Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

Incompressibility in finite fermionic systems: application to stableand exotic nuclei†
E. Khan,∗1 N. Paar,∗2 D. Vretenar,∗2 L.G. Cao,∗3 H. Sagawa,∗4,∗5 and G. Colò∗6150

Recently, it has been inferred that the measurementof isoscalar giant monopole resonances (GMRs) may
probe the incompressibility of the nucleus in a range
below the saturation density, rather than exactly at
the saturation density1) . Recent measurements of isotopic chains2,3) , possibly extended to exotic nuclei4,5) ,are therefore relevant for constraining this more general density-dependent incompressibility. The use of
isotopic chains also provided opportunities to study
pairing and shell eﬀects on incompressibility. It has
been shown that one cannot rule out shell and pairingeﬀects on nuclear incompressibility6,7) . We therefore
focus on the nuclear incompressibility along isotopic
chains and through magicity.
The calculations following fully microscopic approaches based on the energy density functionals(EDFs) to predict the GMR position are usually performed using the constrained Hartree-Fock-Bogoliubov(CHFB) or the quasiparticle random-phase approximation (QRPA) approaches8) . In the present study,
we calculate the GMR energy for the Skyrme EDF
with the CHFB approach. For completeness, results
obtained using Skyrme and relativistic functionals arealso given using the QRPA approach.
The results obtained using the various methods mentioned above are displayed in Fig. 1 for Sn and Pb isotopic chains. The nuclear incompressibility of a ﬁnite
nucleus KA is evaluated using
KA =2A⟨r2 ⟩2g.s.
,m−1140130∗1∗2∗3∗4∗5∗6PbKA (MeV)

Sn12011010090100

120140160180200

A220240260

280300Fig. 1. Nuclear incompressibility in Sn and Pb isotopic
chains calculated using the microscopic Skyrme-CHFB
method (circles), the Skyrme-QRPA method (triangles)
and relativistic QRPA method (squares).

link between the incompressibility and the zero-point
kinetic energy T0 , implying that incompressibility isrooted in the localization properties of the constituentsof the system. The HO model provides KA ≃5T0 ,showing that 140 MeV is a sound estimation of thenuclear incompressibility in stable nuclei.
In order to study the evolution of nuclear incompressibility along isotopic chains, several microscopic
EDF-based approaches have been used. In the case of
exotic nuclei, a decrease in KA is predicted in all themodels because of the emergence of a soft monopole
strength. These results conﬁrm the important role ofthe soft monopole resonance, and attempting to detect
it would be useful. More generally, the behavior of theGMR in exotic nuclei and/or beyond magicity should
be studied experimentally. Measurements of the GMR
around 132 Sn and 210 Pb would be of interest.

(1)

where m−1 is the inverse energy-weighted sum rule and⟨r2 ⟩g.s. is the mean-square radius of the ground state.
First, it should be noted that the three microscopic
methods provide KA values that do not diﬀer by morethan 10%. The nuclear incompressibility KA is almost
constant at 140 MeV around stable nuclei but changes
as a function of A in the case of more exotic nuclei.
The incompressibility of fermionic systems has beenstudied using several approaches. Analytical relations
using simple free Fermi gas and spherical Harmonic
Oscillator (HO) models allowed us to show the direct
†CHFB
RQRPA
QRPAReferences
1) E. Khan, J. Margueron and I. Vidaña, Phys. Rev. Lett.
109, 092501 (2012).
2) T. Li et al., Phys. Rev. C81, 034309 (2010).
3) U. Garg, COMEX3 conference, Mackinac Island (2009).
4) C. Monrozeau et al., Phys. Rev. Lett. 100, 042501(2008).
5) M. Vandebrouck et al., Exp. E456a performed at
GANIL (2010).
6) J. Li, G. Colò and J. Meng, Phys. Rev. C78, 064304(2008).
7) E. Khan, Phys. Rev. C80, 011307(R) (2009)8) N. Paar, D. Vretenar, E. Khan and G. Colò, Rep. Prog.
Phys. 70, 691 (2007)

Condensed from the article in Phys. Rev. C. 87,064311(2013)
Institut de Physique Nucléaire, Université Paris-Sud,
IN2P3-CNRS
Physics Department, Faculty of Science, University of Zagreb
Institute of Modern Physics, Chinese Academy of Science
RIKEN Nishina Center
Mathematics and Physics, University of Aizu
Dipartimento di Fisica, Università degli Studi and INFN
Sez. di Milano- 53 -