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Equation of State of Nuclear Matter, Neutron Rich Nuclei in Laboratories, and Pasta Nuclei in Neutron Star Crusts K. Oyamatsu

Equation of State of Nuclear Matter, Neutron Rich Nuclei in Laboratories, and Pasta Nuclei in Neutron Star Crusts K. Oyamatsu

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408



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+---(n-no)+- 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 Hartree-Fock 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 neutron-rich 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 al-bz 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 P-stability 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 al-by 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 Hartree-Fock models with finite-range

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

non-relativistic phenomenological Skyrine Hartree-Fock 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 A-I in

Fig. 3. The EOS reasonably covers phenomenological EOS’s of contemporary use. For example, the SIII EOS in non-relativistic 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 A-I correspond to EOS A-I 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 (fm-3)

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 (fm-3)



n (fm-3)



Nine typical EOS's corresponding to points A-I 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 crust-core 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 crust-core 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 Thomas-Fermi

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 Hartree-Fock 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

neutron-rich 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

NUCLEAR-PASTA 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 319-1195, Japan



2 Department of Physics, Kyoto University, Kyoto 606-8502, Japan



3 Mosww Institute for Physics and Engineering, Kashirskoe sh. 31, Moscow

115409, Russia



4 Japan Society for the Promotion of Science, Tokyo 102-8471, Japan

5 BRS, Nihon University, Fujisawa, Kanagawa 252-8510, Japan

Using the density functional theory (DFT) with the relativistic mean field (RMF)

model, we study the non-uniform state of nuclear matter, “nuclear pasta”. We

self-consistently 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 low-density nuclear matter is the possibility of the existence of non-uniform structures, called “nuclear pastas” .l

At low densities, nuclei in matter are expected to form the Coulomb lattice embedded in the neutron-electron 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 low-density 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 hadron-quark deconfinement transition, kaon condensation

and liquid-gas 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 liquid-gas 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



Mean-field Model

To study the non-uniform 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



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