On the phase structure of baryonic matter

Nuclear Physics A532 (1991) 634-646 North-Holland

THE

ASE STRUCTURE OF

ARYONIC MATTER

E. HEIDE and P.J. ELLIS School of

t°sics and Astrononko; Unirvrsity of Minnesota, Minneapolis, MN55455, USA Received 13 November 1990 (Revised S April 1991)

Abstract: We have studied the phase structure of baryonic matter in a me -lei which includes nucleons and delta resonances interacting with a- and cv-mesons . In the mean-field approximation, the existence of phase transitions to delta matter and to a baryon-antibaryon plasma was strongly dependent on the values chosen for the equilibrium effective mass and compression modulus. When vacuum fluctuations were included, the physically acceptable solutions only yielded a liquid-gas phase transition. Further, these solutions were restricted to rather large values of the effective mass and compression modulus which did not include the currently accepted values .

l . Introduction e relativistic field theory introduced by Walecka') has been the subject of much study and has been shown to provide a good description of nuclear matter and finite nuclei') . Most of these calculations have been carried out at the mean-field or the one-loop level. Quantum hadrodynamics is, however, a strong coupling theory and the two-loop contributions have been found to be large 3), indicating that the loop expansion is not convergent. In addition ifone seeks to include delta resonances, which certainly play a role in nuclear matter away from equilibrium, one faces the fact that a spin-- lagrangian is not renormalizable, i .e., it requires, in principle, an infinite number of counterterms and hence an infinite number of parameters . In this situation, it seems most sensible to view the theory as phenomenological with an effective lagrangian containing a small number of parameters to be fitted at the tree or one-loop level. In this spirit, Glendenning 4) and Waldhauser et al. 5) have discussed the existence of phase transitions in a field theory which included nucleons and deltas [and heavier baryons in the case of ref. ')] interacting with o-- and co-mesons. They worked in the mean-field approximation and included Q3 and 0r4 terms in the effective lagrangian so as to obtain their chosen values for the equilibrium compression modulus K and effective mass MN . Since considerable uncertainty exists in the values of K and M N , one of our purposes here is to study the phase structure as a function of these variables. One might also expect the negative-energy sea to play a role, so that a second aim is to examine the effect of vacuum fluctuations at the one-loop level. 0375-9474/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

E. Heide, P.J. Eüis / Phase structure

635

We shall study the predictions of this theory up to quite high temperature and density, although it must be borne in mind that at some point the baryon-meson description will become invalid and a quark-gluon plasma will be formed. The density at which this takes place is quite uncertain, although a recent estimate') gives about ten times nuclear matter density at T = 0. The critical temperature 7) is thought to be ^-200 MeV. 2. Theory Our lagrangian takes the form Y=

S

-S-By,to .

B

- MB+g~B~)+/fB+2a,~~a~`Q-

U(tir)

-âFF1 ° +2mw+CTC ,

where F,,,, = o1,,,w -aw,, and U(Q)

= 2i11QQ2+3bMNgaN U3+4CgaN~4

(2

As usual CTC refers to the counterterm contributions . The sum over baryons in eq. (1) involves both nucleons and delta resonances. The latter, being spin particles, require a lagrangian of the Rarita-Schwinger type g) ; however, at the one-loop level considered here, the nucleon and delta contributions to the partition function differ 9) only in their degent-racy factors sc, that we have used a simple schematic notation in eq. (1). Now summing the one-Loop diagrams illustrated in fig. , .sing for example the techniques of ref. '°), we obtain for the grand partition function in Z=-NCV[ U(U) -2m2v2+4E]

2

+V

YB

V

(2 V ,ff )3

(27r)3

4 d3k [In (1+e-ß ( EB-i`B')+ln (1+e - ß' EB+ B ')]

d3k [in (1-e-R~?) +31n (1-e-~

(3)

Fig. 1 . Typical baryon and meson one-loop diagrams ; the dashed and wiggly lines indicate 0- and w-mesons, respectively.

