SUPERDENSE DEGENERATE PLASMA

Go S~ Saakyan and L. Sh. Grigoryan

It is shown that in degenerate matter at densities p > 3.4.1010 g/cm 3 the phenom-

enon of neutronization must be replaced by pionization. With increasing density, the relative number of negative pions in nuclei increases monotonically, whereas the concentrations of neutrons and protons are frozen at the values 0.6 and 0.4. At density 6.1011 g/cm 3 and pressure 6.6,1029 erg/cm 3 there is a phase transition

to the state of continuous nuclear matter. In this phase, the numbers of neu-

trons, protons, and pions are equal to 0.591N, 0.409N, and 0.406N, where N is the number of nucleons, and the state is of the liquid type: the matter is in

a bound state (the chemical potentials of the nucleons are negative) and is in- compressible to pressures ~5.1033 erg/cm 3. A hadron plasma at densities above

the nuclear density is considered qualitatively. An equation of state for a

degenerate plasma is obtained in the complete range of densities.

I. Introduction

During the last two decades, the theory of superdense celestial bodies has been trans-

formed into one of the most important branches of astrophysics. This was due especially to

the discovery of pulsars. The idea of superdense celestial bodies was put forward for the

first time by Landau [I]. The theory of white dwarfs was developed in its essentials by

Chandrasekhar [2] and others [3-5]. In the thirties, Baade and Zwicky [6] and Oppenheimer and Volkoff [7] developed the idea of neutron stars -- configurations consisting predominantly

of neutrons. The discovery of hyperons and other hadrons after 1950 and also the publication

of Ambartsumyan's cosmogonic conception in 1958 [8] served as powerful stimuli for intensive

in~estiga~ions in the theory of superdense celestial bodies [9-12]. A new important stage

in this field was the discovery of the phenomenon of pionization of matter. It was Migdal

in [13] who first drew attention to the role of n mesons in nuclei. Later, in the series of papers [13-18] tbe formation of a pion condensate in infinite nuclear matter was investigated.

�9 n [19] it was established that pionization of matter in fact begins at densities which are

between three and four orders lower than the nuclear. From a limiting energy of the electrons

of order of a few MeV, nuclei in a plasma are populated by w- mesons. The occurrence of the

--mesons is advantageous since it slows down the comparatively rapid growth of the limiting

electron energy and thereby reduces the energy of the system. It is noteworthy that as a result of the appearance of the pions the neutronization of matter comes to a halt at proton

and neutron concentrations equal to 0.4 and 0.6, respectively.

In this paper~ we develop our earlier paper [19]. We have succeeded in making a number

of i~uprovements which~ we believe, may lead to important astrophysical consequences.

2. Improved Weizs~cker Formula

A correct investigation of the state of the pion condensate in nuclear matter is dif-

ficult because it requires good knowledge of the nucleon--nucleon, pion--pion, and nucleon--

pion interactions. The problem is further complicated by the fact that many-particle inter-

actions are here also important. In [20], this complicated theoretical investigation of the

question was put aside in favor of a phenomenological treatment. For a number of particular

reasons, heavy nuclei with A > 200 must contain not only the neutral meson background in-

vestigated in [13] but also negative pions. Introducing into the well-known Weizs~cker

formula corresponding terms to take into account the presence of the z- mesons in the nuclei

and then, by the method of least squares, fitting the theoretical formula to the experimental

nuclear binding energies, we determined all the coefficients in the improved mass formula:

Mc ~ - N.m,,c 2 + N .moC coA + ctA 2" -!- c~ (No i

Erevan S t a t e U n i v e r s i t y . T r a n s l a t e d from A s t r o f i z i k a , Vol . 13, No. 4, pp. 669-684 , October~December~ 1977. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 11, 1977.

396 0 5 7 1 - 7 1 3 2 / 7 7 / 1 3 0 4 - 0 3 9 6 5 0 7 . 5 0 �9 1978 Plenum Publishing Corporation

m c3 I N , - - ( N , - - A~) ]'~ q- c , [N,, - - ( N , - - N , ) ] 4

A A 3 (1)

where Nn, Np, and Nn are, respectively, the numbers of neutrons, protons, and v- mesons, and A = N n + N O is the mass number. The Coulomb term is proportional to (N n -- Nv) 2 since the charge of the nucleus is determined by the protons and v- mesons, The Nv protons that in

accordance with the p-wave pion--nucleon interaction at low energies can be regarded as sur- rounded by a meson cloud are clearly in some sense not identical with the remaining Np -- Nv protons. Therefore, they are not taken into account in the sixth and seventh terms, which represent the symmetry energy of the nucleons. Their exchange energy is separately repre- sented by the eighth term. The last term in (1) is the total energy of the pions.

