The Astrophysical Journal, 539:888-901, 2000 August 20
© 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

 

Pycnonuclear Reactions in Dense Matter near Solidification

Hikaru Kitamura 1
Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545

Received 1999 October 18; accepted 2000 March 21

ABSTRACT

Rates of nuclear fusion reactions in dense matter and their enhancement due to electron screening and internuclear many-body correlation are calculated by explicitly taking into account dielectric functions of relativistic and nonrelativistic electrons, screening potentials based on the Monte Carlo simulations, and interaction free energies in dense electron-screened binary-ionic–mixture fluids. Pycnonuclear reactions are predicted near fluid-solid transitions at high densities, where the total reaction rates in fluid and solid phases take on comparable values that are virtually independent of the temperature. Analytic expressions for the reaction rates are presented and are applied to dense carbon-oxygen matter in the white dwarf interiors near ignition conditions and to dense proton-deuteron matter in giant planets and brown dwarfs. Possibility of a laboratory detection of pycnonuclear reactions in ultrahigh-pressure metallic hydrogen near solidification is explored.

Subject headings: dense matter; nuclear reactions, nucleosynthesis, abundances; planetary systems; stars: interiors; stars: low-mass, brown dwarfs; supernovae: general

     1 Present address: Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan; kitamura@issp.u-tokyo.ac.jp.

1. INTRODUCTION

     In dense matter relevant to the interiors of degenerate stars, rates of nuclear fusion reactions may take on values significantly enhanced over the binary Gamow rates, owing to electron screening and strong internuclear many-body correlation (Schatzman 1948; Cameron 1959; Salpeter & Van Horn 1969; Brown & Sawyer 1997; Ichimaru & Kitamura 1999). Enhancement of the reaction rates plays an important role in various astrophysical phenomena such as carbon ignition in white dwarfs (Nomoto 1982), helium burning in accreting neutron stars (Nomoto, Thielemann, & Miyaji 1985), and deuteron burning in giant planets (Saumon et al. 1996) and brown dwarfs (Burrows & Liebert 1993).

     When the electron density ne is so high that



is satisfied, electrons can be regarded as a uniform background of negative charges. Here ae ≡ (3/4πne)1/3 refers to the Wigner-Seitz radius. With this assumption, Cameron (1959) constructed an approximate effective potential of scattering on the basis of the ion-sphere model (Salpeter 1954; Ichimaru 1982) and computed the penetration factor of the potential barrier with the WKB approximation. It was thereby shown that the nuclear reaction rates in ultradense matter at relatively low temperatures are insensitive to the bombarding energy and take on values proportional to exp(-Cρ), with C being a constant and ρm denoting the mass density; such a process is called a pycnonuclear reaction.

     Enhancement factors of nuclear reactions can be formulated in terms of the quantum-mechanical pair-correlation function at short distances (Alastuey & Jancovici 1978; Ichimaru 1993; Ogata 1997). Within the semiclassical approximation, effective interaction between the reacting nuclei in the plasma may be described by a two-body potential called screening potential (Ichimaru 1993), which is related to the classical pair-distribution function of the plasma. Quantitative calculation of the enhancement factors in the pycnonuclear regime was carried out by Salpeter & Van Horn (1969) on the basis of the WKB approximation, with the screening potentials evaluated for the body-centered cubic (bcc) Coulomb solids. The solid pycnonuclear rates at zero-temperature were thereby derived, which are likewise proportional to exp(-Cρ). These solid screening potentials were, however, applied to high-temperature regime as well, where the nuclei undergo melting and no longer form lattice structures; the reaction rates and the phase transitions were not treated self-consistently.

     First-principles calculations of the screening potentials in both fluid and solid phases were performed through Monte Carlo simulations of the radial distribution functions (Brush, Sahlin, & Teller 1966; Ogata, Iyetomi, & Ichimaru 1991). The reaction rates and the enhancement factors were thereby evaluated by Ogata et al. (1991) for dense binary-ionic–mixture (BIM) fluids and solids on the basis of the contact probabilities obtained by the exact solutions to the Schrödinger equations. The resultant solid pycnonuclear rates are almost identical with those of Salpeter & Van Horn (1969). Though their results for the fluid phase are valid in the entire thermonuclear regime, their validity in the fluid pycnonuclear regime (Cameron 1959) near solidification was not discussed.

