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208 Pb. stable nucleus, lying along the stability valley one-neutron separation energy = S n  7.40 MeV. 11 Li. halo nucleus, lying near or at the n-drip line two-neutron separation energy = S 2n  300 keV. Our mean field calculation. HF-BCS approximation

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


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

Our mean field calculation

  • HF-BCS approximation
  • Skyrme-type interaction MSk73
  • particle-particle channel:
  • Wigner term
  • Finite proton correction

(rms = 0.754 when fitted to 1768 nuclei)

  • -pairing force
  • pp and nn channel
  • state dependent matrix elements
  • energy cutoff at 1 h=41A-1/3
  • different pairing strength for  and 

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

The largest deviations from experiment

are associated to closed shell nuclei

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

3 developed by Goriely et al.

  • vibrations of the nuclear surface
  • pairing vibrations

One has to consider collective degrees of freedom like:

Dynamic vibrations of the surface

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)…

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

are associated with

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

where Yki() are the

backwards-going amplitudes

of the RPA wavefunctions

Analogy between

Experimental observation

spatial (quadrupole) deformations and

pairing deformations

Some details of our calculation:

(t,p) and (p,t) reactions are excellent tools

for probing pairing correlations

Deformation of the surface

of the nucleus.

Distortion of the Fermi surface

(superfluid state).

  • 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%

(neutron) pairing vibrations in even Ca nuclei

The associated average field is not invariant under

rotations in three dimensions,

gauge transformations,

Where are correlation energies expected to be important?

whose generator is the

(nr, na) are pair removal

and pair addition quanta

particle number operator N.

total angular momentum operator I.

In a spherical nucleus

One can parametrize the deformation of the potential in terms of

vibrational spectrum

(e.g. of quadrupole type)

 and  and of the Euler angles 

the BCS gap parameter  and

the gauge angle 

that defines an orientation of the intrinsic frame of reference

in ordinary 3D space.

in gauge space.

In a deformed nucleus

Going from a physical state with

an additional rotational structure is displayed

2+ (one phonon state)

total angular momentum I1

particle number N1

(neutron) pairing rotations in even Sn nuclei

to another physical state with

strong B(E2) due to

high collectivity

total angular momentum I2 ,

particle number N2 ,

exp. values

0+ (g.s.)

there is a change in the energy along the

a permanent (shape) deformation makes

the system more rigid to oscillations

surface vibrations are more

important in spherical nuclei

2+ (vibrational)

relative cross sections display

a linear dependence on the

number of pairs added/removed

from N=28 shell

exp. values

In short:

rotational band.

pairing rotational band.

neutron closed

shell nucleus

harmonic model

For small values of the interaction parameter, the system has

rotational band: it “absorbs”

most of collectivity


g.s.  g.s.

cross sections are much

larger than g.s.  p.v.

cross sections



Q0=0 (spherical nucleus)

=0 (normal nucleus)

by analogy

weak B(E2)



and displays a typical phonon spectrum

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

(surface vibrations).

(pairing vibrations).

In a closed shell nucleus

It corresponds to oscillations

of the energy gap around eq = 0,

of the surface around spherical shape,

In an open shell nucleus

  • permanent pairing deformation (eq  0)
  • most of the pairing collectivity is found in
  • pairing rotational bands

the excited states being states with different

angular momentum.

particle number.

pairing vibrations are more

important in closed shell nuclei

In short:

doubly closed shell nuclei

neutron closed shell nuclei

Pairing 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 X02 systems, X0 being the magic neutron (N0) or proton
  • (Z0) number associated with the closed shell system




(magic) Z = 8

(magic) Z = 20

(magic) Z = 82




(magic) Z = 50

Z = 18

Z = 22

Congresso del Dipartimento di Fisica

Highlights in Physics 2005

11–14 October 2005, Dipartimento di Fisica, Università di Milano

Contribution to nuclear binding energies arising from surface and pairing vibrations

S.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

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

Nuclear masses: the state of the art…

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

Describing the nucleus like a liquid drop

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

2.970 MeV

Finite-range droplet method1……………………………….

0.689 MeV

1654 nuclei fitted

Using microscopically grounded methods

(mean field approximation)

The r-processes nucleosynthesis path

evolves along the neutron drip line region

rms error

HF-BCS calculation with Skyrme interaction2……..

0.674 MeV

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

2135 nuclei fitted

Hartree-Fock Bardeen-Cooper-Schrieffer


0.709 MeV

Extended Thomas-Fermi plus Strutinsky integral

1719 nuclei fitted

We need a formula at least a factor of two more

accurate than present microscopic ones!!

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

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

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


Calculations have been carried out for 52 spherical

nuclei in different regions of the mass table

  • clear reduction of rms errors in closed shell nuclei

(all data in MeV)

a factor of nearly 5 better!!

  • extension to open shell nuclei:

(all data in MeV)