Equation of State of Nuclear Matter, Neutron Rich Nuclei in Laboratories, and Pasta Nuclei in Neutron Star Crusts K. Oyamatsu
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rodlike and spherical nuclear bubbles, occur. In this study, we attempt to
specify the most important EOS parameter that governs these properties.
The energy per nucleon near the saturation point of symmetric nuclear
matter is generally expressed as'
r;,
L
~ = w o +  ( n  n o ) ~ + So+(nno)+ K u s y m ( n  n 0 ) 2 ] Q2. (1)
18ng
3no
18ng
Here W O , no and KOare the saturation energy, the saturation density and
the incompressibility of symmetric nuclear matter. The neutron excess
is defined as cy = 1  2x using proton fraction x. The parameters SO
(the symmetry energy), L (the density symmetry coefficient) and Kusym
characterize the density dependent symmetry energy S ( n ) at n = no;
[
L
= 3no(dS/dn),,,,,
(3)
K~~~~~~
= 9n~(d2S/dn2),=,,.
(4)
From Eq. ( l ) ,the saturation density n, and energy w, of asymmetric
nuclear matter with fixed proton fraction are given, up to the second order
of a , by
One useful empirical parameter to characterize the saturation of asymmetric nuclear matter is the slope, y, of the saturation line near Q = 0
(x = 1/2).' It is expressed as
KOSO
y = 
3noL
(7)
'
2. Macroscopic nuclear model
In constructing a niacroscopic nuclear model, we begin with a simple expression for the bulk energy per n ~ c l e o i i , ~
w=
where
3h2(37r2)2/3
(ny3
10m,n
+ n y ) + (1

+
a2)v,(n)/n a%,(n)/n,
(8)
409
and
are the potential energy densities for symmetric nuclear matter and pure
neutron matter, and m, is the neutron mass. Here, replacement of the
proton mass mp by m, in the proton kinetic energy makes only a negligible difference. Equation (8) can well reproduce the microscopic calculations of symmetric nuclear matter and pure neutron matter by Friedman and Pandharipandd and of asymmetric nuclear matter by Lagaris
and Pandharipande4. Furthermore the expression can also reproduce phenomenological Skyrme HartreeFock and relativistic mean field EOS’s.
We determine the parameters included in Eqs. (9) and (10) in such a way
that they reproduce data on radii and masses of stable nuclei. In the limit
of n $ no and a + 0 (z
1/2), expression (8) reduces to the usual form
(1).2The 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 63 as a typical value
1.586 fm3,which was obtained by one of the authors5 in such a way as to
reproduce the neutron matter energy of Friedman and Pandharipz~nde.~
We describe a spherical nucleus of proton number 2 and mass number
A within the framework of a simplified version of the extended ThomasFermi t h e ~ r yWe
. ~ first write the total energy of a nucleus as a function of
the density distributions n,(r) and n p ( r ) according to
$
where the first, second and third terms on the right hand side are the
bulk energy, the gradient energy with an adjustable constant Fo, and the
Coulomb energy, respectively. The symbol N = A  2 denotes the neutron
number. Here we neglect the contribution to the gradient energy from
IV(nn(r) n P ( r ) ) l 2 ; this contribution makes only a little difference even in
the description of extremely neutronrich nuclei, as clarified in the context
of neutron star matter.5
For the present purpose of exanlining the macroscopic properties of
nuclei such as masses and radii, it is sufficient to characterize the neutron
and proton distributions for each nucleus by the central densities, radii and
surface diffuseness different between neutrons and protons, as in Ref. 5 .
410
We thus assume the nucleon distributions ni(r)(i = n , p ) , where r is the
distance from the center of the nucleus, as
(12)
r
2 Ri.
Here Ri roughly represents the nucleon radius, ti the relative surface diffuseness, and n? the central number density. The density of the outside
nucleon gas, npUt,is greater than zero for nuclei in the inner crust of a
neutron star while it is zero for laboratory nuclei. The proton distribution
of the form (12) can fairly well reproduce the experiniental data for stable
nuclei such as "Zr and '08Pb5
3. Optimal relations among EOS parameters
The EOS paranieters albz and Fo are determined from masses and radii
of stable nuclei in the same way as in Ref. 5 using the empirical values
for nine nuclei on the smoothed Pstability line ranging 25 5 A 5 245 (see
Table A.l in Ref. 5 , which is based on Refs. 6, 7). For fixed slope y and
inconipressibility K O ,such a comparison is made by a usual least squares
fitting, which gives rise to an optimal set of the paranieters alby and Fo.
Here, we set y and KOas 1800 MeV fni3 5 y I 200 MeV fni3 and 180
MeV 5 KOI 3 6 0 MeV; the numerical results for no, wo.So, L and Fo are
obtained for about 200 conibinations of y and KO. All of them reproduce
the input nuclear data almost equally.
As shown in Fig. 1, we find an empirical correlation between So and L,
So
M
B
+ CL,
(13)
with B M 28 MeV and C M 0.075. A similar result, B = 29 MeV and
C = 0.1, was obtained froni various HartreeFock models with finiterange
forces by Fariiie et al..8
The saturation energy of symmetric nuclear matter, wo,always takes
on a vdue of 16.0 it 0.5 MeV. There is a weak correlation between no
and KO as shown in Fig. 2. This is a feature which was found among
nonrelativistic phenomenological Skyrine HartreeFock EOS's (see Fig. 4
of Ref. 9).
In Fig. 3, the uncertainties in L and KOis represented as a band, which
reflects the constraint on (y, K O ) .In this band, L increases with increasing
41 1
40 
z
E
u?
30
+
Dresent results
11
 S0=27.8+0.075L
20
10 
n
0
I
I
I
50
100
150
L (MeV)
Figure 1. The empirical correlation between SOand L .
0.1 70+

