Abstract
Astrophysical dense plasmas are those we find in the interiors, surfaces, and outer envelopes of stellar objects such as neutron stars, white dwarfs, the Sun, and giant planets. Condensed plasmas in the laboratory settings include those in ultrahigh-pressure metal-physics experiments undertaken for realization of metallic hydrogen. We review basic physics issues studied in the past 60 some years on the phase transitions, the interparticle correlations, and the elementary processes in dense plasmas, through survey on scattering of electromagnetic waves, equations of state, phase diagrams, transport processes, stellar and planetary magnetisms, and thermo- and pycnonuclear reactions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Condensed plasmas in nature
Plasmas are any statistical systems containing mobile charged particles. When such a system is condensed, interaction between particles becomes so effective that the system may undergo changes in the internal states or the phase transitions. Thermodynamic properties and rates of elementary processes may likewise be influenced significantly by the interparticle correlations in such a system.
Astrophysical dense plasmas are those we find in the interiors, surfaces, and outer envelopes of stellar objects such as neutron stars, white dwarfs, the Sun, brown dwarfs, and giant planets (Van Horn 1991; Ichimaru 1994). Condensed plasmas in the laboratory settings include (Ichimaru 1994): metals and alloys (solid, amorphous, liquid, and compressed), semiconductors (electrons, holes, and their droplets), various realizations of dense plasmas (shock-compressed, diamond anvil cell, metal vaporization, and pinch compression), and cryogenic, non-neutral plasmas (Davidson 1990) [pure electron- or ion plasmas (Driscoll and Malmberg 1983; Bollinger et al. 1990)] in the electromagnetic traps or on the surfaces of dielectrics such as liquid helium (Grimes 1978).
The physics issues in such condensed plasmas are (Ichimaru et al. 1987): phase transitions, construction of the phase diagrams, and accounting for the stellar as well as magnetic structures. Phase transitions to be considered are: gas to liquid, liquid to solid (Wigner 1935, 1938), insulator to metal (Wigner and Huntington 1935), hadrons to quark–gluon plasmas (Yagi et al. 2005), and para- to ferromagnetism.
Elementary processes involved in those plasmas, then, include (Elementary Processes in Dense Plasmas 1995): scattering of electromagnetic waves (Rosenbluth and Rostoker 1962; Ichimaru 1973), photon transfers and opacities, emission of latent heat through phase transitions, electric and thermal transports, shear moduli of the crystalline solids, and enhanced thermo- and pycnonuclear reactions (Gamow and Teller 1938; Cameron 1959). The rates of these processes may depend sensitively on the changes in microscopic, macroscopic, thermodynamic, dielectric, and/or magnetic states of the matter. These changes of states may be associated with freezing transitions, chemical separations between the compositions, ionization or insulator-to-metal transitions, magnetic transitions, and transitions between normal and superconductive phases.
1.2 Parameters of dense plasmas
We model a plasma at a temperature T as consisting of atomic nuclei (which will be called “ions”) with an electric charge Ze and a rest mass M (=Am N) and electrons with the electric charge -e and the rest mass m. Here, Z is the charge number, and A refers to the mass number with m N denoting the mass of a nucleon.
In certain cases, salient features of a plasma can be clarified through the study of a one-component plasma (OCP) (Ichimaru 1982), as against a two-component, electron–ion plasma characterized in the foregoing paragraph. This model consists of a single species of charged particles with number density n embedded in a uniform background of neutralizing opposite charges. It is then useful to introduce the Coulomb-coupling parameter Γ for such an OCP via
with a = (4πn/3)−1/3 referring to the ion-sphere radius, and k B (=1.38066 × 10−16 erg/K) denotes Boltzmann’s constant; hereafter, lengths will be measured in units of a, unless specified otherwise. This Γ, inversely proportional to T, is the ratio of the Coulomb energy to the average kinetic energy for plasmas obeying the classical statistics. We call OCP as strongly coupled when Γ > 1, weakly coupled when Γ < 0.1, and intermediately coupled when 0.1 ≤ Γ ≤ 1 (Ichimaru 1982).
Coulomb interaction plays the cardinal role in determining the physical properties of plasmas. In the theoretical treatment of plasmas in strong coupling, one cannot resort to a usual scheme of expansion in which the Coulomb interaction is regarded as a weak perturbation. We may also note that the interaction potential adopted for OCP has a simple and unique character: Among the repulsive potentials expressible as inverse power \(r^{-\upsilon}\) of the distance r, OCP constitutes a typical example (\(\upsilon =1\)) of soft cases, while the hard-core case corresponds to the other extreme, \(\upsilon \rightarrow \infty\).
It may, therefore, be said that we are here faced with a charged liquid problem. It is, nevertheless, to be noted that strongly coupled plasmas exhibit a remarkable similarity to hard-sphere systems in a number of significant aspects, such as short-ranged ordering in solidification (Alder and Wainwright 1959). In fact, the short-ranged repulsive forces do play the essential parts as the origin of cohesive forces inducing Wigner crystallization (Wigner 1935, 1938) and ferromagnetic transitions.
As we observe numerically in (1), Γ takes on extremely small values in ordinary gaseous plasmas. We shall find later; however, OCP may undergo the Wigner crystallization when Γ exceeds 172–180. We shall also find that the value of Γ critically affects the enhancement rate of nuclear reactions in dense plasmas (Ichimaru 1993).
We may point out that the “dense plasmas” here refer to “high material-density plasmas,” where Γ > 1 mostly as n takes on an exceedingly large number. Under these circumstances, interparticle correlations seriously affect the phase-related properties such as rates of the elementary processes (Elementary Processes in Dense Plasmas 1995). In this review, we shall elucidate some of the physics issues on those dense plasmas.
In these connections, we also recognize the significance and importance of the activities in the fields of “high-energy density science.” Here, the phase-related properties of “high-energy density plasmas” created as radiation-heated and shock-compressed matter are probed by powerful penetrating X-ray sources (Glenzer and Redmer 2009; Dorchies and Recoules 2016). It represents the warm matter or intense beam science, describing “high kinetic energy density plasmas” mostly with Γ < 1. In this review, we shall touch on some of those recent developments, as well.
2 Scattering of electromagnetic waves by a strongly correlated plasma
2.1 Dynamic structure factor
Incoherent scattering of electromagnetic waves by electron density fluctuations in a plasma, portrayed in Fig. 1, is characterized by the dynamic structure factor, S(k, ω), the Fourier transform of the electron density–density correlation function (Ichimaru 1973). It is defined as follows:
Incoherent scattering of electromagnetic waves
where
refers to the spatial Fourier components of the electron density fluctuations, and N is the total number of electrons in a volume V. In this expression, δ(k,k’) is the three-dimensional Kronecker’s delta, n (= N/V) is the average number density, and r j(t) describes the trajectory of jth electron.
We designate, as in Fig. 1, the wave vector k and the frequency ω of the incoming and outgoing waves as (k 1, ω1) and (k 2, ω2). The differential cross section for scattering into a solid angle dο and a frequency interval dω is expressed in a form proportional to S(k,ω) with k = k 2 − k 1 and ω = ω2 − ω1, that is
Here, σT = (8π/3)(e 2/mc 2)2 = 6.533 × 10−25 cm2 is the cross section of Thomson scattering with c denoting the light velocity in vacuum, θ is the angle between the incident and scattered waves, and we have averaged over directions of polarization of both waves. Scattering of electromagnetic waves thus provides a unique way of monitoring electron–electron density correlations in a plasma.
2.2 Collective vs. individual particles aspects of fluctuations
In the early 1950s, Bohm and Pines advanced a series of papers (Pines and Bohm 1952; Bohm and Pines 1953) dealing with “a collective description of electron interactions,” in which they explicitly advanced:
The density fluctuations may be split into two approximately independent components. The collective component, that is, the plasma oscillation, is present only for wavelengths greater than the Debye length. The individual particles component represent a collection of individual electrons surrounded by co-moving clouds of screening charges; collisions between them may be negligible under the random-phase approximation.
Here, the plasma oscillation has a characteristic frequency (Ichimaru 1973):
the Debye length is the inverse of the Debye wave number (Ichimaru 1973):
In analyzing the dynamic evolution of density fluctuations in a plasma, Bohm and Pines introduced an approximation, called the random-phase approximation (RPA), in which non-linear coupling between two fluctuations may be neglected (Pines and Bohm 1952; Bohm and Pines 1953); as we note in (3), the fluctuations ρk represent a sum of exponential terms with varying phases.
2.3 Dielectric formulation
The collective vs. individual particles aspects of fluctuations described above can be most succinctly described in terms of the dielectric response function, ε(k,ω) (Ichimaru 1973; Pines and Nozières 1966). It is a linear response function of a plasma as an externally applied potential Φext(k,ω) may induce a potential Φind(k,ω); the resultant total potential field Φtot(k,ω) (= Φext(k,ω) + Φind(k,ω)) may then be expressed as follows:
The zeros of the dielectric response function, determined from ε(k,ω) = 0 on the complex ω-plane, that is, ω = ω k + iγ k , give the frequency dispersions and the lifetimes of the collective mode.
In an electron OCP, density fluctuations of an individual electron moving in a trajectory, r j(t) = r j + v j t, are expressed as follows:
those of the dressed electrons may then be given as
In RPA, where collisions between dressed particles are negligible, the dynamic structure factor may be evaluated by superposition of the dressed test charges as follows (Ichimaru 1973; Rostoker and Rosenbluth 1960; Ichimaru 1962):
Here, the expression
is applicable to a non-equilibrium situation such as a beam-plasma system with F(p) denoting the single-particle momentum (p = m v) distribution; the expression
is applicable to a plasma in thermodynamic equilibrium at temperature T.
2.4 Radar backscattering from the ionosphere
For a first example of scattering experiments that demonstrate correlation effects in plasmas, let us take up radar back scattering from the ionosphere, as carried out by Bowles of the NBS (Bowles 1958, 1961), the experiments particularly mentioned in the first paragraph of the Rosenbluth–Rostoker scattering paper (1962).
The F layer of ionosphere, consisting in electrons and ions (mostly oxygen), extends from 200 to 500 km in altitude; average number densities are around 105–106 cm−3 at temperatures about 1500 K.
Observation that Bowles made in 1958 utilizes radar pulses at frequency 42.92 MHz, pulse width of 120 μs, repetition frequency of 25–40 per second, and peak power of 1 MW. Since the radar frequency is far greater than the plasma frequency (5) at 13 MHz, ionosphere is transparent to those radar pulses, and backscatters them at strengths proportional to the local densities of electrons. The density profile so observed is shown in Fig. 2.
Radar-backscattering measurement of the ionospheric electron distribution as a function of the scattering altitude. [1959 Feb. 27, 7:40 pm—Illinois local time] After Ref. Bowles (1958)
We expect that the back-scattered waves would be broadened, on top of a 9 kHz spread in the radar frequencies, by a width of 82 kHz with the thermal motion of the electrons as in (10b). To detect such a spectral distribution, the receiver’s bandwidth was fixed at 9 kHz. When the central frequency of the receiver was set at the outgoing radar frequency, a signal with a maximum strength was obtained. When it was shifted by 15 kHz relative to that radar frequency, however, virtually no signals were observed. This casts an enigma, since it might mean no broadening caused by the scatterers.
To look into the features of broadening more closely, Pineo, et al. of MIT 2 years later performed analogous experiments, in which, however, the outgoing radar frequency was raised to 440 MHz (Pineo et al. 1960). Frequency spectrum of the back-scattered waves has now become detectable as in Fig. 3; we here find that the broadening does actually take place correspondingly, however, to thermal motion of the ions.
Spectral distribution of the back-scattered wave from the ionosphere (~300 km in altitude) in the measurement of Ref. Pineo et al. (1960)
Now, in these experiments, we recognize that the wave number k relevant to scattering is much smaller than the Debye wave number kD of (6), meaning that we are in the collective regime where the effects of correlation are predominant.
In the present case of an electron and ion two-component plasma, the features of the collective vs. individual particles aspects of fluctuations have to be altered significantly from those with the electron OCP described in Sect. 2.2.
First, the presence of “dressed” ions should be noted. Coulomb potential around an ion is screened by co-moving cloud of electrons at half the strength and the remaining half stems from repelled ions. The observed spectrum in Fig. 3 may be interpreted as coming from scattering by those electrons co-moving with ions.
2.5 Ion-acoustic waves
In the present case of ionospheric plasmas, which are of two components, collective oscillations likewise consist in two modes, optical and acoustic. For the treatment of those collective modes, we thus extend the dielectric formulations of Sect. 2.3 to two-component plasmas, in which we use the subscript “e” for the electrons and the subscript “i” for the ions; for simplicity, we assume the charge number Z of an ion to be unity, and n e = n i = n.
In this section, we are concerned with a situation close to the thermodynamic equilibrium, so that the velocity distributions are given by the Maxwellian:
In so doing, we are assuming the possibility that the temperatures of the electrons and the ions may be different.
In these connections, we note that the relaxation times for Maxwellization of electrons and of ions and for temperature equality are in the approximate ratios (Ichimaru 1973):
Since we may take m i/m e ≫ 1 generally for plasmas, use of the Maxwellians with unequal temperatures may be looked upon as reasonable; we find such an unequal temperature plasma in the glow discharges, for instance.
The dielectric response function in the RPA for the Maxwellian plasma is then calculated as
with
Here, the positive infinitesimal η serves to assure the adiabatic turning on of the disturbance and thereby to guarantee a causal response of the system; we let η → + 0 eventually (Ichimaru 1973).
