Abstract
This chapter is concerned with the properties of high-energy-density matter, and with how it differs from ideal plasma s and solids. It introduces the concept of equations of state that relate various thermodynamic variables. After reviewing some simple equations of state and some aspects of ideal plasma s, the chapter proceeds to build up an understanding of high-energy-density matter. It discusses the behavior of the electrons, the degree of ionization, continuum lowering, and Coulomb interactions. This enables the generation of a very simple model for the equations of state of these systems. Some additional features are revealed by a more complex model based on statistical mechanics. The chapter then discusses generalized polytropic indices, the degenerate and strongly coupled regime, tabular equations of state, and equations of state in some physical contexts.
The discussion of energy in Sect. 2.1 was entirely based on the notion of a polytropic gas . The speed of sound waves, which we found by examining fluctuations in density and velocity, was found to depend upon the derivative of pressure with density, and the solution was dependent upon having this derivative be constant. This was our first encounter with what is known as the closure problem. The physical densities of interest involve progressively higher powers of velocity: mass density (ρ), momentum density (ρ u), energy density ( ∝ ρu 2), energy flux density ( ∝ ρu 3), and so on if needed. But the equations for any given density always involve the divergence of a flux, as we discussed with reference to (2.5), and this flux corresponds to the density involving the next higher power of velocity. Thus, for example, the equation for mass density (2.1) involves the mass density flux, which is also the momentum density. The result is that there is, in general, no way to obtain a closed set of equations simply by including new conservation equations for additional quantities. This is the closure problem.
The only way to ultimately obtain a closed set of equations is to find a way to relate all variables in some set of equations to other variables already defined. We did this twice in Chap. 2. In finding the dispersion relation for acoustic waves, we assumed the pressure (a momentum density flux) to be a knowable function of density only. In relating the Euler equations to the more general equation for energy density, we assumed ρ𝜖 = p/(γ − 1) and also assumed the heat flux (and other source terms) to be zero. As another example, sometimes the energy equation is expressed as an equation for temperature, pressure is written as a function of density and temperature, and the heat flux is written as − κ th ∇T, which also produces a closed system of three equations. The relations necessary to obtain closed sets of equations, in order to describe behavior of some specific fluid medium, are among its equations of state.
Readers with a plasma-physics background may find this discussion a bit convoluted, since in plasma physics one typically derives all the fluid equations by taking moments of the distribution of the particles in velocity, v, as is done in graduate courses in plasma physics. The continuity equation is the moment taken with v 0, the momentum equation is the moment with v, the energy equation is the moment with v 2, the heat transport equation is the moment with v 3, and one can keep going. The closure problem then arises because every moment equation contains terms involving the next higher moment. But this point of view does not readily allow for some of the more complicated aspects of high-energy-density fluids, such as complex relations between ρ𝜖 and p, cases where the pressure is tensorial, and viscosity .
To successfully solve the continuity, momentum, and energy equations, even numerically, one must typically understand the relation among internal energy, pressure, density, temperature, and ionization. This chapter discusses the equations of state (EOS) that specify these relations. It begins by discussing simple equations of state, which are often useful in limited regimes and for estimates generally. It then considers the conditions for ideal-plasma theory to be valid, because the equations of state are simpler where it is valid.
What follows is an examination of the various issues that lead high-energy-density plasmas to be more complex than low-density plasmas composed of hydrogen. First is a more extensive discussion of electrons, aimed at understanding their transition from ideal-gas behavior to degenerate behavior. Section 3.4 then takes up issues that also involve the ions. The first of these is the degree of ionization, because high-energy-density plasmas are always somewhat ionized but only occasionally fully ionized. The next is how the ions behave when the Debye length is less than the size of an atom. The third is how strong Coulomb interactions manifest themselves in thermodynamic behavior. By the end of this section we will understand the fundamental elements of the equations of state of high-energy-density plasma. This will enable us, in Sect. 3.5, to develop two models for the EOS of high-energy-density matter, and to see how pressure, internal energy, and ionization vary with density and temperature.
Following this, we consider more specifically the high-pressure, low- temperature conditions where the matter is both strongly coupled and Fermi-degenerate, about which we learned in the decade preceding 2015 that our prior understanding was substantially incorrect. We will at that point be in a good position to consider one approach to generalizing the polytropic indices that will prove useful later in the book. Section 3.6 does this. At this point, we will have completely addressed the problem of what high-energy-density matter is, at the level of conceptual discussions and simple models. Section 3.9 then discusses briefly approaches to equations of state in support of computer simulations while Sect. 3.10 discusses the relation of EOS measurements in the laboratory to astrophysical questions. Specific experimental methods for measuring some aspects of the EOS are discussed in Sect. 4.2 after we explore shock waves.
Before turning to the details, consider this example of the relevance of EOS to astrophysics. Figure 3.1 shows a theoretical phase diagram for hydrogen, and also shows where various interesting objects lie in this diagram. The objects include Jupiter, a typical brown dwarf, and a typical dwarf star. The phase diagram is a model, and the location of the curves depends on the model. These might be wrong, but the range of pressures is correct. The phase diagram of hydrogen includes a region of molecular hydrogen, of atomic hydrogen, and of so-called metallic hydrogen in which the electrons are free to move and to conduct electricity. Metallic hydrogen carries the currents that sustain Jupiter’s large magnetic field. These regions have boundaries, which might on the one hand be gradual transitions and might on the other hand be abrupt phase transitions. In this particular model, the molecular-to-metallic transition is a phase transition. Evidently a thorough understanding of the EOS will be essential to thoroughly understand astrophysical objects.
Phase diagram of hydrogen. The dark curve segment shows a theoretical plasma phase transition. Dotted curves show the theoretical path of various astrophysical objects. Adapted from Saumon et al. (1995)
3.1 Simple Equations of State
Figure 3.2 shows an image of a Type Ia supernova explosion. This explosion is brighter than the entire galaxy that surrounds it, which is not uncommon. Current understanding is that a Type Ia supernova occurs when a white dwarf star, accumulating mass from its environment, reaches a total of just over 1.4 solar masses. This is enough for gravitational forces to overcome the pressure of the degenerate electrons (Sect. 3.4), which initiates the gravitational collapse of the star. However, the star does not fully collapse. Instead, the energy released as collapse begins heats the C and O that make up the white dwarf, which initiates the violent fusion burning that blows the star apart. The properties of the star as its explosion begins are very relevant to this chapter. Its outer layers are accurately described using the polytropic equation of state for an ideal gas (Sect. 3.1.1). To describe its core, one must use the Fermi-degenerate equation of state (Sect. 3.1.3). And the region heated by the fusion burning requires an equation of state for a radiation-dominated plasma (Sect. 3.1.2). These simple models introduce the relevant regimes and concepts; detailed treatment of white dwarf stars and Type Ia supernovae requires more-sophisticated models.
A Type Ia supernova produced the bright spot of emission near the edge of this galaxy. Credit: Jha et al., Harvard Center for Astrophysics
3.1.1 Polytropic Gases
The polytropic equation of state (EOS), often also described as an ideal-gas EOS, is a useful approximation under many circumstances. At a high enough temperature, any material will behave like an ideal gas. In practice, once the temperature is far enough above the value required to fully ionize any material, its behavior is well described by a polytropic EOS. As we shall see, even radiation-dominated plasmas can be described that way. Moreover, for conceptual and analytic calculations we often use a polytropic description even when it is not precisely accurate. A polytropic gas having n degrees of freedom has certain simple and interconnected properties. The pressure is
where for the moment N is the total number density of particles, and k B and T are the Boltzmann constant and the temperature, respectively. The final expression for pressure often applies to a high-energy-density plasma, having an average level of ionization Z, an average atomic mass of the ions in the fluid A, and where the proton mass is m p . This equation also implies that the gas constant is R = ρ(1 + Z)k B /(Am p ). In the discussion that follows in the present subsection, we assume Z and thus R to actually be constant. This is often a poor assumption in high-energy-density physics, as will become clear in later sections. We will see below that (3.1) also fails to apply when Coulomb interactions become too important or if the radiation pressure becomes too high. Equation (3.1) can be recognized as essentially Boyle’s law.
The internal energy, for a system of particles having n degrees of freedom, is
reflecting the basic result from statistical physics that the mean energy of a particle in equilibrium is one-half k B T per degree of freedom. From this, the specific heat at constant volume is
if Z (R) is independent of T. Evaluating the specific heat at constant pressure, c p , is a more complex result from thermodynamics, discussed later in this chapter in Sect. 3.6. The result still depends only on (3.1) and (3.2), and is
where similarly R must be independent of T. Thermodynamic arguments also imply a result for the sound speed, specifically
which defines
This is an important result. Note that for n = 3 one finds the familiar consequence that γ = 5/3. But as the degrees of freedom become larger, γ decreases toward one. The importance of (3.5) was seen in the discussion of sound waves surrounding (2.6)–(2.9), and we note that here the partial derivative is taken at constant entropy , designated here by the subscript s.
Equation (3.5) also implies that
for isentropic (i.e., adiabatic) changes over a range of pressures for which γ is constant.
Equations (3.1), (3.2), and (3.6), imply that ρ𝜖 = p/(γ − 1). In addition, we can obtain the same, self-consistent result by evaluating the internal energy as the integral of the pdV work required to assemble an element of fluid from infinity to some volume V . Note that the conserved mass, M = ρV so dV = −Mdρ/ρ 2. The work is
Calculations using polytropic models can become tricky in the important case of an isothermal system. From the perspective of (3.1), an isothermal system would have p ∝ ρ and thus γ = 1. Then (3.9) would imply that the internal energy is infinite. In contrast, for a system whose particles have only kinetic degrees of freedom (3.2) would imply that γ = 5/3. The key here is that (3.9) describes the adiabatic assembly of the system and such a process is not isothermal. To change compression while maintaining constant temperature requires heat transport, and indeed isothermal systems are those having very rapid heat transport. In a typical case of an isothermal system, one would describe small variations in ρ, such as those due to acoustic phenomena, using γ = 1, but would still evaluate the portion of the internal energy due to thermal motions as (3/2)ρRT.
Thus, the basic properties of polytropic gases involve a self-consistent set of relationships any one of which can be described as an equation of state. In the event that R and thus γ are not constant, however, one no longer has such a simple story. This important and realistic case motivates the discussion in Sect. 3.6.
3.1.2 Radiation-Pressure-Dominated Plasma
The properties of blackbody radiation and of systems in which radiation is important or dominant are discussed in Chaps. 6 through 8. The radiation pressure p R is 1/3 the radiation energy density and may be expressed as
where T is the temperature, c is the speed of light, and σ is the Stefan–Boltzmann constant familiar from blackbody emission. Because this pressure depends upon T to the fourth power, while material pressures depend upon T to the first power, at a high enough temperature the radiation pressure will be completely dominant. This is the case, for example, within matter shocked during supernova explosions and near neutron stars and black holes. The transition temperature can be determined by asking when the radiation pressure equals the material pressure. Assuming (3.1) to be accurate, one finds
in which ρ is in g/cm3. Here, outside the parenthesis in the middle term, k B = 1.6 × 10−9 ergs/keV to find the temperature in energy units (keV). Within the parentheses, the units of energy and temperature in k B must be consistent with those in σ and with the other units used there. For laboratory systems or within stars where ρ is within a few orders of magnitude of 1 g/cm3, keV temperatures are thus required for radiation to dominate. At typical astrophysical densities much lower temperatures would be required, except that such systems tend to be “optically thin” (see Chap. 6), implying that the radiation pressure is far below the value given by (3.10).
To utilize simple equations in describing radiation-dominated plasmas, one desires to determine ∂p/∂ρ for this case—that is, to determine how the radiation pressure varies with plasma density. This is often feasible, because in order for the radiation temperature to remain large enough that the system stays radiation-dominated, the mean free path for the radiation must be small on the scale of the physical system of interest. This in turn implies that the material is strongly coupled to the radiation and will have the same temperature. In addition, because the material is strongly coupled to the radiation, changes in the density of the material involve changes in the volume containing a fixed amount of radiation. This prepares us to identify a polytropic index for the radiation-dominated plasma, as follows.
Standard arguments in statistical mechanics lead to an expression for the pressure of the photon gas as
in which the sum is over all possible states j, the mean occupancy of each state is \(\overline {\sigma _j} = 1/ [ \exp (\epsilon _j /k_BT ) -1]\), and the energy of each state is 𝜖 j . Equation (3.12) makes sense when one recalls that the pressure is the negative of the change in internal energy as volume increases. The energy of a state varies with volume as the wavelength of the light in that state is reduced or increased by the compression or expansion. One can see how by considering the simple example of a cubic box with an edge of length L, in which a given state has an integer number of wavelengths along each side of the box. The wavenumber of each state, k j , is then proportional to 1/L, so one has 𝜖 j ∝ hck j ∝ L −1 ∝ V −1/3, where h is the Planck constant. Thus
and p ∝ ρ 4/3, showing that γ = 4/3 for a radiation-dominated plasma. The Euler equations can be applied to such a system using γ = 4/3. One can obtain the same result more simply by recognizing that the photons have 6 degrees of freedom, thanks to their two possible polarizations. Further details can be found in the chapter on radiation hydrodynamics.