     At moderate densities such that rs ≈ 1, electrons polarize and screen the internuclear potential, which leads to further enhancement of the reaction rates, in addition to enhancement due to internuclear correlation. Such strong electron screening is expected in dense liquid metallic hydrogen at low temperatures (Ichimaru 1993). Even at extremely high densities, such that rs < 0.01, relevant to the white dwarf interiors, polarization of degenerate electrons remains finite owing to their relativistic effects (Ichimaru & Ogata 1991). Screening potentials in dense electron-screened one-component plasmas (OCPs) in the fluid phase were obtained in the past by the Monte Carlo simulations for several combinations of densities and temperatures (Totsuji & Tokami 1984; Ichimaru & Ogata 1991). Systematic calculations of the reaction rates with inclusion of these screening potentials have not yet been performed, however.

     In this paper, we calculate the enhancement factors of nuclear reactions in dense electron-screened BIM by explicitly taking into account those existing Monte Carlo data on the screening potentials. We thereby derive analytic formulae for the reaction rates applicable to the entire regime of BIM; in particular, they are applicable to the fluid pycnonuclear regime near solidification, where the total reaction rates turn out to be proportional to exp(-Cρ) and to be bridged smoothly into those in the solid phase across the solidification lines. Validity of the enhancement factors so obtained is reconfirmed separately through consideration of interaction free energies in electron-screened BIM fluids. These formulae are applied to 12C-12C reactions in white-dwarf progenitors of supernovae and to p-p, p-d, and d-d reactions in giant planets and brown dwarfs. Pycnonuclear reaction rates in dense metallic hydrogen near solidification relevant to ultrahigh-pressure experiments are finally considered.

2. SCREENING POTENTIALS

     We consider a BIM consisting of nuclear species "i" with charge number Zi, mass number Ai, mass fraction Xi, and molar fraction xi (i = 1,2), at mass density ρm and temperature T. The electron density is given by



where mN = 1.6605 × 10-24 g is the average mass per nucleon. The binary scattering potential between nuclei "i" and "j" may be described as



where &epsis;(k,ω) is a dielectric function of the electron liquid at density ne, describing the screening effects. The potential (3) may be characterized by the short-range screening length Ds (Ichimaru 1993),



     The many-body interionic correlation may be described in terms of the screening potential, which is defined in conjunction with the joint probability density gij(r) as



The screening potentials for BIM immersed in a uniform charge background of electrons were obtained by Ogata et al. (1991) through Monte Carlo sampling of gij(r) in the intermediate range of r and its extrapolation to r = 0 with the aid of the Widom expansion (Widom 1963). The expression for hij(r) ≡ Hij(r)/(ZiZje2/aij) so obtained is



where















and



is the ion-sphere radius; Γij = ZiZje2/aijkBT refers to the Coulomb coupling parameter. Equation (6) is a consequence of the ion-sphere scaling; small but finite deviations from the ion-sphere scaling law (Ichimaru, Ogata, & van Horn 1992; Ogata, Ichimaru, & van Horn 1993) will be neglected throughout this paper.

     Monte Carlo samplings of gij(r) were performed also for electron-screened OCP, which may generally be characterized by the two parameters, rs and Γ ≡ (Ze)2/akBT, where Z is the ionic charge number and a = (3Z/4πne)1/3. The screening potential was computed (Ichimaru & Ogata 1991) at rs = 1.4 × 10-3, Γ = 44.9 (a/Ds = 0.313), by taking into account the dielectric function of relativistically degenerate electrons in the random-phase approximation (RPA). The result is



with h0 = 0.8252 and h1 = 0.2312.

     Totsuji & Tokami (1984) likewise performed Monte Carlo simulations of electron-screened OCP for several combinations of rs and Γ by using &epsis;(k,0) in the nonrelativistic RPA. The resultant screening potential at rs = 1 and Γ = 50 (a/Ds = 1.172) may be expressed as



with h0 = 0.1835 and h1 = 0.0588. The potentials (6), (10) and (11) are illustrated in Figure 1. In § 4, validity of the values of hij(0) so obtained by the extrapolation scheme will be reconfirmed separately through consideration of interaction free energies for electron-screened BIM fluids.


Fig. 1   Screening potentials of electron-screened OCP obtained by the Monte Carlo simulations for a/Ds = 0 (Ogata et al. 1991), 0.313 (Ichimaru & Ogata 1991), and 1.172 (Totsuji & Tokami 1984). The filled circle, square, and triangle represent the estimations of h0 based on eq. (59) for the corresponding cases.

