3

III. COMPOSITION OF DENSE MATTER

The composition of dense matter will be discussed separately for the four density domains delineated in Section 1. In the first density domain where density is 102-107 g/cm3, dense matter can in principle exist in any composition, especially at low temperatures. The constituent nuclei of the matter system may belong to any nuclide or a combination of nuclides listed in the nuclear table. A sufficient number of electrons must be present to render the system electrically neutral. Stated in such general terms, not much more needs to be added. However, since dense matter is normally found in the interior of stellar objects, its composition naturally depends on nucleosynthesis related to stellar evolution. A review of the stellar evolution process will suggest the prevalent compositions of dense matter in this density domain and the conditions under which they exist.

A star derives its energy through the fusion of hydrogen nuclei to form 4He. It is called hydrogen burning and its ignition temperature is about 107 K. The resulting helium nuclei are collected at the core of the star. As the helium core becomes hot enough and dense enough (estimated to be 1.5x108 K and 5x104 g/cm3), nuclear reactions based on 4He occur, initiating the helium flash. Helium burning generates the following sequence of relatively stable products: 12C -> 16O -> 20Ne -> 24Mg. These 4He-related processes account for the fact that after 1H and 4He, which are respectively the most and second most abundant nuclear species in stellar systems, 16O is the third most abundant, 12C the fourth, and 20Ne the fifth. Both hydrogen burning and helium burning proceed under high temperatures and high densities. The reason is nuclei are electrically charged and they do not normally come close enough for reactions to take place unless their thermal energy is large enough to overcome the repulsive electrical force. The repulsive force increases with the electric charge Z of the nuclei, which is why reactions of this type begin with low-Z nuclei and advance to high-Z nuclei as core temperature rises and density increases. These are exothermicreactions in which energy is generated.

As helium burning proceeds, the helium-exhausted core contracts sufficiently to initiate 12C + 12C and 16O + 16O reactions, called carbon burning and oxygen burning, respectively. The final products of these processes are 24Mg, 28Si and 32S. Upon the conclusion of oxygen burning, the remanent nuclei may not wait for the further contraction of the core to bring about reactions like 24Mg + 24Mg, since the electrical repulsion between them is very large and the temperature and density needed to initiate such reactions are correspondingly high. Instead, these nuclei react with photons to transform themselves in successive steps to the most tightly bound nuclide in the nuclear table, which is 56Fe, in a process called photodisintegration rearrangement. Therefore, the most likely composition of matter with a density reaching the high end of the first density domain is electrons and 56Fe nuclei.

A. Neutronization

As matter density advances into the second density domain, 107-1012 g/cm3, the ground state composition of the dense matter system changes with density. For example, if a dense matter system composed of56Fe nuclei at 107 g/cm3 is compressed, then as density increases the constituent nuclei transmute in a sequence like Fe -> Ni -> Se -> Ge and so on. Transmutations occur because the availability of high-energy electrons in a dense system is making such reactions possible, and transmutations result if the transmuted nuclei lower the total energy of the matter system. This process is called neutronization because the resulting nuclide from such a reaction is always richer in neutron content than that entering into the reaction.

The physical principle involved can be seen from the simplest of such reactions, in which an electron e is captured by a proton p to produce a neutron n and a neutrino n:

                                                                      (37)

This is an endothermic reaction. For it to proceed, the electron must carry with it substantial amount of kinetic energy, the reason being that there is a mass difference between neutron and proton that, expressed in energy units, is (mn-mp)c2 = 1.294 MeV, where 1 MeV = 106 eV. However, when matter density exceeds 107 g/cm3, electrons with kinetic energies comparable to the mass difference become plentiful, and it is indeed energetically favorable for the system to have some of the high-energy electrons captured by protons to form neutrons. The neutrinos from the reactions in general escape from the system since they interact very weakly with matter at such densities. Reactions based on the same principle as Eq. (37) are mainly responsible for the neutroniztion of the nuclei in the matter system.

Let us denote the mass of a nucleus belonging to a certain nuclide of atomic number Z and mass number A by M(Z,A) and compare the energy contents of different pure substances each of which composed of nuclei of a single species. Each system of such a pure substance will be characterized by a nuclei number density nA. Since nA is different for different systems to be compared, it is best to introduce instead a nucleon number density nB. All systems with the same density have the same nB, where the subscript B refers to baryons, a generic term including nucleons and other nucleon-like particles. Each baryon is assigned a unitbaryon number, which is a quantity conserved in all particle reactions. Thus, nB will not be changed by the reaction shown in Eq. (37) or the neutronization process in general, which makes it a useful parameter. It is related to the nuclei number density by nB = AnA and to the electron number density by ne = nBZ/A.

At zero temperature, the energy density of such a system is given by

                                               (38)

where es is the electrostatic interaction energy density and ee the degenerate electron energy density, which is to be calculated from Eq. (16b) for relativistic electrons. The masses of most stable nuclides and their isotopes have been determined quite accurately and are listed in the nuclear table. One may therefore compute e according to Eq. (38) for all candidate nuclides using listed masses from nuclear table. As a rule, onlyeven-even nuclides, which are nuclides possessing even numbers of protons and even numbers of neutrons, are needed for consideration. These nuclides are particularly stable and are capable of bringing the dense matter system to a low energy state.

The electrostatic interaction energy density es may be estimated from methods discussed in Section 2. Even though electrostatic interaction energy plays a minor role in determining the pressure of the system once the density is high, it has an effect on the composition of the system, which depends on relatively small amount of energy difference. The es may be estimated by evaluating the lattice energy due to the electrostatic interaction energy of a system of nuclei that form into a lattice, while the electrons are assumed to be distributed uniformly in space. The lattice giving the maximum binding is found to be the body-centered-cubic lattice, and the corresponding lattice energy is

                                                        (39)

where rs is the cell radius defined for Eq. (21) and is related to nB by

rs = (4pnB/3A)-1/3

For each nuclide, the energy density of the system evaluated from Eq. (38) is a function of nB. Next, plot the quantity e/nB versus 1/nB as shown in Fig. 4, where the curves corresponding to nuclides 62Fe, 62Ni and 64Ni are drawn. From the curves indicated by 62Fe and 62Ni, it is obvious that a matter system composed of 62Ni nuclei would have an energy lower than that composed of 62Fe nuclei at densities shown. The ground state composition of dense matter is determined by nuclides whose curves form the envelope to the left of all the curves, as illustrated, for example, by the curves for by 62Ni and 64Ni in Fig. 4. 

Two of these curves on the envelope cross at a certain density. This means that a change in composition occurs in the neighborhood of that density. Such a change, called a first order phase transition, begins at a density below the density where the curves cross, when some of the 62Ni nuclei are being rearranged into 64Ni, and which ends at a density above that intersection, when all of the 62Ni nuclei are changed into 64Ni. Throughout a first-order phase transition, the pressure of the system remains constant, which is a requirement of thermal equilibrium. The exact density at which a phase transition begins and the density at which it ends can be found from the tangent construction method shown in Fig. 4. By this method one draws a straight line tangent to the convex curves on the envelope as shown by the dotted lines. The values of nB at the points of tangency correspond to the onset and termination of phase transition. All points on the tangent exhibit the same pressure since the expression for pressure, Eq. (9), may be converted to read

                                                                (40)

Within the range of phase transition, the tangent now replaces the enevelope in representing the energy of the matter system.

The sequence of nuclides constituting the zero-temperature ground state of dense matter in the density range 107 -1011 g/cm3 obtained by this method with es replaced by eL is shown in Table 2. 

As indicated, neutronization begins at a density of 8.1x106 g/cm3, which is quite close to 107 g/cm3, and that is the reason why we choose 107 g/cm3 to mark the beginning of the second density domain. Note that the sequence of nuclides in the table shows a relative increase in neutron content with density, which is indicated by the diminishing Z/A ratios.

The lattice structure can be destroyed by thermal agitation. At each density, there is a correspondingmelting temperature Tm, that can be estimated by comparing the lattice energy with its thermal energy. It is usually taken to be

                                                (41)

where cm @ 50-100. Thus, for Fe matter at r = 108 g/cm3, the melting temperature is about Tm @ 2x108K.

B. Nuclear semi-empirical mass formula

As matter density exceeds 4x1011 g/cm3 or so, the ground state composition of the matter system may prefer nuclides that are so rich in neutrons that these nuclides do not exist under normal laboratory conditions, and their masses would not be listed in nuclear tables. To understand these nuclides, theoretical models of the nucleus must be constructed to deduce their masses and other properties. This is a difficult task since nuclear forces are complicated, and the problem is involved. A preliminary investigation of this problem should rely on as many empirical facts about the nucleus as possible. For the present purpose, the nuclear semi-empirical mass formula, which interpolates all known nuclear masses by means of a single expression, becomes a useful tool for suggesting the possible masses of these nuclides.