r?
,f
v
0
0.165
I
I
* + +
0.160 

*$9*
0.145 
0.1 40
Figure 2.
I

+**
0.155 0.150 
I
* 8 $ $ f $2:

I
I
1
I
The empirical correlation between no and

KO
y for fixed KO. The upper bound (y = 200 MeV fni3) reaches a large
value of L, which increases with increase in h’o.
Figure 4 shows nine typical EOS’s that correspond with points AI in
Fig. 3. The EOS reasonably covers phenomenological EOS’s of contemporary use. For example, the SIII EOS in nonrelativistic theory is quite
similar to EOS I and TM1 EOS in the relativistic mean field is between
EOS B and C.
412
200
I
I
I
I
y=350 [MeV fm3]

Slll
I
50
I
I
250
200
350
300
KO (MeV)
Figure 3. The optimum ( L ,K O )values. The symbols AI correspond to EOS AI in
Fig. 4.
::
0
h
%P
r
v
to
P
0
I
x
I . : : " . " .
s:
I
~ . . ' " ' ' . ' , .:: :
: y=1800 M e V fmg
81
KO= 180 M e V
I t , ,
o
. ,
I
0.1
0.06
n (fm3)
Figure 4.
0.16
, , I::::.::::I::::I::::I:
..
I . . . . , . . . . + . . . . . I.
,
I..
:: y=1800 M e V fm3
:. b=230 M e V
o
0.06
::
I
: : h
y=1800 M e V fms
:: &=380 M e V