According to our ideas, nuclei with A < 200 contain no negative pions. In this region, formula (I) without the additional terms satisfactorily describes the binding energy of a

nucleon as a function, of A and Z. Therefore, for the coefficients c0-c 3 we adopted the well- known values from the literature:

co = 15.75; c~ = 17.8; c,, -- 0 .71; ca = 23 .7 MeV, ( 2 )

The coefficient c 4 of the additional symmetry energy term was first determined from the

binding energies of nuclei with 50 ~ A ~ 257. The pion parameters c~ and c~ were determined

from 200 nuclei with 200 ~ A ~ 257. The value of c 4 was improved in the same process. This

yielded

c 4 . . . . 3 .5; c ; - - 17 .65; c= = 1 ] . 9 6 MeV. ( 2 ' )

The rms error in the binding energies of a nucleon calculated in accordance with formula (I)

for these values of the parameters Ck, c3, and c~ in the considered region is approximately an order of magnitude greater than the experimental errors. If it is assumed that all terms

in (i) make approximately the same contribution to (Ab) 2, then

~c0 = • 0~00i ; ~c l = 0 .007 ; ~c.. = • 0 ,0002; lc~. = • 0 .02; ( 3 )

~c4 + 0 .57 ; 5c~ = = 2 .05; Ac_ = • 0 .19 meV

T a k i n g f o r m u l a ( 1 ) a s b a s i s , o n e c a n s e e t h a t n u c l e i w i t h A 5 2 0 0 c o n t a i n n o m e s o n s [ 2 0 ] . B u t i n n u c l e i w i t h A > 2 0 0 , t h e n u m b e r o f ~ - m e s o n s i n i s o b a r s w i t h t h e l a r g e s t Z i s c o m p a r a t i v e l y s m a l l w h i l e t h e i r n u m b e r i n c r e a s e s w i t h d e c r e a s i n g Z , r e a c h i n g 5 - 7 p a r t i c l e s i n t h e i s o b a r s w i t h t h e s m a l l e s t Z . F o r e x a m p l e , f o r t h e g r o u p o f n u c l e i w i t h A = 2 3 0 a n d Z = 9 3 , 9 2 , 9 1 , 9 0 , 8 8 , we h a v e N~ = 1 , 2 , 3 , 4 , 7 , a n d f o r A = 2 5 4 a n d Z = 1 0 2 , 1 0 1 , 1 0 0 ,

9 9 , 9 8 , we h a v e N~ = 2 , 4 , 4 , 5 , 6 .

3. Pionization in a Degenerate Plasma

In our preceding paper [19] it was shown that pionization of the nuclei takes place in

a degenerate Ae plasma. Under ordinary conditions, for the stability of u- mesons it is

necessary that their chemical potential does not exceed a value of order meC2. This condi-

tion is satisfied only in the isobars of heavy nuclei. In a degenerate plasma, the conditions

for stability of negative pions are more favorable since their number is determined from the

condition that the chemical potential of these particles be equal to the limiting energy of

the electrons: ~ = U e. In [19], the pionization effect was investigated by a modification

of the Weizs~cker formula on the basis of general theoretical considerations. In returning

now to this question, we shall proceed from the semiempirical formula (1). The parameters in it can be regarded as more reliable than ones obtained by theoretical calculations since

they are essentially determined from the experimental data on nuclear binding energies.

We consider the ground state of an Ae plasma consisting of a degenerate electron gas

and identical atomic nuclei with the largest values b(A, Z) of the binding energy of nucleons.

The parameters A, Z, and b(A, Z) of these nuclei are determined by the limiting energy of

the electrons, i.e., they are definite functions of the density. The state of such a plasma

is determined by the system of equations

~. = P. + I"~, p: = p~, Mc ~ = N,,I. . + N . p . + N : ~ . -- A p . - - Z:~ , ( 4 )

3 9 7

where ~k is the chemical potential of the particles. Of course, the results obtained below are also valid at nonzero temperatures if these temperatures do not exceed the degeneracy

temperatures of the particles~ The system (4) must be augmented by the condition of elec- tricai neutrality o~ the plasma:

Z n , = --:-. n , (5)

A

where ne is the density of electrons, n is the number of nucleons per unit volume, n/A is the density of the nuclei, and Z = Np -- Nw.

From ( ! ) we find.