E. Heide, P.J. Eilis / Phase structure

636

where the effective nucleon and delta masses are related by

MN - MN gcr N

M.1 _ Mi 9-J

In the second term of eq. (3), which arises from the nucleon loops, yB is 4(16) for nucleons (deltas) and the effective baryon energy and chemical potential are M*?)!A -' EB = (k-+ , 1A B IA + g. B v In the third term of eq. (3) (which is numerically small), the first part arises from the a-meson loops with d' _ k, U(a) , (6) U= ~ + thrr da,' and the second part arises from the thermal omega contribution where eÛ, = k 2 + mw . Self-consistency is obtained by minimizing In Z with respect to or and v. The former yields an equation for M*B , while the latter gives v=_

(grayNPN+goa.1P.l)

,

(7)

where the baryon density is given by 1 (cl lnZ ) PN + P~ a~. 16 V TV =

P

Y_

71.3

d k

(8)

s( ) Finally, we need to specify the vacuum fluctuation contribution, 4E, in eq . (3). Choosing it in such a way that the coefficients b and c in eq. (2) are not modified, the result is 2) AE=

B

- YBMB

16~r' -

MB

L~ MB )

In

MB + MB -MB - 7 MB

MB

2

MB -MB MB

13 MB - M *)33 25 MB - M B `' +-3 t Mp 12 MB ) 1 2 4 4 ~ 2 - rl1 ?c - 3 (ff,' 'Cr - rr!Q 2 + mcr m~r In mcr - m cr

m, 64 7T` - Ô ( bMN g 3 3

2 M Gr

LT

mT 1

mâ

+ 9 Cg,rNQ~ 2bMN )

2

mâ

4 ( bMNgQN 2 3 m"

)

T) 4 ] ( )

Having obtained in Z the various thermodynamical functions of interest can be calculated in the usual way. For instance, the pressure P - (ß V) -' In Z, the free

E. Heide, P.J. Ellis / Phase structure

63 7

energy density F/ V = -P+ jLp and the internal energy density is E/V= U( Cr )+2mwv 2 +4E+E +

(2

1ff

)

3

2?r 3

d3 kEB[(e P'EB ~`B'+1)- ' +(el3 ( EB"B

j d3k [ é,(e ß° -1)- ' +3e,,,(eo --1)- '] .

(10)

3. Results 3 .1 . MEAN-FIELD CASE

In order to proceed we need to specify the coupling constants of the delta relative to those of the nucleon ; unfortunately, there is rather little information available. We shall adopt universal vector coupling, i.e., gwN = gw, . As regards the scalar coupling, Wehrberger et al. ") have studied delta excitation in electron scattering on '2C and 4°Ca and photoabsorption on 209 Pb; they conclude that g?,/g,N > 5.12) for the baryonic plasma ( p = 0) gw.,/gwN . Also, it has been shown explicitly that the choice M1gaN= MN gff.1,

(11)

yields effective masses for the nucleon and the delta which are always positive and tend to zero in the limit of high temperature . This is very desirable physically and we adopt the choice of eq. (11); Wehrberger et al. suggest a smaller value for g?,/g.N , however, as they point out there are a number of approximations made in the analysis and these could affect the ratio. We then have four constants to determine - gaN, 9-w N, b and c - in the mean field case and additionally, when vacuum fluctuations are included, the mass of the o-meson which we fix at 600 MeV for the most part. Two relations are given by the equilibrium nuclear matter properties : the binding energy/nucleon of 16.3 MeV and the saturation density po = 0 .153 fm-3. The remaining two conditions are given by the equilibrium effective mass M N and the compression modulus K = k2(a2 /akF)(E/A). Johnson et al. '3) have obtained a nonrelativistic effective mass of 0.83 from a study of the nucleon mean field in lead and, interpreting this as the 14) point Landau mass 14), we find M N/ MN = 0.78. However, Jaminon and Mahaux out that, when finiteness effects and Coulomb energy corrections are taken into account, the effective mass is reduced to approximately 0.74, giving MN/ MN = 0.69. '5) gives K = 300 MeV [see also refs. '6, ")]. There is Recent giant monopole work considerable uncertainty in these figures so we shall discuss results obtained for a range of values of K and MN . The parameters we obtain for the mean-field case, where the vacuum fluctuation contributions JE in eqs. (3) and (10) are discarded, will not be listed here since they are in agreement with those tabulated by Waldhauser et al.''), when account

