Saturation of nuclear matter and radii of unstable nuclei

Abstract

We examine relations among the parameters characterizing the phenomenological equation of state (EOS) of nearly symmetric, uniform nuclear matter near the saturation density by comparing macroscopic calculations of radii and masses of stable nuclei with the experimental data. The EOS parameters of interest here are the symmetry energy S_0, the symmetry energy density-derivative coefficient L and the incompressibility K_0 at the normal nuclear density. We find a constraint on the relation between K_0 and L from the empirically allowed values of the slope of the saturation line (the line joining the saturation points of nuclear matter at finite neutron excess), together with a strong correlation between S_0 and L. In the light of the uncertainties in the values of K_0 and L, we macroscopically calculate radii of unstable nuclei as expected to be produced in future facilities. We find that the matter radii depend strongly on L while being almost independent of K_0, a feature that will help to determine the L value via systematic measurements of nuclear size. Comment: 26 pages, 7 figures

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arXiv:nucl-th/0204033v1 12 Apr 2002
Saturation of nuclear matter and radii of
unstable nuclei
Kazuhiro Oyamatsua,b, Kei Iidab
aDepartment of Media Production and Theories, Aichi Shukutoku University,
Nagakute, Nagakute-cho, Aichi-gun, Aichi 480-1197, Japan
bThe Institute of Physical and Chemical Research (RIKEN), Hirosawa, Wako,
Saitama 351-0198, Japan
Abstract
We examine relations among the parameters characterizing the phenomenologi-
cal equation of state (EOS) of nearly symmetric, uniform nuclear matter near the
saturation density by comparing macroscopic calculations of radii and masses of
stable nuclei with the experimental data. The EOS parameters of interest here are
the symmetry energy S0, the symmetry energy density-derivative coefficient L and
the incompressibility K0at the normal nuclear density. We find a constraint on the
relation between K0and L from the empirically allowed values of the slope of the
saturation line (the line joining the saturation points of nuclear matter at finite
neutron excess), together with a strong correlation between S0and L. In the light
of the uncertainties in the values of K0and L, we macroscopically calculate radii
of unstable nuclei as expected to be produced in future facilities. We find that the
matter radii depend strongly on L while being almost independent of K0, a feature
that will help to determine the L value via systematic measurements of nuclear size.
Key words: Dense matter, Saturation, Unstable nuclei
PACS: 21.65.+f, 21.10.Gv
1Introduction
Saturation of density and binding energy, one of the fundamental properties
of atomic nuclei, underlies a liquid-drop approach described by a Weizs¨ acker-
Bethe mass formula [1]
− EB= Evol+ Esym+ Esurf+ ECoul, (1)
Preprint submitted to Elsevier Preprint8 February 2008