2C l c~ g~A2j3 -1- 2C 3 (! - - 2 g - - g~) - ~, = m ~ - - Co + .3A l ia 3

- - c , (1 2 v y,~)2 _}_ 4 c a (1 - - '2 9 - - y,:.)3 _ _ 3 c , (1 - - 2 y �9 ) ' c ' ' 2 - - - - H r . " - - 3 5 x

~ = m c ~ - - c o ~- 2c----L ~- 2c~,gA ~-rs - - 2~- g2A~/3 - - 2 c 3 ( 1 - - 2g - - g ~ ) - - c~ (1 - - 2 g - - y = ) o _ P ' 3 A ' t 3 ' 3

(6)

- - 4 c 4 ( i - - 2 g g~)3--3c4(1--2y--g~)'--c'~ sY,,2

~ = - - 2 c d ] A ~ 1 3 + G + 2ca(1 2 y - - y , , ) - / - 4 c 4 ( l - - 2 g - - y ~ . ) * "-- 2c'zy,~.

I t e r e y = Z/A, and y ~ = N~/A.

where a = 6 . 1 1 4 5 ~ - I ~ MeV.cm.

The chemical potential o f the electrons is equal to

~e G~ I3 .1/3 = = a (yn) , ( 7 )

Substituting (6) and (7) in (4), we arrive at the result

TABLE 1o Parameters of the Ground State of a Superdense Degenerate Plasma

P ( g/era 3 )

6.70I I06

7,289 107

2.723 lOS

! ,376 109

3,952 109

.8.669 10 ~

1,623 i0 t~

2.736 101~

13.388 10 TM

3,574 10 t~

6.742 I0 re

1.138 101I

1,778 101I

2.621 101I

3.697 :IO n

5.032 t0 u

5.806 10 u

6.052 10 ll

2 .84t 10 '4

P ( e r g / c m 3)

5,444 1023

1.292 1025

7.374 1025

6,199 1056

2,450 1027

6.756 1027

1.507 1028

2.922 10 2a

3:818 2028

4.061 10 ~s

8.319 i02e

1:462102~

2.300 1029

31330 1059

4.511 t02~

5.784 l052

6.432 1029.

6.627 I0 ~9

A [ Z N,- -N= IA A

62

63

65

0,450

0.445

0.440

68 0.430

71 0.420

75 0.410

78 0 .400

82 0.390

85 0.385

86 0.382

104 0.347

127 0.314

157 0.283

195 0.253

246 0.226

314 0.200

356 0.188

370 0.184

0.00347

N _

0,001

0.004

0.043 :

o.o8o I 0.114 1

0.147 i

O. 177

0.205

0.218

0,221

0.406

nlaC2--[J, n

(MeV)

9.01

8.61

8.21

7.43

6.67

5.94

5.23

4.55

4.24

4.17

3.29

2.55

1 .44

1 .44

1.05

0.74

0.62

0.58

0.58

iJ, e i (MeV)

0.75

1.65

2.55

4.34

6.12

7.88

9.63

1.36

2.15

2.34

4,76

17.00

19.04

20.88

22.53

23.97

24.62

24.80

51.29

(m,, -~/n)c" (MeV)

9.09

8.79

8.49

7.90

7.32

6.75

6:20

5.66

5.41

5.35

4.57

3.88

3.28

2.77

2.32

1 .94

1 .77

1.72

0.63

398

2 % (1 - - 2 y - y=) + 4c~ (1 .... 2 g - y ,)3 __ 2c , s y= _ c= -k Amc ~ : O,

( S ) = : - - e l 1 [a (yn/ I~ + 2c~yA z'3 - - 2c~. + .',mc2], y2 _ _ _

Y~ ~ " 2c,,A

w h e r e 5m = m n - - mp.

T h e p a r a m e t e r s o f t h e g r o u n d s t a t e o f t h e d e g e n e r a t e Ae p l a s m a d e t e r m i n e d b y t h e s y s - t e m o f e q u a t i o n s ( 8 ) a s f u n c t i o n s o f t h e m a s s d e n s i t y

M 3neIt~ n ( ~ ) F = n A + 4c ~ - c 2 F . - - ~ y % ( 9 )

are given in Table I. The data in the last row refer to continuous nuclear matter.

In isolated nuclei, ~- mesons are present only for A > 200. Under conditions of a degenerate Ae plasma, pions also appear in medium nuclei above O = 3.4"1010 g/cm B. At

this threshold density, the mass number of the most stable nucleus is A = 85. With in-

creasing density, the concentration of mesons in the nuclei increases monotonically,

reaching the limiting value y~ = 0.22 at the end of this phase. At this point, O = O 1 =

6"1011 g/cm3, ~n = ~n -- mn c2 = --0.58 MeV, ~e = 24.80 MeV, and the mass number of the most

stable nucleus is A = 370. There is then a phase transition to the state of continuous nuclear matter with p = p0 = 3-1014 g/em 3, ~ = --0.58 MeV. At this transition, the matter

density has a discontinuity, increasing by a factor of approximately 500. In the interval 6"I0~I ~ P 5 3"1014 g/cm3 the state of the plasma is unstable and such a state cannot be

expected to be realized in stellar configurations. Such a discontinuity can be explained ' after as follows. As the density is raised above the value 01 , the chemical potential ~n

a small increase to ~ = --0.I MeV decreases to the value ~ = --0.58 MeV at p = P0' and then increases monotonically with increasing density. In equilibrium configurations consisting

of degenerate matter at the corresponding densities, this behavior of ~ is impossible.