3.1.3 Fermi-Degenerate EOS
In ordinary plasmas it is the thermal pressure, experienced by the particles through Coulomb collisions, that resists compression of the plasma. This is a classical effect, and the properties of the electrons are described by Boltzmann statistics. But when plasma or other matter becomes dense enough, then quantum mechanical effects involving the electrons create pressure and resist compression. The electrons are subject to the Pauli exclusion principle, which prevents more than one of them from occupying the same quantum state. As we will see, this implies that the most energetic electron in cold, high-density matter can be quite energetic indeed. Matter in which nearly all of the electrons are in their lowest-energy states is described as Fermi-degenerate matter. The EOS of Fermi-degenerate matter is of substantial importance in massive planets, white dwarf stars, and inertial fusion implosions or other high-energy-density experiments that compress solid matter. The fact that electrons are fermions has an impact over a broader range of conditions, as we will see in Sect. 3.5. Fundamental derivations of the electron behavior can be found in any book on statistical physics, including for example Reif (1965) and the relevant volume by Landau and Lifshitz (1987).
Figure 3.3 shows the energy distributions of free electrons in dense matter, for several temperatures. In very cold, dense matter the energy distribution is a step function—all the electrons are in the lowest accessible state. As temperature increases, some of these states are depleted and a tail of electrons develops at higher energy. The energy of the state whose occupancy is 50% is known as the Fermi energy . The Fermi energy at absolute zero, 𝜖 F , is
in which m e is the electron mass, n e is the number density of electrons, and n 23 is the electron density in units of 1023 cm−3. This value (1023 cm−3) is of order both the density of electrons in low-Z plasmas with a mass density near 1 g/cm3 and the density of conduction electrons in a typical metal. In any material there may also be bound electrons, attached to specific atoms. These electrons do not contribute to the electron density n e in (3.14). If we displayed the bound electrons on the scale of Fig. 3.3, they would appear as spikes at negative electron energy. We discuss the degree of ionization (and hence the relative numbers of free and bound electrons) in Sect. 3.4. Equation (3.14) has a number of consequences for physical systems of interest here. It implies that the electrons are not Fermi degenerate in plasmas with densities well below solid density, heated to temperatures of tens to hundreds of eV. In contrast, compressed plasmas at densities of more than 100 times solid density, produced in inertial fusion implosions, have a Fermi energy of hundreds of eV. Such plasmas are often cool enough that the EOS of the electrons is the Fermi-degenerate EOS. The degeneracy temperature , T d , above which the electrons can be approximated as a classical gas, is found by setting k B T d = 𝜖 F .
Electron energy distributions in dense matter. The distribution function, normalized to be 1 at zero energy, is shown against energy, normalized to the Fermi energy , for k B T = 0.01, 1, and 10 𝜖 F . The gray curve shows a Maxwellian distribution for k B T = 10𝜖 F
Despite its obvious differences from an ordinary gas, the equation of state of Fermi-degenerate matter is quite similar to that of an ideal polytropic gas with γ = 5/3. Equation (3.8) applies in both cases, so p = (2/3)ρ𝜖. In addition, while the electron pressure in an ideal gas is p = n e k B T, the electron pressure in Fermi-degenerate matter is p F = (2/5)n e 𝜖 F . Evaluating this one finds
or in practical units
in which A/Z ∼ 2 and the units of density are cgs. The transition from (3.16) to (3.1) occurs approximately when T = T d , although one can see in Fig. 3.3 that the electron distribution still departs significantly from a Maxwellian even at T = 10T d .
3.2 Regimes of Validity of Traditional Plasma Theory
High-energy-density systems are nearly always plasmas, in the sense that they are ionized and that electromagnetic interactions at a distance can play a role in their dynamics, at least in principle. Unfortunately, the theory of plasmas, as covered in traditional texts such as Krall and Trivelpiece (1986), has a range of validity that only partly overlaps the regimes of high-energy-density physics. Even so, plasma concepts have tremendous utility when they are valid. This motivates a discussion of these issues.
Traditional plasma theory faces the challenge of describing a system composed of mobile charged particles and capable of dramatic electrodynamic effects. The particles quickly scurry over to surround any exposed charge, yet also can carry currents that produce magnetic fields which can store immense energy. The eruptions on the surface of the sun are an example of the potential consequences. The shielding of exposed charges is one of the fundamental aspects of plasmas. Yet even as the charges try to cluster about one another, their thermal motions limit the clustering. The competition between these gives rise to a characteristic shielding distance, known as the Debye length. The Debye length is defined in Gaussian cgs units by
in which the sum is over all ion species, the subscript e designates electrons while α designates an ion species, n is a number density, T is a temperature, Z is a number of unit charges, k B is the Boltzmann constant, and e is the electronic charge (4.8 × 10−10 statcoul here). On the one hand, when one considers fast enough timescales, the ions cannot move and the electron Debye length,
(in the same units) becomes relevant. This is the only Debye length defined in the NRL Plasma Formulary, among other references. In addition, traditional plasma texts often assume all plasmas to be pure hydrogen, replacing the 4 with an 8 in (3.18). On the other hand, there are cases in dense plasmas when ion–ion shielding determines the behavior, as for example when the electrons cluster poorly because they are Fermi degenerate (Sect. 3.1.3). Then the ion Debye length,
(in the same units) comes into play.
High-energy-density plasmas, like most plasmas, are quasi-neutral, so that
In addition, in such plasmas collision rates are large (Sect. 2.4) so the temperatures of the particle species are usually equal and designated by T. When this is the case, one can use the standard definition of the effective charge, Z eff, as
again in Gaussian cgs units. This is a form we will use in later discussions. For calculations involving binary collisions, Z eff is the appropriate average charge, while for calculations involving particle counting, Z = n e /n i is the appropriate average charge.
The Debye length arises quite naturally in the most-sophisticated developments of plasma theory. It also can be found from a simple calculation that can be used to highlight the limitations of traditional plasma theory for us. We consider a two-species plasma, in which the ions have charge Z, and we also suppose that the particles are distributed by classical statistics with a common temperature T. This implies that the density of particles with charge q, at a location with a potential ϕ relative to the potential at some reference location is proportional to \(\exp [-q \phi /(k_{B} T)]\). Then the charge density ρ c in the vicinity of an ion at x = 0 is
If we assume that |qϕ|≪ k B T and that the plasma is quasi-neutral, then this becomes
At this point we can write the Poisson equation in spherical coordinates, assuming that the charges are distributed with spherical symmetry, as
which (in cgs units) has the solution
Equation (3.26) displays the standard result that the potential of any given charge falls away exponentially faster in a plasma than it would in vacuum. But what is relevant to our interests is two aspects of this derivation. First, (3.23) only makes sense in the end if there are numerous particles within a sphere whose radius is the Debye length . Second, the key assumption in this derivation is that |qϕ|≪ k B T, which must be violated if the particles are cold enough. These turn out to be related, and we will explore them in turn.
The number of particles in a Debye sphere , in a quasi-neutral plasma, is \(n_{e} (1+1/Z)(4\pi /3) \lambda _{D}^{3}\). The inverse of this, sometimes defined without the numerical coefficients, is a fundamental expansion parameter for traditional plasma theory (see Krall and Trivelpiece 1986). A plasma is known as an ideal plasma when the number of particles in a Debye sphere can be taken to approach infinity. In this case collective effects, involving all the particles, remain, while effects relating to particle correlations vanish. Figure 3.4 shows the number of particles in a Debye sphere in the high-energy-density regime. There are not many. The number varies from tens of particles in the upper-left corner of the regime shown to less than 0.01 particles in the lower-right corner. In the lower-right corner the electrons are Fermi degenerate (Sect. 3.1.3), which reduces even further their ability to shield the ions. The ion density in a typical solid is also shown. It is evident from this figure that high-energy-density plasmas are rarely ideal plasma s.
Contours of the number of particles in a Debye sphere. The contours show 0.01, 0.03, 0.1, 0.3, 1, 3, and 10 particles, and increase to the upper left. (a) A high-Z plasma with \(Z=0.63 \sqrt {T_{eV}}\) (see Sect. 3.4). (b) A low-Z plasma with Z = 4
Now consider the assumption that |qϕ|≪ k B T. We can take a typical value of ϕ to be the electrostatic interaction of two particles at their average spacing. We find the average spacing by giving each particle a spherical volume of radius r av, so that the average spacing is 2r av. Thus we take \( 4 \pi r_{\mathrm {av}}^{3}/3= 1/[n_{e} (1+1/Z)].\) Then we find
so that the assumption becomes
where g is the inverse of the number of particles in a Debye sphere, \(1/g=n_{e} (1+1/Z)(4 \pi /3) \lambda _{D}^{3}\). Thus, the two requirements of the Debye-shielding analysis are intimately connected. It is no surprise that this assumption (3.28) is violated over about half the parameter space shown in Fig. 3.4. The ratio |qϕ|/k B T is often known as the strong coupling parameter , Γ, first introduced in Chap. 1. Salzman (1998) discusses this parameter, which he calls the plasma coupling constant , at more length. Like the Debye length , Γ comes in different flavors depending upon whether one evaluates ion–ion coupling, ion–electron coupling, or electron–electron coupling. To be precise, one must evaluate Z and r av for a specific, chosen set of particles. The most common type of Γ found in the literature is that for ion–ion coupling. Across much of the parameter space of Fig. 3.4, the ions are strongly coupled but the ions and electrons are not. We specifically discuss the regime where the ions are strongly coupled and the electrons are Fermi degenerate in Sect. 3.7. The pressure and energy of the plasma will depart from their ideal-gas values across a large fraction of this parameter space. In order to enable us to understand the actual behavior of matter at high energy density, Sect. 3.3 considers the behavior of electrons across this regime and Sect. 3.4 considers ionization and of energies associated with Coulomb forces . This will enable us, in Sect. 3.5 and beyond, to see the combined effect of all these elements.
3.3 Electrons at High Energy Density
We begin by returning to the behavior of the electrons. We know that at high-enough temperature they behave as an ideal gas, and that they are Fermi-degenerate at low-enough temperature and high-enough density, with the boundary between these regimes running directly across the middle of the parameters of interest for high-energy-density physics. What we have not yet determined is how abrupt the transition is. In the limit that it is very abrupt, one could model the electron effects discontinuously, just switching models at the boundary. In contrast, if the transition is very gradual, then one might have to implement a more-complicated treatment of the electrons to even come close to the correct behavior. So our task here is to determine how abrupt the transition from ideal-gas to Fermi-degenerate behavior actually is.
The detailed properties of the partially degenerate matter at a temperature near the degeneracy temperature involve some straightforward numerical integrals. The ion density range of interest to high-energy-density physics spans 1019 to 1024 cm−3, but reaches ∼1026 cm−3 in compressed inertial fusion capsules. All of the electrons participate in Fermi-degenerate behavior, so this corresponds to a range of electron densities from 1019 to 1026 cm−3, where the upper limit might correspond either to high-Z matter at an ion density of 1024 cm−3 or to low-Z matter compressed for inertial fusion. The electron temperatures of interest span 1–1000 eV. Let us examine the behavior of the electrons over this range of conditions.
The electron density is given by the integral over all momenta, χ e , of the probability that an electron will have a specific momentum. With the electron energy given by \(\mathcal {E}_e = \chi _e^2 / (2m_e)\), this is
in which μ is the chemical potential , which has energy units. Within this integral, the factor equal to \(\chi _e^2 \mathrm{d} \chi _e\) gives the scaling of the density of states while the remaining factor gives the probability that a certain state is occupied by an electron. Equation (3.29) can be put in the useful form
which defines the ratio of electron temperature to degeneracy temperature as Θ. We also define in general \(F_n \left ( \phi \right )= \int _0^{\infty {x^n \left [ \mbox{exp} (x-\phi )+1\right ] ^{-1} \mathrm{d} x}}\). This will have further application below. Our parameter range of interest corresponds to Θ = 10−3 to 104.
The chemical potential is the internal energy required to add a particle to the system at constant entropy and constant volume. For a Fermi-degenerate system the chemical potential is positive; a new particle goes in at the Fermi energy even at zero entropy , so one must invest energy to put a new particle into the system. For a Boltzmann system μ is negative: a new particle can be added at zero energy but to keep entropy constant the internal energy of the system must decrease. The limiting behavior of μ/(k B T e ) is of some interest. In the degenerate regime, μ = 𝜖 F so
In the Boltzmann limit, designating the Boltzmann chemical potential as μ c , one has
so
so μ c is zero when Θ = Θ crit = 0.827. Atzeni and Meyer-Ter-Vehn (2004) give a fit due to Ichimaru that spans both limits. This is
One can vary μ/(k B T e ) and calculate the integral (3.30). Figure 3.5 compares the result of this calculation with the values implied by (3.31) and (3.33). The solid curve shows the actual value, with the gray, dashed curve showing the Boltzmann limit and the black, dashed curve showing the Fermi limit. The result is rather dramatic. The electron chemical potential has the Boltzmann value for Θ > Θ crit, where it abruptly transitions to the degenerate value. This observation is the most important result of this section. If the temperature is further than a factor of two from the T d , then the behavior of the electrons is solidly in the corresponding limit. Only very near the transition need one consider using a more complex model.