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where EBis the nuclear binding energy, Evolis the volume energy, Esymis the
symmetry energy, Esurfis the surface energy, and ECoulis the Coulomb energy.
The sum Evol+Esymcorresponds to the saturation energy of uniform nuclear
matter. Since matter in nuclei is a strongly interacting system, it remains a
challenging theoretical problem to understand the nuclear matter equation
of state (EOS) through microscopic calculations that utilize a model of the
nuclear force duly incorporating low-energy two-nucleon scattering data and
properties of light nuclei [2]. Furthermore, it is not straightforward to empir-
ically clarify the EOS, although constraints on the EOS from nuclear masses
and radii (e.g., Refs. [3–5]), observables in heavy-ion collision experiments per-
formed at intermediate and relativistic energies (e.g., Refs. [6–9]), the isoscalar
giant monopole resonance in nuclei (e.g., Ref. [10]) and even X-ray observa-
tions of isolated neutron stars [11] are available. In this work we will consider
such constraints from nuclear masses and radii.
The EOS of bulk nuclear matter is a function of nucleon density n and pro-
ton fraction x, which are related to the neutron and proton number densities
nnand npas nn= n(1−x) and np= nx. We may generally express the energy
per nucleon near the saturation point of symmetric nuclear matter as [12]
w = w0+
K0
18n2
0
(n − n0)2+
?
S0+
L
3n0(n − n0)
?
α2. (2)
Here w0, n0 and K0 are the saturation energy, the saturation density and
the incompressibility of symmetric nuclear matter, S0is the symmetry energy
S(n) at n = n0, L = 3n0(dS/dn)n=n0is the density symmetry coefficient, and
α = 1 − 2x is the neutron excess. As the neutron excess increases from zero,
the saturation point moves in the density versus energy plane. This movement
is determined mainly by the parameters L and S0associated with the density-
dependent symmetry energy S(n) [3]. Up to second order in α, the saturation
energy wsand density nsare given by
ws= w0+ S0α2
(3)
and
ns= n0−3n0L
K0
α2. (4)
The slope, y, of the saturation line near α = 0 (x = 1/2) is thus expressed as
y = −K0S0
3n0L.
(5)
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Derivation of L and S0from nuclear observables is generally obscured by
the interfacial and electrostatic properties. Among the observables, the masses
and root-mean-square radii of nuclei, which are controlled mainly by the bulk
properties, are expected to be good tracers of L and S0. This expectation
was stressed by an earlier investigation [3] based on two macroscopic nuclear
models. Such macroscopic models are reliable in a range of neutron excess,
α ? 0.3, and mass number, A ? 50. In this range the neutron separation
energy that can be evaluated from a Weizs¨ acker-Bethe mass formula is greater
than 2 MeV, allowing us to preclude the possibility of neutron halo formation
expected at large neutron excess (or small separation energy). In constraining
the EOS from masses and radii of nuclei of neutron excess α ? 0.3 and mass
number A ? 50 within the framework of the macroscopic models, systematic
study allowing for uncertainties in the EOS parameters is indispensable.
In this paper we thus explore a systematic way of extracting L and S0from
empirical masses and radii of nuclei, together with the parameters, n0, w0and
K0, characterizing the saturation of symmetric nuclear matter. We first set
an expression for the energy of uniform nuclear matter, which reduces to the
phenomenological form (2) in the limits of n → n0and α → 0 (x → 1/2). Using
this energy expression within a simplified version of the extended Thomas-
Fermi approximation, which permits us to determine the macroscopic features
of the nuclear ground state, we calculate charges, charge radii and masses of
β-stable nuclei. Comparing these calculations with empirical values allows us
to derive the optimal parameter set for various values of the slope y and the
incompressibility K0. We thus find a strong correlation between L and S0. The
next step is to calculate root-mean-square charge and matter radii of more
neutron-rich nuclei that are expected to be produced in future radioactive ion
beam facilities. The results suggest that the density symmetry coefficient L
may be constrained from possible systematic data on the matter radii in a way
nearly independent of the poorly known K0, while the slope y being deducible
as a function of K0. We finally discuss an empirically allowed region in the
space of the parameters characterizing the EOS (2).
In Section 2 we construct a macroscopic model of nuclei. Optimization as-
sociated with fitting to empirical data for nuclei on the smoothed β-stability
line is illustrated in Section 3. In Section 4 we calculate matter and charge
radii of unstable nuclei. Our conclusions are presented in Section 5. Numerical
tables are given in the Appendix.
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2 Macroscopic description of nuclei
In constructing a macroscopic nuclear model, we begin with the expression
for the bulk energy per nucleon [13],
w =3?2(3π2)2/3
10mnn
(n5/3
n
+ n5/3
p) + (1 − α2)vs(n)/n + α2vn(n)/n,(6)
where
vs= a1n2+
a2n3
1 + a3n
(7)
and
vn= b1n2+
b2n3
1 + b3n
(8)
are the potential energy densities for symmetric nuclear matter and pure neu-
tron matter, and mnis the neutron mass. (Replacement of the proton mass
mpby mnin the proton kinetic energy makes only a negligible difference.) For
the later purpose of roughly describing the nucleon distribution in a nucleus,
we incorporate into the potential energy densities (7) and (8) a low density
behaviour ∝ n2as expected from a contact two-nucleon interaction. Note,
however, that we will focus on the EOS of nearly symmetric nuclear matter
near the saturation density. We will thus determine the parameters included
in Eqs. (7) and (8) in such a way that they reproduce data on radii and masses
of stable nuclei.
In the limit of n → n0and α → 0 (x → 1/2), expression (6) reduces to the
usual form (2) according to
S0=1
6
?3π2
2
?2/3?2
mnn2/3
0
+ (b1− a1)n0+
?
b2
1 + b3n0
−
a2
1 + a3n0
?
n2
0, (9)
1
3n0L=1
9
?3π2
?
2
?2/3?2
b2b3
(1 + b3n0)2−
mnn5/3
0
+ (b1− a1)n2
0+ 2
?
b2
1 + b3n0
−
a2
1 + a3n0
?
n3
0
−
a2a3
(1 + a3n0)2
?
n4
0,(10)
w0=
3
10
?3π2
2
?2/3?2
mnn2/3
0
+ a1n0+
a2n2
1 + a3n0,
0
(11)
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K0= −3
5
?3π2
2
?2/3?2
mnn2/3
0
+
18a2n2
(1 + a3n0)3,
0
(12)
0 =1
5
?3π2
2
?2/3?2
mnn−1/3
0
+ a1+
2a2n0
1 + a3n0
−
a2a3n2
(1 + a3n0)2.
0
(13)
Equation (13) comes from the fact that the density derivative of the energy
per nucleon at n = n0and α = 0 (x = 1/2) vanishes due to the saturation.
Note that the five relations (9)–(13) are not sufficient to determine the six
parameters a1,...,b3, and that the parameter b3, which controls the EOS
of matter at large neutron excess and high density, has little effect on the
saturation properties of nearly symmetric nuclear matter. We will thus set b3
as a typical value 1.58632 fm3, which was obtained by one of the authors [13],
and determine the other parameters from the empirical radii and masses of
stable nuclei.
We now proceed to describe a spherical nucleus of proton number Z and
mass number A within the framework of a simplified version of the extended
Thomas-Fermi theory [13]. We first write the total energy of a nucleus as a
function of the density distributions nn(r) and np(r) according to
E = Eb+ Eg+ EC+ Nmn+ Zmp,(14)
where
Eb=
?
d3rn(r)w(nn(r),np(r))(15)
is the bulk energy,
Eg= F0
?
d3r|∇n(r)|2
(16)
is the gradient energy with an adjustable constant F0,
EC=e2
2
?
d3r
?
d3r′np(r)np(r′)
|r − r′|
(17)
is the Coulomb energy, and N = A − Z is the neutron number. Here we
ignore shell and pairing effects. We also neglect the contribution to Egfrom
the gradient of the proton fraction x [see Eq. (16)]; this contribution makes
only a little difference even in the description of extremely neutron-rich nuclei,
as clarified in the context of neutron star matter [13].
5