We now substitute (11) in (13) to obtain
with
Here, the W function is the error function of a complex argument (Ichimaru 1973; Fried and Conte 1961):
For |Z| < 1, it can be expressed in a convergent series:
where
For large Z, we have an asymptotic series
We now proceed to investigate the collective modes in the long-wavelength regime, k 2 ≪ k 2σ , for electron–ion plasmas, setting the solution to ε(k,ω) = 0 as ω = ω k + γ k .
In the high-frequency regime, such that
the dielectric response function (15) may be expressed with the large Z expansion (19) for both electrons and ions; we thus find
This is basically a space-charge wave of electrons, in a uniform, positive-charge background of ions; it represents the optical mode of the plasma oscillations.
In the intermediate-frequency regime, such that
the dielectric response function (15) may be expressed with the small Z expansion (18) for electrons and with the large Z expansion (19) for ions; we thus find
with
This is the acoustic mode of plasma oscillations representing the density waves of electron-screened ions. These acoustic waves in the ionosphere, however, are strongly damped, since temperature of the electrons is almost equal to that of the ions, or T e ≃ T i; hence, a mild hump observed in Fig. 3.
If, however, one considers the plasma with T e ≫ T i, then the ion-acoustic waves, ωk = sk, with the sound velocity:
become well-defined, relatively undamped oscillations with \( \left| {{{\gamma }}_{k} /{{\omega }}_{k} } \right| \approx 0.015 \). They are the density waves of ions interacting mutually via electron-screened, short-ranged Coulomb forces, analogous to the phonons in ordinary materials.
2.6 Plasma critical opalescence
In the RPA, the dynamic structure factor of the electrons for the electron–ion plasma may be simply obtained by superposing the fields due to the dressed particles (Ichimaru et al. 1962); the result is
where
are the polarizabilities of electrons (e) and ions (i); \( f_{{{\sigma }}} \left( v \right) \) are the one-dimensional normalized velocity distribution functions in the directions of k.
Drift motion of the electrons relative to the ions acts to excite the ion-acoustic waves. If one passes to a sufficiently large value of the drift velocity V d, the ion-acoustic waves become unstable; the boundary between growing and damped waves is specified by
where ωk is determined by
Since in (26), S(k,ω) is proportional to 1/|ε(k,ω)|2, one finds a contribution from the immediate vicinity of ωk, which is
For the case of marginal stability, defined by (28), \( S_{\text{res}} \left( {\mathbf{k}} \right) \) obviously diverges.
We have carried out an explicit evaluation of \( S_{\text{res}} \left( {\mathbf{k}} \right) \) for the case that T e ≫ T i, such that the waves close to k = 0 are the first to grow as one increases V d. The result is (Ichimaru et al. 1962)
where \( V_{\text{e}} = \sqrt {{\text{k}}_{\text{B}} T_{\text{e}} /m_{\text{e}} } \), \( V_{\text{c}} \cong s \), χ is the angle between k and V d, and
The result (31) is valid for k 2 ≪ k 2e ; we remark that it is identical in analytical form to the results obtained for the critical fluctuations in the vicinity of a liquid–gas phase transition, or the critical opalescence (Landau and Lifshitz 1969).
2.7 Observation of plasma waves in warm dense matter
Rader backscattering from the ionosphere described in Sect. 2.4 and the plasma critical opalescence treated in Sect. 2.6 are concerned with the collective properties of plasmas.
X-ray Thomson scattering techniques have likewise been employed for observation of collective modes in warm dense matter (Glenzer et al. 2007). The measurements were performed in solid-density beryllium target that had been heated isochorically with a broadband X-ray source into a state of dense plasma with weakly degenerate electrons. The collective scattering regime of the plasma, that is, k ≪ kD in (4), was then approached through forward scattering (i.e., θ ≪ 1) of the narrow-band chlorine Ly-α X-ray line at 2.96 keV. The characteristic peak associated with the collective plasma oscillations (Ichimaru 1973) has thereby been observed in agreement with the theoretical structure factor, Eq. (2), in which the collisional effects have been appropriately taken into account.
3 Correlation and ordering in condensed plasmas
3.1 Static structure factor
In the scattering experiments described in the foregoing chapter, if we pay no attentions to the frequencies, that is, if we integrate the scattered waves over all the frequencies, then the differential cross section of scattering into a solid angle dο may be expressed as
Here, the static structure factor S(k) defined as
corresponds to the spectral distribution of spatial density fluctuations (Ichimaru 1973), that describes spatial density configurations such as the lattice structures.
Short-ranged crystalline order at nearest-neighbor separations may thus be approached by these scattering techniques (Fig. 4) through Eq. (33).
X-ray (electron) scattering spectroscopy
In fact, von Laue observed such diffraction patterns in 1914 by shining X-rays onto metal; the father-and-son Braggs developed X-ray crystallography in 1915. Both works led to the Nobel Prizes in the respective years.
Recently, the advent of accurate X-ray scattering techniques has made it possible to measure the physical properties of dense plasmas for the study in high-energy density physics (Glenzer and Redmer 2009), as we shall revisit these subjects later in Sect. 3.4.
3.2 Monte Carlo simulation study of the OCP
A Monte Carlo (MC) method is any method making use of random numbers to solve a problem. The power of the MC method lies basically in its ability to carry through multi-dimensional integrations with improved accuracy through the techniques of importance sampling and with increased capacity of modern computers (Ichimaru 1994).
In the Metropolis algorithm (Metropolis et al. 1953), one works with a statistical ensemble at constant temperature, volume, and number of particles, that is, the canonical ensemble. Monte Carlo steps (configurations) are generated by random displacements of particles in the many particle system under study. Configurations so created will be accepted or rejected with the probability of acceptance:
where ∆E denotes the energy increment created by the displacements. A Marcoff chain representing the canonical ensemble is thereby generated, with its thermalization usually monitored through evaluation of the internal energy. The probability (35) may thus lead the statistical system to a canonical distribution at temperature T. In their pioneering work, Brush et al. (1966) performed numerical experiments on strongly coupled OCPs by such an MC method.
3.3 Observation of layered structures and Laue patterns in Coulomb glasses
In conjunction with the aforementioned scattering experiments, we now turn to observation of layered structures and Laue patterns in Coulomb glasses, created by such MC simulations (Ogata and Ichimaru 1989); in fact, solidifications such as crystallization and/or glass transition are intriguing events in the thermal evolution of many particle systems.
Later, in Sect. 4.2, we shall find that an OCP may crystallize into a Coulomb solid when the temperature is lowered into Γ > 180, through comparison of the free energies in the respective states.
When an OCP forms a crystalline lattice with a lattice constant b, the electrostatic energy per particle E M, called the Madelung energy, is expressed as
where αM is the Madelung constant. Table 1 lists the values of the energy constant for various lattices (Foldy 1978).
It has been assumed that an OCP in its ground state forms a body-centered-cubic (bcc) crystalline solid, a conclusion reached through comparison of the Madelung energies of the several cubic-lattice and other structures. Extensive MC simulations have been performed for OCP solids with cubic structures (Brush et al. 1966; Slattery et al. 1982), and the bcc lattice has been shown to have the lowest free energy at finite temperatures as well. As we find in Table 1, however, the differences between energy constants for those cubic lattices are so small that it would be inconceivable to assume a monocrystalline structure formed in a real solid.
In fact, if a rapid quench is applied to a temperature well in excess of Γ = 180, the resultant state might quite possibly be a Coulomb glass, characterized by random polycrystalline structures with long-ranged bond-orientational order, in light of the fact that the differences in energies between various lattices are so minute. The particles may then be viewed as virtually locked around their positions in equilibrium.
We thus follow dynamic evolution of an OCP by MC simulations with 432 particles, starting with a fluid state at Γ = 160, leading to formation of Coulomb glasses at different quenches: (A) an application of a sudden quench to Γ = 400 at c = 0; (B) an application of a gradual quench stepwise with ΔΓ = 10 from Γ = 160 at c = 0 to Γ = 400 at c = 23; (C) a sudden quench to Γ = 300 at c = 0 (Ogata and Ichimaru 1989). Here, c denotes the sequential number of MC configurations measured in units of a million configurations; the sequential number corresponds to an MC time via ωpt = 2.7 × 102c, with ωp referring to the plasma frequency (5).
To study the nature of interlayer correlations, we identify the particles in the three central layers and project their positions normally onto a plane. In Fig. 5 on the left, such projections are exhibited for the quenches (A)–(C), where open circles, closed circles, and crosses denote projections of the particles on upper, middle, and lower layers, respectively. For comparison, analogous projections are shown in Fig. 5 on the right for the most closely packed layers in the face-centered-cubic (fcc), hexagonal-close-packed (hcp), and bcc crystals. We find here that the Coulomb glass with the quench (B) has developed an advanced state of polycrystalline nucleation predominantly with local fcc-hcp configurations over those with (A) and (C).
Normal projections of most closely packed layers: open circles upper layer, closed circles middle layer, crosses lower layer (Ogata and Ichimaru 1989)
Intralayer correlations are investigated in terms of the bond-angle distributions P(θ) (Fig. 6) and the two-dimensional radial distribution functions g(r) (Fig. 7); these are joint probability densities of finding two particles at a separation r. We define “bonds” as those lines connecting two adjacent particles located within r < 2.5 on a layer, “bond angle” as the angle between a pair of such bonds originating from a particle, and “coordination number” (CN) as the total number of bonds originating from a given particle.
Bond-angle distribution functions between intralayer particles in the glasses (a)–(c). In the bottom of the figure, the solid lines indicate the bond angles for the fcc-hcp (hexagonal) lattices; the dashed lines, those for the lattices (Ogata and Ichimaru 1989)
Two-dimensional radial distribution functions between intralayer particles in the glasses (a)–(c). The bottom of the figure shows the peak positions for the fcc-hcp (solid lines) and bcc (dashed lines) lattices (Ogata and Ichimaru 1989)
In Fig. 6, we plot the bond-angle distribution functions between intralayer particles in the quenched states (A)–(C), and compare them with analogous quantities for the fcc-hcp (i.e., hexagonal) and bcc lattices. In the state (B), P(θ) = 0 observed at θ − π/2 and 5π/6 implies an advanced degree of nucleation, while splitting of the peaks at θ − π/3 and 2π/3 indicates disordering effects on the local hexagonal configurations.
In Fig. 7, we plot the two-dimensional radial distribution functions between intralayer particles and compare them with the peak positions for the fcc-hcp and bcc lattices. As seen in the figure, all the particles in the state (B) have CN = 6, implying a little distortion in the local hexagonal configurations. In the states (A) and (C), however, distortion in the hexagonal configurations is substantial, since 89 and 90%, respectively, of the particles have CN = 6, while 5 and 8% have CN = 5, and 6 and 2% have CN = 7.
Finally, we investigate the combined effects between the intralayer and interlayer correlations by a scattering method of Sect. 3.1. We thus inject plane waves with wave vector k 1 to the glasses (A)–(C) in the direction normal to the layered structures, and measure the strength of scattered waves k 2 in the directions specified by (χ, ϕ), where χ is the scattering angle between k 1 and k 2 and ϕ is the azimuthal angle around k 1. The cross section (33) for coherent scattering is proportional to the static structure factor (34), where k = k 2–k 1. We assume that the incident wave numbers have a distribution proportional to exp[–(k1–k0)2/κ2] with k0 = 2π and κ = 0.24. The scattering experiment is thus capable of detecting the coherence in the phases 4πsin(χ/2)k·r j/k over those particles r j contained in a slab of width 4.2/sin(χ/2) in the direction of k.
Figure 8 displays the Laue patterns so obtained for the glass states (A)–(C), and compares them with those of the fcc, hcp, and bcc lattices. We observe the existence of local hexagonal order in (B) and to a lesser extent in (A); a slight involvement of local bcc configurations is likewise detected for all the cases of (A)–(C).
Laue patterns for the glasses (a)–(c) and for the fcc, hcp, and bcc lattices of 432 particles. The polar coordinates consist of 0 ≤ (π − χ)/2 ≤ 0.45π and 0 ≤ ϕ ≤ 2π; the origin corresponds to χ = π. Here, χ is the scattering angle between incident and scattered waves; ϕ is the azimuthal angle around the incident wave (Ogata and Ichimaru 1989)
In light of the analyses described above, we may conjecture the following stages of evolution for the glass transitions in dense plasmas: In a super-cooled OCP without an external field, layered structures (i.e., a one-dimensional order) develop first in an arbitrary direction. Intralayer (i.e., two-dimensional ordering) then follows, which would favor formation of fcc-hcp (i.e., hexagonal) local clusters. Since the bcc lattice has a Madelung energy slightly lower than the fcc or hcp lattice in Coulombic systems (Table 1), a possibility of nucleation remains for bcc clusters. Hence, the resultant state may have a complex polycrystalline structure.
Later, in Sect. 9.3, we shall consider the first-principles calculations of shear moduli for Monte Carlo-simulated Coulomb solids, with inclusion of the Coulomb glasses, and apply the results for improved analyses of the non-radial oscillations in neutron stars.
3.4 X-ray Thomson scattering and time-resolved XANES diagnostic with high-energy density plasmas
Accurate measurements of the states of plasmas including temperature, density, and ionization in dense matter are essential in high-energy density physics (Glenzer and Redmer 2009).
In Sect. 2.7, we remarked on the X-ray Thomson scattering measurements in the collective regime carried out in a beryllium target with solid density that was heated isochorically with a broadband X-ray source into a state of dense plasma with weakly degenerate electrons; the characteristic peak associated with the collective plasma oscillation was thereby observed. The spectrally resolving X-ray scattering technique has also been applied in a number of laboratories to study the properties of such dense plasmas (Sawada et al. 2007; Ravasio et al. 2007).
Solid-to-plasma transition dynamics have been approached with the aid of a recently advanced diagnostic technique such as time-resolved XANES (x-ray near edge spectroscopy) (Dorchies and Recoules 2016). Electronic and structural properties with three different (simple, transition, and noble) types of metals have been investigated through absorption spectroscopy experiments with the aid of ultrafast X-ray free electron lasers (Dorchies et al. 2008; Cho et al. 2011; Katayama et al. 2013; Gaudin et al. 2014).
4 Thermodynamics of the classical OCP and the quantum electron liquid
4.1 Radial distribution functions and correlation energies
The microscopic descriptions of the plasma in terms of the correlation functions and the structure factors are connected directly to thermodynamics that specifies the macroscopic states or the phases of the system.
The radial distribution function g(r) is a joint probability density of finding two particles at a separation r; Fig. 9 shows g(r) evaluated with the aid of the MC methods in Sect. 3.2 for Γ > 1. It is related directly to the static structure factor (34) as
Radial distribution functions of OCP fluids obtained by MC methods with N = 1024 at various values of Γ. The number of the MC configurations generated for each run was 7 × 106; g(r) was calculated with 200 bins in the range 0 ≤ r ≤ L/2, a half of the cubic MC cell with size L = 16.2a. The solid curves represent the results calculated with the improved hypernetted scheme. After Ref. Iyetomi et al. (1992)
The correlation energy U int per unit volume can then be calculated, once either S(k) or g(r) is known, through formulae:
Substituting the RPA structure factors (9) in (37) via (34), we obtain the RPA expression for the normalized correlation energy, u ex ≡ U int/nkB T, as
This is called the Debye–Hückel term (Ichimaru 1973).
4.2 Multi-particle correlation and OCP internal energies
The RPA correlation energy (38) takes an account of binary correlation to the lowest order in Γ and thus is applicable to OCP only for Γ ≪ 1. The expression for the correlation energy next order in the Γ expansion has been accurately obtained as
with γ = 0.57721… denoting Euler’s constant.
Correlation energies beyond RPA require accurate assessments for the triple and higher order correlation effects. These have been approached through various theoretical methods, such as the giant cluster-expansion calculation (Abe 1959), expansion in Γ of the BBGKY hierarchy (O’Neil and Rostoker 1963), multi-particle correlation in the convolution approximation (Totsuji and Ichimaru 1973), and the improved hypernetted-chain integral-equation scheme based on the density-functional formulation of multi-particle correlations (Ichimaru et al. 1987; Ogata and Ichimaru 1987; Iyetomi et al. 1992).
The use of Monte Carlo radial distribution functions (such as those in Fig. 9) in Eq. (37) yields the correlation energies in large Γ regime (Slattery et al. 1982; Ogata and Ichimaru 1987):
The formula for OCP correlation energies covering all the parameter regimes accurately has been obtained by connecting Eqs. (39) and (40) as (Ichimaru 1994)
The free energy, f(Γ), in units of the thermal energy is then given by the following (Ichimaru 1994):
with the ratio between the thermal de Broglie wavelength and the ion-sphere radius,
\( \hbar=1.054573 \cdots \times 10^{-27}\) erg·s denoting the Planck constant. In (42), the excess free energy, f ex(Γ), is calculated through the coupling-constant integration (Ichimaru 1994),
of (40) as
Here, in (44), u ex(η) refers to the correlation energy evaluated in a system where the strength of Coulomb-coupling Γ is replaced by ηΓ.
The excess pressure P ex may be evaluated by differentiation of F ex with respect to volume V at a constant temperature (Ichimaru 1994), that is
The thermodynamic functions for classical OCP solids have been investigated through the MC simulations coupled with analytic study of the anharmonic effects in the lattice vibrations (Dubin 1990). The normalized correlation energy for an OCP bcc-crystalline solid is thus given by the following:
where the first term on the right-hand side reflects the value of the Madelung energy for the bcc lattice (Foldy 1978); it is the electrostatic energy per particle in an OCP that forms a crystalline lattice
The free energy f(Γ) in the classical OCP solid can be evaluated by integrating the correlation energy with respect to the inverse temperature as (Dubin 1990)
under the assumption that the ground-state free energy f(∞) is given by the harmonic lattice value (Pollock and Hansen 1973).
Comparing the free energies between the fluid and crystalline phases, we find that an OCP fluid may freeze (i.e., Wigner transition) into a bcc crystal at Γ = 172 ~ 180. In light of a possible formation of Coulomb glasses in Sect. 3.3, however, how the actual transitions may take place in real plasmas remains a delicate issue.
4.3 Equations of state for quantum electron liquids
Metallic hydrogen is a binary system of itinerant electrons and those protons in a fluid or in a solid state. In the jellium model of metals, we regard those itinerant electrons as forming a quantum electron liquid (Pines and Nozières 1966).
The phase diagrams of such an electron liquid are characterized essentially by two (density and temperature) dimensionless parameters (Ichimaru et al. 1987):
Here
is the Wigner–Seitz radius or the ion-sphere radius introduced earlier. It is also useful to recall the classical Coulomb-coupling parameter (1) related to those dimensionless parameters as
The Hermholtz-free energy and the pressure are expressed as sums of the ideal gas and exchange–correlation parts:
The Gibbs-free energy is then given by the following:
The ideal-gas contribution to the free energy is expressed as a balance between those of the chemical potential, μ(Γ,θ) (=Gibbs-free energy per particle) and of the pressure as
Through numerical investigation of the relevant Fermi integrals (Landau and Lifshitz 1969), we find that the chemical potential may be accurately fitted by the expression:
Similarly, we obtain for the pressure
Both the expressions above exactly satisfy the limiting conditions for the first two terms in the expansions for θ ≪ 1 as well as for θ ≫ 1.
The exchange–correlation energies of electron liquids at finite temperatures θ = 0.1, 1.0, 5.0 were evaluated through a solution to a set of integral equations (Tanaka and Ichimaru 1986). The results were then parameterized in analytic formulae as (Ichimaru et al. 1987)
with
The term (54a) represents the Hartree–Fock contribution (Perrot and Dharma-wardana 1984).
The coupling-constant integration (44) is performed with Eq. (55) to yield
Condition that 4e − d 2 > 0 is satisfied for any θ. Values of the exchange and correlation-free energy in the ground state, given by Eq. (55), agree accurately with those obtained by Green’s function Monte Carlo simulations (Ceperley and Alder 1980) at r s = 2, 5, 10, 20, 50, 100.
It is instructive to examine parameter dependence and sign of the elementary contributions in the specific pressure (52b) in the limit of the quantum degeneracy, θ → 0. The ideal-gas contributions behave as
The Hartree–Fock contributions to the pressure (46) stem from the evaluation (53) in which only the term (54a) is retained, and take on the values
The Coulomb pressure, which represents the large-r s contributions in (46), likewise behaves as
Note both the exchange and Coulomb terms, (56b) and (56c), are negative and proportional to the strength of Coulomb coupling represented by e 2.
These observations are essential in elucidating the origin of cohesive forces in the ferromagnetic and freezing transitions.
4.4 Freezing and ferromagnetic transitions in electron liquid
Thus far, we have considered itinerant electrons or the electron liquid and protons in a fluid state or a solid state. Metallic hydrogen, as we consider subsequently, is a binary system of those electrons and protons.
The electron liquid is a quantum OCP of electrons immersed in a uniform compensating background of positive charges. Electrons are fermions, with spin 1/2, obeying the Fermi statistics. Wave functions of two identical fermions with parallel spins are antisymmetric, that is to say, they change their signs when the positions of the two fermions are interchanged. The values of wave functions vanish when two identical fermions occupy the same position; interpreted physically, identical fermions with parallel spins repel each other. This explains the origin of the spin-discriminating (repulsive) exchange forces between such identical fermions. These exchange forces and the ordinary Coulomb forces, both repulsive, are effective between protons as well as between electrons.
These repulsive forces induce the so-called “exchange” and “Coulomb” holes in the two-particle distribution functions for the electrons (e.g., Ichimaru 1982). The interaction between electrons and such “holes,” which is attractive, then produces negative contributions to the free energy and the pressure, as (56b) and (56c) exemplify. These negative free energies stimulate spin ordering in a ferromagnetic transition and crystalline ordering in a freezing transition.
It has thus been expected that an electron liquid may undergo a magnetic transition, from a spin-non-aligned, paramagnetic phase to a spin-aligned, ferromagnetic phase (Ceperley and Alder 1980; Ichimaru 1997, 2000; Ortiz et al. 1999) near the conditions for Wigner crystallization, a phase transition of dilute electrons into a crystalline state at low temperatures (Wigner 1935, 1938). A magnetic transition takes place basically through competition between the spin-dependent exchange processes, which favor a ferromagnetic state, and the kinetic energies, which favor a paramagnetic state. Analogous situations may exist in the case of a freezing transition, where the repulsive Coulomb forces favor an inhomogeneous distribution such as one in a Wigner crystal, while the kinetic processes favor a uniform distribution characteristic of a fluid state.
5 Phase diagrams of hydrogen
5.1 States of hydrogen
Hydrogen, which we know as a light gaseous substance at ambient temperature and pressure, may exhibit extraordinary features pertinent to the strongly correlated plasmas when it is compressed to densities comparable to or greater than those of ordinary solids. Basically, hydrogen matter is a statistical ensemble consisting of electrons and protons. The protons, with the smallest atomic number (unity) among various chemical elements and thus with de Broglie wavelengths longer than those of other nuclear species, tend to interfere more prominently with each other quantum mechanically under such condensed circumstances.
Dense hydrogen under ultrahigh pressures as found in stellar and planetary interiors may be expected to undergo transformation between phases, such as metallization, crystallization, and magnetization. All these phase transitions are not only of great interest in the condensed-matter physics, but, since hydrogen is the most abundant chemical element in the Universe, their nature crucially affects fundamental issues in astrophysics, such as generation of energy and magnetism in the interiors of stars and planets and energy transport to stellar and planetary surfaces. Thus, the physics of hydrogen constitutes a vital element in the formation, structure, and evolution of these astronomical objects.
A hydrogen atom is a bound state between an electron and a proton. The orbital radius of a bound electron in the ground state is the Bohr radius, given by
The binding energy of a hydrogen atom in the ground state constitutes a unit of energy called the Rydberg and takes on the value:
These provide typical scales of length and energy in the atomic physics of hydrogen.
A hydrogen molecule is a bound state between two hydrogen atoms; in the ground state, the average interproton spacing is 0.742 Å (≈1.4 a B).The dissociation energy and the ionization potential of a hydrogen molecule are 4.474 eV (≈0.33 Ry) and 15.43 eV (≈1.13 Ry), respectively; the dissociation energy of a molecular ion, H2 +, is 2.467 eV (≈0.18 Ry). Ionization thus follows immediately after dissociation in dense hydrogen.
If the number density n of protons is high, so that the Wigner–Seitz radius (50) is less than the Bohr radius, i.e., a < a B (corresponding to n > 1.6 × 1024 cm−3), wave functions of orbital electrons in neighboring hydrogen atoms or molecules significantly overlap each other and so they make conduction electrons; such a process is called pressure ionization. In terms of the Fermi energy E F and the Fermi wave number k F,
of the electrons with number density n e, the Fermi pressure is calculated as
implying a pressure in a multi-megabar range for the pressure ionization.
E. Wigner and H. B. Huntington were the first in 1935 to predict the possibility of such a metallic modification of hydrogen at an extreme pressure; they did so through calculations of the energy of a body-centered lattice of hydrogen as a function of the lattice constant and by comparison of the result with the energy of the molecular form (Wigner and Huntington 1935). Hydrogen is thus expected to undergo a first-order, metal–insulator (MI) transition at an ultrahigh density or in a pressure range of megabars (Ceperley and Alder 1987; Kitamura and Ichimaru 1998; McMahon et al. 2012).
Ultrahigh-pressure metal-physics experiments have been undertaken for laboratory realization of such metallic hydrogen and for elucidation of the equations of state and the transport properties of dense hydrogen. The experimental approaches include diamond-anvil-cell compression (e.g., Mao and Hemley 1989, 1994) and shock compression (e.g., Dick and Kerley 1980; Mitchell and Nellis 1981; Fortov 1995; Fortov et al. 2007). Metallization of molecular hydrogen, though elusive in the diamond-anvil-cell experiments (Mao et al. 1991; Ruoff and Vanderbough 1990; Hemley et al. 1996), was successfully demonstrated in experiments using compression through shock wave reverberation between electrically insulating sapphire (Al2O3) anvils (Weir et al. 1996; Da Silva et al. 1997) as we shall recapitulate in Sect. 6.2.
The giant planets such as Jupiter are thought to consist mostly of metallic hydrogen (Van Horn 1991; Stevenson 1982). The first-order MI transitions predict a discontinuous distribution and resistivity near the surface of Jupiter, implying a large magnetic Reynolds number enough to sustain the prominent magnetic activities (e.g., Stevenson 1982; Kennel and Coroniti 1977). The release of latent heat associated with metal-to-insulator transitions through cooling may possibly account for a considerable fraction of its excess infrared luminosity (Aumann et al. 1969; Hubbard 1980), as we shall revisit in Chap. 6.
Ferromagnetic and freezing transitions in the liquid-metallic hydrogen (Ichimaru 2001) are important issues, not only in condensed-matter physics, but in conjunction with the conspicuous magnetic phenomena in astrophysics, such as those associated with the origin of intense magnetization found in the degenerate stars (e.g., Chanmugam 1992). Liquid-metallic hydrogen relevant to the ferromagnetic transitions may, for example, be expected in an outer layer of a hydrogen-rich white dwarf; we shall explore on these in Chap. 7.