The chemical potential is shown vs Θ = T e /T d . In the classical (Boltzmann) regime, μ is negative
One can evaluate the electron pressure by averaging the energy of each state over the probability that the state is occupied. The general integral for the internal energy density, n e 𝜖 e , where 𝜖 e is the specific internal energy per electron, and the pressure, p e , is
which can be written as
Figure 3.6 shows how the normalized pressure, p/(n e k B T e ), increases with μ/(k B T e ), for Θ < Θ crit. The electron contribution to the pressure and internal energy is classical for Θ > Θ crit. Despite the difference in the pressure, the electrons behave as a gas with γ = 5/3 throughout. Under strongly Fermi-degenerate conditions, the electron pressure and energy completely dominate those of the ions. However, because of the energy associated with ionization, the electrons do not necessarily dominate the internal energy of the plasma throughout our regime of interest. We explore this further in Sect. 3.4.
Normalized electron pressure versus chemical potential . This asymptotes to (2/5)μ/(k B T e ) at large μ/(k B T e ) and approaches 1 as μ/(k B T e ) approaches 0 (and is 1 in the Boltzmann regime)
For various applications, including inertial fusion, it is worthwhile to understand the heat capacity and entropy of electrons. For this purpose it helps to understand F n (ϕ) more thoroughly. One can show that \(F_n^{\prime } \left ( \phi \right )= n \phi ' F_{n-1}\left ( \phi \right )\). In addition, if T e is near zero, then F n (ϕ) = ϕ n+1/(n + 1). It is also useful to know that if ϕ is zero, then F 3/2 = 1.153 while F 1/2 = 0.678. In the Boltzmann limit (3.32) implies that
so in the Boltzmann limit F 3/2 = Θ −3/2 while F 1/2 = (2/3)Θ −3/2.
Turning to the heat capacity , one finds
where C V has units of energy per unit volume per unit temperature and the argument of both F 3/2 and F 1/2 is μ/(k B T e ). In the Boltzmann limit this becomes C V = (3/2)n e k B . In the degenerate limit and for small temperatures, one can expand the integrals to find C V = (3/2)n e k B [π 2 k B T e /(3𝜖 F )]. This is the electronic contribution, which is dominant for strongly degenerate matter. As any book on statistical physics will discuss the ionic contribution at small temperature.
The entropy per unit volume of the electrons, S/V, may be found from
in which − pV is one of the thermodynamic potentials discussed by Landau and Lifshitz (1987) in their volume on statistical physics. Note that holding μ and V constant is not identical to holding n e constant, as was done to find C V . This implies
which for the Boltzmann limit is
or in the degenerate limit where T e ≪ 𝜖 F is
The entropy approaches zero as the temperature approaches absolute zero, as it should.
In the context of inertial fusion, one cares about the relation of pressure and entropy , because the shock waves produced during compression increase the entropy (see Chap. 4). Since the pressure is proportional to F 3/2, while density is proportional to F 1/2, (3.40) can be rearranged to obtain
As T e and S approach zero, this reduces to (3.15). One sees that the pressure is not sensitive to the value of the entropy until the entropy reaches a threshold value given by setting the two terms of this equation equal. This is evident in Fig. 3.6, where we see that the pressure begins to depart from 2n e μ/5 when μ ∼ 5k B T e or Θ ∼ 0.2 so T e ∼ 0.2T d .
The quantity p/p F is known in inertial fusion as the degeneracy parameter . It has important practical consequences as the fusion gain decreases for increasing p/p F . In general p/p F = 1 for degenerate matter and increases with Θ, equaling (5/2) Θ in the Boltzmann regime. The practical importance of this quantity makes it useful to have approximate estimates of p/p F . Atzeni and Meyer-Ter-Vehn (2004) give the following fit for p/p F :
The present section has provided a variety of useful models and limiting cases for the behavior of the electrons. If we pull back to our overall mission, what matters is this: The behavior of the electrons changes quite abruptly from ideal-gas behavior to Fermi-degenerate behavior as T crosses T d . Only if the temperature is within a factor of about 2 from this boundary need one consider their behavior in more detail. Taking the point of view that our goal here is to see the overall behavior, in our further discussions we will treat the transition in electron behavior to be abrupt.
3.4 Ionizing Plasmas
Mid-Z and high-Z ions in high-energy-density plasmas are rarely fully stripped , meaning that all their electrons have been removed. Only as temperatures approach and exceed 1 keV, or as compressions exceed ten times solid density will one encounter completely stripped ions of any except very-low-Z species. When it becomes routine to work far above solid density at temperatures of many keV, the materials may become fully stripped, although the increased role of radiation will provide ample new complications. We discuss some of these in Chap. 7. For the moment, it is clear that we must understand the behavior of partially ionized plasmas, which we will describe as ionizing plasmas , if we are to succeed in understanding high-energy-density phenomena.
One needs to estimate the degree of ionization for a variety of reasons. The most important is that their thermodynamic properties also depend upon ionization, as we discuss in the next two sections. The internal energy of fully stripped ions also includes a major contribution from ionization. While the behavior of actual materials is complicated and difficult to calculate accurately, there are some simple models that can capture aspects of their behavior. These we discuss here.
The electron density is Zn i , but the value of the average charge Z depends upon the temperature. To know Z precisely, one must evaluate the ionization balance to determine the relative populations, N i , of the various ionization states. Then one has Z as a sum over ionization states,
in which the state populations can be either a number or a density, and N is either the total number of ions or the ion density n i , respectively.
We will designate the various ionization states of a given species by their net charge Z i . The electrons in any given ion may reside in the ground state or in an excited state. These of course are designated precisely by the necessary quantum numbers, such as the principal quantum number, n, the quantum number for orbital angular momentum, ℓ, and the spin quantum number, s. In the present discussion, we will occasionally have reason to specify the principal quantum number. We will often, however, ignore excited states and implicitly treat all ions as ground state ions. In most cases this is reasonable. The minimum excited state energy, with n = 2, has an energy above the ground state that is 3/4 of the ionization energy, E i . On the one hand, if the ion is in an environment where E i is well above T e , as is common, then the excited state population is smaller than the ground state population by a factor smaller than \(\exp [-3E_i/(4 k_B T_e)]\), which is fairly small. On the other hand, if E i is small relative to T e , then it is more likely that the electrons striking the ion will deliver its outer electron into one of the indefinite number of free states as opposed to one of the few and definite excited states.
The exact ionization energy required to remove the outermost electron from a given ionization state does depend on the number and arrangement of the remaining electrons, but we will ignore this here and adopt a hydrogenic atom analysis. In such a treatment, all atoms and all ions are treated as hydrogenic systems, having one electron and a nucleus with the appropriate net charge. This approach is more accurate as the net charge on the atom increases (so that the inner electrons are more tightly bound). This approach allows comparatively tractable computational models to work with a wide range of atoms and ionization states, giving qualitatively correct answers. In our work here we will primarily use the ionization energy associated with a hydrogenic atom model, which is energy E i = Z 2 E H , where E H = 13.6 eV and Z is the net charge on the atom after ionization (and thus is consistent with our use of “Z” elsewhere).
The simple view of atomic structure we will use here is distinct from the computational “average atom model” (discussed in Salzman 1998). The computational model provides a physically consistent approach to the definition of an “average atom”, including both bound states and free electrons , that characterizes each element.
The density of ions will play an important role in our discussions of ionization, as this scales the electron density. A factor-of-two estimate of the typical ion density, for a solid, can be made by taking ρ = Z n /4 g/cm−3 and A = 2Z n . Then
This density is indicated in several of the plots in the following.
3.4.1 Ionization Balance from the Saha Equation for Boltzmann Electrons
Determining the exact degree of ionization is a difficult problem involving sophisticated calculations, but we can arrive at a reasonable approximation on very simple grounds. We can expect that the ionization energy of the ions in a plasma will have some typical relation to the electron temperature. If we approximate the ion as a hydrogenic ion of charge Z (after ionization), then the ionization energy E i = Z 2 E H , where E H /k B = 13.6 eV. Thus, we expect Z 2 E H /(k B T e ) ∼ C 2, where C is a constant, so \(Z=C\sqrt { k_B T_e/E_H}\), which is \(Z = 0.27 C \sqrt {T_{eV}}\). The problem is to find C. On the one hand, if we recall that Coulomb processes often are effective at energies of about 3k B T e , as is the case for heat transport (see Chap. 9), then we would say \(C \sim \sqrt {3}\), which is not far from the better estimates discussed next.
More sophisticated estimates of the ionization involve balancing ionization and recombination or assuming that the distribution of ions is in equilibrium. These turn out to be equivalent at high enough densities, but not at low densities. Griem (1997) and Salzman (1998) discuss in detail the dynamics that are involved, in their books. Here, and at more length in Chap. 6, we discuss the basic phenomena that are important for high-energy-density systems. In low-density plasmas, the archetype of which is the solar corona, collisional ionization is balanced by radiative recombination , establishing a steady state known as coronal equilibrium . An additional process, dielectronic recombination , is of increasing importance as the density increases, particularly in the range of densities found in magnetic fusion devices. But at most densities found in high-energy-density systems, the relevant balance is between collisional ionization and collisional (three-body) recombination . In equilibrium, collisional ionization and collisional recombination are equal by the principle of detailed balance.
At high enough density and temperature the distribution of ions, and the distribution of electrons within energy levels, approaches the equilibrium distribution given by the Saha equation , derived in statistical mechanics. For an estimate of the ionization balance we will ignore the distribution of electrons among the excited states , and will focus only upon the distribution of ions among the ionization levels. We work here with the Saha equation to estimate Z. The Saha equation gives the ratio of the population of ions in state j, N j , to those in state k, N k , as
in which in cgs units a o = ħ 2/(m e e 2) = 5.29 × 10−9 cm is the Bohr radius, E jk is the energy required to go from state k to state j. The symbols g j and g k are the degeneracies of the ions. These are the number of distinct states of the ion having the same energy for states j and k, respectively. For a hydrogenic ion with an electron having principal quantum number n, this is 2n 2. Thus, for our assumption of hydrogenic ions in the ground state, discussed above, g j = g k = 2. (To help interpret various references, it may also help to know that E H = e 2/(2a o ) so \(E_H a_o^2 = \hbar ^2 /(2 m_e)\), ignoring a very small center-of-mass correction.) For simple calculations, the only practical choice is to assume that the ions are hydrogenic, so that the ionization energy from state k to state k + 1 = j, in an isolated ion, is \(E_{(k+1)k}=Z_{k+1}^2 E_H\). We will discuss below the consequences of the fact that the ions are not isolated. At a high enough temperature, this has a small effect on the average ionization.
We can determine a characteristic charge, not far from the actual average charge, from this equation as follows. There will be some value, Z bal, not necessarily an integer, for which the ratio N j /N k = 1 for two imaginary ionization states having charge Z bal + 1/2 and Z bal − 1/2. Then Z bal should be close to, but may not equal, the average charge Z. Recalling that n e = Zn i , we can solve (3.47) for Z bal to find
which is
with T eV in eV and n 21 being the electron density in units of 1021 cm−3, and (3.49) assuming g j = g k . One might approximate this as \(Z_{\mathrm {bal}} = 0.63 \sqrt {T_{eV}}\), for Z bal ≤ Z n , where Z n is the nuclear charge.
The first estimate is to assume Z = Z bal, in which case one can either approximate n 21 or solve (3.49), which becomes an implicit equation for Z, through the electron density (with n e = Z bal n i ). In terms of the initial formulation of this problem above, the coefficient in (3.49) corresponds to C ∼ 2.3, which is not far from our initial guess of \(\sqrt {3}\). Figure 3.7 shows how Z bal varies as ion density and temperature vary, solving implicitly for Z bal. If the result were strictly \(0.63 \sqrt {T_{eV}}\), the contours would be vertical. The curve crossing the plot shows where the solution for Z bal does equal \(0.63 \sqrt {T_{eV}}\). One can see that using \(0.63 \sqrt {T_{eV}}\) is accurate to about 50% over most of the parameter space shown, with a greater error at ion densities above 1023 cm−3. One would expect the ions to exist primarily in the one or two states for which ionization and recombination nearly balance, so the value of Z from (3.49) ought to be close to the actual average ion charge in the plasma.
Ionization from the Saha equation. Curves of constant Z bal are shown. The electrons are Fermi degenerate in the region above the line labeled “Degenerate”. The lower curve shows where Z bal equals the approximate value \(0.63 \sqrt {T_{\mathrm{eV}}}\)
One can demonstrate that Z ∼ Z bal, when the ionization energies are as assumed above, as follows. One can use the definition of Z bal to rewrite (3.46), for arbitrary j and k, as
Note that this corresponds to a distribution of ions peaked around Z k ∼ Z bal, since N j < N k for E jk /E H > Zbal2, and N j > N k for \(E_{jk}/E_H < Z_{\mathrm {bal}}^2\). Figure 3.8 shows the ratio N j /N 1 for T e = 1 keV and Z bal = 20. Note that to obtain this one must apply (3.50) repeatedly, obtaining
This gives a sum in the exponent that can be evaluated, as follows:
Normalized relative populations of ionization states, for T e = 1 keV and Z n = 30
Figure 3.8 shows a plot of this distribution, which turns out to be very strongly peaked, with nearly all of the ions having a charge within a few unit charges of Z bal. As it should, the peak of the distribution corresponds almost exactly to Z bal as given by (3.49). One could formally evaluate the average charge using (3.52). For the ratio of ionization state populations, and a nuclear charge Z n , one has
One can show that Z determined by this method is quite close to Z bal.
It is worthwhile to emphasize that the fundamental basis for our estimate of Z here is the Saha equation . However, the Saha equation is not an inviolate law of the universe, even for equilibrium systems. It is a consequence of statistical mechanics when the only important energies are the ionization and excitation energies and when the electrons and ions both follow Boltzmann statistics. As plasmas become denser or colder, energies associated with the interaction of the particles become important. Some aspects of this are discussed in the next section and further below. To some extent, these could be accounted for within the framework of the Saha equation. However, once quantum effects become essential to the behavior of the particles, whether through Fermi degeneracy or through ion–ion correlations, their partition functions change significantly and the Saha equation is no longer the relevant statement of equilibrium. This will be true well before (3.48) and (3.49) find Z bal to decrease to zero and then become imaginary at high enough density or low enough temperature. The curve in the upper left corner of Fig. 3.7 shows where the electrons become Fermi degenerate based on the discussion of Sect. 3.1.3 (and assuming \(Z=0.63 \sqrt {T_{eV}}\), although the curve placement on such a log–log plot is not very sensitive to the specific assumption about Z). It remains worth noting, though, that Z decreases as density increases, even solely as a consequence of Boltzmann statistics. This in turn reflects the presence of n e in (3.47), which arises from the degeneracy of the free electrons themselves.
Following through on the question of when the electrons dominate the internal energy of high-energy-density plasmas, we can compare the total energy of ionization, which is part of the internal energy of the plasma, with the internal energy of the electrons. The ionization energy is the sum of \(Z_{i}^2 E_H\) over the ionization states up to Z. Here we will use the integer part of Z bal as Z for this energy. The electron energy per ion is (3/2)Zk B T e , where we will use Z = Z bal. Figure 3.9 shows the comparison of these two energies. The ionization energy forms a stairstep in such a model, though in reality the fact that several ionization states are present would smooth this out. The important conclusion is that, so long as the ion can keep on ionizing and the electrons are not Fermi degenerate , the ionization energy is the larger contribution to the internal energy. Only once the ions become fully stripped will the electron energy come to dominate. This is a major difference in comparison to low-density laboratory or space plasmas, in which the internal energy can usually be ignored.
The increase of internal energy and ionization energy (the stairstep) and electron kinetic energy (the line) in eV with increasing T e
3.4.2 The Ion Sphere Regime and Coulomb Effects
Equation (3.49) becomes inaccurate in compressed, denser matter with a high nuclear charge and low temperature. One reason is that the electrons become Fermi degenerate . Another reason is that the ions in high-energy-density plasmas do not exist in isolation. Even though plasmas are charge-neutral on a volume-averaged basis, in detail the particles arrange themselves so that a particle with any given charge is closer on average to particles of the opposite charge. As a result, one would have to invest energy to pull the plasma apart, so that the particles were far enough away from one another that their interactions were negligible. That is to say, the potential energy of the plasma is negative relative to vacuum. The introduction of new particles or charges to the plasma, as occurs during ionization, lowers the potential even further. This effect was long known as continuum lowering , but more recently is typically labeled ionization potential depression . It has consequences for the ions or atoms in the plasma—the vacuum energy levels having energies between the plasma potential and vacuum no longer exist. Figure 3.10 shows an energy level diagram to illustrate this point. With regard to ionization, the consequence is that the energy required to ionize is reduced relative to its value in vacuum.
A lowered continuum can eliminate some excited states and reduce the ionization energy
The amount by which the ionization potential is lowered can be evaluated by determining the change in electrostatic potential energy produced by the ionization of an atom or ion. There are two basic approaches to this calculation, corresponding to two regimes of validity. For low-density plasmas, in which the Debye length exceeds the spacing of the ions, one can calculate the changes to the shielding potentials and the corresponding electrostatic energy introduced by ionization. Equivalent treatments of this regime can be found in Zel’dovich and Razier (1966), Griem (1997), and Krall and Trivelpiece (1986). We will discuss only the case most relevant to high-energy-density plasmas, in which the spacing of the ions is more than a Debye length. This has the consequence that the shielding occurs in the vicinity of each ion individually. This will still be true if the electrons are Fermi degenerate , but the electron density will be more uniform in space than it would be otherwise. The fact that the shielding is local around each ion gives rise to the ion-sphere model and variations on it. Figure 3.11 shows that the boundary between the Debye shielding regime and the ion-sphere regime lies at lower densities than those of primary interest in high-energy-density physics.
Boundary between ion-sphere and long-range Debye-shielding regimes of ionization potential depression . The ion density is shown for reference, inferred from \(Z_i= 0.63 \sqrt {T_{eV}}\) and n e
In the ion-sphere model, each ion is assumed to influence only a region within a radius R o given by
in which n i is the particle density of the ions. Beyond this distance, the positive and negative charge densities, as seen by the ion, are equal, so these make no contribution to the electrostatic potential energy. Recalling that the typical ion density is 7.5 × 1022 cm−3, one can see that R o ∼ 10−8 cm ∼ 1 Å for solids, as one would expect since atoms are about 1 Å in size. Within R o , the charge due to the free electrons must balance that of the ion, Z i , and for consistency (with the viewpoint of other ions) the average free electron density must equal that throughout the entire plasma, so
The ion-sphere viewpoint provides the context for modern Thomas-Fermi models, for estimates of continuum lowering, and for an approximate EOS for high-energy-density plasmas, which are our next three topics.
3.4.3 The Thomas–Fermi Model and QEOS
The Thomas–Fermi model provides a way to account for the impact of ion-sphere effects on electron behavior. Based on a few very simple relations, the Thomas–Fermi model accurately includes the effects of ionization, excitation, Fermi-degeneracy, Coulomb interactions, self-consistent electron density structure, and to some extent ion–ion coupling. In various versions it may also include some quantum-mechanical effects such as those of shell structure. There is a nice summary of the Thomas–Fermi model in Salzman (1998) and more detail in Eliezer et al. (1986). The model itself requires nontrivial numerical solution, but we reproduce below some fits due to Salzman that provide a useful way to connect temperature and ionization.
The Thomas–Fermi model is a self-consistent combination of the ion–sphere model and the treatment of the electrons as fermions. The key to its power is that it demands that the electrostatic interaction of the electrons as fermions and the nucleus be self-consistent within this context. It naturally accommodates, in a classical context, the increase in electrostatic energy associated with increasing density or temperature. This allows one to ignore ionization and excitation as separate processes. They are accounted for, on average, by the expansion of the heated electrons or the lowering of the continuum as conditions change. The model can be formulated, in a simple form, as follows. It assumes spherical symmetry.
The electric potential, Φ(r), is given by the Poisson equation,
with the boundary condition that ∂Φ/∂r = 0 at the boundary of the ion sphere, r = R o , which follows from the net charge neutrality of each ion sphere. The electron density is given by the generalization of (3.29) to include a varying potential energy:
Thus, the nature of electrons as fermions is accounted for. The net neutrality of each ion sphere sets a constraint on the density,
which determines the chemical potential . These three equations are all that must be solved to describe the system. Once computational mathematics programs evolve beyond root finding to profile finding, this model may become simple to implement.
Now, supposing we have solved the above equations, we consider how the results may be used. The potential is only defined in the above to within an arbitrary constant, although the choice of this constant will affect the value of μ. It is conventional to choose Φ = 0 at the boundary of the sphere. As a result, the potential throughout the sphere becomes increasingly positive as the density increases. This is how this model captures the effects of ion-electron interactions. The charge state is calculated as
which amounts to assuming that the free electrons flow freely between ions and thus establish the density at the ion-sphere boundary.
Some other thermodynamic quantities are as follows. The electron pressure is
and the electron kinetic energy in each ion sphere is given by
This is not, however, the entire energy, because the Coulomb energy of attraction remains to be accounted for. This can be calculated directly, with the interaction energy of the electrons and the nucleus, per atom, being U en while the energy per atom of the interactions among the electrons is U ee . One has
and
With these definitions, the total specific internal energy is
The sum U en + U ee is the electrostatic energy per atom.
In the limit that n(r) is constant (and thus equal to Z n n i , where n i is the ion density), then one finds \(U_{en} = -(3/2) Z_n^2 e^2/R_o\) and \(U_{ee} = (3/5) Z_n^2 e^2/R_o\). As More et al. (1988) discuss, the assumption of constant density for all electrons is most applicable for either fully stripped ions at high temperature or strongly Fermi-degenerate electrons at high density. Under other conditions, the actual total energy for all electrons will reflect the actual profile of electron density. However, these results are also relevant to the free electrons, assumed in the model to have constant density. Their total electrostatic energy per atom, under this assumption, is given by U net = U en + U ee , evaluated for the constant (free) electron density with Z n equal to the net charge Z. Here again, the actual energy will be different to the extent that the free-electron density is not in fact uniform.
To avoid confusion in connecting this section with others, we should note that the zero of the energy scale here for the electrostatic energies is not consistent with the conventions used in other discussions in this book. In most of those other discussions, the implicit point of view is that a state of zero energy and pressure is a neutral gas nominally at zero temperature (but without quantum effects). At higher temperatures, positive energy is invested to ionize the gas. At high densities, the Coulomb interactions of the ionized gas provide some binding energy and reduce the energy input that would otherwise be required. In contrast, the state of zero energy in conventional Thomas–Fermi models has all the particles dispersed to infinity. In the ion-sphere applications of Thomas–Fermi models, it is further assumed that the ions are shielded from one another and that the potential of the ion-sphere boundary is equivalent to the potential at infinity. To convert the Thomas–Fermi result to the standard scale, one would have to add the ionization energy necessary to ionize the atom in vacuum, up to its ionization state Z, to the internal energy per atom. (See Sect. 3.5 below.) This matters to account for the internal energy properly.
Salzman (1998) provides a more extensive discussion of Thomas-Fermi models, and of many issues involving atomic physics in plasmas. He includes a fit to the ionization produced by such models, attributed to a laboratory report by R.M. More. My students have found this quite useful, and so it is included here for convenience. The fits are provided for T in eV and ρ in g/cm3. Using the notation above when feasible, one has an average charge
in which
in which for T = 0 one has \(x = \alpha \mathcal {R}^\beta \), where α = 14.3139, β = 0.6624, and \(\mathcal {R} = \rho /(Z_n A)\).
For T > 0, has \(x = \alpha \mathcal {Q}^\beta \) but the evaluation of \(\mathcal {Q}\) is rather involved. Take \(T_0 = T/Z_n^{4/3}.\) Then
where a 1 = 3.323 × 10−3, a 2 = 0.971832, a 3 = 9.26148 × 10−5, and a 4 = 3.10165. Then with T F = T 0/(1 + T 0) one has
where b 1 = −1.7630, b 2 = 1.43175, and b 3 = 0.315463. One also defines
where c 1 = −0.366667, and c 2 = 0.983333. Further defining \(Q_1 = A \mathcal {R}^B\), one has at last \(\mathcal {Q}\) as
One can set up these relations in a computational mathematics program so that they quickly can be evaluated to find Z(T) and thus T(Z) for conditions of interest, for example in seeing what can be inferred from experimental data.
Finally, there are a class of computer models known as QEOS models, where QEOS stands for quotidian equation of state, where quotidian means “everyday” or “routinely usable”. The Thomas–Fermi model is often incorporated into these (see for example the description in More et al. (1988)). Such models are likely to include additional terms or equations intended to account for the solid, liquid, and gaseous states and for the transitions between them. They can be a useful way to bridge the wide range of parameters that simulations must deal with.
3.4.4 Ionization Potential Depression
Here we consider the amount of continuum lowering in the ion-sphere regime. There have been several approaches. A simple estimate would be that an electron is free if it has enough energy to reach the boundary of the ion sphere, and that the difference between the potential energy there and at infinity (for an isolated ion of initial charge Z i ) is
The generalization of (3.71), to include the behavior at lower densities where the Debye length , λ D , exceeds R o , is
There are a number of models that have attempted to do better for the ion sphere regime, and at this writing (2014) it has only recently become possible to do well-resolved measurements of continuum lowering in this regime. Some measurements seem to support one or the other of the available models. But all the models are simplifications of various types. One can expect a more definitive understanding to arise over the next few years.
A brief survey of the historic models follows here. (A reader not interested in these details could skip to the discussion of the net ionization with reference to the next figure.) To compare these models we generalize (3.71) as follows:
Griem (1997) makes an approximate calculation of the shift in the energy levels of the ion by determining from the Poisson equation the electrostatic potential surrounding the ion, assuming a constant electron density, and by using the first-order perturbation theory of hydrogenic ions from basic quantum mechanics. One finds the principal quantum number of the highest remaining bound state to be
Zel’dovich and Razier (1966) find the same result from the semiclassical argument that the highest quantum number will be the one for which the semimajor axis of the orbit equals R o . The corresponding reduction in ionization energy is
Thus, such models find C = 1/2 and Z ∗ = Z i + 1 in (3.73). One can expect that this might under-estimate the lowering, because most of the electrons having principal quantum number n c will have larger orbital angular momentum and have orbits that attempt to extend beyond R o . But on the other hand, the actual interaction with the other bound electrons might reduce the lowering.
