Transcript Document

What can we learn from vibrational
states ?
“the isoscalar and the chargeexchange excitations”
ECT* Workshop:
G. Colò
The Physics Opportunities with
16-21/1/2006
Modes of nuclear excitations
MONOPOLE
In the isoscalar
resonances, the
n and p oscillate
in phase
In the isovector
case, the n and
p oscillate in
opposition
of
phase
DIPOLE
QUADRUPOLE
Self-consistent mean field calculations (and extensions)
are probably the only possible framework in order to
understand the structure of medium-heavy nuclei.
The study of vibrational excitation is instrumental in order
to constrain the effective nucleon-nucleon interaction.
Both the non-relativistic Veff
(Skyrme or Gogny) and RMF
Lagrangians are fitted using:
• Nuclear matter properties
(saturation point)
• G.s. properties of a limited
set of nuclei (total binding
energy, charge radii).
The effective interaction defines an energy functional
Density
like
in DFT functional theory
E   Hˆ    Hˆ eff   Eˆ 

Slater determinant
  A (1 (r1 )     A (rA ))
Mean field:
E
ˆ
h
ˆ
 ˆ
density matrix
A
ˆ (r, r' )   i (r ) i (r' )
i 1
Interaction:
2

E
ˆ
V 
ˆˆ
Can we go towards “universal” functionals ?
• Ground-state properties of nuclei
• Vibrational excitations (small- and large-amplitude)
• Nuclear deformations
• Rotations
• Superfluid properties
If pairing is introduced, the energy functional depends on both
the usual density ρ=<ψ+(r)ψ(r)> and the abnormal density
κ=<ψ(r)ψ(r)> (κ=<ψ+(r)ψ+(r)>).
The system is described in terms of quasiparticles.
HF becomes HF-BCS or HFB, RPA becomes
QRPA.
What is the most critical part of the
nuclear energy functional ?
In the nuclear systems there are neutrons and protons.
usual (stable) nuclei
neutron-rich (unstable) nuclei
neutron stars
The largest uncertainities
concern the ISOVECTOR,
or SYMMETRY part of the
energy functional.
The Isoscalar Monopole and the nuclear
incompressibility
The nuclear matter (N = Z and no Coulomb interaction) incompressibility
coefficient, K∞ , is a very important physical quantity in the study of nuclei,
supernova collapse, neutron stars, and heavy-ion collisions.
E/A [MeV]
ρ = 0.16 fm-3
E/A = -16 MeV
ρ [fm-3]
Microscopic link E(ISGMR) ↔ nuclear incompressibility
Nowadays, we give credit to the idea that the link should be provided
microscopically. The key concept is the Energy Functional E[ρ].
K∞ in nuclear matter (analytic)
IT PROVIDES AT THE SAME TIME
EISGMR (by means of selfconsistent RPA calculations)
EISGMR
RPA
Eexp
Skyrme
Gogny
RMF
220
240
260
Extracted value of K∞
K∞ [MeV]
Until 2 years ago:
The extraction of the nuclear incompressibility from the
monopole data was plagued by a strong model
dependence: the Skyrme energy functionals seemed to
point to 210-220 MeV, the Gogny functionals to 235 MeV,
and the relativistic functionals to 250-270 MeV.
SLy4 protocol, α=1/6
Results for the ISGMR…
Cf. G. Colò, N. Van Giai, J. Meyer, K. Bennaceur
and P. Bonche, “Microscopic determination of the
nuclear incompressibility within the non-relativistic
framework”, Phys. Rev. C70 (2004) 024307.
K∞ around 230-240 MeV
Full agreement with Gogny; before we had SC violations
• α=0.3563,
• neglect of the Coulomb exchange
and center-of-mass corrections in
the HF mean field.
We have increased the
exponent in the density
dependence of the Skyrme
force
We have also increased
the density dependence of
the symmetry energy (Kτ)
By-product: decrease of m*
The result of B.J. Agrawal et al.,
is consistent with this plot !