I 
...,....,...,+....,...,,,,.,,...d
0.1
a.16
o
am
at
0.16
n (fm3)
n (fm3)
Nine typical EOS's corresponding to points AI in Fig. 3
41 3
4. Neutron rich nuclei in laboratories
Using the systematically obtained EOS's, we calculate radii and masses of
neutron rich nuclei, and examine how they depend on the EOS parameters
in Eq. (1). Figvres 5 and 6 show that the mass and matter radius of
neutron rich "Ni ( Z / A = 0.35) have clear dependence on L. Some points
in Fig. 6 scatter appreciably due probably to small numerical errors in wo
but the neutron separation energy shows clearer dependence on L.
4.32
4.30
4.28
4.26
4.24
4.22
4.20
4.18
40
80
L (MeV)
Figure 5.
I
I
40
120
Matter radius of 80Ni.
I
I
I
80
120
160
L (MeV)
Figure 6.
160
Mass excess of ""i
I
41 4
5. Nuclei in neutron star crusts
A good estimate of the crustcore boundary density is obtained from a
stability analysis against, proton c1ustering.l' In this analysis, we calculate
the energy change when small variations of particle densities are imposed
on a uniform liquid of protons, neutrons and electrons. The estimate of the
boundary density, n(Q),is defined as the lowest density where the system
is stable against the small density modulations. Figvre 7 shows that n(Q)
is obviously dependent on L.
In the inner edge of a neutron star crust, there may or may not exist
pasta nuclei (rodlike and slablike nuclei and rodlike and spherical nuclear
bubbles). The existence is conjectured to depend on the asymmetric nuclear
matter EOS. We calculate pasta nuclei using the nine typical EOS's in Fig.
4 to see this dependence. The result of a preliminary calculation is shown
in Fig. 8. From Figs. 4 and 8, we see that the density range becomes
narrower as the L value increases. For EOS C, which has extremely large
L and KO,the density range disappears so that pasta nuclei do not exist.
0.141
I
I
I
boundarv densitv between crust and core
I
0.12
0.04 I
0
I
50
I
100
L (MeV)
I
+
150
Figure 7. An estimate of the crustcore boundary density
6. Summary
About 200 sets of the EOS parameters are systematically obtained from
fitting to masses and radii of stable nuclei using a simplified ThomasFermi
model paying attention to large uncertainties in KOand L values.
41 5
A 
B
c
DE
X
c.D(
X
onset of proton clustering
’