An important historic approach is that of Ecker and Kroll (1963) . Their calculation is a statistical mechanical one not structurally unlike the calculation we discuss in Sect. 3.5. One difference is that they treat the electrons (and ions) as a classical gas, so their results ought not to apply when the electrons are degenerate (as they are for much of the ion-sphere regime of interest). Another difference is that they seek to calculate the effects of Coulomb forces by a calculation of the microfields in the plasma, using traditional techniques of plasma physics. A third difference is that their calculation is not based on the ion-sphere model. Even so, they are forced to approximate the calculation and in so doing they choose, as a characteristic distance, the average interparticle distance among all particles,
The also find that it is the charge of the ion resulting from the ionization process (1 + Z i ) that matters, and ultimately they find
where their Z ∗ = (1 + Z i )4/3. Their C EK is of order 1 but is a function that depends on Z i and Z n in some way that is not very clear.
Other approaches are based on the semi-classical Thomas-Fermi model discussed in the previous section. The historic paper of this type is by Stewart and Pyatt (1966) . Their calculation differs in some ways from the now-standard approach to Thomas-Fermi, ion-sphere calculations described in the previous section, but what is important is that they apparently do not consider the effect of electron-electron interactions. They find, in the ion-sphere limit,
and thus conclude that Z ∗ = (Z i + 1) and C = 3/2, so the lowering is more than three times larger than in our simple result above. They confirm that, within their assumptions, the results are correct by numerical integration. Note that the ΔE found in (3.78) equals exactly − U en /Z ∗, the average electrostatic energy per electron for Z ∗ electrons of constant density interacting with a central ion of the same charge in an ion sphere (from the previous section). This raises issues discussed more clearly with regard to the next model.
More et al. (1988) report results of a more modern numerical Thomas-Fermi calculation, consistent with the description in the previous section. For a nuclear point charge of value Z n e, the net electrostatic energy per atom is
where coefficient of (9/10) is the combination of coefficients of 3/2 for the electron-ion interactions and − 3/5 for the electron-electron interactions. The negative of this is the amount of energy that would be required to remove the electrons to infinity from an initial state in which they have a uniform density within the ion sphere. Thus U net is the difference between the energy required to fully strip the atom in vacuum vs in the ion-sphere environment, and so represents the total continuum lowering for all the electrons. But as the electrons ionize successively this cannot be equal for each electron. If one imagines that the bound electrons are localized in orbits at small radii (which is necessary to get the ionization energy right in a semiclassical model), then it is a sensible approximation to treat the net charge of the ion, Q, as though it were a point charge, and to assume the density of the free electrons to be constant. In this case the continuum lowering for the next electron will correspond to the difference in U net as Z n increases by one. Mathematically, for ionization out of state Z i , this implies
which at large Z i is even larger than the Stewart-Pyatt value. Thus in this case we find C = 1.8 and Z ∗ = (Z i + 0.5).
All these estimates for continuum lowering imply that there can be conditions where ΔE is larger than the vacuum ionization energy, producing some ionization even at zero temperature. This effect is known by the somewhat misleading name of pressure ionization . (The name is misleading since only density enters. However, in dense, Fermi-degenerate matter the pressure can be substantial even at zero temperature.) To assess this, one can write the ratio of ΔE to the ionization energy in vacuum ( ∼ (Z i + 1)2 E H ). Specifically, using (3.80) for ΔE one finds
in which the approximation uses the ion density in units of 1024 cm−3. In our hydrogenic model a state will be ionized when the left-hand term here equals 1 and we will designate this ionization Z TF for later reference. This gives
In this model the ionization goes to zero at an ion density of 2.8 × 1023 cm−3. Figure 3.12 shows the ionization as calculated from this model.
The ionization from (3.82)
One can see in this figure that the amount of ionization remains fairly small for compression to only a few times solid density. In the regime where the ions behave like hydrogen, as a crude model one could take Z to be the maximum of the values implied by (3.81) and Z bal from the Saha model (3.49). Near the transition between the two models, this will underestimate Z, because plasma effects will reduce the ionization energy of the next couple of ionization states significantly. In addition, once the electrons become degenerate the thermal ionization is further reduced. We discuss the behavior found from a better, statistical-mechanical model below, in Sect. 3.5.2. By the time T e increases much above 10 eV, most materials of interest in present-day experiments will be in the Saha regime.
However, Fig. 3.12 can be quite misleading, because most ordinary materials, at temperatures of order an eV, do not behave like a simple hydrogenic model would predict. One might say, for example, that conductors have an effective ionization state corresponding to the number of free electrons per atom that exist in the conduction band. In the case of solid-density aluminum (Z n = 13), for example, this is about 3 electrons per atom. One can reasonably describe this as “pressure ionization ”, recalling again that a better term would be “density ionization”. To some degree, the appearance of ionization where the hydrogenic model would not find it could be due to subtle quantum effects. But more important is the classical impact of the existence of multiple electrons around the nucleus. In a classical context, one would say that the inner electrons act to shield the outermost electrons from the nuclear charge. This effect is accounted for in the Thomas–Fermi model, which does find approximately the correct number of free electrons for aluminum. (The difference from Fig. 3.12 presumably reflects the evaluation of self-consistent electron density profiles in the full model.) While all materials behave in similar ways in a global sense, the exact density or temperature where certain transitions occur varies greatly. Quantum effects in the ions may be very important, especially in the regime known as warm dense matter , corresponding to densities of order solid density and temperatures below a few eV. We discuss this regime of Fermi degenerate, strongly coupled matter in Sect. 3.7.
3.4.5 Coulomb Contributions to the Equation of State
The Coulomb interactions discussed in the previous section also contribute to the EOS. The corresponding contribution to the internal energy, for ions of charge Z, is
where 〈Z 2〉 should be an average of j 2 over the distribution of states of charge j, but will be quite close to Z 2 for a narrowly peaked state distribution of average ionization Z. The pressure of N a such atoms in a volume V , such that n i = N a /V , is
3.4.6 The Ions
For most conditions of interest, the ions can be treated as an ideal, classical gas. When this is the case, we can take the ion contribution to the pressure, p i , and specific kinetic energy, 𝜖 ik , to be
and
knowing that the Coulomb binding energy associated with close packing of ions will be included in the accounting for the electrons. Here the ion temperature is T i . In addition, in the context of our convention that the initial material state is a low (or zero) but positive energy state, the ions also contribute energies of ionization and excitation. We ignore excitation here, for reasons discussed near the start of Sect. 3.4, and once again use a hydrogenic model , describing the internal energy of the ions as
where the maximum allowed value of Z is Z n and we define R i as the internal energy per ion and we have evaluated it for our hydrogenic model of the ions. One could do better by including the continuum lowering in the sum making up R i , and even better by using actual ionization energies for the species in question.
3.5 Approximate Equations of State for High-Energy-Density Plasmas
In this section we work with the above results to find an approximate EOS for high-energy-density matter at two levels of sophistication. We begin with results for the regime of classical electrons, which spans most of our region of interest, applying as well the simple expression for Saha ionization \(Z \sim 0.63 \sqrt {T_{eV}}\). We then develop a more sophisticated model, working with the Helmholtz Free Energy. The reader can find an excellent discussion of the departure from an ideal plasma in Krall and Trivelpiece (1986), while Griem (1997) provides a connection with the more-recent literature.
There are two limiting cases where the simple models used here break down. One of them, discussed in Chap. 11, is connected with inertial fusion. At the temperatures of cryogenic fusion fuel, one must consider the nature of the fuel as quantum particles. In particular, one needs to examine the behavior of deuterons as bosons and of tritons as fermions. We consider this issue in Chap. 11. The other case is that of Fermi-degenerate, strongly coupled matter. New effects appear when matter is pushed into this regime, which was not known at the time of writing of the first edition of this book. We discuss this regime in Sect. 3.7.
3.5.1 The Simplest EOS Model
Here we combine the results of the previous two sections to assemble an EOS for the regime across which the Saha model is valid, but throughout which the matter may be ionizing and Coulomb effects may not be negligible. This spans most of our parameter space of interest. One could replace the thermal components of pressure and internal energy with results for Fermi-degenerate matter, when T e < 𝜖 F , to have a crude model that also spanned the Fermi-degenerate regime. For simplicity, we have chosen here not to do this. A more complex model, able to deal with the degenerate regime more accurately, is presented in the following subsection.
We can represent the electrons as a classical gas so that the electron thermal pressure is n i Zk B T and the electron internal thermal energy is 3n i Zk B T/2. We can anticipate that these values will become inaccurate when the electrons become Fermi degenerate , and we know that pressure ionization may increase Z at temperatures below some value when the ion density exceeds 1024 cm−3. However, the full impact of ionization is more complex, as we discuss in Sect. 3.5.2. In most cases, the electrons dominate the pressure and the kinetic energy. Based on the discussion above, we can represent the pressure as
and the internal energy as
Here we have expressed the results in terms of the ion density n i = ρ/(Am p ). In this simplest model, we give formulae for these results for two cases: an ionizing regime in which \(Z = 0.63 \sqrt {T_{eV}}\) and a regime with fully stripped ions, for which Z = Z n . With n i in cm−3, we have:
For the ionizing regime
and
and for the fully stripped regime
and
We can evaluate the specific heat at constant volume, c V , for these cases as well. For the ionizing plasma case of (3.91), we find
while for the fully stripped case of (3.93), we have
We will first explore the implications of the first two equations for an element of high enough Z to keep ionizing. Then we will consider carbon as an example of an element that can be fully stripped. Considering a high-Z element, it is informative to compare the pressure and energy from (3.90) and (3.91) with their ideal-gas equivalents, which are p = n i (1 + Z)k B T e and ρ𝜖 = (3/2)n i (1 + Z)k B T e , respectively. Figure 3.13a, b shows the ratio of the more-complete estimates in (3.90) and (3.91) to these ideal-gas values. One can see that the model for the pressure fails badly in the Fermi-degenerate region, which is no surprise. In actuality, pressure and internal energy both increase in that region as one moves down and to the right. Otherwise, the pressure across the space of Fig. 3.13a is typically between 50% and 100% of the ideal-gas value. Thus, the ideal-gas value is not too bad but it may overestimate the pressure. In contrast, Fig. 3.13b shows that the internal energy is typically twice or more the ideal gas value, as indeed one would expect from Fig. 3.9.
(a) The pressure for an ionizing high-Z element is shown, normalized to the ideal-gas pressure. The contours increase from the lower right, and are at 0, 0.1, 0.3, 0.5, and 0.99. (b) The internal energy density for an ionizing high-Z element is shown, normalized to the ideal-gas value. The contours increase from right to left, and are at 3, 5, and 6
With an increased internal energy and a decreased pressure, the value of γ evaluated from ρ𝜖 = p/(γ − 1) must decrease. Figure 3.14 shows the values of γ obtained from (3.90) and (3.91). Here again, the quantitative value (∼1.25) should not be taken too seriously but the qualitative point, that γ should be reduced substantially compared to the ideal-gas value of 5/3, should be real. Standard EOS evaluations for xenon (A = 130, Z n = 54), for example, give γ ∼ 1.2 to 1.3. We will see in Chap. 4 that this implies increased compression by shocks.
The value of γ inferred from the data shown in Fig. 3.13 for an ionizing high-Z element is shown. The contours are labeled. The value never reaches 1.3
Now consider carbon, an element with six electrons that can become fully stripped at modest temperatures. Using our estimate that \(Z = 0.63 \sqrt {T_e}\), Carbon will ionize fully at T e = 91 eV. At higher temperatures, the internal energy still includes the energy of ionization, but this contribution does not increase any further. To estimate the properties of carbon, we use (3.90) and (3.91) until T e = 91 eV, then (3.92) and (3.93) at higher temperatures. This produces Figs. 3.15 and 3.16. The pressure shows a structure similar to that of the ionizing case. In contrast, the internal energy shows more structure, as Fig. 3.15b shows. At temperatures where carbon is not fully stripped, the ionization energy is a dominant factor and the internal energy substantially exceeds the ideal-gas value. As ion density increases, though, this effect becomes smaller. Then, once the temperature has increased enough to fully strip the material, the internal energy decreases toward the ideal-gas value.
(a) The pressure for carbon is shown, normalized to the ideal-gas pressure. The contours increase from the lower right, as labeled. (b) The internal energy density for carbon is shown, normalized to the ideal-gas value. The contours are labeled
The value of γ inferred from the data shown in Fig. 3.15 for carbon is shown. The contours are labeled
This behavior leaves its footprint on the inferred γ, shown in Fig. 3.16. At temperatures below 91 eV, one sees behavior very like that of Fig. 3.14. The inferred value of γ is generally near 1.25. Then, once the element becomes fully stripped, γ begins to increase, although in this model it does not reach the ideal-gas limit of 5/3 even at a temperature of 1 keV. In optically thick media, at true LTE this will begin to become artificial, because the coupling to the radiation will begin to reduce γ toward 4/3 by the time T e reaches 1 keV.