This can be seen by considering the condition of thermodynamic equilibrium along the radius

of sufficiently dense configurations in which there is a free neutron gas:

~/gOo (r)" Iz,, (r) ----- const, (lO)

where g00(r) is the time component of the metric tensor and r is the distance from the center !

o f the star. Since g00(r) increases with the distance from the center, W n must decrease monotonically. The relation (i0) is valid until the interface between the continuous nuclear

matter and the Ae plasma. On this interface there must obviously be thermodynamic equilib-

rium between the neutrons in the nuclear matter and the neutrons in the nuclei of the Ae

plasma. The mean distance between the particles is here fairly small, and the equilibrium

is established instantaneously by the tunnel effect. Thus, it follows from the requirement of monotonicity and continuity of ~n that the region of densities 6.1011 < P < 3 "1014 g/cm3

is not realized in corresponding stellar configurations. Strictly speaking, these arguments

are valid if the incompressibility of the nuclear matter is sufficiently high. In configura-

tions with a central supermassive core (with mass appreciably exceeding the solar mass) it will evidently be necessary to take into account the compressibility of the nuclear matter.

In such cases, the transition from the Ae plasma to the phase of continuous nuclear matter

takes place at a higher density corresponding to values of the chemical potential in the in-

terval --0.58 < ~ < w0.1 MeV. Note however that in all cases a discontinuity of the density

involving a factor of about 500 is inevitable on the transition from the one phase to the

other.

Hitherto, we have based our treatment on the semi-empirical formula (I), in which we

have not taken into account the presence of terms of order c~N~/A 3. For ordinary nuclei,

their neglect is fully justified. In continuous nuclear matter and in nuclei of an Ae plasma, in which there are comparatively many z- mesons, the contribution of such terms may be appreciable. From completely general considerations one would expect that c~ = c3, which

we confirmed by numerical calculations when improving the Weizs~cker formula (i). By the

same logic, it is natural to expect that c~N~/A 3 c4[N n (Np N~)]4/A 3, i.e c' = C 4 To elucidate the role of these terms in our calculations, we first omit in (i) the term

with c4, and then take into account also the term c~N~/A 3, assuming c 4' ~ c 4. In the first

3 9 9

case~ the parameters o~ continuous nuclear matter (see the last row in Table 1) are changed

by A~ n = 0.38, A~ e = 2.38 MeV, Ay = 0.0005, Ay~ = 0.03, and in the second case, when the

term with c~ is also taken into account, A~ n = --0.08, A~ e = --0.72 MeV, Ay = --0.0001, Ay~ =

0.01. Thus, the neglect of terms of order c�88 3 does not have an appreciable effect on

the results given above.

The results we have obtained agree qualitatively with the results of [19]. There are

however some differences related to the improvement in the pion condensate parameters. In

[19] we established a significant suppression of the neutronization effect. According to

the results of the present paper, a comparatively small neutronization effect occurs only

~or !a e < 12 MeV, i.e., p < i0 I0 g/cm 3. When the ~- mesons appear in the nuclei, it is

suspended. Indeed, as can be seen from the data in the table, the concentration of protons

Np/A = y + Yw remains constant, equal to 0.4, when there is a further increase in the den-

sity. In [19], we noted that there is an appreciable raising of the threshold for the ap-

pearanoe of the Aen phase of matter (plasma with free neutron gas) as a result of the pion-

ization phenomenon~ Now, because of the enhanced pionization effect this phase is completely

eliminated~ i.e., the phase of a degenerate plasma cont aining a free

neutron gas does not exis t~

4. Continuous Nuclear Matter

We now consider continuous nuclear matter, which is of particular interest for the

theory of superdense celestial bodies. By continuous nuclear matter we understand a plasma

with density equal to the density in ordinary nuclei: n O = 1.7.1038 cm -3. The equilibrium

state of such a plasma is determined by the equations

p. --_ p.p ~t IL -: I~%, n + n: --:- rip. (11)

The chemical potentials of the particles can be obtained from Eqs. (6) and (7) if the Coulomb

and surface energy terms in them are omitted. Substituting these expressions in (11), we

find

t y o z ~ ( 2 c ~ - - A m c e ~ 4 c ~ f f = ) 3, c = ~ - 2 c ~ y ~ - - A m c 2 - - 2 c a ( 1 - - 2 y - - - y : ) 4 c 4 ( 1 - - 2 y - y ~ ) 3 ~ 0 . ( 1 2 )

a 3 n o " , . .