The nuclear mass is given by the expression

                   (42)

where mp, me and mn denote the proton, electron, and neutron mass, respectively, and B(Z,A) the binding energy of the nucleus. The nuclear semi-empirical mass formula for even-even nuclides expresses the binding energy in the following form:

                     (43)

where the coefficents have been determined to be aV = 15.75 MeV, aS = 17.8 MeV, aC = 0.710 MeV, aA = 23.7 MeV and aP = 34 MeV. This mass formula is not just a best-fit formula since it has incorporated many theoretical elements in its construction. This feature, hopefully, will make it suitable for extension to cover unusual nuclides.

Applying Eqs. (42) and (43) to the computation of the matter energy density given by Eq. (38), one finds matter composition with density approaching 4x1011 g/cm3. The general result is that as matter density increases, the constituent nuclei of the system become more and more massive and at the same time become more and more neutron-rich. In general, nuclei become massive so as to minimize the surface energy which is given by the term proportional to aS in Eq. (43), and they turn neutron-rich so as to minimize the electrostatic interaction energy within the nuclei, given by the term proportional to aC in Eq. (43). In general, the nuclear semi-empirical mass formula is not believed to be sufficiently accurate for application to nuclides whose Z/A ratios are much below Z/A @ 0.3. For these nuclides, we must turn to more elaborate theoretical models of the nucleus.

C. Neutron drip

When the nuclei are getting very large and very rich in neutron content, some of the neutrons become very loosely bound to the nuclei; as density is further increased, unbound neutrons begin to escape from the nuclei. The nuclei appear to be immersed in a sea of neutrons. This situation is called neutron drip, a term suggesting that neutrons are dripping out of the giant nuclei to form a surrounding neutron sea. At zero temperature, this occurs at a density of about 4x1011 g/cm3. When it occurs, matter becomes a two-phase system, with one phase consisting of the nuclei and the other consisting of neutrons (with possibly a relatively small admixture of protons). The nuclei may be visualized as liquid drops suspended in a gas consisting of neutrons. These two phases coexist in phase equilibrium. The energy densities of these two phases are to be investigated separately. If the surface effects around the nuclei may be neglected then these two phases are assumed to be uniform systems, each of which exhibits a pressure and whose neutron and proton components reach certain Fermi energies [cf. Eq. (26)]. At phase equilibrium, the pressures of these two phases must be identical, and the Fermi energies of the respective neutron and proton components of these two phases must also be identical. The composition of matter in the density domain of neutron drip is established by the system that is in phase equilibrium and at the same time reaching the lowest possible energy state.

The evaluation of the Fermi energies for the neutron and proton components depends on the nature of the nuclear interaction. Since the nuclear interaction is not as well known as the electrostatic interaction, the results derived for post-neutron drip densities are not as well established as those found for the first density domain. At the present time, there are several forms of effective nuclear interactions constructed specifically to explain nuclear phenomenology, which are quite helpful for this purpose. Effective nuclear interaction is to be distinguished from realistic nuclear interaction which is derived from nuclear scattering data and is regarded as a more fundamental form of nuclear interaction.

D. Liquid drop model

A nucleus in many respects resembles a liquid drop. It possesses a relatively constant density over its entire volume except near the surface, and the average interior density is the same for nuclei of all sizes. A model of the nucleus that takes advantage of these features is the liquid drop model. It considers the total energy of the nucleus to be the additive sum of its bulk energy, surface energy, electrostatic energy, and translational energy. The bulk energy is given by multiplying the volume of the nucleus by the energy density of a uniform nuclear matter system with nuclear interaction included. The surface energy is usually taken to be a semi-empirical quantity based on calculations for nuclei of finite sizes and on experimental results on laboratory nuclei. In a two-phase situation the surface energy must be corrected for nucleon concentration outside the nucleus. The electrostatic energy involves the interaction among all charged particles inside and outside the nucleus. The position of the nuclei may again be assumed to form a body-centered-cubic lattice. Both electrostatic lattice energy and electrostatic exchange energy contribute to it. The translational energy is due to the motion of the nucleus.

At high temperatures, the bulk energy must be computed according to a system of partially degenerate nucleons. This is usually done by employing effective nuclear interaction, in which case the nucleons are assumed to be uncorrelated, and the Fermi-Dirac distribution of Eq. (32) may be directly applied to the nucleons. The problem is much more complicated if the realistic nuclear interaction is employed, in which case particle correlations must be included, and for this reason the computation is much more elaborate. The surface energy shows a reduction with temperature. This result can be extracted from finite nuclei calculations. The lattice energy also shows temperature modification, since nuclei at the lattice points agitate with thermal motions. The same thermal motion also contributes to the translational energy of the nuclei, which may be assumed to possess thermal velocities given by the Boltzmann distribution.

The results of a study based on the liquid drop model of the nucleus is depicted in Fig. 5, where the variation of the size of a nucleus (its A number) with matter density and temperature is shown. The study is done with an effective nuclear interaction called the Skyrme interaction and for a matter system having an overall electron fraction Ye = 0.25. A wide range of temperatures, expressed in terms of kBT in units of MeV, is included. Note that the temperature corresponding to kBT = 1 MeV is T = 1.16x1010 K. The dashed lines indicate the A numbers of the nuclei at various densities and temperatures. The solid line forms the boundary separating a one-phase system from a two-phase system. The conditions for a two-phase system are included in the plot under the solid line. The two-phase condition disappears completely when matter density exceeds 2x1014 g/cm3. Just before that density, there exists a density range where the nuclei in the system merge and trap the surrounding neutrons into a form of bubbles. This range is indicated by the cross-hatched area in the plot and is labeled "bubble".

E. Neutron matter

In general, when matter density exceeds 2x1014 g/cm3, all nuclei are dissolved into a homogeneous system, and for a range of densities above that, matter is composed almost entirely of neutrons. The admixture of protons is negligibly small ( about one percent), because all protons must be accompanied by an equal number of electrons which, being very light, contribute a large amount of kinetic energy to the system. Hence, all ground state systems tend to avoid the presence of electrons. Such a nearly pure neutron system is called neutron matter.

The average density of the atomic nucleus, quite independent of its species, is approximately 2.8x1014 g/cm3, and therefore many of the methods employed for the study of the nucleus may be applied to the study of neutron matter. As we have mentioned before, the distribution of mass density of a nucleus has been found to be quite uniform over its entire volume and that uniform mass density is very nearly the same from one nuclide to the other. For a heavy nucleus, its volume is extended so as to maintain a mass density common to nuclei of all sizes. This remarkable fact is described as nuclear saturation. Nuclear substance possessing this mass density is called nuclear matter, and its mass density, called nuclear density, is found to be 2.8x1014 g/cm3.

Neutron matter, however, is not nuclear matter. Neutron matter consists nearly entirely of neutrons, whereas nuclear matter consists of neutrons and protons in roughly equal fractions; and while nuclear matter forms bound units or nuclei, neutron matter is an unbound system. The ground state composition of an extended system is not given by that of nuclear matter but neutron matter. Neutron matter does not exist terrestrially; its existence is only inferred theoretically. Neutron matter may be studied in analogy to nuclear matter. To accomplish this there are methods based on realistic nuclear interaction expressed in the form of nuclear potentials, the best known of which are the Brueckner-Bethe-Goldstone method and the constrained variational method. There are also methods based on the use of effective nuclear interaction. These methods combine the Hartree-Fock method with some form of effective nuclear potential. Though they are considered less fundamental than the two methods mentioned before, they usually yield more direct and accurate results at nuclear density where they are designed to perform. Since neutron matter exists for a range of densities, the choice of an applicable method depends also on how well these methods may be extended to cover such a wide range of densities. In this respect, a phenomenological model called the relativistic mean field model seems very attractive. We shall return to these methods in the next section when the equation of state of neutron matter is discussed.

It has been seriously proposed that known nuclear interaction forces would make neutron matter superfluid and the proton component in it superconductive when its temperature is below a critical temperature Tc which is estimated to be as high as 109-1010 K. These phenomena would have profound effects on the transport properties of neutron matter, and will be discussed in more detail in Sections 5 and 6.

Since neutrons, like electrons, are fermions and obey Pauli's exclusion principle, many features of a dense degenerate electron system are exhibited by neutron matter. In spite of having a strong nuclear interaction, these features still play dominant roles in the neutron system. Thus, results based on a dense degenerate neutron system serve as useful guides to judge the properties of neutron matter over a wide range of densities. Such guidance is particularly needed when hyperons and other massive particles begin to make their appearance in dense matter, because the precise nature of their interactions is still poorly known.

F. Baryonic matter

As matter density advances beyond 1015 g/cm3 and into the fourth and last density domain, many new and unexpected possibilities may arise. It is fairly sure that light hyperons appear first. Hyperons are like nucleons in every aspect except that they are slightly more massive and carry a non-zero attribute called hypercharge, or strangeness. They are L(1115), and S(1190), where the numbers in the parantheses give their approximate masses, expressed as mc2 in units of MeV. L is electrically neutral, but there are three species of S with charges equal to +1, 0, and -1 of the proton charge. More exact values of their masses may be found in a table of elementary particles. These particles are produced in reactions such as

               (44a)