63 8

E. Neide, P.J. Ellis / Phase structure

is taken of the small differences in the chosen equilibrium binding energy and density. A convenient way to display our results is to plot the free energy density as a function of the chemical potential. Since F/ V = P + A (aP/aA ),-, a first-order phase transition requires F/ V to be multivalued for those values of 14 which correspond to the metastable region. (A second-order phase transition would require a discontinuity in the slope of the curve.) As a representative case, we show in fig. 2 results for MN/ M N = 0.75 and a number of values of the compression modulus K at temperature T = 0. The free energy is rather insensitive to K, particularly for small JA, it is, however, much more sensitive to M *N/ MN . A similar point has been made with respect to the binding energy/nucleon' g). All the cases clearly exhibit a liquid-gas phase transition . In addition, as K decreases a second phase transition develops from nucleon-dominated matter to delta-dominated matter due to the strong attraction of the scalar field. This occurs at a density of 3-4po. The plot of nucleon density/baryon density in fig. 3 shows that for values of K below about 320 eV a definite phase transition occurs, whereas for larger values there is a smooth evolution to delta-dominated matter. When K becomes lower than 275 MeV (dashed curves in figs. 2 and 3), the coefficient c of the a4 term in eq. (2) becomes negative. Physically, this is unreasonable since the potential U -> -00 for large or so that the energy spectrum has no lower bound. In practical terms, for K - 210 MeV we find that the energy/baryon develops a cusp and shows a second minimum at 1

.0

K = 275 MeV K = 300 MeV K = 325 MeV K = 350 MeV K = 400 MeV

0 e00.0 700.0 6W.0 W0.0 400.0 300.0 200.0

Mean Reld M*/M = .75 Temperature = 0 MeV

100 .0 0.0

800.0

900.0

1000.0

1100.0

1200 .0

1300 .0

Chemical potential (MeV)

1400.0

1500.0

1600.0

Fig. 2. The free energy as a function of the chemical potential in the mean-field approximation at T = 0, for an effective mass MN/ MN = 0.75 and various compression moduli K.

E. Heide, P.J. Ellis / Phase structure

k ,u 9.

1.0 0.9

K = 250 MeV --~; (unphysical)

639

K = 275 MeV K = 300 MeV K = 325 MeV K = 350 MW K = 400 MeV

0.8 0.7 0.6

os b .92

0.4

02

z

--------------------------------------------------------

0.3

0.0

Mean Meld - M"/M = .75 Temperature = 0 MeV 800.0

900.0

1000.0

1100.0

1200.0

1300.0

Chemical potential (MeV)

1400.0

1500.0

1600.0

Fig. 3 . The ratio of nucleon to baryon number densities as a function ofthe chemical potential at T = 0, for M*/M, = 0.75 and various K.

==2po with an energy below that of nuclear matter. While there is a small region of K-space where the thermodynamic quantities show no abhorrent behavior at the densities considered here, it is difficult to have confidence that this band is physically meaningful . We therefore exclude cases where c < 0 here and for other effective masses, where similar trends are observed . This is in agreement with the conclusions of Waldhauser et al 's) who discuss in some detail the unphysical results obtained when c < 0. For larger values of M *N/ MN the onset of both the phase transition to delta matter and the unphysical behavior (c < 0) occurs for smaller values of K. The situation is illustrated in table 1, where we show the minimum value of K such that c > 0 and the maximum value for which a delta phase transition is obtained (in the region up to K = 400 MeV that w;; have examined). Thus for MN/ MN = 0.9 there is no phase transition in the physical region, while for a value of 0.65 a phase transition always exists in the range of K-space that we have examined . Fig. 4 shows the effect of increasing temperature for M N/ MN = 0.75 and K = 275 MeV. The free energy becomes single valued by T = 25 MeV so that the liquidgas and nucleon-delta phase transitions have disappeared ; for both transitions the critical temperature Tc == 15 MeV. Glendenning 4) and Waldhauser et al. 5 ) have pointed out that a phase transition to a baryon-antibaryon plasma may occur at relatively high temperature . We see that such a transition indeed appears at T

E Heide, P1 M / Phase structure

640

TABLE I

Minimum, values of the compression modulus (in MeV) for stable solutions and maximum values for which delta and baryon-anti baryon phase transitions occur (the maximum compression modulus considered is 400 MeV)

0.65 0.70 0.75 0.80 0.85 0.90

K ..j,,

K.,a, W)

K,,,z,,, (BB)

370 320 275 235 185 120

400 400 320 235 -

400 400 380 250 -

150 MeV and for higher temperatures the discontinuity intercepts the ordinate axis and a bifurcation takes place. This behavior is also shown in fig. 5 where we give the antibaryon/baryon ratio. Here we have plotted in addition the result for K = 400 MeV and T = 165 Met, for which parameters single-valued behavior is obtained. In fact a phase transition oniy occurs for K below 380 Met. 7W.0 80.0 W0.0

Z.

40&0 WO 2W.0 W&O 0.0 _1O&O

0.0

I

200.0

400.0

600.0

800.0

1000.0

Chemical potential (MeV)

1200.0

1400 .0

Fig. 4. The free energy as a function of the chemical potential for various temperatures with MN/ MN ": 0.75 and K = 275 MeV .

E. Heide, P.J. Ellis / Phase structure arr O v O 84

64 1

1.0 0.9 0.8 0.7 0.8 0.5 0.4 0.3

Ô ~.

z

0.2 0.1 0.0

0.0

200.0

400.0

800.0 800.0 1000.0 1MA Chemical potential (MeV)

1400.0

1800.0

Fig. 5. The ratio of antibaryon to baryon number densities as a function of the chemical potential. The values of T, M*,/ M, and K are indicated .

As is clear from table 1, when we vary M N/ MN the maximum values of K for which an antibaryon-baryon phase transition occurs are very similar to those discussed previously for the delta phase transition. We remark that if one chooses universal scalar coupling, 9.N = g~A, no delta phase transition occurs in the physically acceptable region . Also the antibaryon-baryon transition is suppressed; it occurs for smaller values of K than listed in table 1 and requires higher temperatures than we have discussed here. This suppression can, however, be alleviated by the inclusion of more massive baryons in the theory which give a significant effect only in the region where the phase transition takes place. With this choice of equal coupling constants, negative effective masses are encountered, as we have pointed out, and for this reason we have retained the choice of eq. (11). Finally in fig. 6 we plot the energy/baryon as a function of density for MN/MN = 0.75 and K = 300 MeV. At T = 0 the presence of the deltas results in a slight flattening ofthe curve at 3-4po . As the temperature increases, E/B increases and the minimum shifts to higher densities . By T =155 MeV we have a minimum at -p o with a relatively small number of antibaryons and another at much higher density with a larger antibaryon/baryon ratio, the number of nucleons here is small in comparison to the number or deltas.

E. Heide, P.J. Ellis / Phase structure

642 1400.0

Mean Reld .75 K /300 MeV

12mo 1000.0

T =155 MeV

0

T=100 MeV -

1___"

T = 50 MeV T=0MeV

------------

0.0

o'

0.0

02

~

s 0.4

L

0.6

0.e

1.0

1.2

Baryon number density (1/fm"S)

I

1.4

1.6

Fig . b. The enemy per baryon as a function ofbaryon density at various temperatures for M *,.l M = 0.75 and K = 300 MeV.

3.2. VACUUM FLUCTUATION CASE

We now include vacuum fluctuations in eqs. (3) and (10) ; first we consider the case where the summation over B includes only nucleons . We have only succeeded in obtaining a satisfactory fit to nuclear matter for quite a limited set of values of the equilibrium compression modulus and effective mass. There are indicated in table 2, along with the corresponding parameters that we have obtained . For smaller values of K and MN/ MN there are difficulties with the solutions; however, the situation is more complicated than in the mean-field case. An example is shown for N/ MN = 0.8 and K = 300 MeV in fig. 7, where the plot of the binding energy/nucleon as a function of density shows a second minimum which is lower than the nuclear matter minimum . This second minimum corresponds to a larger value of the scalar field, and this branch, at densities beyond the equilibrium point, frequently yields delta-dominated matter. We have examined the effect of doubling the mass of the o-meson while maintaining a fit to the nuclear matter properties, but this does not significantly change the situation . The type of result shown in fig. 7 is unique for K - 275 MeV with this effective mass. For K = 300 MeV, however, two other sets of coupling constants can be found which obey the condition 9C_g4 N < 1 .

