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IV. EQUATION OF STATE OF DENSE MATTER

The equation of state of a substance is a functional relation between the pressure generated by the substance and its density and temperature. It reflects the composition and internal structure of the substance. The equation of state of dense matter is of prime importance to astrophysical study of compact stellar objects like white dwarf stars and neutron stars. It will be discussed in details for the first three density domains. Our knowledge about dense matter in the fourth density domain is insufficient to provide accurate quantitative evaluation of it properties at this time.

A. The First Density Domain

For the first density domain (102-107 g/cm3), we shall concentrate our discussion on a matter system composed of 56Fe nuclei, which represents the most stable form of matter system in that domain and shall be referred to as Fe matter. The equation of state of Fe matter at zero temperature in this density range is determined basically by the degenerate electrons in the system. Electrostatic interaction among the charged particles plays a relatively minor role, which has been demonstrated in Section 2. Nevertheless, the degenerate electron pressure must be corrected for electrostatic effects. For Fe matter with densities below 104g/cm3, electrostatic interaction must be computed on the basis of the Thomas-Fermi-Dirac method which is discussed in Section 2, and for Fe matter with densities above 104 g/cm3, the electrostatic interaction may be computed on the assumption that electrons are distributed uniformly around the nuclei.

> Figure 6 shows the ground state equation of state of dense matter at zero temperature in a log-log plot over the first three density domains. The portion below 107 g/cm3 belongs to the equation of state for Fe matter in the first density domain. The numerical values of this equation of state are given in Table 3 under the heading BPS. The adiabatic indices are listed under G. The equation of state in this density domain should be quite accurate since its composition is well established.

At high temperatures, degenerate electrons become partially degenerate in a way described by Eq. (32). Finite temperature equations of state will not be shown here. General behaviors of the isotherms and adiabats should be similar to those shown in Fig. 3. At densities below 104 g/cm3, where the energy correction due to electrostatic interaction is important, the Thomas-Fermi-Dirac method must be extended to include thermal effects. Interested readers are referred to references listed in the Bibliography for further details.

B. The Second Density Domain

The ground state composition of matter in the second density domain (107-1012 g/cm3) is described in Section 3. The composition varies with density due to neutronization of the constituent nuclei. As the composition changes from one form to another, a first-order phase transition is involved and, over the density range where phase transition occurs, the pressure remains constant. Therefore, in a detailed plot of the equation of state in this density domain, pressure rises with density except at regions of phase transition where it remains constant. The equation of state appears to rise in steps with increasing density, but since the steps are quite narrow, the equation of state in this density domain may be approximated by a smooth curve.

The pressure of the matter system is due entirely to the degenerate electrons. In establishing the ground state composition of the matter system electrostatic interaction energy in the form of lattice energy has been included, but it is quite negligible as far as the pressure of the system is concerned. The pressure in this density domain does not rise as rapidly with density as it does in the first density domain because the ground state composition of the system shows a gradual decline in the Z/A values with density. The composition is quite well established up to a density of 4 x 1011 g/cm3, which marks the onset of neutron drip. The composition consists of nuclei found under normal laboratory conditions or their nearby isotopes, and their masses can be either measured or extrapolated from known masses with reasonable certainty. The ground state equation of state at zero temperature in this density range is shown in Fig. 7. The numerical values of the equation of state before neutron drip, together with the atomic number Z and mass number A of the constituent nuclei at these densities, are listed in Table 3 under the heading BBP.

C. The Third Density Domain

The third density domain (1012-1015 g/cm3) begins properly with the onset of neutron drip, which occurs at a density of 4 x 1011 g/cm3. With the onset of neutron drip, matter is composed of giant nuclei immersed in a sea of neutrons. Nucleons inside the nuclei coexist with nucleons outside the nuclei forming a two-phase system. Theoretical studies of such a system rely heavily on the nuclear liquid drop model which gives a proper account of the different energy components in a nucleus. The nuclear liquid drop model is described in Section 3. At each density, matter is composed of a particular species of nuclei characterized by certain Z and A numbers, which exist in phase equilibrium with the neutron sea outside which has a much lower Z/A ratio. Such a two-phase system constitutes the ground state composition of matter in this density range.