The rates of nuclear process such as thermonuclear and pycnonuclear reactions are influenced significantly by the state or the phase that a dense matter may assume (e.g., Ichimaru 1993). A huge enhancement of the reaction rates arising from internuclear Coulomb correlation in dense matter, albeit ineffective for the solar nuclear reactions or for the ICF experiments, provides a physical mechanism vital to supernovae. Experimental and theoretical progress in ultrahigh-pressure metal-physics may make a “supernova on the Earth” scheme utilizing enhanced pycnonuclear reactions in ultradense metallic hydrogen an attractive and possibly even realizable prospect for fusion studies; we shall take up on this subject in Chap. 8.
5.2 Equations of state for hydrogen
Metallic fluid hydrogen consists of itinerant electrons (fermions with spin ½) and itinerant protons (classical, or fermions with spin ½) with strong e-i coupling. Metallic solid hydrogen consists of itinerant electrons (fermions with spin ½) and a bcc array of protons (classical) with harmonic and anharmonic lattice vibrations with strong e-i coupling (e.g., Ichimaru 1994; Kitamura and Ichimaru 1998).
Equations of state in the molecular fluid insulator phase of hydrogen consist of an ideal Bose gas, short-range repulsive interaction between molecular cores, attractive (dipolar) van der Waals forces, molecular rotation (roton), intramolecular vibration (vibron), and the ground-state energy of an H2 molecule (E H2) (Kitamura and Ichimaru 1998; Hansen and McDonald 1986; Hansen and Verlet 1969; Hansen 1970).
Equations of state in the molecular solid insulator phase of hydrogen consist of cohesive energy with a hcp structure, lattice vibration (phonon), roton, vibron, and the ground-state energy of a H2 molecule (E H2) (Kitamura and Ichimaru 1998; Hirschfelder et al. 1954).
In addition, we take into account contributions of atomic hydrogen to the equations of state, including the repulsive hard sphere, the attractive van der Waals, and the ground-state energy (EH) of H atoms (Kitamura and Ichimaru 1998; Dargarno 1967; Victor and Dargarno 1970).
5.3 Phases of hydrogen matter
The phase diagrams for the MI transitions in hydrogen matter may be determined through the explicit formulation of the equations of state in the metallic (solid, paramagnetic fluid, and ferromagnetic fluid) phases as well as in the insulator (molecular solid, molecular fluid, atomic, and molecular fluid) phases, as listed in the previous section. Thus, we consider a matter consisting of atomic, molecular, and ionized hydrogen, which may be characterized by the temperature T, the total number density of protons n p, the degree of ionization <Z>, and the degree of dissociation αd. The number densities of ions, plasma electrons, neutral atoms, and molecules are then given by
For a fluid phase, the total Helmholtz-free energy, F tot, may thus be expressed as a sum of molecular (57a), atomic (57b), metallic (57c), and intermolecular (57d) contributions in the following (Kitamura and Ichimaru 1998):
In this formulation, interaction between plasmas and neutral particles is taken into account through excluded-volume effects and changes in the levels of bound electrons: In the former effects, specific volume for the plasma particles (i.e., ions and electrons) is effectively reduced by the presence of neutral atoms and molecules, so that the term (57c) contains a normalized density, \( {\bar{{\rho }}}_{\text{m}} = {{\rho }}_{\text{m}} /\left( {1 - {{\eta }}} \right) \), where η is the packing fraction:
with d a and d m denoting the effective hard-sphere diameters of a hydrogen atom and molecule (Lebowitz and Rowlinson 1964). Thus, the presence of neutral atoms has been effectively taken into account in (57d).
When metal and insulator phases coexist, the energy level of an electron bound in a molecule or in an atom in a dense plasma may be lowered, or may even disappear, owing to the screening action of plasma electrons (e.g., Ichimaru 1994). The extent to which such a modification may take place depends on the ratio between the Bohr radius and the short-range screening distance D s of the plasma defined in terms of the dielectric response function εe(k, 0) of the electrons as
Thus, the ground-state energy of a hydrogen atom in a plasma may be expressed as
where
This screening function has been obtained through the numerical solution to a Schrödinger equation for an electron in a Yukawa potential, —(e 2/r) exp (—r/D s). As x increases from zero, the value of \( f_{\text{s}} \left( x \right) \) decreases from \( f_{\text{s}} \left( 0 \right) = 1 \), meaning that an atomic or a molecular level is lowered; \( f_{\text{s}} \left( x \right) \) vanishes at x = 1.17, where a bound state disappears.
Of significance in these connections is the essential difference between the two screening lengths defined by Eq. (59) and by
with μ denoting the chemical potential of the screening electrons; the usual Debye–Hückel screening distance, the inverse of kD in (6), is a version of this D L. Since D 2L has been defined in terms of the isothermal compressibility of the electrons, the latter quantity may take on a negative value at low densities (i.e., in the strong Coulomb coupling), when D L would become an ill-defined quantity. On the other hand, D s in (59) remains a well-defied quantity, since one generally proves that:
from a causality requirement with a density–density response function (e.g., Ichimaru 1982). Since D s characterizes short-range behavior of the screened Coulomb forces, it plays an essential part in calculating the rate of nuclear reactions in dense plasmas, as we shall recapitulate in VIII.
A ground-state energy, E H2, of a hydrogen molecule in plasma may likewise be calculated as
The last numeral, 4.747, represents the dissociation energy in units of eV. We remark that this number does not contain the contribution of zero-point energies of the vibrons; the latter has been taken into account already in the term (57a).
When the values of the mass density ρm and the temperature T are given, the chemical equilibrium of the system may be determined through the condition that the total free energy, Eq. (57a–d), be minimized with respect to <Z> and αd. With the state of the matter so determined, we may calculate the values of the thermodynamic quantities in a standard way: the pressure \( = - \left( {\partial F/\partial T} \right)_{{V,\langle Z \rangle,{{\alpha }}_{\text{d}} }} \), the entropy \( S = - \left( {\partial F/\partial V} \right)_{{{\text{V}},{\langle \text{Z}\rangle},{{\alpha }}_{\text{d}} }} \), etc.
Coexistence curves between the MI transitions are derived from the general conditions for the phase equilibrium (Landau and Lifshitz 1969), that is
The mass densities, ρM and ρI, are those of metallic and insulating hydrogen along the coexistence curves.
5.4 Metal–insulator transitions
The MI transitions in dense hydrogen have attracted the interest of many investigators since the pioneering work of Wigner and Huntington (1935). Friedel and Ashcroft (1977) carried out approximate electron band calculations that predicted a band crossing at ρm = 0.82 g/cm3. Analogous calculations were performed by Chacham and Louie (1991) for the bandgap of solid molecular hydrogen in the hcp structure, which predicted that the orientationally ordered phase undergoes metallization due to an indirect band overlap at ρm = 0.8 g/cm3 and an orientationally disordered phase at ρm = 1.06 g/cm3.
In conjunction with shock-compressed states determined by the Hugoniot relations, Ross et al. (1983) and Holmes et al. (1995) obtained the so-called shock equations of state with the aid of adopted model potentials and predicted a transition to a bcc metal above 5 Mbar. Equations of state in dense hydrogen were investigated with model intermolecular potentials to predict the conditions for pressure ionization (Brovman et al. 1972; Ebeling et al. 1991; Saumon et al. 1995). Quantum Monte Carlo simulations were performed for MI transitions in the ground state (Ceperley and Alder 1987) as well as at elevated temperatures (Magro et al. 1996). Recent progress in the computer simulation studies on the properties of hydrogen and helium under extreme conditions has been extensively reviewed (McMahon et al. 2012).
The coexistence curves for the MI transitions in hydrogen (Kitamura and Ichimaru 1998) obtained through the theoretical procedures of the foregoing section are shown in Fig. 10. The diagrams have been constructed through the explicit calculations of the equations of state for hydrogen in the metallic (solid, paramagnetic fluid, ferromagnetic fluid, and partially ionized atomic and molecular fluid) phases as well as in the insulator (molecular solid, molecular fluid, and atomic and molecular fluid) phases. In Fig. 10, we observe the first-order MI transitions exhibited in a high-density regime of hydrogen.
Phase diagram of hydrogen: The dotted curves are isobars at the designated pressure values. <Z> denotes the degree of ionization; α d, the degree of atomic dissociation. C MI and C GL are the critical points associated with metal–insulator and gas–liquid transitions; T GLS, the gas–liquid–solid triple point; T spf, the triple point for the solid-paramagnetic-ferromagnetic phases; C msg, the critical point for the ferromagnetic transitions. See the text for designations of examples cited for hydrogen matter in terrestrial laboratory and astrophysical setting. After Ref. Kitamura and Ichimaru (1998)
Summarizing the theoretical predictions in Fig. 10 for the MI transitions, we note: In the coexistence conditions between the metal and insulator phases, all the thermodynamic quantities except for the pressure, the temperature, and the chemical potential are discontinuous. The density, the entropy, and the enthalpy are greater in the metallic phase than in the insulator phase. These discontinuities vanish at the critical point C MI, where log P (bar) = 3.477; log T (K) = 4.320; log ρm (g/cm3) = −2.721 (Kitamura and Ichimaru 1998).
An insulator-to-metal transition (i.e., metallization) proceeds in the direction of decreasing the chemical potential, accompanied by increase of the temperature and/or decrease of the pressure. It is thus an endothermic process, analogous to our familiar vaporization and melting transitions; the entropy and the enthalpy increase. Release of the latent heat is expected in such a metal-to-insulator transition.
Looking from somewhat different directions, we note: When hydrogen in an insulator phase is compressed to a state of high density, such that average interparticle spacing between protons becomes comparable to or less than the orbital radii of the bound electrons, on the order of the Bohr radius, electrons begin to assume itinerant states due to overlapping of wave functions between adjacent electrons; aforementioned pressure ionization may thus result.
Analogous process of metallization may take place when the temperature is raised above the atomic or molecular binding energies of electrons; it is thermal ionization.
In Fig. 10, we also exhibit approximate parameter domains for the hydrogen matter appropriate to the solar interior, the Jovian interior, inertial-confinement-fusion (ICF) studies, ultrahigh-pressure metal experiments (shock or diamond-anvil-cell compression), proton–deuteron (p-d) fusion in ultradense metallic hydrogen, and hydrogen at standard (STD) conditions. (The “rock” shown here for the core of Jupiter is not of hydrogen, however.)
6 Transport processes in dense plasmas
6.1 Electric and thermal resistivity
Electric and thermal resistivity arises as a consequence of scattering between electrons and ions in a plasma. A proper treatment of e–i interaction, which would diverge in a classical treatment at short distances, is, therefore, essential. In particular, quantum diffraction of electrons in the vicinity of ions, called “incipient Rydberg state (IRS)” effects (Ichimaru 1992, 1994; Tanaka et al. 1990), plays a major part in the theory of Coulomb resistivity for dense matter especially close to MI transitions.
We begin with the parameterized formulae of the resistivity in fully ionized plasmas with the ionic charge number Z, the ionic number density n i, and the electronic number density n e. Electric resistivity and thermal resistivity, ρE and ρT, arising from e-i scattering are expressed as (Kitamura and Ichimaru 1995)
where \( C_{\text{P}} \) and \( C_{\text{P}}^{\left( 0 \right)} \) are the specific heat (per electron) at constant pressure for the plasma and that for the ideal-gas electrons, respectively. In dense plasmas, due to the presence of strongly coupled ions, the values of \( C_{\text{P}}^{\left( 0 \right)} /C_{\text{P}} \) can be significantly smaller than unity; this effect would thus act to enhance the thermal transport.
Formulae (62) and (63) define the generalized Coulomb logarithms, L E and L T. In the classical (θ ≫ 1) and weak-coupling (Γ ≪ 1) regime, both L E and L T approach the Debye–Hückel limiting values:
where
and γ = 0.57721… is Euler’s constant. It was derived by Kivelson and DuBois (1964) with the aid of the quantum-mechanical Lenard–Balescu–Guernsey equation (e.g., Ichimaru 1992).
In the degenerate (θ ≪ 1) and strong-coupling (Γ ≫ 1) regime, the interparticle correlations are described by the ion-sphere model (e.g., Ichimaru 1982), in which one considers an ion surrounded by a uniform electronic charge sphere of the ion-sphere radius (50), as depicted in Fig. 11. In this model, the potential of scattering around an ion is expressed as
Coulomb energy in the ion-sphere model
where the position of the ion is set at the origin, r = 0.
The resistivity is proportional to the transport cross section, Q m(k F), for the electrons with wave number k F = (3π2 n e)1/3. This quantity may be evaluated from the phase shifts, obtained through numerical solutions to Schrödinger equation with the potential U(r).
The Born approximation is applicable for E F > Z 2Ry. In this regime, the values of Q m(k F) obtained through the phase shift analyses, in fact, show good agreement with the results in the Born approximation (Kitamura and Ichimaru 1995), which can be expressed in a fitting formula:
This formula is applicable for Z ≤ 26.
On the basis of the results, (64) and (65), applicable to both ends of the limit, we may express the Coulomb logarithms as follows:
where αE = 1, αT = 75/13π2 = 0.5845…, and
In these formulae, A E = 0.42, B E = 0.063, A T = 0.38, B T = 0.049, C = 6 × 10−4, D = 2, and K = 2.5, which have been determined through fit to computed results for hydrogen plasmas at 0.01 ≤ θ ≤ 10, 0.05 ≤ Γ ≤ 43.441, and x b ≤ 1.5 (Tanaka et al. 1990).
Here and in (66),
is a dimensionless IRS parameter, proportional linearly to e;
Thus, representing a ratio between a binding energy of a hydrogen atom and a kinetic energy of a free electron, the quantity x 4b measures the strength of Coulomb coupling between electrons and ions relevant to the MI transitions.