Ksurf = cK with c ~ -1 (cf. Ref. [1]).
KA = K (non rel.)(1+cA-1/3) + Kτ (non rel.) δ2 + KCoul (non rel.) Z2 A-4/3
KA = K (rel.)(1+cA-1/3) + Kτ (rel.) δ2 + KCoul (rel.) Z2 A-4/3
KCoul should not vary much from the non-relativistic to the
relativistic description. But since both the terms which
include K and Kτ contribute, a more negative Kτ can lead to
a the extraction of a larger K (and vice-versa).
Remember: Kτ is negative and depends on the density
dependence of the symmetry energy !
[1] M. Centelles et al., Phys. Rev. C65 (2002) 044304
CONCLUSION FROM THE ISGMR
Fully self-consistent calculations of the ISGMR using
Skyrme forces lead to K∞~ 230-240 MeV.
Relativistic mean field (RMF) plus RPA: lower limit for K∞
equal to 250 MeV.
It is possible to build bona fide Skyrme forces so that the
incompressibility is close to the relativistic value.
→ K∞ = 240 ± 10 MeV.
To reduce this uncertainity one should fix the density
dependence of the symmetry energy.
How to experimentally discriminate between models ?
E ~ A-1/3
δE/E = δA/3A
Even if we take a long isotopic chain of stable, spherical isotopes:
Sn → δE/E is of the order of 3%, that is, 0.45 MeV (≈ 2σexp).
If we are able to measure outside this range (that is, we consider
unstable nuclei) we can have a larger variation of the monopole
energy and be able to see the effect of the symmetry term.
A word about the energies which are required
The most recent experiments on stable
nuclei employ α particles at ≈ 400 MeV,
which means 100 MeV/u (e.g., at RCNP,
Osaka).
However, previous experiments at lower
energies (of the order of 60 MeV/u) had
given positive results, although maybe
with larger background and less
accurate determination of the details of
the structure of the vibrational mode.
Speculations…
If neutron-rich nuclei are
develop a “halo” or “skin”,
think that this “excess” of
can vibrate independently
core at a lower frequency.
able to
one may
neutrons
from the
The low-energy peak would give
access to the compressibility of
low-density neutron matter.
This idea is familiar to solid-state
physicists !
Problem: calculations SO FAR are consistent
with the idea that only light nuclei develop a halo
and halo excitations are not collective.
Low-energy quadrupole
I.Hamamoto, H. Sagawa and X.Z.Zhang, PRC 55, 2361
• The GQR is lower than the
systematics (62A-1/3) by about 10%
• Implications for the effective mass
since E ÷ (m/m*)1/2.
• The neutron content is much larger
(about 50%) than N/Z
• It cannot be separated by low-lying
pure neutron strength
• The low-lying quadrupole, and to some extent, the
“usual” GQR, do not have the standard isospin. The
low-lying strength is half IS and half IV. To reproduce it
amounts to testing the energy functional in a very
different situation compared to standard nuclei.
• Relationship with the evolution of the effective mass
far from stability.
• Low-energy should make the quadrupole a better
physics case for EURISOL.
Folding model calculation [D.T. Khoa et al., NPA 706 (2002), 61]
S isotopes:
30,32
S
38,40
S
Use of microscopic
(QRPA) transition
densities.
Pairing far from stability
If the collective modes involve excitations not so far from
the Fermi surface, in open-shell isotopes pairing is
obviously important.
2hω
1hω
0hω
Do we have a theory for pairing ?
Example of an effective
pairing force.
Surface pairing: ρ0 = ρsat
Mixed pairing: ρ0 = 2ρsat
n-rich side: the
big dispersion
of the pairing
gaps will have
an effect on 2+
excitations
T. Duguet et al., nucl-th/0508054
F.Barranco, R.A.Broglia,
G. Colò, G.Gori,
E.Vigezzi, P.F. Bortignon
(2004)
Diagonalizing the v14 interaction
within the generalized BCS (on a
HF basis) account for only half of
the experimental gap in 120Sn.