Once the ground state composition is established, the equation of state can be found as before by establishing the electron fraction Ye of the system and proceeding to evaluate the electron pressure. The electron pressure remains the main pressure component of the system until neutron pressure takes over at higher densities. Electron number density, however, will be kept fairly constant throughout the neutron drip region. It is the neutron number density and not the electron number density that is rising with increasing matter density. The suppression in electron number density with increasing density has kept the system pressure relatively constant in the density range between 4 x 1011-1012 g/cm3. This is shown in Figure 7. Eventually enough neutrons are produced and the system pressure is taken over entirely by the neutron pressure. Then the equation of state shows a rapid rise subsequent to 1012 g/cm3.

The numerical values of this portion of the equation of state are listed in Table 3 under the heading BBP. They are computed by means of the so called Reid soft-core potential, which is determined phenomenologically by fitting nucleon-nucleon scattering data at energies below 300 MeV, as well as the properties of the deuteron. It is considered one of the best realistic nuclear potentials applicable to nuclear problems. The computation is done in the elaborate pair approximation which includes correlation effects between pairs of nucleons. In practice, this usually means solving the Brueckner-Bethe-Goldstone equations for the nucleon energy of each quantum state occupied by the nucleons. The summation of the nucleon energies for all occupied states yields the energy density of the system. Computations with the realistic nuclear potential are very involved, and several additional corrections are needed to achieve agreement with empirical results. Extension of the method to include finite temperature calculations has not been attempted.

A second form of approach is described as independent particle approximation, in which case all particles are uncorrelated and move in the system without experiencing the presence of the others except through an overall nuclear potential. The success of the method depends on the adequacy of the effective potential that is prescribed for interaction between each pair of nucleons. A large variety of nuclear effective potentials has been devised. Potentials are usually expressed in functional forms depending on the separation between the interacting pair of nucleons. These potentials depend not only on the particle distance but also on their spin orientations. Some even prescribe dependence on the relative velocity between the nucleons. One effective nuclear potential deserving special mention is the Skyrme potential which, like the Reid soft-core potential, belongs to the class of velocity-dependent potentials. Computations based on effective nuclear potentials seem to constitute the only viable method in dealing with the neutron drip problem at finite temperatures. Some results on finite temperature equations of state in the neutron drip region have been obtained by means of the Skyrme potenial. The difficulty in working with a neutron drip system lies in the treatment of phase equilibrium, which must be handled with delicate care. The nuclear liquid drop model serves to reduce much of that work to detailed algebraic manipulations. Still, quantitative results of finite temperature equations of state in the neutron drip region are scarce.

D. Neutron Matter Region

As matter density increases towards 1012 g/cm3, the constituent nuclei become so large and their Z/A ratio so low that they become merged with the neutron sea at a density of approximately 2 x 1014 g/cm3, where the phenomenon of neutron drip terminates, and the ground state of the matter system is represented by nearly pure neutron matter. Since neutron matter is so similar to nuclear matter, all successful theories that describe nuclear matter properties have been applied to predict the properties of neutron matter. The ground state equation of neutron matter that is determined by the pair approximation is listed in Table 2 under BBP. It terminates at a density of 5 x 1014 g/cm3, which is the upper limit of its applicability. There are also results obtained by the constrained variational method and the independent particle method with the Skyrme potential. These different equations of state are compared in Fig. 8. It gives us some idea as to how unsettled the issue remains at the present time.

The numerical results of the ground state equation of state obtained by the constrained variational method are listed in the Table 3 under the heading FP. They are evaluated from a form of realistic nuclear potential by a method that solves the nuclear many-body problem by means of a variational technique. The finite temperature equation of state evaluated by this method is also available.

The numerical results of the ground state equation of state evaluated from the Skyrme potential are listed in Table 3 under the heading SKM. Being an effective potential, the Skyrme potential contains adjustable parameters that are established by fitting nuclear properties. As the potential is tried out by different investigators, new sets of potential parameters are being proposed. The results given here are based on a recent set of parameters designated as SKM in the literature.