These equations are valid only at temperatures appreciably lower than the degeneracy tem-

perature of the particles~ ~5oI011. Solving (12), we find

,%-~ - m , c ~- - - 0 . 5 8 ; I~ - m p c ~" ~ 5 0 . 5 8 ; IL := ?u : 5 1 . 2 9 MeV; (13)

.V,, = 0o591; yp -:= 0 . 4 0 9 ; y.~ - - 0 . 4 0 6 ; y~ - 0 . 0 0 3 5 ,

where Yk = nk/n0 are the concentrations of the particles. If we also take into account the T 4 3 contribution of the term c4Nw/A in the chemical potentials, assuming c~ z c 4, we obtain

P.,~ - - m , , c " . . . . 0 . 6 6 ; F% -- It, ---- 5 0 . 5 7 MeV; U. - : 0 . 5 8 0 ; y~, ~- 0 . 4 2 0 ; y~ 0 . 4 1 7 . ( 1 3 ' )

Thus, if our ideas about the presence of ~ - mesons in ordinary heavy nuclei are cor-

rect~ then the results (13) obtained on the basis of formula (i) provide a justification

for asserting that continuous nuclear matter is in a condensed state similar to that of a

liquid~ i.eo, we are dealing with a bound state of the system. Indeed, we can readily show

that

? m,, c 2 := (!5~ - - m , c~) .... ~ y , l~e :~ 0 . 6 3 MeV, ( 1 4 ) n

t where pc 2 iS the density of the total energy. With allowance for the term with c4, we should

have (p/n ~ ran)C2 = ~0.70 MeV. Thus, at near-nuclear densities we cannot re-

tain the idea that we are dealing with a gas or that this gas con-

~ists predominantly of neutrons.

Since nuclear matter is in a bound state, it does not have intrinsic pressure (formally,

the pressure is negative). Such a state can be realized in superdense stars at sufficiently

high central densities. Nuclear matter withstands hydrodynamic pressure of the masses by

400

'virtue o f its relatively high incompressibility, if the particle~ are packed together only slightly more closely, large internal stresses arise which compensate the external pressure. The relative change of some volume element under the influence of pressure is

A V 3Ar o P . . . . . . . K " ( 1 5 )

V r 0

w h e r e r 0 = 1 . 1 2 . 1 0 - 1 3 cm i s t h e mean d i s t a n c e b e t w e e n t h e n u c l e o n s , K i s t h e m o d u l u s o f h y d r o s t a t i c c o m p r e s s i o n , and 1 /K i s t h e c o e f f i c i e n t o f h y d r o s t a t i c c o m p r e s s i o n . I n n u c l e a r p h y s i c s , one u s e s , n o t K, b u t t h e p a r a m e t e r K' = 4 ~ r 3 K / 3 , w h i c h w i t h o u t a l l o w a n c e f o r t h e c o n t r i b u t i o n o f t h e 7 - m e s o n s i s e q u a l t o [21 , 22] 0

-9- r~ drr~ - - m. ~ 15 ReV. (16 )

S u b s t i t u t i n g (16) i n ( 1 5 ) , we f i n d

[Srol_ P P2 = 1.22.10 a4 e r g / c m 3 . (17)

ro Pc

T h u s . a t p r e s s u r e s P < P2 t h e n u c l e a r m a t t e r i s i n c o m p r e s s i b l e and f o r P > P~ i t i s c o m p r e s - s i b l e , i . e . , t r a n s f o r m e d i n t o a r e a l g a s . I n c a l c u l a t i o n s o f t h e c o r r e s p o n d i n g s t e l l a r c o n - f i g u r a t i o n s i t i s n e c e s s a r y t o t a k e i n t o a c c o u n t t h i s c i r c u m s t a n c e and a l s o t h e d i s c o n t i n u i t y o f t h e d e n s i t y on t h e t r a n s i t i o n t o t h e p h a s e o f c o n t i n u o u s n u c l e a r m a t t e r . I t i s i n t e r e s t - i n g t o n o t e t h a t a l r e a d y i n 1932 L a n d a u , i n h i s s t u d y [1] on s u p e r d e n s e c e l e s t i a l b o d i e s , c o n s i d e r e d t h e e x i s t e n c e o f a " h i g h l y c o n d e n s e d c e n t r a l r e g i o n . . . . , s u r r o u n d e d by m a t t e r i n t h e o r d i n a r y s t a t e . . . . j u s t as a l i q u i d and i t s v a p o r a r e s e p a r a t e d " .

