ELSEVIER

1 December 1994

Physics Letters B 340 (1994) 1-5

PHYSICS LETTERS B

Nuclear symmetry energy and structure of dense matter in neutron stars

M a r e k K u t s c h e r a H Nlewodntczar~skt Instttute of Nuclear Physws, ul Radzzkowsktego 152, 31-342 Krak6w, Poland

Received 10 May 1994; revised manuscript recewed 23 September 1994 Editor' C. Mahaux

Abstract

Consequences for neutron star matter of the behaviour of symmetry energy which results in disappearing of protons at high densities are explored. It is shown that interactions responsible for disappearance of protons tend to separate protons and neutrons at lower densities. Two separation mechanisms are considered: A bulk separatmn of protons and neutrons and formation of a neutron bubble around a single proton. The latter one corresponding to trapping of protons in the neutron background bubbles is more hkely to occur in neutron star matter. In this case protons form polarons which are localized.

The nuclear symmetry energy Es(n) (strictly its interaction part) determines the proton concentration x (n) in the neutron star matter. There exists some con- troversy as far as high density behaviour of both quan- tities is concerned. Various equations o f state (EOS) predict a similar behaviour of Es (n) and x (n) at lower densities. In particular, they reproduce the empirical value of the symmetry energy at saturation density, Es (no) = 34 4- 4 MeV [ 1 ]. At higher densities, how- ever, calculations based on a relativistic mean-field ap- proximation predict that x(n) increases with density [2] whereas more sophisticated many-body calcula- tions [3,4] predict that x(n) = 0 at high densities.

This controversy should be resolved in favour o f the latter possibility [ 3 ]. Realistic nuclear matter calcula- tions [3,4] show that the symmetry energy becomes negative at high densities and this leads to disappear- ance of protons, x(n) = O.

In this paper the physical origin of this behaviour of E~ (n) is briefly discussed and its consequences for neutron star matter are explored. It is shown that the

interactions which make the proton fraction vanish- ing, are likely to induce density inhomogeneities at lower densities. Nature of these inhomogeneities and their influence on the neutron star matter is studied. Throughout this paper units are such that h = c = 1.

The proton fraction x of r -s table nucleon matter, neglecting muons and the electron rest mass, is given by the equation

(37r2nx) 1/3 ---- tzN(n, X) -- IXp (n, x) , (1)

where/ZN and tZp is the neutron and proton chemical potential, respectively. The right-hand side (rhs) of Eq. ( 1 ) is closely related to the symmetry energy, as is shown below.

The energy per particle of nucleon matter can be parametrized as [ 5 ]

E ( n , x ) = T (n , x ) + Vo(n) + ( 1 - 2x)aV2(n), (2)

where T ( n , x ) is the kinetic energy contribution; V0 and V2 are interaction contributions. The nuclear sym-

0370-2693/94/$07 00 (~) 1994 Elsevier Science B V. All rights reserved SSDI 0370-2693(94)01266-0

M. Kutschera / Physws Letters B 340 (1994) 1-5

metry energy is

e , ( n ) = ½) + (3)

and the difference of proton and neutron chemical po- tentials is

tZN(n, X) -- tZP (n, x)

= g ~ (3~rr2n)2/3[ (1 -- x)2/3 _ X2/3 ]

+ 4 ( 1 - 2x) ½(n ) . (4)

Thus, both the symmetry energy, Es(n), and proton fraction, x(n) , of fl-stable matter are determined by the function ½(n ) which measures deviation of the interaction energy of pure neutron matter from that of symmetric nuclear matter.

The general behaviour of Es(n) is similar for real- istic nucleon interactions [ 3,4 ] : Es (n) increases near no then saturates at a few times no and finally decreases reaching negative values at high densities where the interaction term ½(n ) becomes negative. The proton fraction vanishes for sufficiently negative ½ (n).

In mean-field models of neutron star matter [2] the interaction contribution to the symmetry energy is V2(n) = Sn. The coefficient S is positive and hence the symmetry energy increases monotonically from its empirical value at no. The reason of the discrepancy is neglecting of tensor forces in the mean-field approx- imation [6]. Since these play a crucial role in deter- mining ½ ( n ) the mean-field result is not to be trusted at high density.

The physical mechanism of disappearance of pro- tons from the neutron star matter was explained long ago by Pandharipande and Garde [ 3 ]. They observed that it is due to greater short-range repulsion in isospin singlet nucleon pairs than in isospin triplet pairs.

