The nuclear matter incompressibility K from isoscalar compression modes

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The nuclear matter incompressibility K? from
isoscalar compression modes

WCI-3 workshop, College Station, TX February
12-16, 2005
G. Colò, V. Kolomietz, S. Shlomo
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Outline

• Introduction
• Definitions nuclear matter incompressibility
  coefficient K?
• Background isoscalar giant monopole resonance,
• isoscalar giant dipole resonance
• Hadron excitation of giant resonances
• Theoretical approaches for giant resonances
• Hartree-Fock plus Random Phase Approximation
  (RPA)
• Comments self-consistency ?
• Relativistic mean field (RMF) plus RPA
• Discussion
• ISGMR viz. ISGDR
• Non-relativistic viz. Relativistic
• Symmetry energy (Colò)
• Memory effects (Kolomietz)

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Introduction

The nuclear matter (N Z and no Coulomb
interaction) incompressibility coefficient, K8 ,
is a very important physical quantity in the
study of nuclei, supernova collapse, neutron
stars, and heavy-ion collisions, since it is
directly related to the curvature of the nuclear
matter (NM) equation of state (EOS), E E?,
E/A MeV
? 0.16 fm-3
? fm-3
E/A -16 MeV
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History

• ISOSCALAR GIANT MONOPOLE RESONANCE (ISGMR)
• 1977 DISCOVERY OF THE CENTROID ENERGY OF THE
  ISGMR IN 208Pb
• E0 13.5 MeV (TAMU)

• This led to modification of commonly used
  effective nucleon-nucleon interactions.
  Hartree-Fock (HF) plus Random Phase Approximation
  (RPA) calculations, with effective interactions
  (Skyrme and others) which reproduce data on
  masses, radii and the ISGMR energies have
• K8 210 20 MeV (J.P. BLAIZOT, 1980).
• ISOSCALAR GIANT DIPOLE RESONANCE (ISGDR)
• 1980 EXPERIMENTAL CENTROID ENERGY IN 208Pb AT
• E1 21.3 MeV (Jülich), PRL 45 (1980) 337
  19 MeV, PRC 63 (2001) 031301
• HF-RPA with interactions reproducing E0 predicted
  E1 25 MeV.
• K8 170 MeV from ISGDR ?
• T.S. Dimitrescu and F.E. Serr PRC 27 (1983) 211
  pointed out If further measurement confirm the
  value of 21.3 MeV for this mode, the discrepancy
  may be significant.
• ? Relativistic mean field (RMF) plus RPA with NL3
  interaction predict K8270 MeV from the ISGMR N.
  Van Giai et al., NPA 687 (2001) 449.

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Hadron excitation of giant resonances

Theorists calculate transition strength S(E)
within HF-RPA using a simple scattering operator
F rLYLM Experimentalists calculate cross
sections within Distorted Wave Born Approximation
(DWBA)
or using folding model.
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DWBA-Folding model description

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(No Transcript)
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Hartree-Fock (HF) - Random Phase Approximation
(RPA)

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• Are mean-field RPA calculations fully
  self-consistent ?
• NO ! In practice, one makes approximations.
• Mean field and Vph determined independently ? no
  information on K8.
• In HF-RPA one
• 1. neglects the Coulomb part in Vph
  
  
• 2. neglects the two-body spin-orbit
  
  
• 3. uses limited upper energy for s.p. states
  (e.g. Eph(max) 60 MeV)
  
• 4. introduces smearing parameters.
• Main effects
• change in the moments of S(E), of the order of
  0.5-1 MeV note
• spurious state mixing in the ISGDR
• inaccuracy of transition densities.

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• Commonly used scattering operators
• for ISGMR
• for ISGDR
• In fully self-consistent HF-RPA calculations the
  (T0, L1) spurious state (associated with the
  center-of-mass motion) appears at E0 and no
  mixing (SSM) in the ISGDR occurs.
• In practice SSM takes place and we have to
  correct for it.
• Replace the ISGDR operator with
• (prescriptions for ? discussion in the
  literature)

NUMERICS Rmax 90 fm
?r 0.1 fm (continuum RPA) Ephmax
500 MeV ?1 ?2 Experimental range
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Relativistic Mean Field Random Phase
Approximation
The steps involved in the relativistic mean field
based RPA calculations are analogous to those for
the non-relativistic HF-RPA approach. The
nucleon-nucleon interaction is generated through
the exchange of various effective mesons. An
effective Lagrangian which represents a system of
interacting nucleons looks like
It contains nucleons (?) with mass M s, ?, ?
mesons the electromagnetic field non linear
self-interactions for the s (and possibly ?)
field. Values of the parameters for the most
widely used NL3 interaction are ms508.194 MeV,
m?782.501 MeV, m?763.000 MeV, gs10.217,
g?12.868, g?4.474, g2-10.431 fm-1 and
g3-28.885 (in this case there is no
self-interaction for the ? meson). NL3 K8271.76
MeV, G.A.Lalazissis et al., PRC 55 (1997)
540. RMF-RPA J. Piekarewicz PRC 62 (2000)
051304 Z.Y. Ma et al., NPA 686 (2001) 173.
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K8 from the ISGMR in 208Pb Skyrme calc.
Non fully s.c. 210 MeV
Fully s.c. 235 MeV
G. Colò and N. Van Giai, NPA 731 (2004) 15.
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Relativistic RPA Values of K8 of the order of
250-270 MeV were extracted.
T. Nikic et al., PRC 66 (2002) 064302.
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A. Kolomiets, O. Pochivalov, and S. Shlomo, PRC
61 (2000) 034312
ISGMR fr2Y00 SL1 interaction, K8230 MeV Ea
240 MeV
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ISGDR
S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65,
044310 (2002)
SL1 interaction, K8230 MeV Ea 240 MeV
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Nuclear matter properties calculated from RMF
theory with NL3 parameters and from the
non-relativistic HF calculations
ISGMR centroid energy (in MeV) obtained by
integrating over the energy range ?1-?2 with the
strength function smeared by using G/2 1 MeV.
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CONCLUSION
Fully self-consistent calculations of the ISGMR
using Skyrme forces lead to K8 230-240
MeV. ISGDR At high excitation energy, the
maximum cross section for the ISGDR drops below
the experimental sensitivity. There remain some
problems in the experimental analysis. It is
possible to build bona fide Skyrme forces so that
the incompressibility is close to the
relativistic value. Recent relativistic mean
field (RMF) plus RPA lower limit for K8 equal to
250 MeV. ? K8 240 20 MeV.
sensitivity to symmetry energy.