Transcript Highligh in Physics 2005

Congresso del Dipartimento di Fisica
Highlights in Physics 2005
11–14 October 2005, Dipartimento di Fisica, Università di Milano
ntribution to nuclear binding energies arising from surface and pairing vibrati
†
*
S.Baroni ,
#
F.Barranco ,
†
*
P.F.Bortignon ,
†x
*
R.A.Broglia ,
†
*
G.Colò ,
†
E.Vigezzi
* Dipartimento
† INFN – Sezione di Milano
di Fisica, Università 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
208Pb
Weizsacker formula (1935)…………………………………. 2.970 MeV
Finite-range droplet
method1……………………………….
11Li
0.689 MeV
1654 nuclei fitted
Using microscopically grounded methods
(mean field approximation)
0.674 MeV
ETFSI….…………..……………………………..………………….. 0.709 MeV
Extended Thomas-Fermi plus Strutinsky integral
• particle-particle channel:
- -pairing force
- pp and nn channel
- state dependent matrix elements
- energy cutoff at 1 h=41A-1/3
- different pairing strength for  and 
• Wigner term
• halo nucleus, lying near or at the n-drip line
• two-neutron separation energy = S2n  300 keV
(correcting the absence of T=0 np pairing in the model)
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
(rms = 0.754 when fitted to 1768 nuclei)

HF-BCS calculation with Skyrme
• HF-BCS approximation
• Skyrme-type interaction MSk73
• stable nucleus, lying along the stability valley
• one-neutron separation energy = Sn  7.40 MeV
The r-processes nucleosynthesis path
evolves along the neutron drip line region
rms error
interaction2……..
Our mean field calculation
• Finite proton correction
The largest deviations from experiment
are associated to closed shell nuclei
1719 nuclei fitted
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
3 developed
We need a formula at least a factor of two more
accurate than present microscopic ones!!
A remarkable accuracy, but one is still not satisfied!!
by Goriely et al.
For a better prediction one has to go beyond static mean field approximation.
One has to consider collective degrees of freedom like:
• vibrations of the nuclear surface
• pairing vibrations
What are pairing vibrations?
Dynamic vibrations of the surface
• Microscopic description, Random Phase Approximation (RPA)
• Vibrations: coherent particle-hole excitations
Oscillations in the shape of the nucleus
are associated with
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)…
The correlation energy associated to zero point fluctuations has the expression:
Analogy between
spatial (quadrupole) deformations and
Some details of our calculation:
• 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%
total angular momentum operator I.
0+ (g.s.)
an additional rotational structure is displayed
weak B(E2)
a permanent (shape) deformation makes
the system more rigid to oscillations
particle number operator N.
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.
Going from a physical state with
total angular momentum I1
particle number N1
2+ (vibrational)
6+
4+
2+
}
rotational band: it “absorbs”
most of collectivity
surface vibrations are more
important in spherical nuclei
exp. values
g.s.  g.s.
cross sections are much
larger than g.s.  p.v.
cross sections
(surface vibrations).
(pairing vibrations).
It corresponds to oscillations
of the surface around spherical shape,
of the energy gap around eq = 0,
the excited states being states with different
angular momentum.
In short: pairing vibrations are more
important in closed shell nuclei
doubly closed shell nuclei
neutron closed shell nuclei
Calcium
(neutron) pairing rotations in even Sn nuclei
and displays a typical phonon spectrum
• permanent pairing deformation (eq  0)
• most of the pairing collectivity is found in
pairing rotational bands
(magic) Z = 8
neutron closed
shell nucleus
relative cross sections display
a linear dependence on the
number of pairs added/removed
from N=28 shell
rotational band.
pairing rotational band.
For small values of the interaction parameter, the system has
Q0=0 (spherical nucleus)
=0 (normal nucleus)
• no stable pairing distortion
• high collectivity of pairing vibrational modes
Calculations have been carried out for 52 spherical
nuclei in different regions of the mass table
exp. values
harmonic model
to another physical state with
total angular momentum I2 ,
particle number N2 ,
there is a change in the energy along the
by analogy
Oxygen
(nr, na) are pair removal
and pair addition quanta
 and  and of the Euler angles 
strong B(E2) due to
high collectivity
0+
In an open shell nucleus
(neutron) pairing vibrations in even Ca nuclei
One can parametrize the deformation of the potential in terms of
In a deformed nucleus
In a closed shell nucleus
pairing deformations
rotations in three dimensions,
gauge transformations,
whose generator is the
2+ (one phonon state)
vibrational spectrum
(e.g. of quadrupole type)
In short:
(t,p) and (p,t) reactions are excellent tools
for probing pairing correlations
Deformation of the surface
Distortion of the Fermi surface
of the nucleus.
(superfluid state).
The associated average field is not invariant under
Where are correlation energies expected to be important?
In a spherical nucleus
the corresponding collective mode is called
pairing vibration
Experimental observation
where Yki() are the
backwards-going amplitudes
of the RPA wavefunctions

…there exist vibrational modes based on
fields which create or annihilate pairs of particles
(magic) Z = 20
particle number.
Results
Pairing vibration calculations details
• calculations carried out in the RPA
Lead
(magic) Z = 82
• 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
• clear reduction of rms errors in closed shell nuclei
(all data in MeV)
Tin
(magic) Z = 50
Argon
Z = 18
Titanium
Z = 22
• extension to open shell nuclei:
a factor of nearly 5 better!!
(all data in MeV)