction

new interaction

corrected

MF b.e.

+

(Q)RPA

SV, PV

correlations

REFIT

Results ground state energy

HF+BCS

MF

binding energies

interaction

interaction

new interaction

corrected

MF b.e.

+

(Q)RPA

SV, PV

correlations

REFIT

corrected & refitted

MF b.e.

linear

Spettro vibraz. Pb208 ground state energy

How was the pairing strength calculated in closed shell nuclei?

Spettro vibraz. Pb208 (gs) ground state energy

Spettro vibraz. Pb208 (indep) ground state energy

Spettro vibrazionale di pairing per 208Pb

INDEPENDENT

PARTICLE

MODEL

INDEPENDENT

PARTICLE

MODEL

Spettro vibraz. Pb208 (extra energy) ground state energy

INDEPENDENT

PARTICLE

MODEL

INDEPENDENT

PARTICLE

MODEL

Spettro vibraz. Pb208 (due fononi) ground state energy

Curva dispersione ground state energy

Dispersion curve

Modello di pairing (0+) ground state energy

Pairing model ( = 0+)

{

In harmonic approximation:

Dispersion relation:

Modello di pairing (2+) ground state energy

Pairing model

In harmonic approximation:

Dispersion relation: