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II. BASIC THEORETICAL METHOD

Matter may first be considered as a homogeneous system of atoms without any particular structure. Such a system may be a finite portion in an infinte body of such substance, so that boundary effects on the system are minimized. The system in the chosen volume possesses a fixed number of atoms. At densities of interest all atoms in it are crushed, and the substance in the system is best described as a plasma of atomic nuclei and electrons. We begin by studying pure substances each formed by nuclei belonging to a single nuclear species, or nuclide. The admixture of other nuclides in a pure substance can be accounted for and will be considered after the general method of investigation is introduced.

The physical properties of the system do not depend on the size of the volume, since it is chosen arbitrarily, but depend on parameters such as density, which is obtained by dividing the total mass of the system by its volume. The concept of density will be enlarged by introducing a host of density-like parameters, all of which are obtained by dividing the total quantities of these items in the system by its volume. The term density will be qualified by calling it mass density whenever necessary. In addition, parameters such as the electron number density and the nuclei number density will be introduced.

Each nuclide is specified by its atomic number Z and mass number A. Z is also the number of protons in the nucleus and A the total number of protons and neutrons there. Protons and neutrons have very nearly the same mass and also very similar nuclear properties; they are often referred to collectively as nucleons. Each nuclide will be designated by placing its mass number as a left-hand superscript to its chemical symbol, such as 4He and 56Fe.

Let the system consist of N atoms in a volume V. The nuclei number density is

and the electron number density is

which is the same as the proton number density, because the system is electrically neutral. The mass of a nuclide is given in nuclear mass tables to high accuracy. To determine the mass density of a matter system we usually do not need to know the nuclear mass to such accuracy and may simply approximate it by the quantity Amp, where mp=1.67x10-24 g is the proton mass. The actual mass of a nucleus should be slightly less than that because some of the nuclear mass, less than one percent, is converted into the binding energy of the nucleus. The mass density r of the matter system is given simply by

For example, for a system composed of electrons and helium nuclei, for which Z = 2 and A = 4, a mass density of 100 g/cm3 corresponds to a nuclei number density of nA = 4xl024 cm-3 and an electron number density ne = 2nA.

Like all material substances, dense matter possesses an internal pressure that resists compression, and there is a definite relation between the density of a substance and the pressure it generates. The functional relationship between pressure and density is the equation of state of the substance and is a very important physical property for astrophysical study of stellar objects. The structure of a stellar object is determined mainly by the equation of state of the stellar substance. In this section, the method for the derivation of the equation of state will be illustrated.

Unlike a body of low-density gas, whose pressure is due to the thermal motions of its constituent atoms and is therefore directly related to the temperature of the gas, dense matter derives its pressure from the electrons in the system. When matter density is high and its electron number density is correspondingly high, an important physical phenomenon is brought into play, which determines many physical properties of the system. We shall illustrate the application of this phenomenon to the study of dense matter with densities lying in the first density domain, 102 - 107 g/cm3.

A. Pauli's Exclusion Principle

It is a fundamental physical principle that all elementary particles, such as electrons, protons, neutrons, photons, and mesons, can be classified into two major categories called fermions and bosons. The fermions include electrons, protons, and neutrons, while the bosons include photons and mesons. Atomic nuclei are regarded as composite systems and need not fall into these classes. In this study, one fermionic property, Pauli's exclusion principle, plays a particularly important role. The principle states that no two identical fermions should occupy the same quantum state in a system. Let us first explain the meaning of identical fermions. Two fermions are identical if they are of the same type and have the same spin orientation. The first requirement is clear but the second deserves further elaboration. A fermion possesses intrinsic angular momentum called spin with magnitude equal to 1/2, or 3/2, or 5/2,.... basic units of the angular momentum (each basic unit equals Planck's constant divided by 2p). The electron spin is equal to 1/2 of the basic unit, and electrons are also referred to as spin-half particles. Spin is a vector quantity and is associated with a direction. In the case of electron spin, its direction may be specified by declaring whether it is oriented parallel to or antiparallel to a chosen direction. The fact that there are no other orientations besides these two is a result of nature's quantal manifestation, the recognition of which laid the foundation of modern atomic physics. These two spin orientations are simply referred to as spin-up and spin-down. All spin-up electrons in a system are identical to each other, as are all spin-down electrons. The term identical electrons is thus defined. Normally, these two types of electrons are evenly distributed, since the system does not prefer one type of orientation over the other.

We come now to the meaning of a quantum state. Each electron in a dynamical system is assigned a quantum state. The quantum state occupied by an electron may be specified by the electron momentum, denoted by p, a vector that is expressable in Cartesian coordinates as px, py, pz. When an electron's momentum is changed to p' in a dynamical process, it moves to a new quantum state. The momentum associated with the new quantum state p' must, however, differ from p by a finite amount. For a system of uniformly distributed electrons in a volume V, this amount may be specified in the following way. If each of the three momentum components are compared, such as px and px', they must differ by an amount equal to h/L, where h is Planck's constant and L = V1/3. In other words, in the space of all possible momenta for the electrons, each quantum state occupies a size of h3/V, which is the product of the momentum differences in all three components.

B. Fermi Gas Method

According to Pauli's exclusion principle, all electrons of the same spin orientation in a system must possess momenta that differ from each other by the amount specified above. The momenta of all electrons are completely fixed if it is further required that the system exist in its ground state, which is the lowest possible energy state for the system because there is just one way that this can be accomplished, which is for the electrons to occupy all the low-momentum quantum states before they occupy any of the high-momentum quantum states. In other words, there exists a fixed momentum magnitude pF; and in the ground state configuration, all states with momentum magnitudes below pF are occupied while the others with momentum magnitudes above pF are unoccupied. Graphically, the momenta of the occupied states plotted in a three-dimensional momentum space appear like a solid sphere centered about the point of zero momentum. The summation of all occupied quantum states is equivalent to finding the volume of the sphere, since all states are equally spaced from each other inside the sphere. Let the radius of the sphere be given by pF, called the Fermi momentum, then the integral for the volume of the sphere with the momentum variable expressed in the spherical polar coordinates is given by:

where p is the magnitude of p, p = |p|. The number of electrons accommodated in this situation is obtained by dividing the above momentum volume by h3/V and then multipling by 2, which is to account for electrons with two different spin orientations. The result is proportional to the volume V which appears because the total number of electrons in the system is sought. The volume V is divided out if the electron number density is evaluated, which is given by

Equation (5) may be viewed as a relation connecting the electron number density ne to the Fermi momentum pF. Henceforth, pF will be employed as an independent variable in establishing the physical properties of the system.

The total kinetic energy density (energy per unit volume) of the electrons can be found by summing the kinetic energies of all occupied states and then dividing by the volume. Each state of momentum p possesses a kinetic energy of p2/2me, where me is the electron mass, and the expression for the electron kinetic energy density is

The average kinetic energy per electron is obtained by dividing ee by ne:

where pF2/2me is the kinetic energy associated with the most energetic electrons in the system, and here we see that the average kinetic energy of all electrons in the system is just six-tenths of it. This is an important result to bear in mind. It tells us that pF is a representative parameter for the system and may be used for order of magnitude estimates of many of the energy related quantities. The method described here for finding the energy of the system of electrons may be applied to other fermions such as protons and neutrons and is referred to generally as the Fermi gas method.

It is now instructive to see what would be the average electron kinetic energy in a matter system that is composed of 4He nuclei at a density of 100 g/cm3. From the electron number density that we have computed before, ne = 8x1024 cm-3, we find that

where eV stands for electron-volts. In evaluating expressions such as Eq. (8), we shall follow a scheme that reduces all units needed in the problem to two by inserting factors of h or c at appropriate places. These two units are picked to be electron-volts (eV) for energy and centimeters (cm) for length. Thus, instead of expressing the electron mass in grams it is converted into energy units by multiplying by a factor of c2, and mec2 = 0.511x106 eV. The units for h can also be simplified if they are combined wtih a factor of c, since hc = 1.24 x10-4 eV-cm. This scheme is employed in the evaluation of Eq.(8) as the insertions of the c factors are explicitly displayed.

The average kinetic energy per electron given by Eq.(8) is already quite formidable, and it increases at a rate proportional to the two-thirds power of the matter density. For comparison, we estimate the average thermal energy per particle in a system, which may be approximated by the expression kBT, where kB is Boltzmann's constant and T the temperature of the system in degrees-Kelvin (K). The average thermal energy per particle at a temperature of 106 K is about 100 eV. Consequently, in the study of compact stellar objects whose core densities may reach millions of g/cm3 while the temperature is only 107 K, the thermal energy of the particles may justifiably be ignored.

Electrons belonging to the ground state of a system are called degenerate electrons, a term that signifies the fact that all electron states whose momenta have magnitudes below the Fermi momentum pF are totally occupied. Thus, whenever electrons are excited dynamically or thermally to states with momenta above pF, leaving states with momenta below pF empty, such a distribution of occupied states is called partially degenerate. Partially degenerate electrons will be discussed later in connection with systems at high temperatures.

C. Pressure

The internal pressure P of a system of particles is given by the theromdynamic expression:

where E is the total energy of the system and N its particle number in the volume V. For example, in the case of the electron gas, E= eeV and N = neV. The evaluation of the derivative in Eq. (9) is best carried out by using pF as an independent variable and converting the derivative into a ratio of the partial derivatives of E and V with respect to pF, while keeping N fixed. For a non-interacting degenerate electron system, the pressure can also be derived from simple kinematical considerations, and the result is expressed as

where v is the velocity of the electron, v = p/me. By either method, the electron pressure is evaluated to be

Pressure due to a degenerate Fermi system is called degenerate pressure.

Pressure generated by the nuclei may be added directly to the electron pressure, treating both as partial pressures. They contribute additively to the total pressure of the system. Since nuclei are not fermions (with the sole exception of the hydrogen nuclei, which are protons), they do not possess degenerate pressure but only thermal pressure, which is non-existent at zero temperature. In the case of the hydrogen nuclei, the degenerate pressure they generate may also be neglected when compared with the electron pressure, since the degenerate pressure is inversely proportional to the particle mass; and the proton mass being 2000 times larger than the electron mass makes the proton pressure 2000 times smaller than the electron pressure (bearing in mind also that the proton number density is the same as the electron number density). Thus the pressure from the system P is due entirely to the electron pressure:

D. Relativistic Electrons

Because the electrons have very small mass, they turn relativistic at fairly low kinetic energies. Relativistic kinematics must be employed for the electrons when the kinetic energy approaches the electron rest energy mec2. For a non-interacting degenerate electron system, its most energetic electrons should reach this energy threshold when the electron number density is ne = 2x1030 cm-3, which translates into a helium matter density of r = 3 x107 g/cm3.

In the relativistic formalism, the evaluation of the electron number density of the system remains unchanged since it depends only on pF. The evaluation of the electron energy density must, however, be modified by replacing the individual electron kinetic energy from the expression p2/2me to the relativistic expression

so that

The electron pressure may also be found from Eq. (9) without modification, but if Eq. (10) is used, the electron velocity must be modified to the relativistic form, which is

In summary, the physical quantities of a non-interacting degenerate electron system are

where le = h/mec is the electron Compton wavelength having the dimension of a length and t = pF/mec, which is dimensionless.

E. Equation of State

The mass density of dense matter given by Eq. (3) may be expressed in terms of the electron number density as:

where Ye, called the electron fraction, is the ratio of the number of electrons to nucleons in the sytem. For a pure substance, it is just Ye = Z/A, but for a mixed substance, where a variety of nuclides are present, it is given by

where the subscript i designates the nuclide type and xi denotes the relative abundance of that nuclide in the system. For example, Ye = 1.0 for a pure hydrogen system, Ye = 0.5 for a pure 4He system, and Ye = 26/56 = 0.464 for a pure 56Fe system. All pure substances that are composed of nuclides lying between He and Fe in the atomic periodic table have electron fractions bounded by the values 0.464 and 0.5. Since pressure depends on ne of a system while its mass density depends on ne/Ye, it is clear that matter systems with the same Ye would have the same equation of state.

In Fig. 1, typical equations of state of dense matter at zero temperature are plotted for the cases of Ye = 1.0 and Ye = 0.5. The pressure is due entirely to that of a non-interacting degenerate electron system and is computed from Eq. (16c). Pressure is expressed in units of dynes per square centimeter (dyn/cm2). From the approximate linearity of these curves on a log-log plot, it is apparent that the pressure and density are related very nearly by a power law. It is therefore both convenient and useful to express the equation of state in the form

where

is called the adiabatic index of the equation of state. G should be approximately constant in the density range belonging to the first density domain. We find G = 5/3 at the low-density end of the density domain, and it decreases slowly to values approaching G = 4/3 at the high-density end of the regime. The magnitude of G is a measure of the "stiffness" of the equation of state, in the language of astrophysics. A high G means the equation of state is stiff (the substance is hard to compress). Such information is of course of utmost importance to the study of stellar structure.

The numerical values of the equation of state of dense matter composed of helium nuclei at zero temperature are listed in Table l. In spite of the fact that this equation of state is obtained by neglecting electrostatic interaction, it should still be applicable to the study of white dwarf stars which are expected to be composed predominantly of dense helium matter. As we shall see below, the electrostatic interaction is not going to be important at high densities. The exact composition of the star is also not important in establishing a representative equation of state for the substance forming the white dwarf star, since all nuclides from He to Fe yield very similar Ye factors; therefore any mixture of these nuclides would give very nearly the same equation of state. The maximum core density of the largest white dwarf star should not exceed 107 g/cm3.

F. Electrostatic Interaction

In a plasma of electrons and nuclei, the most important form of interaction is the electrostatic interaction among the charged particles, which could modify the total energy of the system. Several estimates of the electrostatic interaction energy will be given here beginning with the crudest.

Each electron in the plasma experiences attraction from the surrounding nuclei and repulsion by the other electrons. Its electrostatic interaction does not depend on the overall size of the system chosen for the investigation, since it is electrically neutral and should depend only on the local distribution of the electric charges. The positive charges carried by the protons are packed together in units of Z in the nucleus, while the negative charges carried by the electrons are distributed in the surrounding space. To investigate the electrostatic energy due to such a charge distribution, let us imagine that the system can be isolated into units, each occupied by a single nucleus of charge Z and its accompanying Z electrons. The volume of the unit is given by Z/ne, since 1/ne is the volume occupied by each electron and there are Z electrons in the unit. The shape of the unit may be approximated by a spherical cell with the nucleus residing at its center. In each spherical cell, the electrons are distributed in a spherically symmetric way about the center. In other words, if a set of spherical polar coordinates were introduced to describe this cell, with the coordinate origin at the nucleus, the electron distribution can only be a function of the radial variable r and not of the polar angle variables. With such a charge distribution, the cell appears electrically neutral to charges outside the cell, and all electrostatic energy of such a unit must be due to interactions within the cell. The radius of the cell rs is given by the relation (4p/3)rs3 = Z/ne. A more accurate treatment would call for the replacement of the spherial cell by polygonal cells of definite shapes by assuming that all the nuclei in the system are organized into a lattice structure, but the energy corrections due to such considerations belong to higher orders and may be ignored for the time being.

The determination of the radial distribution of the electrons within the cell is complicated by the fact that it cannot be established by classical electrostatic methods, because in this case with cell size in atomic dimensions, Pauli's exclusion principle plays a major role. As a first estimate we may assume the electrons are distributed uniformly within the cell. The interaction energy due to electron-nucleus interaction is

and that due to electron-electron interaction is

giving a total electrostatic energy per cell of

In these expressions, the electric charge is expressed in the Gaussian (c.g.s.) units, so the atomic fine-structure constant is given by a=2pe2/hc = (137.04)-1. Dividing Eq. (23) by Z gives us the average electrostatic energy per electron, which, if expressed as a function of the electron number density of the system, is

Note that the interaction energy is negative and thus tends to reduce the electron pressure in the system. Quantitatively, for an electron number density of ne = 8x1024 cm-3 and a corresponding helium matter density of 100 g/cm3, we find es = 65 eV which is about 20% of the average electron kinetic energy. At higher densities, the relative importance of the electrostatic interaction energy to the total energy of the system is actually reduced.

G. Thomas-Fermi Method

The assumption of uniform electron distribution in a cell, though expedient, is certainly unjustified, since the electrons are attracted to the nucleus; and thus the electron distribution in the neighborhood of the nucleus should be higher. The exact determination of the electron distribution in a cell is a quantum mechanical problem of high degree of difficulty. The best known results on this problem are derived from the Thomas-Fermi method, which is an approximate method for the solution of the underlying quantum mechanical problem. The method assumes the following:

1. It is meaningful to introduce a radially dependent Fermi momentum pF(r) which determines the electron number density, as in Eq. (5), so that