There are also attempts to formulate the nuclear interaction problem in a relativistic formalism. This seems to be quite necessary if the method is to be applicable to matter density in the 1015 g/cm3 region. A simplified nuclear interaction model based on the relativisitic quantum field theory seems quite attractive. It is called the relativistic mean field model. In this model, nuclear interaction is described by the exchange of mesons. In its simplest form, only two types of mesons, a scalar meson and a vector meson, both electrically neutral, are called upon to describe nuclear interaction at short range. The scalar meson is called the s-meson and the vector meson the w-meson. Employed on a phenomenological level, the model needs only two adjustable parameters to complete its description, and when it is applied to study the nuclear matter problem, it successfully predicts nuclear saturation. These two parameters may then be adjusted so that saturation occurs at the right density and with the correct binding energy. The model is then completely prescribed and may then be extended to study the neutron matter problem. The equation of state thus obtained for neutron matter at zero temperature is plotted in Fig. 7, together with others for comparison. The numerical results are listed in Table 2 under the heading RMF.

The relativistic mean field model shows great promise as a viable model for dense matter study. In the present form it has some defects. For example, it over-predicts the value of the compression modulus of nuclear matter. Also, the inclusion of a few other types of mesons may be needed to improve on the description of nuclear interaction. In particular, a charge vector meson that plays no role in the nuclear matter system contributes nevertheless to the neutron matter system and should be included. Since empirical results on dense matter are limited, it is difficult to fix the added parameters due to the new mesons in a phenomenological way. Future improvements of this model may have to be based on new results coming from experimental heavy ion collision results.

The diagonal line labelled causality limit in Fig. 7 represents an equation of state given by P = c r. Any substance whose equation of state extends above the this line would give rise to a sound speed exceeding the speed of light c, which is not allowed because it would violate causality; thus the causality limit defines an upper limit to all equations of state. It is drawn here to guide the eye.

The pressure of neutron matter rises rapidly with increasing density. Such an equation of state is called stiff. This feature is already exhibited by a freeneutron system that is devoid of interaction. The equation of state of a free neutron system is plotted alongside the others in Fig. 8 for comparison. It is labeled free neutron. The stiffness of the free neutron equation of state had led scientists in the 1930s to suggest the possible existence of neutron stars in the stellar systems almost as soon as neutrons were discovered. Neutron stars were detected a quarter of century later.

E. Pion Condensation Region

The stiffness of the equation of state of neutron matter could be greatly modified should pions be found to form a condensate in neutron matter. A pion condensate occurs in a matter system only if the interaction between the nucleons and pions lower the interaction energy sufficiently to account for the added mass and kinetic energy of the pions. The possibility of pion condensation has been suspected for a long time since pion-nucleon scattering experiments have revealed a strong attractive interaction between pions and nucleons, and this attractive mode could produce the pion condensation phenomenon. However, a quantitatively reliable solution of this problem is yet to be established. Nuclear interactions are hard to deal with, and most theoretical results are not considered trustworthy unless they can be collaborated by some empirical facts. In the pion condensation problem, no empirical verification is available, and many of the estimates about pion condensation, though plausible, cannot be easily accepted.

The effect of pion condensation on the equation of state of neutron matter is to lower the pressure of the system, because it adds mass to the system without contributing the comparable amount of kinetic energy. The softening of the equation of state is a reflection of the presence of an attractive interaction that brings about pion condensation and serves to lower the pressure. Some estimates have suggested that pion condensation should occur at a neutron matter density of around 5 x 1015 g/cm3. No quantitative results, however, will be presented here for the pion condensation situation.

F. Baryonic Matter

As the density of neutron matter increases, the Fermi energy of the neutron system increases accordingly. If we neglect interactions among the particles, then as soon as the neutron Fermi energy reaches the energy corresponding to the mass difference of the neutron and L, the ground state of the matter system favors the replacement of the high-energy neutrons by L. L is the least massive baryon heavier than the nucleons. Based on an estimate in which interactions are neglected, L should appear in neutron matter at a density of 1.6 x 1015 g/cm3. On a scale of ascending mass, the next group of baryons after the nucleons an L is the s, which consists of three charged species, and after that is the D, which consists of four charged species. Similar estimates put the appearance of s at 2.9 x 1015 g/cm3 and the appearance of D at 3.7 x 1015 g/cm3.

The above estimates emphasize the mass of the baryons. It turns out that the electric charge of the baryon is also important. For example, when a s- appears, it replaces not only an energetic neutron but also an energetic electron. Again, based on estimates in which interactions are neglected, s- appears at a density of 1015 g/cm3 and D- at 1.3 x 1015 g/cm3, which are lower values than those given above.

To study baryonic matter properly, particle interactions must be included. Unfortunately, our knowledge of baryon interaction is still quite limited, and the interaction strengths among the different species of baryons are not yet established with sufficient accuracy. Therefore, much of our understanding of the baryonic matter is based on theoretical conjectures of the nature of baryon interaction.

One of the best methods appropriate to the study of baryonic matter is the relativistic mean field model mentioned before. The model considers all particle interactions to be mediated by two mesons, the scalars-meson and the vector w-meson, both electrically neutral. Particle interaction is established once the coupling constants of these mesons with different baryons are known. There are theoretical justifications to believe that the w-meson interacts with all baryons at equal strength, in the same manner that the electric field interacts with all electric charges equally no matter which particles carry them. In analogy with the electric charge, each baryon is associated with a unit of baryonic charge and its anit-particle with a negative unit of baryonic charge. The baryonic force mediated by the w-meson behaves very much like the electric force in the sense that like charges repel and unlike charges attract. It is repulsive between baryons but attractive between baryons and anti-baryons. If this is the case, the coupling constants of the w-meson with all baryons in the relativistic mean field model may be taken to be the same as that adopted for the nucleons. One possible fallacy in this reasoning is that the coupling constants in the relativistic mean field model represent effective coupling constants that incorporate modifications due to the presence of neighboring particles. Therefore, even though the concept of a baryonic charge to which the w-meson couples is correct, the effective w-meson coupling constants with different baryons in the relativistic mean field model need not be the same.

For the s-meson, the best estimate of its coupling constants is probaby by means of the quark model which assigns definite quark contents to the baryons and s-meson. If the quarks are assumed to interact equally among themselves, and given the fact that particle attributes such as electric charge and hypercharge must be conserved in the interaction process, it is possible to deduce that the s-meson couples equally between nucleons and D, and also equally between L and s, but its coupling with L and s is only two-thirds of the coupling with the nucleon and D.

Incorporating these couplings in the relativistic mean field model, we can deduce the equation of state for baryonic matter. It shows a slight decrease in pressure compared with that due to neutron matter when matter density exceeds 1015 g/cm3. The effect of admixing other baryons in a neutron matter system is relatively minor on the system's equation of state. On the other hand, the effect of particle interaction on the equation of state is quite significant. Therefore, a proper understanding of the equation of state of matter in the 1015 g/cm3 range awaits better knowledge of particle interactions as well as methods in dealing with a many-body system.

G. Quark matter

In a quark model of the elementary particles, the nucleon is viewed as the bound state of three quarks, which interact via the exchange of gluons as determined by quantum chromodynamics. The size of the nucleon is therefore determined by the confining radius of the quark interaction. When matter density reaches the point that the average separation of the nucleons is less than its confining radius, the individual identity of the nucleon is lost, and the matter system turns into a uniform system of quarks forming quark matter.

Quarks are fermions and like electrons and neutrons obey Pauli's exclusion principle. There are different species of quarks, and the known quarks are given the names of up quark (u), down quark (d), strange quark (s), and charm quark (c). There may be others. Inside a nucleon, their effective masses are estimated to be (expressed in mc2) under 100 MeV for the u and d quarks, approximately 100-200 MeV for the s quark, and approximately 2000 MeV for the c quark. If quark interaction is neglected, the equation of state of quark matter may be deduced in the same manner as it is for a degenerate electron system. The treatment of quark interaction turns out to be not as difficult as it is for neutrons. In a quark matter, the effective quark interaction is relatively weak, and perturbative treatment of the interaction is possible. This interesting feature of quantum chromodynamics prescribes that the effective interaction of the quarks should decrease in strength as the quarks interact in close proximity to each other, and this has been verified to be true in experiments. The equation of state of quark matter at zero temperature computed with quark interaction has been obtained. One version of it is plotted in Fig. 7 for illustration.

The transition from baryonic matter to quark matter is a first-order phase transition that may be established by a tangent construction method on the energy density of the baryonic matter and that of the quark matter in a manner similar to that discussed in Section 3 for the neutronization process and illustrated in Fig.4. The onset of transition from neutron matter to quark matter has been estimated to begin at densities around 1.5 to 4 x 1015 g/cm3.

In Table 3, only the basic ground state equation of state of dense matter is presented. It serves to suggest the possible behavior of matter compressed to high densities. It would be too elaborate to detail all aspects of the equation of state at finite temperatures. At the present time, the study of dense matter is being actively pursued.