The term inside the braces in the formula (66), a steeply increasing function of x b, describes enhancement of scattering due to the strong e-i Coulomb coupling. In the low-density (r s ≫ 1) limit, however, the IRS effects should vanish, since the probability of finding electrons within a Bohr radius of an atom is small; the factor exp(−Cr Ds ) accounts for such an effect.
Those analytic formulae retain the following features: In the classical (θ ≫ 1) and weak-coupling (Γ ≪ 1) case, Eq. (66) reproduces the Debye–Hückel result (64) since ζDH ≪ 1 and ζBorn ≪ 1. In the degenerate (θ ≪ 1) and strong-coupling (Γ ≫ 1) case, L E ≈ αE/2ζBorn and L T ≈ αT/2ζBorn, since 1 ≪ ζBorn < ζDH. The transport cross section Q m(kF) obtained from ρE via a Drude formula:
is thus proportional to \( Q_{\text{m}}^{\text{Born}} \left( {k_{\text{F}} } \right) \) of (65); the Wiedemann–Frantz relation (Ichimaru 1992)
is satisfied.
6.2 Ultrahigh-pressure metal-physics experiments
Weir, Mitchell, and Nellis (Weir et al. 1996) compressed molecular fluid hydrogen to pressures ranging 0.93–1.80 Mbar by shock wave reverberation between insulating Al2Os anvils (Fig. 12), and thereby measured the pressure and the electric resistivity attained in seven runs of hydrogen compression/metallization experiments with the data as exhibited in Fig. 13. These authors then interpreted the experimental data in terms of a continuous transition from semiconducting to metallic diatomic fluids associated with a closure of a semiconductor bandgap, E g, near 1.4 Mbar. Such an interpretation, however, contradicts against any of the theoretical predictions (Wigner and Huntington 1935; Ceperley and Alder 1987; Saumon et al. 1995; Magro et al. 1996), which would foresee first-order insulator-to-metal transitions from molecular to monatomic hydrogen. On the basis of the equations of state described in Chap. 5 and the electric resistivity of dense hydrogen near the MI transitions in the preceding section, it has been shown (Kitamura and Ichimaru 1998) that those experimental results can be interpreted consistently with the phase diagram (Fig. 10) of hydrogen exhibiting the first-order MI transitions.
Shock-compression experiment: a time-resolved side-on radiograph of a laser-shocked D2 cell. Position of the pusher (Al) and the evolving shock front are measured as functions of time. After Ref. Da Silva et al. (1997)
Values of the pressure and the electric resistivity measured in seven runs of hydrogen compression/metallization experiments (Weir et al. 1996)
In interpreting the experiments, it is instructive to examine the relevant time scales in the compression and metallization processes. Let the thickness of the compressed hydrogen be ξ. Typical values of ξ in the experiments (see Fig. 12) are on the order of 100 μm. A hydrodynamic or compression time, τH, estimated as ξ divided by a sound velocity, may take on a value on the order of 10 ns. An electronic or ionization time, τE, estimated as ξ divided by the Fermi velocity, that is, the Planck constant times the Fermi wave number k F, may take on a value of about 60 ps. Since τE ≪ τH, we find that metallization develops “instantaneously” in a manner analogous to electric breakdown.
The shock-metallization experiments may thus be portrayed in two sequential stages:
Compression In a typical experiment, hydrogen (with a total mass of about 20 mg) is compressed from the state (P ≈ 1 bar, T ≈ 20 K) through reverberating shock imparted by an Al-Al2O3 impactor. An impactor with a mass of 2–3 g and a velocity of 5–6 μm/ns, which are typical experimental parameters, carries a kinetic energy on the order of 50 kJ. The hydrogen pressure takes on its maximal value when the state of hydrogen reaches the insulator side of the MI transitions; the pressures measured in the experiments (P = 1.0–1.5 Mbar) correspond to these maximal values. The time for such a compression is several times τH. Here, change in the enthalpy stems mostly from the hydro-mechanical contribution in Fig. 14.
Enthalpy changes stemming from different physical origins: thermal and hydro-mechanical
Metallization As metallization progresses in a time τE subsequent to that maximal-pressure state, the entropy, the temperature, and the enthalpy of the hydrogen increase. Efficiency of the metallization involves delicate matching between these changes and the dynamic conditions of the compression cell. We remark that shock compression by itself does not give optimum conditions. For efficient metallization, an additional process such as injection of intense, ultrashort pulses into the compressed hydrogen would have to be involved. Here, change in the enthalpy stems mostly from the thermal contribution in Fig. 14.
Figure 15 displays the plots of compressed insulator and final metallized states assessed for the experiments. Theoretical adiabats are also exhibited therein to guide the eyes. We thus find that experimental results are consistent with the phase diagram (Fig. 10) of hydrogen exhibiting the first-order MI transitions.
Plots of the initial (insulator) and final (metallized) states assessed for the experiments (Weir et al. 1996). The dashed and chain curves represent the theoretical adiabats starting from the initial conditions: P = 1 bar, T = 20 K (molecular fluid); and P = 1 bar, T = 10 K (molecular solid), respectively (Kitamura and Ichimaru 1998)
6.3 Jovian interiors and excess infrared luminosity
The interiors of giant planets (Jupiter, Saturn, Uranus, Neptune) offer important objects of study in the condensed-matter physics of hydrogen. Models for the internal structures of these planets were proposed on the bases of the thermodynamic and transport properties of the interiors, the surfaces, and the atmosphere coupled with the observational data such as gravitational harmonics (Stevenson 1982; Hubbard and Marley 1989).
Typically, Jupiter has the radius R J ≈ 7 × 104 km, some 11 times that of the Earth and approximately 1/10 of the solar radius, and the mass M J ≈ 1.90 × 1030 g, some 300 times that of the Earth and approximately 1/1000 of the solar mass. Model ranges of the mass density, the temperature, and the pressure of its interiors, consisting of the central “rock”, the “metal” hydrogen with a few percent (in molar fraction) admixture of helium, are displayed in Fig. 10.
The visible luminosity of the bright planet Jupiter, in fact, originates from solar radiation reflected from its surface, with albedo at 0.35. Jupiter has been known to emit radiation energy in the infrared range, approximately 2.7 times as intense as the total amount of radiation that it receives from the Sun. By observation through terrestrial atmospheric transmission windows at 8−14 μm (Menzel et al. 1926) and 17.5 − 25 μm (Low 1966), Jupiter has been known to be an unexpectedly bright infrared radiator. This feature has been reconfirmed quantitatively by a telescope airborne at an altitude of 15 km and through flyby measurements with Pioneer 10 and Pioneer 11 spacecraft. For Jupiter, the effective surface temperature determined from integrated infrared power over 8 to 300 μm was 129 ± 4 K, while the surface temperature calculated from equilibration with the absorbed solar radiation was 109.4 K (Hubbard 1980); the balance needs to be accounted for by internal power generation, hence, the issue of excess infrared luminosity of Jupiter.
Over the evolution period (~4.6 × 109 year) of Jupiter, the estimated excess luminosity, L ex ≈ 4.6 × 1017 W, would amount to the total released energy of approximately 6.5 × 1034 J. To account for the source of such an excess infrared luminosity, theoretical models such as “adiabatic cooling” (Hubbard 1968; Graboske et al. 1975; Stevenson and Salpeter 1976) and “gravitational unmixing” (Stevenson and Salpeter 1976; Smoluchowski 1967) have been considered. Here, we apply the phase diagram (Fig. 10) and the electric resistivity calculations of Sect. 6.1 to the issues of Jovian internal structure and luminosity.
The first-order MI transitions in hydrogen predict existence of a boundary layer near the surface of Jupiter across which the mass density and resistivity change discontinuously (Fig. 16). Assuming the temperature of the boundary layer at 6.5 × 103 K (Van Horn 1991; Kitamura and Ichimaru 1998), we calculate the density in the outer insulator side to be 0.34 g/cm3, and that of the inner metal side, 0.54 g/cm3.
Jovian model showing three different phases
Some 4.6 billion years ago when our solar system was formed, temperature was so high that hydrogen in Jupiter was in the ionized metallic states. The outer insulator side has then been formed over the evolution period through metal-to-insulator transitions.
The mass of the outer (molecular hydrogen) layer so formed was estimated to be M ins ≈ 0.1 × M J (Hubbard and Marley 1989). Hence, the total amount of latent heat released through the MI transitions, that is, the thermal contributions in Fig. 14, is \( \left( {{\text{M}}_{\text{ins}} /m_{\text{p}} } \right){{\Delta }}s_{\text{MI}} {\text{k}}_{\text{B}} T_{\text{MI}} \), where the entropy increment \( {{\Delta }}s_{\text{MI}} = 1.1 \) and \( T_{\text{MI}} = 6.5 \times 10^{3} {\text{K}} \) (Kitamura and Ichimaru 1998). These approximate calculations may thus indicate that the total latent heat would amount to 1.1 × 1034 J, accounting possibly for about 1/6 of the energy in the excess infrared luminosity.
A final solution to these issues of transformation and transfer of energy, however, should await for further investigations into the internal structure and evolution of Jupiter.
7 Stellar and planetary magnetisms
7.1 Preliminary
Astrophysical magnetic phenomena include those related to degenerate stars (e.g., Chanmugam 1992), solar flares (e.g., Parker 1979; Tsuneta 1995), and giant planets (e.g., Stevenson 1982). Surface magnetic fields of magnetic white dwarfs range 106 − 109 gauss. Average strengths of magnetic activities in the solar chromosphere are on the order of 50 gauss. Hydrogen is the major constituent in astronomical objects such as stars and planets. Stellar and planetary magnetisms may thus be strongly influenced by the phase transformations in dense hydrogen, such as metallization and magnetization.
Highly conductive liquid-metallic hydrogen in motion is capable of distorting and amplifying the magnetic-field configurations of stars and planets. The magnetic Reynolds number, R m, in magnetohydrodynamics is a number representing the ratio between the convective effects of a conductive fluid dragging and stretching the magnetic lines of force and the dissipative effects due to decay of electric current by the resistivity (e.g., Ichimaru 1992). The larger the R m, the more effectively are the magnetic activities and field strengths sustained and amplified by the fluid motion.
A ferromagnetic transition, should it take place, may provide a considerable strength of spontaneous magnetic field. In this section, we shall consider how the issues of stellar and planetary magnetisms may be related with such phase transitions in dense hydrogen.
7.2 Jovian magnetic activities
The dominant field contribution of the planet Jupiter for the external observer is the dipole of magnitude 4.2 gauss·R 3J and a tilt of ~10° to the rotation axis (Smith et al. 1976). Closer to the planet, however, the multipole contributions are so large that an additional dipole term at a depth of ~2 × 104 km appears to be implied (Elphic and Russel 1978).
The first-order MI transitions in hydrogen may predict existence of a boundary layer inside Jupiter across which the mass density and the resistivity change discontinuously, as Fig. 10 implies. With the estimates of Jovian parameters across the MI boundary in Fig. 16 and Sect. 6.3, we calculate the electric resistivity ρE of the metallic hydrogen inside the MI discontinuity to be 1.37 × 10−4 Ω·cm (Kitamura and Ichimaru 1995), only about 140 times greater than that of copper. The magnetic Reynolds number associated with this resistivity and Jupiter’s self-rotation may then be estimated as
where \( {{\omega }}_{\text{J}} \) (=15.2 rad/day) represents the angular velocity of Jupiter’s self-rotation.
This value is to be compared with a corresponding estimate ~1.0 × 108 for the solar magnetic activities (e.g., Ichimaru 1996). This comparison implies that prominent magnetic activities may be amply sustained near Jupiter by the presence of highly conductive metallic hydrogen inside its MI boundary. We remark, on the other hand, that there cannot be expected any electrically conductive (i.e., ionized) material in the frigid (T ≈ 129 K) conditions of the Jovian surface and in its atmosphere, a feature drastically in contrast with those in the solar chromosphere at T = ~6000 K, where a considerable amount of ionized gas may exist. The first-order MI transitions in dense hydrogen may thus be looked upon as an element of physics that is essential to the Jovian magnetic activities.
7.3 Ferromagnetic and freezing transitions in metallic hydrogen
In addition to the metal–insulator transitions treated previously, another class of phase transitions may be found for hydrogen in metallized states. As the cases with itinerant electrons or the electron liquids (Ceperley and Alder 1980; Ichimaru 2000), the protons in metallic hydrogen may be in a Wigner crystalline state as well as in a paramagnetic or a ferromagnetic fluid state. In this section, we consider the issues of ferromagnetic and/or freezing transitions and thereby elucidate the associated phase diagrams for metallic hydrogen (Ichimaru 2001).
Theoretical approaches to these issues begin with evaluations of the free energies as in Eq. (57). In the present case, the degrees of ionization and molecular dissociation are to be set at <Z> = 1 and αd = 0; instead, the spin polarization, ζ = (n ↑ − n ↓)/n, with n ↑ and n ↓ denoting the partial number densities of spin up and down protons (where n = n ↑ + n ↓) enters as a new parameter.
We thus consider the total free energy, f tot (n, T; ζ), in place of Eq. (57). The degrees of spin polarization and the resultant magnetic states are determined through minimization of the total free energy with respect to the variation of ζ.
Phase diagrams of hydrogen describing magnetization and solidification of metallic hydrogen are obtained in Fig. 17 for higher density and finite temperature regime (Ichimaru 2001). Table 2 lists the values of the physical parameters at the fluid–solid critical point (C FS) and the magnetic critical point (M C) in the phase diagrams of Fig. 17.
Phase diagrams of metallic hydrogen describing partial spin ordering and Wigner crystallization. The thick solid curve depicts the phase boundary between fluid and solid, with C FS designating the associated critical point. The dashed and chain curves describe the conditions at a constant strength magnetization, with the numerals denoting the decimal exponents of the field strength B M in G; M C designates the magnetic critical point. The diamond markers plot the observed surface-field strengths (B surface) vs. the effective surface temperatures (T surface) for the magnetic white dwarfs. After Ref. Ichimaru (2001)
7.4 Nuclear ferromagnetism with magnetic white dwarfs
The white dwarf represents a final stage of stellar evolution, corresponding to a star of about one solar mass compressed to a characteristic radius (R WD) of approximately 5000 km and an average density of some 106–107 g/cm3. Its interior consists of a multi-ionic condensed matter composed of C and O as the main elements and Ne, Mg, Si, …, Fe as trace elements. Observationally, a class of white dwarfs (DA) possesses hydrogen-rich atmosphere and envelopes.
Of all the isolated white dwarfs surveyed, only about 3–5% have observable magnetic fields that are in the range ~1–500 MG. The surface magnetic fluxes of these white dwarfs, ~BR 2WD = (1024–5 × 1026) gauss·cm2, are regarded as similar in magnitude to those of the magnetic Ap stars, a spectroscopic type of stars that exhibit intense hydrogen lines. Such Ap stars may, therefore, be looked upon as probable progenitors of the magnetic white dwarfs (Chanmugam 1992; Weisheit 1995).
The ferromagnetic transitions in dense metallic hydrogen as elucidated in Fig. 17 may offer an ingredient of physics relevant to the mechanisms for the origin of strong magnetization in the magnetic white dwarfs (Ichimaru 1997). The magnetic critical point occurs at a temperature of 6.6 × 105 K and the strengths of induced magnetization may range as high as 3 × 107 G along the B max line of Fig. 17. The plots in Fig. 17 of the magnetic-field strengths (B surface) vs. the effective blackbody temperatures (T surface) on the surfaces of 25 magnetic white dwarfs observed (Weisheit 1995) have shown that except for a few cases (with the field strengths exceeding the theoretical maximum by a factor of 1–15) the observed field strengths fall within the predicted range and that the surface temperatures are lower than the critical temperature by approximately two orders of magnitude.
The hydrogen densities (~104–106 g/cm3) required for magnetization may be expected in an outer shell of a hydrogen-rich white dwarf at some 70–90% of the stellar radius, where the mass densities may take on values lower by three or more orders of magnitude than those in the core. Solidification of metallic hydrogen, on the other hand, may not take place in such a white dwarf, since all the T surface values exceed the temperature at C FS.
For those white dwarfs with super-strong magnetic fields exceeding 3 × 107 G, a maximum field strength obtainable by the ferromagnetism alone in metallic hydrogen, a separate amplification mechanism such as a “dynamo” would have to be called for (e.g., Chanmugam 1992), where the nuclear ferromagnetism in Fig. 17 may provide “seed” fields for such an amplification.
The electric resistivity ρE calculated for the metallic hydrogen at the critical conditions (M C) takes on the value, 6.0 × 10−12 Ωcm (Kitamura and Ichimaru 1996). The magnetic Reynolds number associated with a rotating white dwarf is thus evaluated as
where ωWD is the angular velocity of a white dwarf. Comparing this number with the corresponding estimates, ~1.2 × 1012 and ~1.0 × 108, for Jovian and solar magnetic activities (Ichimaru 1997) (where the ferromagnetic seed fields are not available), respectively, we may conclude that considerable strengths of the magnetic fields may be sustained around a white dwarf by the presence of highly conductive, ferromagnetic hydrogen in the outer shell.
These observations may possibly account for some of the questions raised by Weisheit (1995) in regard to apparent non-correlation between surface magnetic fields, temperatures, and rotation periods with magnetic white dwarfs.
8 Nuclear fusion in metallic hydrogen
8.1 Preliminary
Nuclear reactions in hydrogen and its isotopes, deuterium, and tritium are the principal subjects in the development of fusion reactors. Condensed-matter effects including thermodynamics and phase transitions drastically affect the rate of nuclear reactions in ultradense hydrogen (Ichimaru 1993; Ichimaru and Kitamura 1999).
In metallic substances under ultrahigh pressures, electrons act to weaken or screen Coulomb repulsion between the atomic nuclei; the effects become so conspicuous that rates of nuclear reactions at relatively low temperatures take on values independent of the temperature. Cameron (1959) coined the term “pycnonuclear” reactions (from the Greek πψκυοσ, meaning “compact, dense”) to describe such nuclear processes. These reactions are thought to be applicable in a white-dwarf progenitor of a supernova.
In addition to such screening effects by electrons, the strong spatial correlation between atomic nuclei in dense matter due to their Coulomb repulsion, the very cohesive effect manifested in the freezing processes, tends to enhance the reaction rates through the effective reduction in overall internuclear repulsion; this enhancement factor in a dense fluid increases steeply as the temperature is lowered. The reaction rates in a solid, on the other hand, increase sharply with the temperature, as the rates depend sensitively on amplitudes of atomic vibrations. Hence, a maximum enhancement may be attained near the conditions of freezing.
In a white-dwarf supernova progenitor, enhancement by a factor of 30–40 orders of magnitude may be anticipated in the rate of nuclear reactions (Salpeter and Van Horn 1969; Ichimaru 1993; Ichimaru and Kitamura 1999). Virtually, no enhancement is expected, however, in high temperature, relatively low-density states with the solar interior or with the inertial-confinement fusion (ICF) plasmas.
Construction of the overall phase diagrams describing the MI and fluid–solid transitions is, therefore, pertinent to nuclear fusion in metallic hydrogen under ultrahigh pressure as well; such may possibly lead to a scheme of a “supernova on the Earth” for fusion studies.
8.2 Thermonuclear and pycnonuclear reactions
We consider dense binary-ionic substances with mass density ρm, pressure P, and temperature T, consisting of nuclear species with charge number Z i, mass number A i, and molar fraction x i (i = 1, 2). The number density of the nuclei of the species “i” and that of electrons are then given by
where \( m_{\text{N}} = 1.6605 \times 10^{ - 24} {\text{g}} \) denotes the average mass per nucleon. The ion-sphere radius of (50) may here be extended to encompass the binary-ionic systems, so that we define
Events of scattering between the nuclei, i and j, via the potential, W ij(r), with relative velocity v and reduced mass
may be described by the wave functions,
for the colliding pairs at an internuclear separation r; here \( {\text{P}}_{\ell } \left( {\cos \vartheta } \right) \) denote the Legendre polynomials (e.g., Landau and Lifshitz 1965). The wave functions obey the Schrödinger equation:
where E = (μij/2)v 2 denotes the center-of-mass energy.
The penetration probabilities, p(E), proportional to |Ψij(r N)|2, of the colliding nuclei to a nuclear reaction radius r N may be obtained by the solution to the Schrödinger equation. With the bare Coulomb potential,
substituted in Eq. (71), the essential parameters characterizing Coulomb scattering in the short ranges are the nuclear Bohr radius,
and the Gamow energy,
The events of scattering are governed primarily by the effective potential between the nuclei in the short-range domain, where the potential may be regarded as isotropic and Coulombic, irrespective of the interparticle configurations. Calculation of the reaction rates may be facilitated by the observation that the major contributions to the penetration probabilities arise from the s-wave (l = 0) scattering between the reacting nuclei, as wave functions in a spherically symmetric potential with azimuthal quantum number l are proportional to r l in the short ranges.
Since one can generally assume that the nuclear reaction radius r N < r *ij , the s-wave scattering is the major contribution to the reaction rates; hence, r N ≈ 0 may be taken for the calculation of the penetration probabilities, p(E). In these connections, one notes the short-range cusp condition with Coulomb scattering:
The usual boundary conditions in a treatment of these scattering problems assume an incident plane wave in the z-direction. The asymptotic (r → ∞) form of the Coulomb wave function is then expressed as follows (e.g., Landau and Lifshitz 1965):
where
is the angular function of the scattering wave,
the radial coordinate r with respect to the scattering center represents the distance between the reacting nuclei, and Γ(z) refers to the gamma function defined as follows:
With a normalization, such that
over a spherical volume Ω with a radius far greater than 2a ij, the incident flux at z → –∞ is given by the following:
The penetration probability or the square of the wave function at the origin then takes on the values:
This function approaches unity in the high-energy limit, E ≫ E G, as it should.
In the low-energy (E ≪ E G) processes of astrophysical interest, the penetration probability of the Coulomb barrier vanishes exponentially as follows:
Hence, one singles out this exponential factor, representing a purely Coulombic effect, from the cross section σij(E) for the nuclear reactions between species i and j, and thereby introduces the cross-sectional factor, S ij(E), via (Salpeter 1952; Barnes 1971; Fowler 1984)
Cross-sectional factors, therefore, quantify the proficiency of reactions intrinsic to the nuclei.
For non-resonant nuclear reactions, S ij(E) are functions slowly varying with E and may be expressed as follows:
For isotopes of hydrogen, we list in Table 3 the values of the cross-sectional factors and the energies Q ij released per reaction for the reactions:
In a dense metallic system, electrons act to screen the Coulomb repulsion between atomic nuclei. The potential W ij(r) of scattering in Eq. (71) now deviates from the purely Coulombic form (72) and takes on values:
For non-relativistic electrons (r s ≥ 0.1), we employ the local-field corrections in εe(k,0) (Ichimaru and Utsumi 1981, 1983) accounting for strong Coulomb-coupling between electrons, and evaluate the short-range screening distance D s of the electrons, defined by Eq. (59), as
with
For relativistic electrons (r s < 0.1 and θ ≪ 0.1), the screening parameter in Eq. (59) has been computed (Ichimaru and Utsumi 1981, 1983) with Jancovici dielectric function (Jancovici 1962) as follows:
It is noteworthy that the screening length (in units of a) takes on a finite value 5.8 in the limit of high densities (r s → 0), while the non-relativistic Thomas–Fermi length (e.g., Pines and Nozières 1966) diverges in the same limit. Due to such relativistic effects, the electron screening may thus remain considerable in dense stellar materials.
Substituting (77), rather than (72), in (71), we find that the pycnonuclear penetration probability is now given by
Here
is the Coulombic energy associated with the screening distance. It should be remarked here that p(E) in Eq. (80) takes on a non-vanishing value:
in the limit, E → 0. This is a feature unique in the pycnonuclear reactions, markedly different from the thermonuclear case of Eq. (75); the limiting value (82) decreases exponentially as the reduced mass increases. It offers the very reason why the p–d pycnonuclear reactions should be pursued in metallic hydrogen.
In Fig. 18, we plot and compare the penetration probabilities, the Boltzmann distributions,
Dashed and dotted curves: thermonuclear (d–t, at 108 K) vs. solid curves: pycnonuclear (p–d, at 450 K with n e = 2.4 × 1024 cm−3) reactions; p(E) denotes the penetration probability through the Coulomb barrier between a pair of reacting nuclei with a center-of-mass energy, E(K), in temperature units; f(E), the Boltzmann distribution; C(E) = p(E)·f(E), a contact probability between a pair of reacting nuclei; A·C(E), enhanced contact probability with A (=5.7 × 1042) denoting the enhancement factor (93)
and the contact probabilities,
as functions of log E(K), the logarithmic energy in temperature units, for the d–t thermonuclear reactions at T = 108 K and for the p–d pycnonuclear reactions at T = 450 K with the electron density, n e = 2.4 × 1024 cm3.
The rates of thermonuclear reactions stemming from those scattering processes may be calculated by the thermal averages <σij(E)v> over the Boltzmann distribution:
The result yields the Gamow rates (Gamow and Teller 1938; Thompson 1957) of the thermonuclear reactions:
Here, S ij refers to a thermal average of the cross-sectional factor, δij denotes Kronecker’s delta distinguishing between the cases with i ≠ j and with i = j;
refers to the Gamow exponent. The reaction rate (84) contains a factor exp(−τij). The magnitude of this Gamow exponent increases with the charge numbers and/or with the reduced masses. The thermonuclear reaction rates vanish exponentially as the temperature decreases.
The integration leading to the Gamow rate contains in its integrand a product between a steeply rising cross-section σij(E) and a steeply decreasing Boltzmann distribution f B(E) as functions of E. The product thus exhibits a Gamow peak at the energy:
The radius r TP of the classical turning point for a colliding pair with the Gamow peak energy is given by
The S ij in Eq. (84) is thus to be evaluated as S ij(E GP). In Fig. 18, we observe a Gamow peak at E ≈ 3 × 108 K for the contact probability in the d–t thermonuclear reactions.
For the pycnonuclear reactions, the screening temperature, derived from setting E s = E GP, may be expressed as follows (Ichimaru 1993; Ichimaru and Kitamura 1999).
In the weak screening regime, such that T > T s, the rate of reactions is calculated through a perturbative modification of Gamow’s thermonuclear rate. Thus, the enhancement factor due to such weak screening is
The rate of thermonuclear reactions is then calculated as the product between the Gamow rate (84) and the enhancement (88).
In the strong screening regime T < T s, the pycnonuclear counterpart to this product is given by the following (Ichimaru 1993; Ichimaru and Kitamura 1999):
In this case, the classical turning point takes place at r TP = D s. The screening distance, given by (78) or (79), is virtually independent of T in a dense material, because θ < 0.1. Contrary to the Gamow rate (84), which changes sharply with the temperature through τij, the pycnonuclear rate (89) is practically independent of the temperature; the strong exponential decrease with D s, the charge product Z i Z j, and the reduced mass should be noted.
8.3 Solar processes and inertial-confinement fusion
In the Sun, nuclear reactions take place most vigorously near the core, where the mass density and the temperature of the metallic hydrogen are estimated to be 56.2 g/cm3 and 1.55 × 107 K (see Fig. 10).