3.5.2 An EOS Model Based on the Helmholtz Free Energy
The assumptions of the prior section are that each ion species and the electrons behave as a classical gas and that there is some energy of Coulomb interaction that lowers the internal energy. This applies across much of the high-energy-density regime. A more broadly applicable set of altered assumptions are that the electrons behave as fermions within each ion sphere and that the Coulomb effects also act to lower the ionization energies. We can analyze such a system using very standard statistical mechanics. We begin with a very brief summary of the statistical mechanical context. Our point of view will be that there are N a atoms within a volume V . The statistical analysis is based on the assumption that we know the probability that the entire system will be found in some state having total energy E T . For particles obeying Boltzmann statistics, this is proportional to \(e^{-E_T/(k_B T)}\), with the constant of proportionality established so that the sum (or integral) over all possible states, \(\mathcal {S}\), yields the correct total number of particles. The discussion in this section in part follows that of More et al. (1988) and also (unpublished) work by Igor Sokolov.
The Helmholtz free energy is then given by
The Helmholtz Free Energy is useful in determining quantities of interest. The units of free energy are energy units, though it may be expressed per particle, per unit mass, per unit volume, or as a total for a system of particles. We did the latter above. From this free energy one can find the pressure, the internal energy density (ρ𝜖), heat capacity at constant volume C V , the entropy density (ρs), and the electron chemical potential , μ e on the assumption that T e = T i = T, as follows:
and
Equation (3.98) first relates the internal-energy density to F and to entropy density S, then gives a simpler combined expression. (One property of F is that S = ∂F/∂T at constant N a and V .)
For a system having independent species labeled A, B, …, each of which has a total number of particles N A , N B , …, one has
in which the partition function of species k is given by \(\mathcal {P}_k\). The partition functions are in turn sums over the probability that given states are occupied, but here we will use results for the values of such sums and will not derive them. Using Stirling’s formula, N! = (N/e)N, where here e is the base of the natural logarithm, one finds
where one assumes all species are equilibrated to the same temperature T.
The specific case of a single atomic species that is ionizing, which is our focus here, deserves some further discussion. One could write down a partition function for all possible states of the atom including all degrees of ionization. The corresponding value of N in these equations would be the total number of atoms, N a . Thus N a ! would appear in the denominator. However, such a sum would include many very unlikely states, such as states near complete ionization having total energies very far above k B T. Since we will in fact be interested only in equilibrium and perhaps near-equilibrium states, we divide the states of the atom into distinct ionization states, each of which has a population N j and (in principle) a range of energy levels accessible to it. And we will find the equilibrium populations by seeking a minimum of F. We then, for the ions, have a sum like that of (3.103), in which there is a term for each ionization state with its corresponding population.
How to handle the electrons is a somewhat more subtle question. The standard derivation of the Saha equations assumes that the electrons freely sample the entire volume, and implicitly takes the electron density to be constant throughout. In an ion-sphere environment, however, this is not really a sensible assumption. In the Thomas–Fermi model, for example, the chemical potential is determined within an individual ion sphere. Even in the classical limit, the electron partition function in an ion-sphere environment varies with the ionization state. We will approach this as follows. We will assume that the distribution of the electrons corresponds quite closely to having local quasi-neutrality, so that N ej = jN j electrons associated with atoms having ionization j and the corresponding electron partition function. This leads us to write
where F e (j) is the free energy of the electrons in an atom of ionization state j, which we will later evaluate for the two limiting cases.
It is standard practice to recognize that the partition function of each species can often be factored into terms representing distinct physical mechanisms, such as the energies associated with translation and rotation. In the present context, the question of how to treat the contribution to the free energy of the electrostatic energies that produce continuum lowering is also significant. In a low-density plasma, this represents averages over many atoms, and thus reflects the average state of ionization of the system. In the ion-sphere limit, however, each ionic species has its own electrostatic energy and its own electron chemical potential (here μ e ), reflecting the specific properties of that ion species. In effect, the statistical analysis here must be viewed as an average over a large number of distinct atoms or over an ensemble of possible states of a given atom. The implication is that we must write the partition function of the species having ionization j as
where
in which the ion mass is M, the Planck constant is h, the Boltzmann constant is k B , and the ion-sphere radius is R o . Note that the electronic charge is (italic) e, while e as in (3.104) is the base of the natural logarithm. We expand \(\ln (\mathrm{e} )\) to 1 in the following to minimize confusion. Here as in Sect. 3.4.1 we consider only ground-state electrons so that g j = 2. The total ionization energy of state j is the sum over the ionization energies, in vacuum, of all lower ionization states,
with E k(k−1) being the energy to ionize from state (k − 1) to k, as it was above. Here also E 0 = 0.
Motivated by the results shown in Sect. 3.1.3, we will consider the electron free energy F ej in only its two limits, adding to the subscript when needed to discriminate between them. For the degenerate limit we have
which depends on the electron density of an ion sphere having j free electrons. For the classical limit we have
in which the electron mass is m e . Note that neither of these electron free energies depends on N j . The partition function of the bound electrons is accounted for by g j in \(\mathcal {P}_{\mathrm {trans}}\), and there is no further electron contribution for the neutral state so F ed(0) = F ec(0) = 0. The Fermi energy 𝜖 F is defined above in Sect. 3.1.3. For the degenerate case, (3.110) gives the correct value for the chemical potential , as is seen below. But it is significant that the Fermi energy in an ion sphere depends on j, and must be evaluated using an electron density of n e = jN a /V , as shown explicitly in (3.110).
Regarding the classical case, note that the electron partition function is defined by integrating over all the states corresponding to a single ion sphere, of volume V/N a , and then recognizing that the total volume corresponding to the N j such ion spheres is V N j /N a . Then when the partition function corresponding to this volume is divided by N ej = jN j one obtains the final result in (3.111). In instead one were to demand that the electron density in this term be constant independent of ionization state, which is done implicitly in the standard derivation of the Saha equation , then the analysis below would obtain Eq. (3.47) to describe the ionization states.
With all the above, we have for the Helmholtz free energy
It is worth noting that one would obtain the same result by defining E (k+1)k to include the contributions from continuum lowering from (3.80). Equation (3.112) differs from the results shown in More et al. (1988) in two ways, both reflecting the considerations about how to include the electron effects. They have the Coulomb energy depending on 〈j〉2, with 〈 〉 denoting an average over ionization states, while here we have energy proportional to 〈j 2〉 through the average of U net. This is, in most cases, a small difference. Similarly, they define F ej in terms of an average charge, while the average shown here is a more complex one.
3.5.2.1 Ionization from the Helmholtz Free Energy
The equilibrium ionization corresponds to a minimum in the free energy. We thus find the variation in F with respect to a change in the populations of two adjacent ionization states, N j and N j+1 and set this equal to zero. Because we associated the electrons with each ionization state above, there is no separate contribution from them. An ionization event reduces N j and increases N j+1 by one. We have
Since δN j = −δN j+1 this implies
from which
We apply this relation recursively to obtain N j /N 0, which will prove most useful below. Since E o = U net(0) = 0 this gives
We can proceed to evaluate the average ionization. Just as we discussed in Sect. 3.4.1, we can now find
where N 0/N 0 is of course 1. We can note that the denominator here is equal to the ratio N a /N 0, which will also prove useful below. A complication for high densities is that continuum lowering may remove one or more of the lower ionization states. For such cases one must modify this calculation accordingly, setting N j to zero for the quenched ionization states and using a reference state other than state zero in (3.116). For this purpose it is helpful to realize that one can multiply the numerator and denominator of (3.117) by any quantity whatsoever without changing its validity.
To this point these equations are completely general, within the validity of their assumptions. Thus, for example, one could use known values of the ionization energies and a general treatment of F e to determine the thermodynamic quantities needed for a simulation and to accurately capture the behavior across the transition from classical to degenerate behavior. This is the approach taken by some EOS models for computations (for the regime of ionized matter), such as the PROPACEOS model (developed by PRISM Scientific). Or one could develop more general models in the same spirit, to include for example electronic excitation.
If instead we assume hydrogenic ions and so take E (j+1)j = (j + 1)2 E H , then we obtain the distributions shown in Figs. 3.17 and 3.18. Here in evaluating (3.124) we have applied the form of F e appropriate to whether or not any given ionization state is degenerate. The trends seen in the Saha regime we explored in Sect. 3.4.1 remain and are labeled in Fig. 3.17: over much of the space the ionization increases gradually with temperature by thermal ionization. In the degenerate regime, ionization is suppressed until the density becomes so large that pressure ionization sets in. At high enough temperatures, the carbon becomes fully ionized. In Fig. 3.18, one can see similar trends, although the large thermal ionization at low density and high temperature makes them less evident.
Ionization level shown against electron temperature and ion density for carbon with Z n = 6
Ionization level shown against electron temperature and ion density for ionizing high-Z matter having Z n = 80
Degeneracy acts to strongly reduce ionization , as we can see by considering the purely degenerate case, when all ionization states of the atom are degenerate. (In the non-degenerate case, despite the modest effects of continuum lowering, we obtain results very similar to those of Sect. 3.1.3). In the degenerate limit (3.115) can be written
which has the correct limit for large j, when the change in electron free energy should equal the chemical potential , μ e = 𝜖 F . This gives, with g j+1 = g j ,
We can view these population balance equations as having three factors—an ionization factor, a Coulomb factor, and an electron factor. In order to produce a population distribution that is peaked above whatever minimum is allowed by continuum lowering, the other factors must exceed the factor involving the ionization energy, so that the net argument of the exponent is positive for some ionization states. As a practical matter, the Coulomb factor cannot do this for states beyond those already quenched by continuum lowering. In the classical limit the electron factor can be negative enough for the average Z to be peaked well above zero, just as we saw in Sect. 3.4.1. In contrast, in (3.119) the electron factor has the wrong sign to produce a distribution peaked above the minimum.
3.5.2.2 Thermodynamic Properties from the Helmholtz Free Energy
We can now write expressions for the thermodynamic parameters of interest, working with (3.97) through (3.101). We have the pressure,
the internal energy
the heat capacity at constant volume
and so on, where for any function D
Here to be consistent with usage elsewhere in the text we write N a /V = n i , noting that here this is the total density of particles having a nucleus, including neutral atoms.
We have chosen in the above to write the sums in terms of F e (j), and not yet to evaluate F e , for the following reason. As the ionization increases, the electron density in the ion sphere increases. As a result, an ion sphere can transition from the classical limit to the degenerate limit in consequence of becoming more ionized. An accurate EOS model would evaluate F e (j) using the methods discussed in Sect. 3.1.3. To obtain the approximate numerical results given below, we apply a sharp transition from F ec to F ed as the Fermi temperature rises above T with increasing density. For the limiting cases is worth noting that
and that
which gives the intuitive result for the internal energy of the electrons. One also finds
One can evaluate the above quantities numerically. For (hydrogenic) carbon, Figs. 3.19 and 3.20 show plots for the present model like those shown in Figs. 3.15 and 3.16 for the simpler model above. The ideal-gas pressure and internal energy are based on the calculated ionization. The density scale is shifted here, to show more of the behavior at higher densities where degeneracy may matter. In the Saha regime, at relatively low density and high temperature, the two sets of figures are similar, showing a moderate pressure decrease that is due to Coulomb effects, and an internal energy that decreases from near twice the ideal gas value (because of energy in ionization) as temperature increases. There also remains a substantial region of reduced γ, although the values are larger than they were in the simpler model. The figures show more structure, some of which reflects the explicit calculation of the ionization. One sees this effect most strongly as temperature increases at the lowest density. The curve in all these figures showing the boundary for electron degeneracy is drawn on the assumption that \(Z = 0.63 \sqrt {T_{eV}}\), which ignores the decrease in ionization as the electrons become degenerate, so the actual transition to degenerate behavior occurs at a few times higher density, as the contours in the figures suggest.
(a) The pressure for carbon is shown, normalized to the ideal-gas pressure. The contours increase from the lower right, as labeled. (b) The internal energy density for carbon is shown, normalized to the ideal-gas value. The contours are labeled. The abrupt changes at some boundaries reflect either the threshold for pressure ionization or the finite grid of the calculation. The dashed, gray, diagonal lines show the location where k B T e = 𝜖 F , evaluated using \(Z = 0.63 \sqrt {T_{eV}}\). The solid diagonal lines, at about three times higher density, show the approximate location where Fermi degeneracy becomes important in this calculation, where the internal energy begins to increase
The value of γ inferred from the data shown in Fig. 3.15 for carbon is shown. The contours are labeled. The dashed, gray, diagonal line shows the location where k B T e = 𝜖 F , evaluated using \(Z = 0.63 \sqrt {T_{eV}}\). The solid diagonal line, at about three times higher density, shows the approximate location where Fermi degeneracy becomes important in this calculation, where the internal energy begins to increase
The degenerate regime shows additional effects reflecting the additional physics included in the present model. Both pressure and internal energy increase strongly above ideal-gas values at low temperature and high density. This reflects the Fermi pressure and Fermi energy of the electrons freed by continuum lowering. At higher temperatures, there is a modest reduction of pressure and internal energy by a combination of degeneracy and Coulomb effects. The Coulomb aspect was already present in the Saha model.
3.6 Generalized Polytropic Indices
Both ionizing and radiating plasmas, unfortunately, have pressures and internal energies that change in complex ways until after the plasma is fully ionized or completely radiation dominated. As a result, the assumption of constant polytropic index is a poor one for such systems. In this case, the question is whether there is any fairly simple way to treat the behavior of the system that might still allow simple models to be developed. Fortunately, one is able to do so, and several approaches are worked out in the literature. The best choice depends on the application. Here we identify three generalized polytropic indices for specific contexts of interest.
