Compressibility of nuclear matter and breathing mode of finite nuclei
in relativistic random phase approximation
Zhongyu Ma,
*
Nguyen Van Giai,
†
and Hiroshi Toki
Research Center for Nuclear Physics, Osaka University, Osaka 567, Japan
Marcelle L’Huillier
Division de Physique The ´orique, Institut de Physique Nucle ´aire, F-91406 Orsay Cedex, France
Received 6 December 1996
Isoscalar monopole modes in finite nuclei are studied in the framework of relativistic models currently used
in ground state calculations. Response functions in the random phase approximation are calculated with
nonlinear models for the first time. It is found that some effective Lagrangians having a bulk compression
modulus in the range 280–350 MeV can predict correctly breathing mode energies in medium and heavy
nuclei. It is pointed out that the parametrization NL1 ( K
=211 MeV) leads to an anomalous behavior of the
monopole response. S0556-28139705405-8
PACS numbers: 21.10.Re, 21.60.Jz, 21.65.+f, 24.30.Cz
The issue of determining the value of the compression
modulus K
of nuclear matter is of great importance for
obtaining the nuclear equation of state. From the experimen-
tal side, the main information at our disposal comes from
energy systematics of the breathing mode measured in many
nuclei across the Periodic Table. Yet, it is only possible at
the present stage to ascertain the value of K
to belong to the
200–350 MeV interval 1 if one tries to avoid model-
dependent analyses and deduce K
from an A
-1/3
expansion
of the finite nucleus compressibility K
A
. Thus, one has to
introduce some degree of model dependence in order to es-
tablish a link between the energy of the isoscalar giant
monopole resonance GMR in finite nuclei and the nuclear
matter incompressibility. This has been done already for
quite some time in the nonrelativistic framework, and the
commonly accepted value of K
=21030 MeV was de-
duced by Blaizot 2 by calculating nuclear matter properties
with effective interactions which could describe satisfacto-
rily the GMR in Hartree-Fock random phase approximation
RPA models.
In recent years the relativistic many-body theory has met
great success in predicting ground state properties of finite
nuclei including unstable ones up to the nucleon drip lines.
Since the early work of Walecka 3 several parametrizations
of effective Lagrangians containing self-interaction terms of
the meson fields or density-dependent coupling constants
have been proposed 4–9, all of them aiming at a good
description of nuclear ground states in a relativistic mean
field, i.e., Dirac-Hartree framework. It turns out that the val-
ues of K
that they predict can span a wide range, from
about 200 MeV up to above 500 MeV. However, one does
not know what monopole energies in finite nuclei these non-
linear or density-dependent models would give. Only a few
attempts have been made to predict with relativistic models
the GMR energies in nuclei, either by using the relativistic
RPA RRPA method 10,11 or the constrained Dirac-
Hartree method 12. In both cases, however, the investiga-
tions were limited to the linear model with the parameter set
of Horowitz and Serot 13HS which corresponds to K
=545 MeV and consequently it was found that GMR ener-
gies in medium and heavy nuclei are overestimated.
The purpose of this work is to examine the more recently
proposed effective Lagrangians including nonlinear terms
from the point of view of their GMR predictions in nuclei.
The RRPA is the appropriate framework to extend the rela-
tivistic mean field description to the nuclear excitations. In-
deed, in a way similar to the nonrelativistic case, the RRPA
can be seen as the small amplitude limit of the time-
dependent Dirac-Hartree theory and therefore the same ef-
fective Lagrangian should be able to describe ground states
and giant resonances as well. Effective Lagrangians often
used successfully in Dirac-Hartree calculations all belong to
the class of non-linear models with self-interaction terms in
the field and also sometimes in the field. We are thus led
to calculate the linear response function of nuclei in RRPA
with nonlinear models. We shall discuss the GMR energies
obtained with three nonlinear models often found in the lit-
erature, namely the NL-SH model of Sharma et al. 7, TM1
of Sugahara and Toki 8, and NL1 of Reinhard et al. 4.
We start from an effective Lagrangian of the form
L=
¯
i
- M
N
-g
-g
-g
a
a
+
1
2
-U
-
1
4
W
W
+U
+
1
2
m
2
a
a
-
1
4
R
a
R
a
-
¯
e
A
1
2
1 -
3
-
1
4
F
F
, 1
where
*Permanent address: China Institute of Atomic Energy, P.O.
Box 275/18, Beijing 102413, People’s Republic of China.
†
Permanent address: Division de Physique The ´rique, Institut de
Physique Nucle ´aire, F-91406 Orsay Cedex, France.
PHYSICAL REVIEW C MAY 1997 VOLUME 55, NUMBER 5
55 0556-2813/97/555/23854/$10.00 2385 © 1997 The American Physical Society