An important result of this behaviour of Es(n) is that the proton fraction of the neutron star matter is low at all densities [ 3,4]. It eventually vanishes at a density no such that rhs of Eq. (4) vanishes at no for x -- 0. For this to occur ½(nv) must be negative. Maximum proton fraction found e.g. in Ref. [4] is 12% for the UV14 + UV/I EOS and 5% for the UV14 + TNI EOS. One may conclude that protons are a dilute component of the neutron star matter and become impurities close to no. As is shown below interesting effects can occur in this case.

The behaviour of the symmetry energy suggests that already at lower densities, where the protons are still present, the energy of nuclear matter can be lowered by separating protons from neutrons. This diminishes the interaction contribution due to stronger proton- neutron repulsion. Such a separation can be achieved in two ways, either by bulk separation of protons and neutrons or by producing a neutron bubble around each proton. In the former case the neutron star matter is unstable with respect to formation of domains with high proton concentration immersed in pure neutron matter. In the latter case protons behave as polarons in solids which are localized by potential wells due to crystal deformation.

To explore these possibilities a simple second-order polynomial form of the function V2(n) is used. V2(n) is required to fit the experimental value of E~(n0) and to display the same high density behaviour as found in realistic calculations, i.e. V2(nz ) = 0 at some density nz (nz < no) and ½(n) < 0 for n > nz. The value of nz, however, depends on a model of nuclear interactions [3,4] and we shall treat it as a parameter. As an example we use the function Vo(n) = 246.5n 2 - 108n-26.4 which fits tabulated values for the UV14+ TNI interactions from Ref. [4]. Here n is in fm -3 and all functions are in MeV. Calculations below are performed with ½(n ) -- -181.9n 2 + 99n + 6.1 for nz = 0.6fm -3 and ½(n) = -138.8n 2 + 87.5n + 6.7 for nz = 0.7 fm -3.

The condition of bulk separation instability can be obtained by performing a standard analysis of small proton and neutron density oscillations [7]. It is dis- cussed in Ref. [6]. Existence of such an instability is rather obvious from Eq. (2) which shows that at densities n > nz the sum of interaction terms I,~ + (1 - 2x)2½ has the lowest value for x = 0. When n > no pure neutron matter can coexist with pure pro- ton matter (neglecting Coulomb interactions) since t ze (n ,x = O) > tzp(n,x = 1) and IzN(n,x = O) < IzN(n,x = 1) = I~e(n,x = 0).

In case of fl-stable nuclear matter the separation instability means that pure neutron matter coexists with nucleon matter containing some proton fraction Xc. The coexistence requires that the pressure in both phases is the same, PN(nN,X = O) = Pc(nc, Xc), the neutron chemical potential is also the same, tzN(nN,x = O) = tzN(nc,Xc), and the proton chemi- cal potentials satisfy the inequality/xj,(nN, x = 0) >

M Kutschera/Phystcs Letters B 340 (1994) 1-5 3

4001" (a) ? ~_.550 82

soo

2 5 0 t

i .......... 1500 25 ' ' 0 5 0 X

o 2 nN(frn -'~)

A074! c ?~_E 0 71 v

080 085 090 095 1 O0 nN(fnq -3)

Fig I. (a) The chemical potential of protons (dashed line) and neutrons (solid line) as functions of proton fracnon x for indicated values of pressure (in MeV fm-3). (b) The proton fraction of nuclear matter in coexistence with neutron matter as a function of neutron matter density for the same equation of state as in Fig. la. (c) The density of nuclear matter in coexistence with neutron matter as a functaon of neutron matter density corresponding to the proton fraction in Fig lb

/xp (no, Xc). The last condition ensures that protons do not diffuse into neutron matter. In the neutron star matter separation instability occurs at some critical pressure Pl for which the coexistence conditions are satisfied for the first time.

The coexistence conditions are implemented in a simple way by constructing isobars [7] which are plots of neutron and proton chemical potentials as functions of proton fraction x under constant pressure, Fig. la. For a given pressure, P, the density of neu- tron matter, n~, and the neutron chemical potential, tzN(nN, X = 0), are fixed. First two coexistence con- ditions can be solved to give the baryon density, n~, and the proton fraction, Xc, of nucleon matter which is in equilibrium with neutron matter at this pressure. The isobar plots, Fig. la, are then used to check if the proton chemical potentials satisfy the above inequal- ity. P1 is the lowest value of pressure for which the last condition is satisfied.

