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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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Our mean field calculation
(rms = 0.754 when fitted to 1768 nuclei)
(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.
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)…
are associated with
The correlation energy associated to zero point fluctuations has the expression:
where Yki() are the
of the RPA wavefunctions
spatial (quadrupole) deformations and
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
(neutron) pairing vibrations in even Ca nuclei
The associated average field is not invariant under
rotations in three dimensions,
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
(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
total angular momentum I2 ,
particle number N2 ,
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
relative cross sections display
a linear dependence on the
number of pairs added/removed
from N=28 shell
pairing rotational band.
For small values of the interaction parameter, the system has
rotational band: it “absorbs”
most of collectivity
cross sections are much
larger than g.s. p.v.
Q0=0 (spherical nucleus)
=0 (normal nucleus)
and displays a typical phonon spectrum
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
the excited states being states with different
pairing vibrations are more
important in closed shell nuclei
doubly closed shell nuclei
neutron closed shell nuclei
Pairing vibration calculations details
(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)………………………………….
Finite-range droplet method1……………………………….
1654 nuclei fitted
Using microscopically grounded methods
(mean field approximation)
The r-processes nucleosynthesis path
evolves along the neutron drip line region
HF-BCS calculation with Skyrme interaction2……..
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
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
Calculations have been carried out for 52 spherical
nuclei in different regions of the mass table
(all data in MeV)
a factor of nearly 5 better!!
(all data in MeV)