The derivation of shock behavior in Chap. 4 depended on the explicit expression ρ𝜖 = p/(γ − 1). Accordingly, we define γ as the shock index via this relation. This is what we used in Sects. 3.5.1 and 3.5.2 to plot γ based on our EOS models. This index gives correct results for the change in properties across a shock front. For sound-wave applications, the derivation in Chap. 2 makes it clear that the relevant index (under isentropic conditions) is
However, our EOS does let us take this derivative directly so there will be some work to do to figure out how to evaluate this. For heat-transport applications, we need to find a thermodynamically correct generalization of (2.31)), which will define a heat-transport index, γ h . For an energy density flux −∇⋅H, we will find an equation of the form of (2.31),
We seek to know how this equation relates to our typical, known EOS equations. To apply the thermodynamic analysis, note that ∇⋅H = ρdq/dt, where an increment of specific heat input is dq. If γ s as defined by (3.127) is constant, then γ = γ s = γ h .
Finding useful expressions for these quantities, and in particular a useful equation for heat transport, takes one into the realm of thermodynamic functions. It is easy to get lost in the forest where one seemingly can take the partial derivative of anything with respect to everything. Our job here is not to visit all the trees in this forest, but rather to develop specific equations that we will use later. Remarkably, aside from some patience, all the fundamental information we need to do this is a pair of equations from the first and second laws of thermodynamics,
where d𝜖, dq, and ds are the specific internal energy, heat input, and entropy , respectively, and two mathematical relations, specifically
and
As is usual in thermodynamic calculations, at any given moment we express the thermodynamic functions in terms of two independent variables chosen from the three quantities ρ, p, and T. We proceed at first by expressing 𝜖 as 𝜖(p, ρ); so from (3.129) we find
We also have, as ds is an exact differential,
The specific heats involve the use of T and ρ or T and p as the thermodynamic variables. Equation (3.132) implies that the specific heat at constant volume is
while the specific heat at constant pressure is found by writing 𝜖 as 𝜖(T, ρ) in (3.129) and then differentiating, to obtain
Note that we can evaluate the coefficients in (3.132), (3.134), and (3.135) from any EOS like those above that relates p, 𝜖, ρ, and T. Multiplying (3.135) by (∂T/∂ρ) p , and using the definition of c V , one finds
while from the chain rule
This then gives for the heat input per (3.132)
To simplify this further note that
obtained by substituting from (3.136) and (3.137) into (3.132) and taking (∂p/∂ρ) s . The isentropic sound speed and thus γ s can be found from (3.139) and (3.135), by
This expression for the sound speed is readily evaluated from expressions for p and 𝜖.
We pursue the heat-transport coefficient as follows. We substitute for c p in (3.138), which becomes
or
using (3.131). This is the form we were seeking. The quantity in square brackets has the form of (2.31), as desired. It also shows that the sound speed in (3.128) is the isentropic sound speed. One can convert to an expression for the total heat input by multiplying (3.142) by ρ. Thus one finds
So long as (∂p/∂ρ) T = p/ρ and (∂𝜖/∂ρ) T = 0, one has γ s = γ h . This is the case for simple ideal gasses but not in the regimes where more complex effects are important or in the radiating plasmas considered in Chap. 7.
We can then apply our simplest model—(3.140) and (3.143)—to find γ s and γ h . This produces rather messy expressions, but they are readily evaluated by computer. Figure 3.21 show the results for carbon at 1 g cm−3 , which has an ionizing regime followed by a fully ionizing regime. The shock index, γ, increases slowly in the ionizing regime and then more quickly once the carbon is fully ionized. It will eventually approach 5/3. The heat-flow index, γ h , is close to γ in the ionizing regime but jumps abruptly to 5/3 when the carbon becomes fully ionized. This is sensible—beyond that point energy is not being absorbed by further ionization. These trends in γ and γ h remain present at lower density. The sound-speed index, γ s , starts much higher than γ and decreases to approach it with increasing temperature. Then upon full ionization the index jumps abruptly to near 5/3 (and thus the sound speed jumps too). At lower density, γ s remains close to γ in the ionizing regime but still jumps upon full ionization. At higher density the behavior becomes more complex, but the model becomes poor at low temperature when the electrons in fact become degenerate. The major conclusion here is that the sound speed and heat-flow rate can vary substantially across high-energy-density systems, especially as they transition to a fully ionized state.
Values of the shock γ (solid black), sound-speed γ s (dashed), and heat flux γ h (gray), for carbon at 1 g cm−3
3.7 The Degenerate, Strongly Coupled Regime
At the time of the writing of the first edition of this book, the community believed that material physics would become simple as density increased. The electrons would be strongly degenerate and tend toward constant density, and the ions would be happily isolated in their ion spheres, not unlike the monads of Leibniz. The model described above, based on the Helmholtz Free Energy, would work well. But we have learned since that this model also fails at high enough density. We came to know this because experiments began to observe behavior showing that late-twentieth-century models were qualitatively incorrect. They observed changes in material structure that were entirely unexpected, and that the transition to a liquid state occurred at very different temperatures than had been predicted. We came to understand what is happening qualitatively by means of advanced computer simulations that only become possible during this same period.
Figure 3.22 illustrates the fundamental reason why unanticipated behavior arose. When one fills some volume with spheres, there remain gaps between them. The models above assumed that the electron-density at and beyond the ion-sphere boundaries was constant, but in fact what happens is that the electrons cluster in the gaps. Once the ions are incompletely shielded, they are affected by other ions as well as by the clusters of electronic charge. This enables the formation of chemical bonds, not unlike ionic bonds. The difference relative to ordinary chemistry is that these bonds have transition energies in the keV range as opposed to the eV range. These interactions are now sometimes described as “kilovolt chemistry.”
Charge clustering is a key feature of Fermi-degenerate, strongly coupled matter
One might describe Fermi-degenerate , strongly coupled matter at high energy density as a “quasi-solid”, because it can have various crystal structures and can transition between them, despite the presence of additional freed electrons. Table 3.1 shows some of the standard structural designations. Modeling these structures is challenging, as one must account for the quantum mechanical effects including the interactions of several (or more) ions. The theoretical calculations employ two methods. One of these is Density Functional Theory , a quantum-mechanical approach that represents the electron density using functions. The other is Molecular Dynamics , which models the interactions of multiple molecules (or ions) from first principles, using potentials to describe the forces between them. In the regime of interest, the potentials must be quantum-mechanical and may come from calculations using Density Functional Theory. Only in the twenty-first century have computers become capable enough to enable these calculations for high-energy-density conditions. The interested reader will have to seek further discussion of these methods elsewhere, as this is a topic rather afar from the primary focus of the present book. Simple models that capture the essential physics have not yet emerged, and by analogy with chemistry may not.
Figure 3.23 is based on the results of calculations by Martinez-Canales et al. (2012), using Density Functional Theory, for carbon. One sees that the structure, at low temperature, is predicted to change three times between 10 and 200 Mbar pressure. Experiments with carbon, at this writing, have reached pressures of 50 Mbar (Smith et al. 2014). There remains much to be learned about the behavior of quasi-solids at high energy density.
Phase diagram of Fermi-degenerate, strongly coupled carbon, from calculations using Density Functional Theory
3.8 The EOS Landscape
At this point it may be useful to summarize what we have learned about the equation of state in high-energy-density systems. Figure 3.24 provides this summary. The specific lines in the figure are drawn for an ionizing plasma , assuming A = 2Z n , but the relative orientation of the various elements in this log–log space is not sensitive to these assumptions. At the upper left is the ideal-plasma regime. Examples are hot enough coronal plasmas , as for example in the laser-heated zone in front of a dense target, or the plasma generated in Z pinches during their implosion (see Chap. 9). At the right pressure ionization becomes important, as occurs when solids are sufficiently compressed. Throughout the lower right region the electrons are Fermi degenerate , which determines the pressure needed to compress solid-density matter including the fuel for inertial fusion.
The landscape of EOS for high-energy-density plasmas
In between these limits is the realm of many experiments in the early twenty-first century. Here the matter is partly ionized but probably is not fully stripped, the ions live in the privacy of their own ion spheres but represent much of the internal energy in the system, and the electrons are a Fermi gas whose pressure is reduced by Coulomb interactions. The internal energy is roughly twice that of an ideal gas, because of the energy invested in ionization. As a result, γ is in the vicinity of 4/3. Across much of this regime, a model based on the Saha equation should work well. But the Saha model fails at high density and temperature because of the combined effects of degeneracy and continuum lowering, as we have seen.
This completes our discussion of specific models of equations of state. In the following chapters, we will typically take γ = 4/3 or 5/3 for our examples. We will not need to distinguish among the different polytropic indices until we work with radiation hydrodynamics in Chap. 7. But it should be clear from the above that γ can be substantially less than 5/3, that these dense plasmas are not ideal-gases, and that it is not so easy to know just what the equation of state is. The next section discusses tabular equations of state.
3.9 Tabular Equations of State
The chapter thus far has made it evident that equations of state in the dense-plasma regime are complicated. The appeal of using a polytropic index , at the expense of detailed accuracy, is quite clear. Indeed, this will be our approach throughout much of the text. But if one is to try to simulate these systems with computers, then one would hope to be more accurate. It is evidently a great challenge to accurately simulate the behavior of materials at high energy density. One has Coulomb energy corrections, degenerate electrons, pressure ionization , and ionization potential depression , among other effects. To be fully accurate one would need to include several effects that we mentioned but did not incorporate, such as the impact of bound electrons. One would also need to handle the transitions between regimes more accurately. But the actual problem is worse than this, because high-energy-density matter nearly always evolves out of and is adjacent to matter that is not at high energy density, but rather is in a solid or liquid state. So realistic computations must also be able to account for these states of matter and for their transition to hotter and perhaps denser conditions. A particularly difficult example at this writing is that of the behavior of the wires in Z-pinch plasmas (see Chap. 9). These begin as solids, ablate and (perhaps) explode, creating the material that the Z pinch accelerates inward. Modeling these dynamics is a severe challenge.
One approach to addressing these issues for simulations is to use a tabular EOS. The idea behind a tabular EOS is that one can work with experimental data, molecular dynamics simulations, and the best possible models. From them one can construct a table giving two of the thermodynamic variables (ρ, p, 𝜖, and T) as a function of the other two. As is true of all the models we have discussed, this is necessarily done in equilibrium. Then a computer code can interpolate from the tables to find the properties it needs with adequate accuracy.
One challenging aspect of constructing such a table is the need for thermodynamic consistency. The table will show how some thermodynamic quantities vary when others are held constant. These variations must be thermodynamically consistent. As one does work on the material or adds heat to it, the changes of state that result must be consistent with the first law of thermodynamics. If this were not the case, then the computer code using the table would mysteriously create or absorb energy in an unphysical way. Achieving thermodynamic consistency in practice, while merging models that cover adjacent regimes, can be very difficult. One can check for thermodynamic consistency by applying the first law of thermodynamics to the table. One way to do this is to evaluate the local deviation from the first law of thermodynamics. Landau and Lifshitz (1987) show in Vol. 5 that one can write the first law of thermodynamics as d(ρ𝜖)/dV + p − T(dp/dT) = 0. One can evaluate this quantity throughout a candidate EOS table and display the results as curves or a contour plot.
The most widely used EOS tables are the SESAME tables, available from the Los Alamos National Laboratories. These tabulate specific pressure (pressure per unit density) and specific energy as functions of density and temperature, over several orders of magnitude in density and in temperature. Figure 3.25 shows two examples based on these tables. In each case, we have used the equation of state to plot γ. The range of temperatures in the table is shown. The densities shown are solid density (dashed) and 0.1 g/cm3, which are relevant to laboratory work in high-energy-density physics. One sees first that the behavior at low temperatures is quite different. This reflects the presumed development of a gaseous state (and perhaps even clusters) at low densities, with many degrees of freedom, which forces γ close to 1. In contrast, the solid becomes more ordered as temperature decreases. From traditional thermodynamics, one would expect γ to approach 3 at low temperatures if the solid forms a lattice with tightly bound planes. In the tables, γ sometimes exceeds 3 at low temperatures.
For polyethylene (C1H1) on the left and xenon (A = 131, Z = 54) on the right, these figures show the inferred γ from the SESAME table. The lower curve is at 0.1 g/cm3 density, while the upper curve is at solid density. Credit: Carolyn Kuranz
At the highest temperatures, the materials seem to approach γ = 5/3, which would correspond to a fully stripped, ideal gas. We comment more on this below. At intermediate temperatures, between a few eV and 100 eV for polyethylene and a few eV and 1000 eV for xenon, γ is reduced. This is as expected from the previous discussion in this chapter. Indeed, the result for xenon is not far above the value we inferred for an ionizing, high-Z material. The value of γ for polyethylene, on the other hand, is not so far below 5/3. One might be skeptical as to whether this decrease is in fact large enough.
These figures also provide one example of the limitations of these tables. If the high-temperature states (above 1 keV) were truly in equilibrium, as is assumed, then the presence of the radiation field would be driving γ to 4/3. So these tables ignore the radiation field. The problem is that they have to make some specific assumptions, though in this case they do not assume true equilibrium. Real systems do vary greatly with regard to the coupling of the radiation field and the matter. There is no way that one table can account for this. Any given computer code may or may not handle it well.