In Figs. la, lb and lc results corresponding to nz = 0.6fm -3 are shown. The critical pressure is P1 = 100 MeV fm -3. The neutron matter of this pressure has a density nN = 0.75 fm -3 and coexists with nu- clear matter of density nc = 0.69 fm -3 and proton frac- tion Xc = 0.095. For higher pressure neutron matter coexists with nuclear matter of higher proton fraction, Xc >~ 0.095.

One can estimate bulk properties of the inhomoge- neous phase assuming uniform electron distribution and neglecting surface effects. From Eq. ( 1 ) the elec- tron density, ne, is obtained. Overall charge neutrality

gives the fraction of volume, a, occupied by proton clusters, a = ne/Xcnc, from which the average pro- ton fraction, 2 ~ CtXcnc/nN, is obtained. For P1 the average proton fraction is $ ~ 0.002. With increas- ing pressure the value of x¢ increases while the proton fraction :~ decreases. Protons disappear (i.e. ~ = 0) at pressure P2 -- 182MeV fm -3. Corresponding neutron matter density is nN = 0.96 fm -3 and parameters of nuclear matter are Xc = 0.5 and nc = 0.76 fm -3. Thus, in a pressure range PI < P < P2 the neutron matter coexists with proton clusters. The proton fraction, xc, and nucleon density, no, of proton clusters as functions of coexisting neutron matter density, nN, are shown in Figs. lb and lc, respectively. For other values of nz we find a general behaviour that P1 increases with nz whereas 2 decreases.

The two-phase core exists inside neutron stars for which the central pressure eeent is such that e l <~

/°cent < P2- For neu t ron stars wi th ecent > P2 the two-

phase matter forms a shell with thickness generally decreasing as/°cent increases. For a given equation of state (i.e. given nz ) one finds the radius of the core and thickness of the shell as functions of the neutron star mass. This subject is discussed elsewhere.

The bulk separation instability may be physically relevant in case of relatively low values nz < 0.6 fm -3 and for rather soft l,~(n). For higher values of nz and harder 1~(n), a different mechanism of separation of protons and neutrons is more likely to operate. It is based on the observation [ 8 ] that for protons at densi- ties slightly lower than nv, which can be treated as im-

M Kutschera/Physics Letters B 340 (1994) 1-5

purities in the neutron matter, the coupling to phonons becomes important. It can affect strongly the effec- tive mass of the proton. This is because the relevant energy scale for the impurity is its vanishing kinetic energy. Thus any phonon contribution can change the effective mass significantly.

The coupling of a proton impurity to phonons can be derived [8] by considering the energy of a slow proton (k ~ 0) in a uniform neutron matter which is

k 2 Ep(k) = ~,m, + / z p ( n , 0 ) . (5)

Allowing for small oscillations of neutron density n + 6n(r , t ) we find [8] the proton-phonon in- teraction Hamiltonian which has the form of the deformation-potential coupling in solids [9] with a density-dependent coupling constant

0/t/,p ~(n) = n - - (6)

This coupling constant increases with density, Fig. 2 in [ 8 ], and at n ,-~ no its values are in the strong coupling regime. The functions o-(n) for different values of nz are very close to each other.

The deformation-potential electron-phonon cou- pling is responsible for formation of polarons in cova- lent crystals [9]. For strong coupling this interaction is known to localize electrons which form small po- larons [ 10]. A similar effect is expected to occur for a proton impurity: At sufficiently high density, i.e. for high enough o-, it should be localized.

We have studied possibility of localization of a pro- ton impurity in a Thomas-Fermi approximation for a few simple models of nucleon matter which display vanishing of x(n) at high densities [8,11 ]. We have found that localizations can occur at nl -'~ 1 fm -3. This result is, however, model-dependent. It is sensitive to the value of the proton effective mass m. due to nu- clear forces which is not well known at high densities. The localization corresponds to trapping of a proton in a potential well due to a bubble in the neutron back- ground distribution induced by the proton.

From analysis of localization of a single proton one can draw important conclusions for neutron star matter containing a few percent proton fraction. The deformation-potential coupling tends to localize the proton inside a bubble of the volume Vp = A/n with

0 05

0 O0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 200 . 500 400

P(M~V/f~ °) Fig 2 The proton fraction as a function of pressure Solid hne is for umform nuclear matter and dashed lines are for neutron matter with localized protons. Short- and long-dashed lines correspond respectavely to A/~l and A/~2

A ,-~ 1 [ 10]. For low proton concentration, x(n) be- low 10%, we do not expect the wave functions of individual protons to overlap. Thus the neutron star matter at such densities that the deformation-potential constant or is in the strong coupling regime is com- posed of neutrons with localized protons which look like raisins in the pudding.