The remaining part comes from renormalization due to
the particle vibration coupling.
it is possible to treat on the same footing
and
CONCLUSION
Probably EURISOL can be able to provide answers to the
problem of pairing (i.e., how to treat in a unified way the
“usual” like-particle pairing in nuclei with usual N/Z ratios
and the pairing in n-rich systems) by means of other
experiments like TRANSFER reactions.
However, low-lying excited states are sensitive BOTH to
particle-hole correlations and pairing correlations.
Charge-exchange excitations
They are induced by charge-exchange reactions, like (p,n) or (3He,t), so
that starting from (N,Z) states in the neighbouring nuclei (N,Z±1) are
excited.
A systematic picture of these states
is missing.
However, such a knowledge would
be important for astrophysics, or
neutrino physics
“Nuclear matrix elements have to be
evaluated with uncertainities of
less than 20-30% to establish the
neutrino mass spectrum.”
K. Zuber, workshop
decay, 2005
on
double-β,
(p,n)
Z+1,N-1
(n,p)
Z,N
Z-1,N+1
t
• Isobaric Analog Resonance (IAR)
 L =0,  S =0
Z
N
Strict connection with the isospin symmetry : if H commutes with isospin,
the IAR must lie at zero energy. BCS breaks the symmetry and only selfconsistent QRPA can restore it.
H includes parts which provide explicit symmetry breaking: the Coulomb
interaction, charge-breaking terms in the NN interaction, e.m. spin-orbit.
• Gamow-Teller Resonance (GTR)
 L =0,  S =1
jl
1
2
 t
jl
Z
N
1
2
Can the energy difference GT-IAR provide
a measure of the neutron skin ?
Sn nuclei
D. Vretenar et al.
Phys. Rev. Lett. 91, 262502 (2003)
Hartree-Bogoliubov/pn-quasiparticle RPA
Ex(GT)-Ex(IAR):
depends on spin-orbit potential which is
reduced for large N-Z
Effective NN force at 0 momentum transfer
W.G. Love and M.A. Franey, PRC 24, 1073
Below 100 MeV/u there is a
transition between the region
of dominance of the non spinflip component and that of the
spin-flip component – this can
be exploited by EURISOL.
Non spin-flip: IAR, isovector monopole, dipole…
The IV monopole (r2τ)
We are still waiting to know where it lies… We miss an
idea about a really selective probe. Yet it can give access
to:
• isospin mixing in the ground-state
• symmetry energy
Can we see the problem ?
Courtesy of R. Zegers
Self-consistent CE RPA based on Skyrme have been available for many
years.
On the other hand, essentially all the calculations made for open-shell
systems are phenomenological QRPA based on Woods-Saxon plus a
simple separable force with adjustable gph and gpp parameters.
→ Need of a self-consistent QRPA !
p
n-1
n-1
p
p
p
Based on HF-BCS. A zero-range DD pairing force is
employed:
• p-h channel : Skyrme
n
n
• p-p channel : we have a residual proton-neutron
interaction which exists in the T=0 and T=1 channels.
In the T=1 channel we can take the same force used
for BCS due to isospin invariance
IAR energies in
104-132Sn
Exp: K. Pham et al., PRC 51 (1995) 526.
S. Fracasso and G. Colò, “The fully self-consistent
charge-exchange QRPA and its application to
the Isobaric Analog Resonances”, Phys. Rev. C72
(2005).
CONCLUSION
• In the charge-exchange sector, the energy below
about 60 MeV/u seems more favourable for the non
spin-flip excitations, in contrast with the fact that the GT
“window” is above 100 MeV/u. Complementarity of
EURISOL with respect to higher-energy facilities.
• The charge-exchange modes have been always quite
elusive in this channel, with the exception of the IAR.
• If RIA starts, certainly emphasis will be given to these
kind of studies (JINA: Nuclear Astrophysics).
• Inverse kinematics ?