5. P l a s m a a t D e n s i t i e s a b o v e t h e N u c l e a r D e n s i t y

U n f o r t u n a t e l y , t h e c o r r e c t s t u d y o f t h e p r o p e r t i e s o f a p l a s m a a t d e n s i t i e s a b o v e t h e n u c l e a r i s a t p r e s e n t i m p o s s i b l e b e c a u s e we do n o t h a v e s u f f i c i e n t i n f o r m a t i o n a b o u t t h e s t r u c t u r e and i n t e r a c t i o n s o f p a r t i c l e s a t d i s t a n c e s r ~ 10 - 1 3 cm. B e l o w , we g i v e some

q u a l i t a t i v e a r g u m e n t s and e s t i m a t e s o f t h e p r o p e r t i e s o f s u c h a p l a s m a .

The f i r s t e x t e n s i v e s t u d y o f t h e p r o p e r t i e s o f p l a s m a a t d e n s i t i e s a b o v e t h e n u c l e a r u n d e r t h e a s s u m p t i o n t h a t t h e b a r y o n s f o r m an i d e a l g a s was made i n [ 9 ] . T h e s e r e s u l t s a l s o r e m a i n t r u e i n t h e c a s e o f a r e a l g a s i f one a s s u m e s t h a t t h e i n t e r a c t i o n e n e r g i e s o f t h e b a r y o n s o f d i f f e r e n t s p e c i e s a r e a p p r o x i m a t e l y t h e same . T h u s , t h e r e was e s t a b l i s h e d t h e s u c c e s s i v e a p p e a r a n c e o f ~ , A, and o t h e r h y p e r o n s and r e s o n a n c e s a t d e n s i t i e s a b o v e t h e n u c l e a r . I n [ 9 ] , t h e m o s t s e r i o u s d e p a r t u r e f r o m r e a l i t y was made i n t h e e s t i m a t e o f t h e r o l e o f t h e 7 - m e s o n s , a v a l u e a p p r o x i m a t e l y 500 t i m e s h i g h e r t h a n t h e n u c l e a r d e n s i t y b e i n g o b t a i n e d f o r t h e t h r e s h o l d o f t h e i r p r o d u c t i o n . A l l o w a n c e f o r n u c l e a r i n t e r a c t i o n s b e t w e e n t h e p a r t i c l e s i n t r o d u c e s i m p o r t a n t c h a n g e s i n t o t h e s e r e s u l t s .

I n t h e p r e c e d i n g s e c t i o n , we h a v e shown t h a t i n a p l a s m a a t n u c l e a r d e n s i t y n e u t r o n s , protons, and ~- mesons are present with approximately equal concentrations. There are no

negative muons, and the electron concentration is ~3%. As the density is increased, the

concentrations of protons and mesons must increase to the threshold for the appearance of

the following new particle. The concentration of pions in this region cannot differ ap- preciably from the concentration of the nucleons since a pronounced increase in their number

is prevented by the repulsive forces which act between them.

Let us estimate the threshold values of the densities above which the other particles

acquire stability in the nonrelativistic range of energies. We assume that all species of

baryons interact in the same way, so that

9

P~ (18) Fk = ~ (Pk) V, (Pk) :~ mkc~ + V(O) + ~ ,

w h e r e ~k i s t h e c h e m i c a l p o t e n t i a l , E k i s t h e l i m i t i n g e n e r g y , Pk i s t h e l i m i t i n g momentum, Vk(p ) i s t h e m o m e n t u m - d e p e n d e n t p o t e n t i a l , and m E i s t h e e f f e c t i v e mass o f t h e b a r y o n . F o r b a r y o n s , t h e n o n r e l a t i v i s t i c a p p r o x i m a t i o n i s v a l i d a t d e n s i t i e s n < 4"1040 cm - B . We now c a l c u l a t e t h e s t a b i l i t y t h r e s h o l d s o f t h e A, Z - , ~o , and Z+ h y p e r o n s u n d e r t h e a s s u m p t i o n

t h a t i n t h e p l a s m a me ~ 0 .5m k . T a k i n g i n t o a c c o u n t ( 1 8 ) , f r o m t h e r e l a t i o n PA = Pn we f i n d

401

that the threshold density is

In the presence of a pion condensate, the limiting energy of the electrons will evidently not grow strongly with increasing density. Assuming that ~e differs little from its value

=51 MeV in nuclear matter, we find that the threshold for the appearance of Z- hyperons is

n ~ - ~ 3 n ~ 3 l m " ( m ~ c ' - - r n " c ' - - P ' ) c ' I s t 2 ~ l ' ! " l O 3 9 c m ~ 3 " a " (20)

Similarly, for Z ~ and E +

nv~-~2.0-10 a9 cm -3, nv+ ~ 3 . 3 " I 0 39 cm -3. (21)

At :even higher densities in the plasma, the other hyperons and baryon resonances become

stable. Comparing these results with the results of [9, i0], we note that not only the

particle stability thresholds but also the order in which they follow one another in the density scale are changed~ In the considered approximation, the concentrations of baryons

with the same electric charges are related by simple analytic formulas, which are given in

[9, 103.