F
H
X
G I
0.04
I
0.06
0.08
0.10
0.12
0.14
nucleon number density ( f ~ n  ~ )
Figure 8. The density range of pasta nuclei
As for symmetric nuclear matter, the saturation density no has a weak
KOdependence while the saturation energy, WO, is essentially constant.
There is a strong correlation between SO and L : So = 28 + 0.075L
(MeV). Although the L value can not be determined from stable nuclei,
the empirical upper bound of L is found to be an increasing function of
KO. This is a consequence of the empirical constraint on the slope of the
saturation line. Taking the uncertainties in L and KO into account, we
systematically obtain various asymmetric matter EOS’s which reasonably
cover typical Skyrme HartreeFock EOS’s and relativistic mean field EOS’s.
For neutron rich nuclei in laboratories, we find that the matter radii
and masses have clear L dependence almost independent of KO.For nuclei
in a neutron star crust, the boundary density between the core and the
crust has clear L dependence. The density range of pasta nuclei becomes
narrower with L, and vanishes with the largest L value. From these results,
we conclude that future systematic nieasurenients of the matter radii of
neutronrich nuclei could help deduce the L value, which in turn could give
useful information about the presence of pasta nuclei in neutron star crusts.
41 6
References
1. K. Oyamatsu, I. Tanihata, Y . Sugahara, K. Sumiyoshi, H. Toki, Nucl. Phys.
A 634 (1998) 3.
2. K. Oyamatsu and K. Iida, Prog. Theor. Phys. 109 (2003) 631.
3. B. Friedman, V.R. Pandharipande, Nucl. Phys. A 361 (1981) 502.
4. I.E. Lagaris, V.R. Pandharipande, Nucl. Phys. A 369 (1981) 470.
5. K. Oyamatsu, Nucl. Phys. A 561 (1993) 431.
6. M. Yamada, Prog. Theor. Phys. 32 (1964) 512.
7. H. de Vries, C.W. de Jager, C. de Vries, At. Data Nucl. Data Tables 36 (1987)
495.
8. M. Farine, J.M. Pearson, B. Rouben, Nucl. Phys. A 304 (1978) 317.
9. J.P. Blaizot, Phys. Rep. 64, 171 (1980).
10. C.J. Pethick, D.G. Ravenhall, C.P. Lorenz, Nucl. Phys. A 584 (1995) 675.
COULOMB SCREENING EFFECT ON THE
NUCLEARPASTA STRUCTURE
TOSHIKI MARUYAMA~,TOSHITAKA TATSUMI~,
DMITRI N. VOSKRESENSKY3, TOMONORI TANIGAWA4?l,
SATOSHI CHIBA’, TOMOWKI MARUYAMA’
1 Advanced Science Research Center, Japan Atomic Energy Research Institute,
Tokai, Ibaraki 3191195, Japan
2 Department of Physics, Kyoto University, Kyoto 6068502, Japan
3 Mosww Institute for Physics and Engineering, Kashirskoe sh. 31, Moscow
115409, Russia
4 Japan Society for the Promotion of Science, Tokyo 1028471, Japan
5 BRS, Nihon University, Fujisawa, Kanagawa 2528510, Japan
Using the density functional theory (DFT) with the relativistic mean field (RMF)
model, we study the nonuniform state of nuclear matter, “nuclear pasta”. We
selfconsistently include the Coulomb interaction together with other interactions.
It is found that the Coulomb screening effect is significant for each pasta structure
but not for the bulk equation of state (EOS) of the nuclear pasta phase.
1. Introduction
One of the most interesting features of lowdensity nuclear matter is the possibility of the existence of nonuniform structures, called “nuclear pastas” .l
At low densities, nuclei in matter are expected to form the Coulomb lattice embedded in the neutronelectron seas,so as to minimize the Coulomb
interaction energy. On the other hand, another possibility has been discussed: the stable nuclear shape may change from sphere to cylinder, slab,
cylindrical hole, and to spherical hole with increase of the matter density,
and “pastas” are eventually dissolved into uniform matter at a certain nucleon density close to the saturation density, ps = 0.16 fmP3. The existence
417
418
of such “pasta” phases, instead of the ordinary crystalline lattice of nuclei,
would modify several important processes in supernova explosions by changing the hydrodynamic properties and the neutrino opacity in the supernova
matter. Also expected is the influence of the “pasta” phases on star quakes
of neutron stars and pulsar glitches via the change of mechanical properties
of the crust matter.
Several authors have investigated the lowdensity nuclear matter using
various models.
Roughly speaking, the favorable nuclear shape is determined by a balance between the surface and the Coulomb energies, as has
been shown by previous studies, where the rearrangement effect on the density profile of the charged particles, especially electrons, by the Coulomb
interaction is discarded. However, the proper treatment of the Coulomb
interaction should be very important, as it is demonstrated in Ref. 9; the
screening of the Coulomb interaction by the charged particles may give a
large effect on the stability of the geometrical structures.
We have been recently exploring the effect of the Coulomb screening in
the context of the structured mixed phases in various first order phase transitions such as hadronquark deconfinement transition, kaon condensation
and liquidgas transition in nuclear matter. We treat the nuclear “pasta”
phases as a part of our project, since they can be considered as structured
mixed phases during the liquidgas transition in nuclear matter.
Our aim here is to study the nuclear “pasta” structures by means of a
mean field model, which includes the Coulomb interaction in a proper way,
and we figure out how the Coulomb screening effect modifies the previous
results without it.
2. Density Functional Theory with the Relativistic
Meanfield Model
To study the nonuniform nuclear matter, we follow the density functional
theory (DFT) with the relativistic mean field (RMF) model.1° The RMF
model with fields of mesons and baryons is rather simple for numerical
calculations, but realistic enough to reproduce main nuclear matter properties. In our framework, the Coulomb interaction is properly included in
the equations of motion for nucleons, electrons and the meson mean fields,
and we solve the Poisson equation for the Coulomb potential VcOulselfconsistently with them. Thus the baryon and electron density profiles, as
well as the meson mean fields, are determined in a way fully consistent with
the Coulomb potential. Note that our framework can be easily extended to