(12)

E. Heide, P.J. Ellis / Phase structure

643

TABLE 2

Coupling constants for the vacuum-corrected cases MN/M,

K

N-vacuum corrections 0 .80 0 .80 0 .85 0 .85 0 .85 0 .90 0 .90 0 .90 0 .90

370 400 270

(gfNMN/mT) 2

96.04 96.04

181 .0

c

0.8074 x -0 .6268 x 0.8989 x 0.3693 x -0.4309 x 0.2213 x 0.1397 x

10-3 10-3 10-2 10-2

10-2 10- ' 10 - ' -0.1210x 10- ' -0.3250 x 10 - '

0.1333 x 10 - ' 0.1740x 10 - ' 0.2175 x 10"' 0.2958 x 10 - ' 0 .6368 x 10- ' 0.6816 x 10 - ' 0.7597 x 10 - ' 0.2428 0.5365

-0.5293 x 10 -2 0.1802 x 10 - ' 0.1407 x 10 - ' -0.1230 x 10 - ' -0.3264 x 10 - '

0 .6134 x 10 - ' 0 .6869 x 10- ' 0.7352 x 10- ' 0 .2422 0 .5360

400 190 200 300 400

29.07 29.07

132 .0 97 .53 66.61

62.65 29.07 29.07 29.07 29.07

135 .0 135 .6 132 .0 97 .50 66.61

400 195 200 300 400

b

176.7 161 .7 152.7 135 .7 139.4

62.65 62.65 62.65 29.07 29.07

300

N +,à -vacuum corrections 0 .85 0 .90 0 .90 0 .90 0.90

(9.NMN/m.) 2

900.0

200.0

100.0 WIR

W

0.0

-100.0

-200.0

Vacuum corrected Me/M= .8 K = 300 MeV T=0MeV 0.0

0.2

0.4

0.8

0.8

1.0

Baryon number density (1/fm"*3)

12

1.4

Fig. 7 . The energy per baryon as a function ofbaryon density at T = 0 with vacuum fluctuations included. Here MN/ MN = 0.8 and K = 300 MeV.

E. Neide, PJ. Ellis / Phase structure

64 4

This condition implies that U(o,) + DE tends to -oo for large a. We would argue that this implies that the theory is fundamentally unsound; however, as a practical matter the theory is not well defined since we have two different sets of coupling constants (which yield closely similar binding energy/nucleon curves) as well as a third set which is ph2,sically unrealistic . Note that the parameters used by Glendenning '9) lie in this region . In almost all cases we find that as we reduce the equilibrium values of MN/ MN and K below the values in table 2, we enter a region where three solutions are to be found. One of these exhibits an energy minimum below that of nuclear matter, another obeys eq. (12), while the third may or may not satisfy eq. (12). If the equilibrium parameters are reduced further then only the first of these possibilities is retained. We conclude that only the parameter sets of table 2 yield a satisfactory theory. If we include both nucleon and delta vacuum fluctuations according to eq. (9), the admissible region of parameter space is reduced further, as indicated in table 2. Unfortunately neither nucleon nor nucleon-plus-delta vacuum fluctuations yield physical solutions in the region of the desired values, M N/ MN = 0.69 and K = 300 MeV. Further, for the parameters of table 2, no phase transitions, other than the liquid-gas transition, occur. The system simply evolves smoothly and, with o, .0

1-

.0~

0.0 h

-100 .0

®

0.0

1

200.0

1

400.0

600.0

800.0

Chemical potential (MeV)

1000.0

1200.0

Fig. 8. The free energy density as a function of the chemical potential at various temperatures, for MN/ Mr, = 0.85 and K = 300 MeV. The full curves give the vacuum corrected result and the dashed curves refer to the mean-field case.

E. Heide, P.J. Ellis / Phase structure

645

increasing T, increasing numbers of deltas and antibaryons are present. By including baryons of higher mass in the theory it might be possible to drive the scalar field to larger values so that additional phase transitions could appear. We have investigated this possibility by including all well-established baryon resonances up to a mass of 1775 MeV, assuming universal vector coupling and a scalar coupling which scales with mass. This, however, did not lead to the appearance of additional phase transitions . Picking a representative case where we are able to obtain physical solutions both with and without vacuum fluctuations, we show the free energy at various temperatures for M N/ MN = 0.85 and K = 300 MeV in fig. 8. The results obtained with vacuum fluctuations are indicated by a single full line since there is no significant difference between the nucleon and nucleon-plus-delta cases; the mean field results are given by the dashed lines. All curves exhibit a liquid-gas phase transition at low temperature, but are otherwise rather featureless . Clearly the inclusion of the vacuum fluctuations produces very little effect (provided of course that the parameters are chosen to reproduce the same equilibrium properties). This is to be expected near the saturation point since the leading vacuum correction is of O(0") 4 which can be approximated fairly well by ~3 and o- terms of opposite sign. The differences between the curves increase somewhat at high temperature and density, but still remain relatively small due to the reduced importance of the vacuum corrections in this domain . 4. Conclusions