The chemical potential of the localized proton is lower than that of a plane-wave proton,

/ / , loc(n) = t z p ( n , 0 ) + Alz(n). (7)

This formula is valid also at low proton fraction, x < 0.1, as long as the wave functions of individual pro- tons do not overlap. In Ref. [ 8] we have calculated the correction A/~(n) for the Friedman-Pandharipande- Ravenhall EOS [ 12,13] which is essentially the same as the one used here with nz = 0.7 fm -3. The function A/.t(n) is well approximated by a second-order ex- pression. We use here two parametrizations, Atz(n) l = -570.7n 2 + 4 5 7 . 4 n - 58.9 and A/z(n)2 = -303.1 n 2 + 147.7n + 46.6, corresponding to curves labelled 1 in Fig. 8 of Ref. [8] which differ in the gradient term contribution. The chemical potential of the localized proton, /Zinc, has a weaker density dependence than /zp (n, 0) which allows localized protons to survive up to higher densities. In Fig. 2 we show the proton frac- tion x (n) for both parametrizations of A/z (n). Local- ized protons survive to higher pressure than those in uniform and bulk-separated matter.

Localized protons in a neutron star core could form a lattice due to the Coulomb interaction at low tem- perature. The lattice energy WL ~ e2(nx) 1/3 for lo- calized protons is of the order of 0.5 MeV. The melt- ing temperature can be estimated [ 14] to be of order 10 keV. Thus, a nuclear solid phase can exist inside cores of sufficiently cool neutron stars. Formation and

M Kutschera/Physics Letters B 340 (1994) 1-5

presence o f a sol id core can be impor tant for neutron

star quakes.

The loca l ized pro ton state can affect s ignif icantly

magne t ic proper t ies o f the neutron star mat ter [ 15].

In particular, there can exist a magne t ic instability,

as the spins o f loca l ized protons can be a l igned with

no energy cost. Magne t i c phase o f the neutron star mat ter cou ld contr ibute to the magnet ic m o m e n t o f a

neutron star [ 16].

This w o r k was par t ia l ly supported by K B N grants

2 0204 91 01 and 2 0054 91 01.

References

[1] W.D. Myers, At. Data Nucl. Data Tables 17 (1976) 411; H.V. Groote, E.R. Hilf and K Takahashi, ibid. 17 (1976) 418, P.A Seeger and W.M. Havard, ibid. 17 (1976) 428, M Bauer, ibid. 17 (1976) 442; J. Janecke and B.P. Eynon, ibid. 17 (1976) 467.

[2] C Horowltz and B Serot, Nucl. Phys. A 464 (1987) 613,

N.K. Glendenning, Z Phys. A 326 (1987) 57; 327 (1987) 295, B.D. Serot and H Uechi, Ann. Phys (NY.) 179 (1987) 272.

[ 3 ] V.R. Pandharipande and V K Garde, Phys Lett. B 38 (1972) 608.

[4] R.B. Wiringa, V Fiks and A. Fabrocini, Phys Rev C 38 (1988) 1010.

[5] I.E. Lagaris and V.R. Pandharipande, Nucl Phys A 369 (1981) 470

[6] M. Kutschera, Z. Phys. A 348 (1994) 263 [7] G. Baym, H.A. Bethe and C J. Pethick, Nucl. Phys A 175

(1971) 225 [8] M. Kutschera and W. W6jclk, Phys Rev C 47 (1993) 1077. [9] J. Bardeen and W Shockley, Phys. Rev. 80 (1950) 72.

[10] Y. Toyozawa, Prog Theor Phys. 26 (1961) 29 [11] M. Kutschera and W. W6jclk, Acta Phys. Pol. B 21 (1990)

823. [12] B. Friedman and VR Pandhanpande, Nucl Phys A 361

(1981) 502 [13] J.M. Lattimer, Ann. Rev. Nucl. Part. Sci. 31 (1981) 337 [ 14] W.L. Slattery, G.D. Doolen and H.E. Dewitt, Phys. Rev A

21 (1980) 2087. [15] M. Kutschera and W. W6jcik, Phys. Lett. B 223 (1989) l l [16] M. Kutschera and W. W6jcik, Acta Phys. Pol. B 23 (1992)

947.