In the meson family, besides the pions only the negative K- mesons can become stable in

the interior of superdense stars. By analogy with the pions, we can expect that the kaon

becomes stable in a plasma at mean distances between the baryons that are of order of the

kaon Compton wavelength, i.e.

l ( r n x c ' ] ~ n x - ~--~-- x ~ ] ~ 4 . !0 a9 cm -3 (22)

The concentrations of the particles above the threshold for production of K- mesons will be

determined by the relations

~b ~ = P . ' ~b- = ~, ,+ ~e, ~b- = I t . - ~e, ~ x = ~== ~ , (23)

where ~b is the chemical potential of the baryons. Of course, this system must be augmented by the condit:ion of electrical neutrality of the plasma. At such densities, the K meson ap-

pears if its potential energy is sufficiently high, and the corresponding condensate will evidently be in the s state. The negative muon may also become stable provided the limiting

energy of the electrons is not frozen due to the presence of the pion and kaon condensates.

Just above the threshold (22) the concentrations of all particles except the leptons

are obviously of about the same order:

n, ~ np ~ n.~ ~ nvo "~ n,5. "~ n= ~ n K, �9 �9 . (24)

Experiments on the elastic scattering of electrons on nucleons have shown that the

nucleons are extended formations with rms electric radius equal to i e = 0.8 F. Analysis of

the experiments 0n the scattering of high-energy neutrinos on nucleons have led in their

turn to the conclusion that nucleons consist of several particles, which are called partons.

There are good grounds for identifying these with the quarks. The radius l of the region

in which the partons are confined must of course be less than i e. It would seem that l 0.5 F, to which the density n B 1040 cm -3 corresponds. According to the existing ideas,

not only nucleons but all hadrons consist of partons (quarks). Accordingly, at densities

n > n 3 the physical picture in the plasma must he radically changed. Under these conditions, the concept of individual hadrons becomes meaningless. The plasma goes over into a phase in which the roles of individual particles are played by the constituents of the hadrons,

ioe.~ the partons.

6.~__quation of State

It is expedient to specify the equation of state of the plasma in the form 0 = 9(P).

For an Ae plasma in the nonrelativistic region, it has the form

402

20 r

"~5[ 81 I ! I

iOt" ~ , l

'~0 25 30

.'2"" 4."

tgP r , - , afO 35

Fig. 1. Equation of state of a degenerate

plasma. The pressure is measured in units of erg/cm 3 and the density is g/cm 3. The part

of the curve up to the point A represents the state of the Ae plasma. The dashed vertical

section AB corresponds to the discontinuity

of the density on the transition to the phase

of continuous nuclear matter. The section BC describes the state of incompressible nuclear

matter, CD a real hadron gas, and DE the state of an incompressible parton fluid. The upper

and lower lines beyond the point E represent the two alternative limiting equations of

state corresponding to ideal and real parton gases. The approximate dependence beyond the

point C is indicated by the dashed curve.

~-, = 3.423. l O - S p ais, P ~ 2 .1 .10 '-''3 e r g / c m 3 �9 (25)

In the region in which the electron gas is relativistic, the data of Table 1 can be approx-

imated by the formula

f~=1.059.10-1l (1 : 3 .996 ,10-SP 14 4.088-10-~5P12 1.568-10-22P34) .P 3'~, 2 . 1 - 1 5 : 3 ~ P - . ~ . P 1. (26)

At pressures greater than PI = 6"6"1029 erg/cm3' there is a phase transition to the nuclear

phase of the plasma. Here, up to P ~ P2 = 1034 erg/cm3 (see (17)) the plasma is virtually

incompressible, i.e.

,~, ~ 2.84-1019 g/era 3 , P1 -% P < P2. (27 )

The velocity of longitudinal sound waves in the nuclear matter is given by

F K - C l -~ - |// --

9 (28)

and is equal to c I = 4.109 cm/sec. The velocity of elastic transverse waves is lower than

that of the longitudinal ones.

At pressures P2 < P < P ' the plasma is a real gas of hadrons, P3 being the pressure ~ 3

above which the plasma is in the parton phase. At the transition point n 3 ~ 1040 cm -3 and

the limiting energy of the baryons is ak = mk c2, so that

2 - 23 ~ -~ na~ ~ ~ 5- 1036 erg/cm 3 �9 (29)

An estimate in accordance with the relativistic formula gives approximately the same result.