We have examined the phase structure of baryonic matter in a model which includes nucleons and delta resonances interacting with o' and W mesons . As is expected for any system of fermions which is self-bound, all approximations gave rise to a liquid-gas phase transition . The density and temperature at the critical point of this transition were, in many cases, well approximated by the simple expressions of Kapusta 20). These expressions assume, among other things, a groundstate energy which is parabolic in the density about the minimum . If higher powers ofthe density are also significant, as we found in some cases, the accuracy necessarily deteriorates . In the mean-field approximation, the existence of additional phase transitions was highly dependent on the values chosen for the equilibrium effective mass M** / MN and compression modulus K. Phase transitions to delta matter and to a baryon-antibaryon plasma occurred in very similar regions of this parameter space. As the assumed equilibrium M N/ MN or K were reduced the phase transitions began to occur and further reduction eventually led to a potential with the unphysical property that it diverged to -co for large values of the scalar field. Thus it seems reasonable to question whether these phase transitions are real physical effects or simply harbingers of this unphysical behavior .

64 6

E. Heide, P.J. Ellis / Phase structure

When vacuum fluctuations were included, we were able to find physically acceptable solutions for a much more limited range of values of the equilibrium MN/ MN and K and these were such that no delta or baryon-antibaryon phase transitions occurred . Our physically acceptable solutions did not encompass the currently O,3 accepted estimates of M N/ MN = 0.69 and K = 300 MeV. Thus the inclusion of and &4 terms in the potential does not appear to accomplish the desired task of allowing one to obtain the required values of the effective mass and compression modulus . We thank J. Kapusta for a number of helpful discussions . A grant for computing time from the Minnesota Supercomputer Institute is gratefully acknowledged. This work was supported in part by the US DOE under contract No. DE-FG0287ER40328. efererces 1) J.D. Walecka, Ann . of Phys. 83 (1974) 491 2) B.D. Serot and J.D. Walecka, in Advances in nuclear physics, vol . 16, ed. J.W. Negele and E. Vogt (Plenum, New York, 1986) 3) R.J. Furnstahl, R.J. Perry and B.D. Serot, Phys. Rev . C40 (1989) 321 4) N.K. Glendenning, Nucl. Phys. A469 (1987) 600, and references therein 5) B. %Wadhauser, J Theis, J.A. Maruhn, H. Stöcker and W. Greiner, Phys . Rev. C36 (1987) 1019 6) A.K. Holme, E.F. Staubo, L.P. Csernai, E. Osnes and D. Strottman, Phys. Rev. C40 (1989) 3735 7) G.F. Bertsch, in Trends in theoretical physics, vol. 1, ed. P.J. Ellis and Y.C. Tang (Addison-Wesley, Reading, 1989) p. 79 8) W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61 9) D.K. Griegel, Phys. Rev. C43 (1991) 904 10) J.1. Kapusta, Finite-temperature field theory (Cambridge Univ. Press, Cambridge, 1989) 11) K. Wehrberger, C. Bedau and F. Beck, Nucl. Phys. A504 (1989) 797 12) P. Levai, B. Lukacs, B. Waldhauser and J. Zimanyi, Phys. Lett. B177 (1986) 5 13) C.H. ~ohnson, D.J. Horen and C. Mahaux, Phys. Rev. C36 (1987) 2252 141 M. Jaminon and C. Mahaux, Phys. Rev . C40 (1989) 354 15) M.M. Sharma, W.T.A. Borghols, S. Brandenberg, S. Crona, A. van der Woude and M.N. Harakeh, Phys. Rev. C38 (1988) 2562 16) N.K. Glendenning, Phys. Rev. C37 (1988) 2733 17) W.D. Myes, Proc. XVIII Int . Workshop on gross properties of nuclei and nuclear excitations, Hirschegg, Austria, 1990 18) B.M. Waldhauser, J.A. Maruhn, H. Stöcker and W. Greiner, Phys . Rev. C38 (1988) 1003 19) N.K. Glendenning, Nucl. Phys. A493 (1989) 521 20) J. Kapusta, Phys. Rev. C29 (1984) 1735