Terrestrial inertial-confinement fusion researchers are presently attempting to realize conditions analogous to those near the solar core by the compression of deuterium–tritium (d–t) fuel to mass densities and temperatures on the order of 3–60 g/cm3 and ~108 K; hence, a “sun on the Earth.” At such a high temperature, required for the thermonuclear reactions, one anticipates an assortment of dynamic instabilities to overcome in the compression of hydrogen matter.
One of the proton–proton chains, the fundamental nuclear processes in the solar interiors, consists in
which altogether yields
The cross-sectional factors S ij and the Q ij values are (Bahcall and Ulrich 1988)
The p–p chain, starting with p(p,e +νe)d, involves a β process and thus is extremely slow; the rate of this chain is controlled by these slow processes.
8.4 Enhancement of nuclear reactions in metallic fluids
As a progenitor of type I supernova, a white dwarf with interiors consisting of a carbon–oxygen mixture can be considered a kind of binary-ionic mixture, with a central mass density of 107–1010 g/cm3 and a temperature of 107 to several times 109 K (Starrfield et al. 1972; Whelen and Iben 1973). Nuclear runaway leading to supernova explosion may take place when the thermal output due to nuclear reactions exceeds the rate of energy losses.
In addition to the electronic screening effects manifested in the pycnonuclear penetration probability (80), strong Coulomb correlation between atomic nuclei in dense matter acts to enhance the reaction rates through an effective reduction of the internuclear repulsion (e.g., Ichimaru 1993; Ichimaru and Kitamura 1999). It is the very correlation effect responsible for the freezing transitions considered earlier in Sects. 4.2, 4.4, and 5.3, and is closely related to the Coulombic chemical potentials in dense plasmas treated in Sects. 4.2 and 4.3.
The increment of the Coulombic free energy before and after the reaction is expressed as
where μCoul(Z i) denotes the Coulombic chemical potential of a charge Z i. The increment before and after the nuclear fusion ΔF ij is a negative quantity, expressing a cohesive effect in Coulombic matter; its magnitude increases proportionally to the cubic root of the matter density.
It is instructive at this stage to recall the ion-sphere model (e.g., Ichimaru 1982) as illustrated in Fig. 11. We construct an ion sphere by picking a particle (an ion with the electric charge Ze) in the plasma and by associating with it a sphere of neutralizing charges that would exactly cancel the point charge of the ion. This sphere has the radius a and the electric charge density −3Ze/4πa 3. The electrostatic energy of the ion sphere is then calculated to be −0.9(Ze)2/a; the increment before and after the nuclear fusion is thus −1.057(Ze)2/a in the ion-sphere model.
The enhancement factor for the rate of nuclear reactions in dense metallic fluid is expressed approximately as
in the ion-sphere model, it takes on a huge value, exp(1.057Γ), with a strongly coupled plasma, as Fig. 19 explains.
Origin of the cohesive force and the enhancement factor in a Coulombic matter
The quantity ΔF ij may be calculated more accurately in terms of the ion–ion correlation energies as
where the so-called “linear mixing law” for the interaction energies, applicable fairly accurately to a dense Coulombic matter (e.g., Ichimaru 1994), has been adopted. In (90), we employ a semiclassical expression (48) for the ion–ion correlation energy applicable for Γ ≫ 1 with spin-independent, quantum-statistical corrections (Ichimaru 1997):
where
Γi is the electron-screened Coulomb-coupling parameter for the nuclei with charge Z i, that is
with a e = (3/4πn e)1/3, and
As we have noted in Sect. 8.2, the rate of nuclear reactions is proportional to the statistical averages of penetration or contact probabilities, |Ψij(0)|2, which are in fact the joint probability densities, g ij(r), between nuclei “i” and “j” evaluated at a nuclear-force distance, r N ≈ 0. Enhancement factors for thermonuclear or pycnonuclear rates have been calculated through the quantum-statistical treatments of such joint probability densities (Ichimaru 1993; Ichimaru and Kitamura 1999; Alastuey and Jancovici 1978; Ogata et al. 1991); the results are expressed compactly as
where
with
C1 = 1.1858, C2 = −0.2472, and C3 = —0.07009 (Ichimaru 1991). Note that the enhancement factor (93) depends sensitively on ρm and T; it increases with ρm and sharply as T decreases.
In Table 4, we list the rate of nuclear reactions P ij (expressed in power per unit mass) and the enhancement factor A ij computed for a hydrogen plasma appropriate to the solar core, an ICF deuteron–triton plasma with equal molar fractions, dense 12C matter expected in a white dwarf, and cases of metallic hydrogen with equal molar fractions of protons and deuterons. We observe huge enhancement factors for the “white dwarf” and “liquid metal” cases. As we find in Fig. 18, the net contact probability, A·C(E), for the “liquid metal” case may assume a magnitude comparable to that for the “ICF” case.
8.5 “Supernova on the Earth”
Possibilities of combined utilization of the pycnonuclear proton–deuteron (p–d) reactions at lower temperatures and their enhancement due to the strong Coulomb correlation, both applicable in ultradense metallic hydrogen near freezing conditions, have led to a proposal of a “supernova on the Earth” scheme for nuclear fusion studies (Ichimaru 1991, 1993; Ichimaru and Kitamura 1998, 1999). The idea is to bring a p–d mixture to a liquid-metallic state near solidification, as illustrated in Fig. 10.
For a concrete example, let us consider an experimental scheme of compression and metallization of a p-d mixture with equal molar fractions to a final state: ρm ≈ 6 g/cm3, T ≈ 459 K, P ≈ 50 Mbar (the case “liquid metal 1” in Table 4). As illustrated in Fig. 18, a considerable rate of enhanced pycnonuclear reactions may be expected in such a p–d mixture.
To achieve such an end, we may start from a H2–D2 mixture in a low-entropy, molecular solid (insulator) state at ~1 bar and ~10 K. As shown in Fig. 15, we find that the initial state here is connected by an adiabat to a state with P ≈ 2.4 Mbar, T ≈ 160 K, ρm ≈ 0.84 g/cm3, on the insulator side of the MI coexistence curves. A reverberating shock imparted by a low-speed (~1 μm/ns) impactor with a kinetic energy ~20 kJ, as depicted schematically in Fig. 20, may thus be utilized for compression of such a H2–D2 mixture with a total mass of ~4.8 mg in a volume of 32 mm3 (2 mm × 16 mm2), say. The compression may bring the mixture to a state at ~2.4 Mbar and ~220 K on the insulator side of the MI coexistence curves; the enthalpy increment ΔW in such a compression process may amount to ~0.9 kJ. Since the impactor speed is lower, we expect a departure from the adiabat to an extent lesser than those experienced in the Livermore shock-compression experiments (Weir et al. 1996; Da Silva et al. 1997); thus, the shock compression proposed here may be looked upon as a technical extension in line with these experiments.
Schematic diagram of compression/metallization experiments for hydrogen
The ingredient indispensable to such a compression/metallization scheme may then be an injection of a super-intense, ultrashort laser pulse into the compressed hydrogen, at the instant of compression, to ensure an efficient metallization. We estimate the enthalpy ΔW necessary for the metallization is approximately 11 J. Since the laser pulse width must be significantly shorter than the time for metallization τE (~10 ps), the required laser power should exceed ~1.1 TW. In this regard, the scheme may still be looked upon within the range of technical feasibility.
The quasi-adiabatic compression exerted continuously by the slow impactor may bring the resultant liquid-metallic hydrogen finally into a further compressed state at ~50 Mbar and ~450 K, say, near the freezing conditions; ΔW in this compression would be ~3.0 kJ. Contrary to the ultrahigh-temperature ICF plasmas, the dense hydrogen in a liquid-metallic state near solidification is a stable object; no dynamic instabilities are expected.
In this final state, the estimated fusion power would be ~5.2 W/g (Table 4), a detectable level for the rate of nuclear reactions. Recent experimental and theoretical progress in ultrahigh-pressure metal physics may make such a scheme of detecting the astrophysical enhanced pycnonuclear reactions, for the first time in terrestrial laboratory, an attractive and realizable project for fusion studies (Ichimaru and Kitamura 1995, 1998).
With such a feasibility experiment successfully conducted, a further extension into a parameter regime of still higher pressure and density would eventually lead to a fusion scheme with net power production. For example, if compression to the case “liquid metal 2” (at a pressure of 112 Mbar, still far below an ICF pressure) in Table 4 is realized with an increase of the impactor mass and energy, we might expect a burst of power production at a rate ~34.7 GW/g. Actual duration for such a nuclear burning would depend sensitively on thermal evolution of the hydrogen fuel (Kitamura and Ichimaru 1996). Assuming that the reactions might last for ~1 ms at this rate, we would find a total thermal output of ~0.17 MJ, which would correspond to a burning ratio of ~2 × 10−4.
9 Phase diagrams of nuclear matter
9.1 Deconfinement of quarks from nucleons
The quark–gluon plasma is a phase of matter, whose elementary constituents are the quarks, antiquarks, and gluons that make up the strongly interacting particles. It is a new phase in the sense that it has not yet detected in the laboratory. To probe the densest states of nuclear matter, the nuclear physics community has embarked on a large-scale program of studying collisions of ultrarelativistic heavy ions, called RHIC (=Relativistic Heavy Ion Collider) at Brookhaven National Laboratory and CERN (e.g., Yagi et al. 2005; Baym 1995). Recently, the STAR Collaboration (2017) at RHIC reported the first measurement of the rotation of the quark–gluon plasma produced in heavy ion collisions, providing crucial information for theoretical models that try to explain how such a plasma is formed.
A quark–gluon plasma is expected at high densities, deep in the cores of neutron stars. It is also said to be the oldest phase of matter, the form of the matter that filled up the early Universe at temperatures of trillions degrees, until the first few microseconds after the big bang.
In constructing the phase diagrams for nuclear matter, relative to the phase diagrams of hydrogen (Fig. 10), it is instructive to draw analogy between metallization of hydrogen atoms, treated in Sect. 5.4, and deconfinement of quarks from nucleons. G. Baym thus advanced phase diagrams of nuclear matter, 20 some years ago (Baym 1995), noticing thereby the correspondence between hadrons and insulators on one hand, and between quark–gluon plasmas and metals on the other. He then noted the gas–liquid phase transitions in hadrons and discontinuous deconfinement transitions in nuclear matter.
9.2 Phases of nuclear matter
Recently, Yagi, Hatsuda, and Miake (2005) advanced schematic phase diagrams of nuclear matter shown in Fig. 21, in line with the earlier predictions of Baym (1995).
Schematic phase diagram of nuclear matter. “Hadron,” “QGP”, and “CSC” denote the hadronic phase, the quark–gluon plasma and the color-superconducting phase, respectively; ρnm denotes the baryon density of the normal nuclear matter. Possible locations at which we may find the various phases of nuclear matter include the hot plasma in the early Universe, dense plasma in the interior of neutron stars, and the hot/dense matter created in heavy ion collisions (HICs) (Yagi et al. 2005)
Here, “Hadron,” “QGP”, and “CSC” denote the hadronic phase, the quark–gluon plasma, and the color-superconducting phase, respectively; ρnm denotes the baryon density of the normal nuclear matter. Possible locations at which we may find the various phases of nuclear matter include hot plasmas in the early Universe, dense plasmas in the interiors of neutron stars, and the hot/dense matter created in heavy ion collisions (HICs).
In Fig. 21, we also note the critical point associated with the discontinuous deconfinement transitions, analogous to the critical point C MI in Fig. 10 associated with discontinuous metal–insulator transitions in hydrogen (Kitamura and Ichimaru 1998).
9.3 Structure of a neutron star
A neutron star is an astronomical object of about one solar mass compressed to a radius of approximately 10 km with an average density well in excess of 0.1 billion tons/cm3, comparable to the mass density of atomic nuclei, ρnm.
The neutron star, like the planet Jupiter in Fig. 16, may have a three-part structure, consisting of the ultradense central core of quark–gluon plasmas, the main body of condensed neutron liquids, and the outer layer of Wigner-crystallized Coulomb solids consisting mostly in Fe nuclei, as shown in Fig. 22.
Structure of a neutron star
In conjunction with such structures, McDermott et al. (1985, 1988) were the first to analyze non-radial oscillations of neutron stars, with the predictions of the bulk and interfacial modes, associated with the non-vanishing shear modulus of the crustal solid, with characteristic periodicity on the order of milliseconds. First-principle calculations of shear moduli for Monte Carlo-simulated Coulomb solids, with inclusion of the Coulomb glasses of Sect. 3.3, were presented (Ogata and Ichimaru 1989, 1990); the results have been applied for improved analyses of the non-radial oscillations (Strohmayer et al. 1991).
The density of the central core of a neutron star may, therefore, exceed 1 billion tons/cm3. It is above the discontinuous deconfinement transitions, and thus, the core may consist in the phase with the quark–gluon plasmas.
References
R. Abe, Prog. Theor. Phys. 22, 213 (1959)
A. Alastuey, B. Jancovici, Astrophys. J. 226, 1034 (1978)
B.J. Alder, T.E. Wainwright, J. Chem. Phys. 31, 459 (1959)
H.H. Aumann, C.M. Gillespie Jr., F.J. Low, Astrophys. J. Lett. 157, L69 (1969)
J.N. Bahcall, R.K. Ulrich, Rev. Mod. Phys. 60, 297 (1988)
C.A. Barnes, Adv. Nucl. Phys. 4, 133 (1971)
G. Baym, in Elementary Processes in Dense Plasmas: Proc. Oji International Seminar, eds. S. Ichimaru and S. Ogata (Addison-Wesley, Reading, 1995), p. 3
D. Bohm, D. Pines, Phys. Rev. 92, 609 (1953)
J.J. Bollinger, S.L. Gilbert, D.J. Heinzen, W.M. Itano, D.J. Wineland, in Strongly Coupled Plasma Physics, ed. by S. Ichimaru (North-Holland/Yamada Science Foundation, Amsterdam, 1990), p. 117.