There are other problems with the use of EOS tables in particular, and equilibrium models in general, in simulations of real systems. Real systems are almost never in equilibrium. They are often in steady state, or nearly in steady state, but not in equilibrium. A good example is a plasma that expands from a hot surface but is not actively heated. The expanding plasma cools, and after a time its properties slowly evolve. Even so, on the scale of tens of ns that often applies, the ions and electrons may not recombine and the plasma certainly will not reach its equilibrium state. The EOS table, on the other hand, presumes the plasma is instantaneously in equilibrium. Thus, if it reaches a condensation temperature, the table will make it condense, no matter how unrealistic this may be. This, and theoretical equilibrium phase changes in general, can be a source of abrupt density changes in simulations that are completely unreal. There are times when an ideal-gas model with fixed γ provides a much more realistic approach to simulating a time-varying system. The main point is that one must pay attention, think about what one sees, and not assume that the code reveals truth.
In addition, you may have noticed that some of the equations above would produce regimes where the pressure from a given model became negative. This happens with the models used for the EOS tables as well. In some cases, this is sensible. For example, the only realistic way to incorporate tension in a material, in the context of a hydrodynamic model, is by adding negative terms to the pressure. If the material is tightly enough bound and cold enough, it may be sensible in this sense to treat the pressure as negative. However, the existence of negative pressure regions in EOS tables can create serious problems when simulating real, nonequilibrium systems. In the example of the previous paragraph, for example, the plasma expanding from a surface may have a temperature and density that would correspond to a condensed state with tension in equilibrium, yet in actual fact may be more accurately treated as an ideal-gas. In some contexts, it is sensible to modify the EOS tables to destroy the tension regimes and maintain positive pressure. When the EOS table works well, it will do a better job of reproducing the dynamics than any simpler model can. But it cannot be counted on to always work well. It is very often sensible to compare simulations using EOS tables for various similar materials and also using a fixed γ to help determine which aspects of the observed dynamics are due to the specifics of the EOS table.
Finally, tables do not typically exist for novel materials, such as low-density foams. These materials are not microscopically uniform. They are unlikely to behave like a uniform, low-density material. There is some discussion of foam behavior in Zel’dovich and Razier (1966), but it applies only to foams that are compressed very gently by comparison to the behavior of typical high-energy-density systems. Indeed, in high-energy-density experiments to date with foams, the uniform-density models fail to accurately predict phenomena such as shock-wave propagation. Whether in the end new tables or some other approach proves the best for working with them remains to be seen.
3.10 Equations of State in the Laboratory and in Astrophysics
A moment’s thought will show that equation of state (EOS) properties are quite important in astrophysics. In gravitationally bound objects, such as planets, white-dwarf stars, or neutron stars, the interior pressure is determined primarily by gravity. However, to know the density, and hence the volume of the material in any given pressure range one must know the equation of state. Direct astronomical measurements can determine the mass, and sometimes the size, of such objects, and may be able to learn about the surface composition from spectroscopy. But there is usually neither direct nor indirect information relating to the interior. (An exception is the Sun, for which seismology is possible and productive, producing data that greatly constrain the EOS.)
Assuming that one knows the EOS, one can construct a model of a planet in which the known mass of the planet is distributed in radius as gravitational pressure and the EOS dictate, based on assumptions about what the composition of the planet is. Uncertainties in the EOS make this more difficult. In the case of Jupiter, for example, it is an interesting question whether an entire planet of its size and mass might be composed of hydrogen or whether there must be an ice and rock core. This certainly has implications for theories of planet formation. With sufficient knowledge of the hydrogen EOS, one will be able to answer this question. At the turn of the twenty-first century, such knowledge was insufficient.
In addition, the EOS affects one’s ability to understand magnetic fields, as we discussed briefly with reference to Fig. 1.4. Planetary magnetic fields are produced by interior currents, known as dynamos. The theory of planetary dynamos unfortunately requires complex three-dimensional calculations. Nonetheless, the possibilities for magnetic field generation are constrained by the locations where the planetary interior is conducting, and this is constrained by the EOS. Here again Jupiter provides an interesting way to frame the puzzle. Jupiter has an extremely strong magnetic field, producing very-large-scale effects within the solar system. At the surface of Jupiter, hydrogen is an insulator. The nature of the hydrogen EOS will determine how close to the surface of Jupiter currents can flow and what volume of the planet can participate in the dynamo. This will constrain the possibilities for the production of Jupiter’s magnetic field.
3.10.1 The Astrophysical Context for EOS
To illustrate the importance of EOS, consider Jupiter in more detail. Figure 1.4 showed a schematic of its interior based on one specific model (for more discussion, see Saumon and Guillot 2004). Jupiter has an outer envelope of dielectric molecular H2, believed to transition to metallic atomic hydrogen at a radius of 0.75R J and pressure of p ∼ 2 Mbar, and ending in an ice–rock core, in this model, when the pressure reaches ∼40 Mbar. There is uncertainty about whether such an ice–rock core actually is present, and if so how massive it is. In part this reflects uncertainty in the EOS of H. The mass of Jupiter is M J ∼ 10−3 M S , (where M S is the mass of the sun) and its intrinsic radius is R J = 7.2 × 104 km. Some model calculations for the interior of Jupiter are shown as temperature–pressure (T–p) profiles as a function of age in Fig. 3.26. Profiles for the brown dwarf Gl229B are also shown in this figure. Under these conditions, molecular hydrogen (H2) dissociates to atomic hydrogen and ionizes deeper in the mantle, changing from a dielectric to a conductor. The pressure and temperature in the mantle of Jupiter near the surface are in the range of 1–3 Mbar at temperatures of a fraction of an eV. Deeper in the interior, the pressure and temperature increase, rising to ∼80 Mbar at a couple of eV at the center. (The corresponding numbers for the brown dwarf Gl 229 are similar in the mantle, but it has four orders of magnitude higher pressures in the core, p core ∼ 105 Mbar.)
Temperature–pressure profiles in Jupiter and brown dwarf GI 229B, for various ages, from models in Hubbard et al. (1997)
One of the key questions about the interior of Jupiter is whether there is a sharp boundary between the molecular hydrogen mantle and the monatomic hydrogen core, caused by a first-order plasma phase transition. This has significance for the exact internal structure, as the discontinuities caused by such a phase transition tend to inhibit convective heat transport, modifying the thermal profile of the planetary interior. This also affects the degree and rate of gravitational energy-release due to sedimentation of He and heavier elements. Jupiter and Saturn’s atmospheres are observed to contain less helium than is believed to have been present at their formation. This is thought to be due to a H–He phase separation. The presence of a helium-poor outer region, and helium-rich inner region is important, both because it has implications for the amount of heavier elements contained deeper in the interior of the planet, and also because of the gravitational energy released as heat during helium sedimentation. Helium sedimentation is required to explain Saturn’s intrinsic heat flux, and may also be significant in Jupiter. The important point in the present context is that all of the detailed issues of hydrogen behavior are quite uncertain at present. The nature of the transition from molecular to monatomic hydrogen, the existence of a metallic phase, the possibility of a H–He phase separation, and other factors are not known.
The EOS of elements heavier than H and He, relevant to Earth-like planets, is even more complex at ultrahigh pressures. To illustrate this, we show in Fig. 3.27 a plot of a number of different theoretical models for the behavior of Al at very-high pressures and compressions, p > 10 Mbar, ρ/ρ o > 3. These models calculate the shock Hugoniot , which is the locus of the points in pressure and density that can be reached from a single initial condition by means of shock waves of varying strength. The various models (see Avrorin et al. 1987; Hicks et al. 2009) exhibit significant differences. The simplest and most widely used of the models is the statistical Thomas–Fermi model with quantum corrections (TFQC), shown by the solid curve. This model does not include atomic shell structure, but rather treats the electron states as a continuum. The self-consistent field (SCF), Hartree–Fock–Slater (HFS), and INFERNO models treat the electron shells quantum mechanically, but differ in their handling of close-packed levels corresponding to energy bands. The semiclassical equation of state (SCES) model treats both the discrete electron shells and the energy bands semiclassically. The ACTEX model is an ionization equilibrium plasma model which uses effective electron–ion potentials fitted to experimental spectroscopic data. These models typically include the nuclear component using the ideal-gas approximation. An exception is a Monte Carlo treatment of the thermal motion of the nuclei implemented in one of the versions of the semiclassical equation of state model (SCES).
Various theoretical models of the shock Hugoniot of Al, as described in the text. Note the considerable uncertainty, which only experiments can resolve. Adapted from Avrorin et al. (1987)
The oscillations in the theoretical pressure versus compression curves shown in Fig. 3.27 result from the pressure ionization of the K- and L-shell electrons of Al. At pressures of 100–500 Mbar, ionization of the L-shell electrons occurs as the high compression forces neighboring atoms sufficiently close together to disrupt the n = 2 electron orbital. When the shock places the material in a state where these electrons are becoming free, more of the energy flowing through the shock must go into internal energy. This leads to a larger density increase, exactly as we discuss in Sect. 4.1. Hence, at the onset of pressure ionization of a new shell in a model, the postshock density increases more rapidly with postshock pressure, behavior known as a “softer” EOS. This pressure-ionization effect on the EOS is qualitatively similar to that due to molecular dissociation of N2 and D2, which has been experimentally observed at lower pressures (see Nellis 2006). Once ionization from the shell is complete, the effect is a “hardening” of the EOS, as the fraction of the energy flowing through the shock that is converted to internal energy decreases. This is why, above ∼1 Gbar, some of the p − ρ curves turn back toward lower compression. A similar softening–hardening oscillation is predicted at pressures of 3–5 Gbar due to ionization of the K-shell electrons, though the magnitude of the effect is smaller due to the lower number of K electrons. How real such oscillations in the Hugoniot are is unclear at this writing. If the actual process of liberating new electrons develops more gradually than it does in the model, this may smooth out the response and avoid the oscillation.
3.10.2 Connecting EOS from the Laboratory to Astrophysics
The EOS describes the equilibrium properties of any large aggregation of atoms of a given type. Even microscopic quantities of matter typically include enormous numbers of atoms. As a result, measurements using aggregations of matter that are very small on a human scale can provide results which apply directly to aggregations of matter on a planetary or stellar scale. In this sense, it is straightforward to make a laboratory measurement that applies directly to astrophysics.
Unfortunately, however, laboratory measurements can only achieve a limited range of pressures and densities by comparison with those existing in astrophysical systems. It would be desirable to be able to scale the equation of state in pressure and density, so that laboratory measurements could be applied to a wider range of astrophysical conditions. This is possible but unnecessary in the case of simple equations of state, such as an ideal-gas or a radiation-dominated system. In more-complex cases, however, the dynamics of the material is specific to the material conditions. The chemical structure of a material is not easily scaled to other conditions, and processes such as dissociation and ionization occur only at specific energies. Thus, laboratory measurements can only address astrophysical issues in EOS at pressures they can actually achieve.
Given the technologies of the early twenty-first century, it seems likely that the pressures employed for EOS studies during this period, using planar targets, will be in the range of 1 to less than 100 Mbar. It may prove feasible, using implosions , to access pressures of a Gbar or even more. These are suitable for addressing issues in planetary equations of state. One can expect this to be the primary focus of such studies.
3.11 Homework Problems
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3.1
Inertial fusion designs typically involve the compression of DT fuel to about 1000 times the liquid density of 0.25 g cm−3. Assuming that this compression is isentropic and that the fuel remains at absolute zero, determine the energy per gram required to compress this fuel. Compare this to the energy per gram required to isentropically compress the fuel to this same density, assuming the fuel is an ideal gas whose final temperature is to be the ignition temperature of 5 keV.
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3.2
Generalize the derivation of the Debye length in Sect. 3.2 to a plasma with an arbitrary number of ion species, each of which may have a distinct temperature.
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3.3
Examine the behavior of the integrals for Fermions. Argue conceptually that the contribution of the denominator in (3.29) at large μ/(k B T e ) is a step function. Evaluate this integral numerically to determine how rapidly it becomes a step function as μ/(k B T e ) increases.
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3.4
Examine the limiting behavior of the internal energy of Fermi degenerate electrons. Show, in the limit as T e → 0, that n e 𝜖 e = (3/5)n e 𝜖 F .
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3.5
What is the relation of heat capacity and entropy ? Derive 3.38 and 3.40 and discuss their differences.
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3.6
Make plots comparing Z bal from (3.49) with the estimate \(20\sqrt {T_e}\) as a function of T e , for ion densities of 1019, 1021, and 1023 cm−3. Discuss the results.
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3.7
Carry out the evaluation of the average charge, Z, in (3.53) and compare the result to Z bal, for T e = 1 keV, Z n = 30, and n i = 1021 cm−3.
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3.8
Plot the ratio of ΔE to the ionization energy versus ion density for the various models described in Sect. 3.4.4. Discuss the results.
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3.9
The value of R i used in Sect. 3.4.6 ignores the internal energy in excited states (as well as the energy lost by radiation during ionization, which would properly have to be treated by more general equations). Again assuming hydrogenic ions, estimate what fraction of the internal energy is present in excited states, and how this varies with Z.
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Drake, R.P. (2018). Properties of High-Energy-Density Plasmas. In: High-Energy-Density Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-67711-8_3
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