Congresso del Dipartimento di Fisica

Highlights in Physics 200511–14 October 2005, Dipartimento di Fisica, Università di Milano

Contribution to nuclear binding energies arising from surface and pairing vibrationsS.Baroni*†, F.Barranco#, P.F.Bortignon*†, R.A.Broglia*†x, G.Colò*†, E.Vigezzi†

* Dipartimento di Fisica, Università di Milano † INFN – Sezione di Milano# Escuela de Ingenieros, Sevilla, Spain xThe Niels Bohr Institute, Copenhagen, Denmark

In the table of nuclei one can encounter very different systems:

• stable nucleus, lying along the stability valley• one-neutron separation energy = Sn 7.40 MeV

11Li

208Pb

• halo nucleus, lying near or at the n-drip line• two-neutron separation energy = S2n 300 keV

The r-processes nucleosynthesis path evolves along the neutron drip line region

The need of a mass formula able to predict nuclear masses with an accuracy of the order of magnitude of S2n 300 keV seems quite natural

We need a formula at least a factor of two moreWe need a formula at least a factor of two moreaccurate than present microscopic ones!!accurate than present microscopic ones!!

For a better prediction one has to go beyond static mean field approximation.For a better prediction one has to go beyond static mean field approximation.

One has to consider collective degrees of freedom like:One has to consider collective degrees of freedom like: • vibrations of the nuclear surfacevibrations of the nuclear surface• pairing vibrationspairing vibrations

Pairing vibration calculations detailsPairing vibration calculations details• calculations carried out in the RPA• separable pairing interaction with constant matrix elements• L = 0+, 2+ multipolarities taken into account (only L = 0+ for lightest nuclei)• pairing interaction parameter calculated in double closed shell nuclei, solving a dispersion relation and reproducing the experimental extra binding energies observed in X02 systems, X0 being the magic neutron (N0) or proton (Z0) number associated with the closed shell system

Calculations have been carried out for 52 spherical nuclei in different regions of the mass table

ResultsResults

• extension to open shell nuclei:extension to open shell nuclei:

What are pairing vibrations?What are pairing vibrations?

…there exist vibrational modes based on fields which create or annihilate pairs of particles

the corresponding collective mode is called pairing vibration

Oscillations in the shape of the nucleus

a change in the binding field of each particle (i.e. with a field which conserves the number of particles

and arising from ph residual interaction)…

are associated with

Oxygen (magic) Z = 8 Calcium (magic) Z = 20 Lead (magic) Z = 82

Tin (magic) Z = 50 Argon Z = 18 Titanium Z = 22

• clear reduction of rms errors in closed shell nucleiclear reduction of rms errors in closed shell nuclei

doubly closed shell nuclei neutron closed shell nuclei

a factor of nearly 5 better!!

(all data in MeV)

(all data in MeV)

Our mean field calculationOur mean field calculation• HF-BCS approximation• Skyrme-type interaction MSk73

• particle-particle channel:

• Wigner term

• Finite proton correction

3 developed by Goriely et al.

(correcting the absence of T=0 np pairing in the model)

- -pairing force- pp and nn channel- state dependent matrix elements- energy cutoff at 1 h=41A-1/3

- different pairing strength for and

(rms = 0.754 when fitted to 1768 nuclei)

The largest deviations from experimentThe largest deviations from experimentare associated to closed shell nucleiare associated to closed shell nuclei

Where are correlation energies expected to be important?Where are correlation energies expected to be important?

In a spherical nucleusvibrational spectrum

(e.g. of quadrupole type)

In a deformed nucleusan additional rotational structure is displayed

0+ (g.s.)

2+ (one phonon state)

strong B(E2) due to high collectivity

a permanent (shape) deformation makes the system more rigid to oscillations

surface vibrations are more surface vibrations are more important in spherical nucleiimportant in spherical nuclei

In short:

In a closed shell nucleus

by analogy

• no stable pairing distortion• high collectivity of pairing vibrational modes

In an open shell nucleus • permanent pairing deformation (eq 0)• most of the pairing collectivity is found in pairing rotational bands

In short: pairing vibrations are more pairing vibrations are more important in closed shell nucleiimportant in closed shell nuclei

0+

2+ (vibrational)

}rotational band: it “absorbs”

most of collectivity2+4+6+

weak B(E2)

Q0=0 (spherical nucleus)

(surface vibrations).

The associated average field is not invariant under

whose generator is the

One can parametrize the deformation of the potential in terms of

that defines an orientation of the intrinsic frame of reference

there is a change in the energy along the

For small values of the interaction parameter, the system has

to another physical state with

and displays a typical phonon spectrum

It corresponds to oscillations

Going from a physical state with

Analogy betweenAnalogy between

Deformation of the surface of the nucleus.

Distortion of the Fermi surface (superfluid state).

rotations in three dimensions, gauge transformations,

and and of the Euler angles the BCS gap parameter and the gauge angle

total angular momentum I1 particle number N1

total angular momentum I2 , particle number N2 ,

rotational band. pairing rotational band.

=0 (normal nucleus)

(pairing vibrations).

of the surface around spherical shape, of the energy gap around eq = 0,

in ordinary 3D space. in gauge space.

the excited states being states with different

angular momentum. particle number.

total angular momentum operator I. particle number operator N.

spatial (quadrupole) deformations spatial (quadrupole) deformations andand pairing deformationspairing deformations

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

A remarkable accuracy, but one is still not satisfied!!A remarkable accuracy, but one is still not satisfied!!

rms errorrms error

Weizsacker formula (1935)………………………………….

Describing the nucleus like a liquid drop

Finite-range droplet method1………………………………. 0.689 MeV0.689 MeV1654 nuclei fitted

2.970 MeV2.970 MeV

1 P.Möller et al., At. Data Nucl. Data Tables 59 (1995) 1852 S.Goriely et al., Phys. Rev. C 66 (2002) 024326-1

Using microscopically grounded methods(mean field approximationmean field approximation)

HF-BCS calculation with Skyrme interaction2……..

ETFSI….…………..……………………………..…………………..

0.674 MeV0.674 MeV2135 nuclei fitted

Extended Thomas-Fermi plus Strutinsky integral

Hartree-Fock Bardeen-Cooper-Schrieffer

0.709 MeV0.709 MeV1719 nuclei fitted

Experimental observationExperimental observation

(t,p) and (p,t) reactions are excellent tools for probing pairing correlations

(neutron) pairing vibrations in even Ca nuclei

(neutron) pairing rotations in even Sn nuclei

exp. values

harmonic model

relative cross sections displaya linear dependence on the

number of pairs added/removedfrom N=28 shell

neutron closed shell nucleus

(nr, na) are pair removal and pair addition quanta

exp. values

g.s. g.s. cross sections are much larger than g.s. p.v.

cross sections

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

Dynamic vibrations of the surfaceDynamic vibrations of the surface

The correlation energy associated to zero point fluctuations has the expression:

Some details of our calculation:

where Yki() are the backwards-going amplitudes

of the RPA wavefunctions

• Microscopic description, Random Phase Approximation (RPA)• Vibrations: coherent particle-hole excitations

• Skyrme-type interaction MSk7 with a pairing force• 2+ and 3- multipolarities are taken into account• states with h < 10 MeV and with B(E) 2%