K.L. Bowles, Phys. Rev. Lett. 14, 45 (1958)
K.L. Bowles, J. Res. N. B. S. 65D, 1 (1961)
E.G. Brovman, Yu. Kagan, A. Kholas, Sov. Phys. JETP 35, 783 (1972)
S.G. Brush, H.L. Sahlin, E. Teller, J. Chem. Phys. 45, 2102 (1966)
A.G.W. Cameron, Astrophys. J. 130, 916 (1959)
D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45, 566 (1980)
D.M. Ceperley, B.J. Alder, Phys. Rev. B 36, 2092 (1987)
H. Chacham, S.G. Louie, Phys. Rev. Lett. 66, 64 (1991)
G. Chanmugam, Annu. Rev. Astron. Astrophys. 30, 173 (1992)
B.J. Cho et al., Phys. Rev. Lett. 106, 167601 (2011)
L.B. Da Silva et al., Phys. Rev. Lett. 78, 483 (1997)
A. Dargarno, J. Chem. Phys. 12, 143 (1967)
R.C. Davidson, Physics of Nonneutral Plasmas (Addison-Wesley, Redwood City, 1990)
R.D. Dick, G.I. Kerley, J. Chem. Phys. 73, 5264 (1980)
D. Dorchies, V. Recoules, Phys. Rep. 657, 1 (2016)
F. Dorchies et al., Appl. Phys. Lett. 93, 121113 (2008)
C.F. Driscoll, J.F. Malmberg, Phys. Rev. Lett. 50, 167 (1983)
D.H.E. Dubin, Phys. Rev. A 42, 4972 (1990)
W. Ebeling, A. Förster, V.E. Fortov, V.K. Gryaznov, A.Ya. Polishchuk, Thermodynamic Properties of Hot Dense Plasmas (Teubner, Stuttgart, 1991)
Elementary Processes in Dense Plasmas: Proc. Oji International Seminar, eds. S. Ichimaru, S. Ogata (Addison-Wesley, Reading, 1995)
R.C. Elphic, C.T. Russel, Geophys. Res. Lett. 5, 211 (1978)
L.L. Foldy, Phys. Rev. B 17, 4889 (1978)
V. E. Fortov, et al., Phys. Rev. Lett. 99 (2007)
V. E. Fortov, in Elementary Processes in Dense Plasmas: Proc. Oji International Seminar, eds. S. Ichimaru and S. Ogata (Addison-Wesley, Reading, 1995), p. 389
W.A. Fowler, Rev. Mod. Phys. 56, 149 (1984)
B.D. Fried, S.D. Conte, The Plasma Dispersion Function (Academic, New York, 1961)
C. Friedel, N.W. Ashcroft, Phys. Rev. B 16, 662 (1977)
G. Gamow, E. Teller, Phys. Rev. 53, 608 (1938)
J. Gaudin et al., Sci. Rep. 4, 4724 (2014)
S.H. Glenzer, R. Redmer, Rev. Mod. Phys. 81, 1625 (2009)
S.H. Glenzer et al., Phys. Rev. Lett. 98, 065002 (2007)
H.C. Graboske et al., Astrophys. J. 199, 265 (1975)
C.C. Grimes, Surf. Sci. 73, 397 (1978)
J.-P. Hansen, Phys. Rev. A 2, 221 (1970)
J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, 2nd edn. (Academic Press, London, 1986)
J.-P. Hansen, L. Verlet, Phys. Rev. A 184, 151 (1969)
R.J. Hemley et al., Phys. Rev. Lett. 76, 1667 (1996)
J.O. Hirschfelder, C.F. Curtis, R.B. Bird, Molecular Theory of Gases and Liquids, Chap. 13 (Wiley, New York, 1954).
N.C. Holmes, M. Ross, D.A. Young, Phys. Rev. B 52, 15853 (1995)
W.B. Hubbard, Astrophys. J. 152, 745 (1968)
W.B. Hubbard, Rev. Geophys. Sp. Phys. 18, 1 (1980)
W.B. Hubbard, M.S. Marley, Icarus 78, 102 (1989)
S. Ichimaru, Ann. Phys. 20, 78 (1962)
S. Ichimaru, Basic Principles of Plasma Physics (Benjamin, Reading, 1973)
S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982)
S. Ichimaru, J. Phys. Soc. Jpn. 60, 1437 (1991)
S. Ichimaru, Statistical Plasma Physics—Vol. I: Basic Principles. (Addison-Wesley, Redwood City, CA, 1992/Westview Press, Boulder, 2004)
S. Ichimaru, Statistical Plasma Physics—Vol. II: Condensed Plasmas. (Addison-Wesley, Redwood City, CA, 1994/Westview Press, Boulder, 2004)
S. Ichimaru, Rev. Mod. Phys. 65, 255 (1993)
S. Ichimaru, Phys. Plasmas 3, 233 (1996)
S. Ichimaru, Phys. Lett. A 235, 83 (1997)
S. Ichimaru, Phys. Rev. Lett. 84, 1842 (2000)
S. Ichimaru, Phys. Plasmas 8, 48 (2001)
S. Ichimaru, H. Kitamura, Phys. Plasmas, 6, 2649 (1999). (Erratum 7, 3482 (2000))
S. Ichimaru, K. Utsumi, Phys. Rev. B 24, 7385 (1981)
S. Ichimaru, K. Utsumi, Astrophys. J. Lett. 269, L51 (1983)
S. Ichimaru, D. Pines, N. Rostoker, Phys. Rev. Lett. 8, 231 (1962)
S. Ichimaru, H. Iyetomi, S. Tanaka, Phys. Rep. 149, 92 (1987)
S. Ichimaru, H. Kitamura, J. Phys. Soc. Jpn. 64, 2270 (1995)
S. Ichimaru, H. Kitamura, in Proc. Internat’l Conf. Superstrong Fields in Plasmas, ed. by M. Lomtano et al. (AIP Conf. Proc. 426, Varrena, Italy, 1998) p. 541
H. Iyetomi, S. Ogata, S. Ichimaru, Phys. Rev. A 46, 1051 (1992)
B. Jancovici, Nuovo Cimento 25, 428 (1962)
T. Katayama et al., Appl. Phys. Lett. 103, 135006 (2013)
C.F. Kennel, F.V. Coroniti, Ann. Rev. Astron. Astrophys. 15, 389 (1977)
H. Kitamura, S. Ichimaru, Phys. Rev. E 51, 6004 (1995)
H. Kitamura, S. Ichimaru, J. Phys. Soc. Jpn. 65, 1250 (1996)
H. Kitamura, S. Ichimaru, J. Phys. Soc. Jpn. 67, 950 (1998)
M.G. Kivelson, D.F. DuBois, Phys. Fluids 7, 1578 (1964)
L.D. Landau, E.M. Lifshitz, Quantum Mechanics—Non-Relativistic Theory, 2nd edn. (Pergamon Press, Oxford, 1965)
L.D. Landau, E.M. Lifshitz, Statistical Physics, 2nd edn. (Addison-Wesley, Reading, 1969)
J.L. Lebowitz, J.S. Rowlinson, J. Chem. Phys. 41, 133 (1964)
F.J. Low, Astron. J. 71, 391 (1966)
W.R. Magro, D.M. Ceperley, C. Pierleoni, B. Bernu, Phys. Rev. Lett. 76, 1240 (1996)
H.K. Mao, R.J. Hemley, Science 244, 1462 (1989)
H.K. Mao, R.J. Hemley, Rev. Mod. Phys. 66, 671 (1994)
H.K. Mao, R.J. Hemley, M. Hanfland, Phys. Rev. Lett. 65, 484 (1991)
P.N. McDermott, C.J. Hansen, H.M. Van Horn, R. Buland, Astrophys. J. Lett. 297, L37 (1985)
P.N. McDermott, H.M. Van Horn, C.J. Hansen, Astrophys. J. 325, 725 (1988)
J.M. McMahon, M.A. Morales, C. Pierleoni, D.M. Ceperley, Rev. Mod. Phys. 84, 1607 (2012)
D.H. Menzel, W.W. Coblentz, C.O. Lampland, Astrophys. J. 63, 177 (1926)
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)
A.C. Mitchell, W.J. Nellis, Rev. Sci. Instrum. 52, 347 (1981)
T. O’Neil, N. Rostoker, Phys. Fluids 8, 1109 (1963)
S. Ogata, S. Ichimaru, Phys. Rev. A 36, 5451 (1987)
S. Ogata, S. Ichimaru, Phys. Rev. Lett. 62, 2293 (1989)
S. Ogata, S. Ichimaru, Phys. Rev. A 42, 4867 (1990)
S. Ogata, H. Iyetomi, S. Ichimaru, Astrophys. J. 372, 259 (1991)
G. Ortiz, M. Harris, P. Ballone, Phys. Rev. Lett. 82, 5317 (1999)
E.N. Parker, Cosmical Magnetic Fields (Clarrendon, Oxford, 1979)
F. Perrot, M.W.C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984)
V.C. Pineo, L.G. Kraft, H.W. Briscoe, J. Geophys. Res. 65, 2629 (1960)
D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952)
D. Pines, P. Nozières, The Theory of Quantum Liquid, vol. I (W. A. Benjamin, New York, 1966)
E.L. Pollock, J.-P. Hansen, Phys. Rev. A 8, 3110 (1973)
A. Ravasio et al., Phys. Rev. Lett. 99, 135006 (2007)
M.N. Rosenbluth, N. Rostoker, Phys. Fluids 5, 776 (1962)
M. Ross, F.H. Ree, D.A. Young, J. Chem. Phys. 79, 1487 (1983)
N. Rostoker, M.N. Rosenbluth, Phys. Fluids 3, 1 (1960)
A.L. Ruoff, C.A. Vanderbough, Phys. Rev. Lett. 66, 754 (1990)
E.E. Salpeter, Phys. Rev. 88, 547 (1952)
E.E. Salpeter, H.M. Van Horn, Astrophys. J. 155, 183 (1969)
D. Saumon, G. Chabrier, H.M. Van Horn, Astrophys. J. Suppl. 99, 713 (1995)
H. Sawada et al., Phys. Plasmas 14, 121702 (2007)
W.L. Slattery, G.D. Doolen, H.E. DeWitt, Phys. Rev. A 26, 2255 (1982)
E.J. Smith, L. Davis Jr., D.E. Jones, in Jupiter, ed. by T. Gehrels (University of Arizona Press, Tucson, 1976), p. 788
R. Smoluchowski, Nature 215, 691 (1967)
S.J. Starrfield et al., Astrophys. J. 176, 169 (1972)
D.J. Stevenson, Ann. Rev. Earth Planet. Sci. 10, 257 (1982)
D.J. Stevenson, E.E. Salpeter, in Jupiter, ed. by T. Gehrels (University of Arizona Press, Tucson, 1976), p. 85
T. Strohmayer, S. Ogata, H. Iyetomi, S. Ichimaru, H.M. Van Horn, Astrophys. J. 375, 679 (1991)
S. Tanaka, S. Ichimaru, J. Phys. Soc. Jpn. 55, 2278 (1986)
S. Tanaka, X.-Z. Yan, S. Ichimaru, Phys. Rev. A 41, 5616 (1990)
The STAR Collaboration, Nature 548, 62 (2017)
W.B. Thompson, Proc. Phys. Soc. (London) B70, 1 (1957)
H. Totsuji, S. Ichimaru, Prog. Theor. Phys. 50, 753 (1973)
S. Tsuneta, in Elementary Processes in Dense Plasmas: Proc. Oji International Seminar, eds. S. Ichimaru and S. Ogata (Addison-Wesley, Reading, 1995), p. 447
H.M. Van Horn, Science 252, 384 (1991)
G.A. Victor, A. Dargarno, J. Chem. Phys. 53, 1316 (1970)
S.T. Weir, A.C. Mitchell, W.J. Nellis, Phys. Rev. Lett. 76, 1860 (1996)
J. Weisheit, in Elementary Processes in Dense Plasmas: Proc. Oji International Seminar, eds. S. Ichimaru and S. Ogata (Addison-Wesley, Reading, 1995), p. 61.
J. Whelen, I. Iben, Astrophys. J. 186, 1007 (1973)
E.P. Wigner, J. Chem. Phys. 3, 764 (1935)
E.P. Wigner, Trans. Faraday Soc. 34, 678 (1938)
E. Wigner, H.B. Huntington, J. Chem. Phys. 3, 764 (1935)
K. Yagi, T. Hatsuda, Y. Miake, Quark Gluon Plasma (Cambridge University Press, Cambridge, 2005)
Acknowledgements
The author thanks R. Abe, N. Ashcroft, G. Baym, D. M. Ceperley, L. B. Da Silva, R. Davidson, H. E. DeWitt, W. Ebeling, V. E. Fortov, J.-P. Hansen, T. Hatsuda, R. J. Hemley, W. B. Hubbard, N. Itoh, H. Iyetomi, B. Jancovici, H. Kitamura, W. Kohn, K. Makishima, J. Meyer-ter-Vehn, S. Ogata, T. O’Neil, C. Pethick, D. Pines, R. Redmer, M. N. Rosenbluth, N. Rostoker, T. Tajima, S. Tanaka, H. Totsuji, K. Utsumi, H. M. Van Horn, J. Weisheit, and M. Yamada for pertinent discussions on these and related subjects.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ichimaru, S. Phase transitions, interparticle correlations, and elementary processes in dense plasmas. Rev. Mod. Plasma Phys. 1, 6 (2017). https://doi.org/10.1007/s41614-017-0008-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41614-017-0008-z