403

I n t h i s n a r r o w r a n g e o f p r e s s u r e s ( s e e Fig . 1) o n e c a n u s e t h e a p p r o x i m a t i o n

~ 5.10- ~SpO.Ss, p~ < p.%/< p~. (3O)

We now consider the situation when the baryons in the plasma are densely packed together and form conti~uous parton (quark) matter, This evidently occurs at hadron densities just greater than n 3 ~ 1040 cm -3 �9 The binding energy of the partons in the hadrons is extremely high~ and this is why it has not yet been possible to separate them in high-energy nuclear reactions. By analogy with nuclear matter, it is natural to expect that here an incompres- sible patton liquid is initially formed. Let us attempt to estimate the range of pressures within which the incompressible parton phase is realized.

In accordance with what we have said above, the transition to the parton liquid phase in stellar configurations is realized at P = P3" In the parton phase, in which the densities are approximately i00 times greater than the nuclear, we expect the velocity of sound waves to differ little from the velocity of light. Assuming c I = c, we find from (28)

K'= 4~13 K-- ~c~ - q , 3 n

where n is the density of the partons and ~ is their mean energy. In all probability, ~ ~

I 000 MeVo For the relative deformation, we obtain

~l P

P4 P~ = 3 K ~ 1 0 as e r g / c m 3.

Thus~ for the phase of incompressible parton matter

(31)

7~ ~ 3-1016 g / c m 3 , P3 ~ P "~T P~. ( 3 2 )

When P > P4 ~ the plasma agains becomes compressible, and the mean distances between the partons are less than the mean distances between partons in hadrons. In this region, one can expect strong repulsive forces between the partons. If these ideas are valid, then in

the region P >> P4 the maximally hard equation of state holds [10]:

Because of relativistic effects is also possible:

"; ~ P S - , p > p~. (33)

(Lorentz contraction of the fields) the other limiting case

, ,~3P/c ~, P> P4" (34)

The intermediate case is also possible: p = aP/c 2, 1 < a < 3, especially at pressures around

P4o

Figure i is a graph of the equation of state. The part of the curve p(P) for P < 1034

erg/em 3 can be assumed to be more or less reliable.

7 . S~ary of Results

At plasma densities p > 3.4.1010 g/cm 3, the neutronization of matter is replaced by the pi~nization phenomenon. In nuclei of a degenerate Ae plasma negative pions appear, their

concentration increasing with increasing limiting energy of the electrons and reaching 22% of the n~berof nucleons at the end of this phase. With the appearance of the pions, the ConcenZration of:protons in the nuclei is frozen at the level 0.4, i.e., essentially, the

neutronization effect in the degenerate plasma is absent.

At pressure P z 1030 erg/cm 3, there is a phase transition from the Ae state to the state

of continuous nuclear matter. The density has a discontinuity involving a factor of about 500: irom S'I0 II g/cm 3 to 3"1014 g/cm 3. This discontinuity is due to the requirement that

the chemical potential of the neutrons he continuous. At the transition point, ~n -- mn c2 = -~0.58 MeV, and ~p and ~e have discontinuities. Because of the pionization effect, the Aen phase of matter, which follows the Ae phase according to the earlier ideas of [I0], is now

completely eliminated~

404

The phase of continuous nuclear matter is characterized by the parameters

['e = 2.84 1014g/ca 3 ; ~,, -- m,,c ~ = -- 0.58 MeV;

,~p -- mpc ~ : -- 50.58 MeV; ,% ---- ~ =: 5129 MeV; y, = 059; y~ - 0.41; y~ ---- 0.41; y~ ---- 0.0035

It can be seen that the idea of nuclear matter consisting predominantly of neutrons is not true. The binding energy of the particles is here negative (see (14)), and therefore the matter in this phase is in a liquid state and does not have intrinsic pressure. Such a plasma in the pressure range i030 ~ P ~ 1034 erg/cm 3 is incompressible.

In the density range 3.10 la < p5 < 3-1016 g/cm 3 the plasma is a real gas of hadrons. At densities p = 1015, 2.1015 , 3.101 , 6-1015 g/cm 3 the A, Z-, zo Z+ and hyperons and K- mesons successively become stable in the plasma. For p > 3-1016 g/cm 3, the hadrons decay

and we are dealing with a parton (quark) plasma whose state is initially similar to that of a liquid and then to that of a gas. The appearance of other baryons and resonance particles before the transition to the patton phase cannot be ruled out.

We have obtained the equation of state p = p(P) of the degenerate superdense plasma plotted in Fig. i. In deriving the equation of state, we took into account the incompres- sibility of the plasma in certain pressure ranges.

We are grateful to Academician V. A. Ambartsumyan for his interest, numerous discussions, and valuable comments. We are also grateful to the participants at the seminar of the De- partment of Theoretical Physics at the Erevan State University for discussions.

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