2. The maximum electron kinetic energy, given by pF2/2me, now varies radially from point to point; but for a system in equilibrium, electrons at all points must reach the same maximum total energy, kinetic and potential, and thus

where F(r) is the electrostatic potential energy, and m, the chemical potential, a constant independent of r (m replaces pF in establishing the electron number density in the cell). 3. The electrostatic potential obeys Gauss' law

where d(r) is the mathematical delta-function. From these three relations the electron distribution may be solved. Since the first relation is based on Eq. (5), which is derived for a system of uniformly distributed electron gas, its accuracy depends on the actual degree of variation in electron distribution in the cell. The second relation can be improved by adding to the electrostatic potential a term called the exchange energy which is a quantum mechanical result. The improved method which includes the exchange energy is called the Thomas-Fermi-Dirac method.

The solution of these three relations for ne(r) is quite laborious, but it has been done for most of the stable nuclides. After it is found, the electron pressure in the system can be evaluated by employing Eq. (16c), setting pF in the expression equal to pF(rs), which is the Fermi momentum of the electrons at the cell boundary. There is no net force acting at the cell boundary, and the pressure there is due entirely to the kinetic energy of the electrons. Thus, the electron pressure can be evaluated by Eq. (16c). Note that, this condition would not be fulfilled by an interior point in the cell. Denoting the degenerate electron pressure evaluated for non-interacting electrons by Pe, the pressure P of the matter system corrected for electrostatic interaction by means of the Thomas-Fermi-Dirac method may be expressed as

where x is a dimensionless parameter related to rs and Z:

where a0 = (h/2p)2/e2me = 5.292x10-9 cm is the Bohr radius and F(x) a rather complicated function that is displayed graphically in Fig. 2. The dashed lines represent the correction factor derived from the Thomas-Fermi method, and it holds for all nuclides. The correction factor derived from the Thomas-Fermi-Dirac method depends on Z, and curves for two extreme cases are shown. Curves for nuclides of intermediate Z should fall between these two curves. The correction curve based on the assumption of uniform electron distribution is shown by the dotted curve for comparison. It is computed for the case of Fe matter. All curves tend to converge for large values of x. The relation between matter density and x is given by

The values of F(x) for all the curves shown in Fig. 2 are less than unity, and the corrected pressure is always less than that given by the non-

The electrons, on the other hand, must be handled by the Fermi gas method. At finite temperatures electrons no longer occupy all the low-energy states; instead, some electrons are thermally excited to high-energy states. Through quantum statistics it is found that the proabability for occupation of a quantum state of momentum p is given by the function:

where e(p) = p2/2me and m is the chemical potential, which in the present non-interacting case is given by the Fermi energy eF:

It accounts for the energy needed to introduce an addition particle into the system. f(p) is called the Fermi-Dirac distribution, which is the distribution of occupied quantum states in an identical fermion system in thermal equilibrium. Since the distribution that we shall employ depends on only the magnitude of the momentum and not its direction, it shall henceforth be written as f(p). The finite temperature parameters of a non-interacting electron system are to be evaluated from

where the range of the p-integration extends from zero to infinity. Note that, for relativistic electrons, e(p) in Eq. (34b) should be replaced by ek of Eq. (13) and v in Eq. (34c) should be changed from v = p/me to that given in Eq. (15). The integrals in these equations cannot be evaluated analytically, and tabulated results of these integrals are available.

It is also useful to evaluate an entropy density (entropy per unit volume) for the system. The entropy density due to the electrons may be expressed in terms of quantities already evaluated as

The entropy density due to the nuclei is

where eA is the energy density of the nuclei. The entropy density has the same dimensions as Boltznman's constant kB.

The equation of state relates the pressure of the system to its density and temperature. Since the pressure is determined by two independent parameters, it is hard to display the result. Special cases are obtained by holding either the temperature constant or the entropy of the system constant as the density is varied. An equation of state that describes a thermodynamic process in which the temperature remains constant is called an isotherm, and an equation of state that describes an adiabatic process for which the entropy of the system remains constant is called an adiabat. In the case where the total number of particles in the system remains unchanged as its volume is varied, constant entropy means the entropy per particle remains a constant, which is obtained by dividing the entropy density given by either Eq. (35) or Eq. (36) by the particle number density. In Fig. 3, typical isotherms and adiabats for a system of (non-interacting) neutrons at high densities are shown.