3. ENHANCEMENT FACTORS

     Enhancement factors Aij of the reaction rates between nuclei "i" and "j" in the fluid phase due to the interionic many-body correlation may be formulated as (Ogata et al. 1993)



where the integrals represent the path integrals over all trajectories from the origin back to the origin in an imaginary time βℏ (β = 1/kBT),



and



are actions, and μij = AiAjmN/(Ai + Aj) is the reduced mass.

     With the aid of the energy conservation relation in the reversed potential (Alastuey & Jancovici 1978),



we can rewrite equation (13a) in the form of



We likewise obtain



Here







bij represents the aphelion of the trajectory, wij(r) = Wij(r)/(ZiZje2/aij), and R = aij/r is a dimensionless density parameter for the ions, with r = ℏ2/(2μijZiZje2) denoting the nuclear "Bohr radius."

     In the WKB approximation (Alastuey & Jancovici 1978), only a trajectory (t) that minimizes the action is retained in the path integral. The values of bij in equations (15a) and (15b) are thus determined through the conditions, ∂S/∂bij = 0 and ∂S0/∂bij = 0, respectively. The value of bij that minimizes S0 is equivalent to the classical turning radius rTP of the binary reaction. Denoting the corresponding minimum values of S and S0 as Smin and S, respectively, the enhancement factor (12) can be evaluated as







3.1. BIM Without Electron Screening

     We begin with the case of BIM in the uniform background of the electrons, with Wij(r) = ZiZje2/r. In this case we can easily obtain I0 = π/2, rTP/aij = 3Γij/τij, and S/ℏ = τij, where



refers to the Gamow exponent (Ichimaru 1993).

     At high temperatures such that 3Γij/τij ≪ 1, we may regard h1 in equation (6) as a small perturbation and evaluate S with the unperturbed path, bij/aij = 3Γij/τij, and hence



The enhancement factor in the thermonuclear regime is thus obtained from equation (18) as



which coincides with the earlier results (Alastuey & Jancovici 1978; Ogata et al. 1993; Ichimaru & Kitamura 1999). If only the first term is retained on the right-hand side, we have ξij = βHij(0).

     As 3Γij/τij increases, the second term inside the curly bracket in equation (15a) becomes larger as compared with the first term. Consequently, at low temperatures 3Γij/τij ≫ 1, the asymptotic value of Smin may be obtained through the condition



which gives the fluid pycnonuclear reactions in the zero-energy limit (Cameron 1959). With the aid of the screening potential (eq. [6]), solution to equation (22) is obtained as bij/aij = 1.50 at Γij = 172, which gives I = 1.381, and



The density dependence (R)1/2 on the right-hand side accounts for the characteristic exponent (-Cρ) of the pycnonuclear rates (Cameron 1959). When equation (22) is solved with hij(r) evaluated at Γij = 180 instead of Γij = 172, we have bij/aij = 1.51 and I = 1.378, hence the coefficient 3.383 in equation (23) becomes 3.386; such slight Γij-dependence near solidification may be neglected. The enhancement factor in the pycnonuclear regime is thus obtained as



     The enhancement factors in the intermediate temperature range have been evaluated numerically through minimization of equation (15a). By combining the results with the asymptotic expressions, equations (21) and (24), we obtain an analytic formula,



where







     Ogata et al. (1991) evaluated the path integrals in equation (12) through exact solutions to the Schrödinger equations using the screening potential (eq. 6) and presented a fitting formula for ξij as (Ogata et al. 1993)



This formula is written in the form of an expansion in a power series of 3Γij/τij, and is not applicable to the pycnonuclear regime near solidification, 3Γij/τij > 2. Alternatively, Ichimaru & Kitamura (1999) proposed a formula,



     This formula was constructed so as to reproduce the numerical data of the semiclassical calculations by Ogata (1997) based on the path-integral Monte Carlo method in the regime 3Γij/τij < 2, but does not satisfy the asymptotic behavior (eq. [24]) for 3Γij/τij > 2.

     The total reaction rate (per unit volume) in BIM may be written in a form



Here EGP = τijkBT/3 is the Gamow peak, and Sij(E) refers to the nuclear cross section factor (Fowler, Caughlan, & Zimmerman; Ichimaru 1993). For nonresonant reactions at low energies, Sij(E) is a smooth function of E and can be expressed as



Experimental values of the cross section factors for various reactions are summarized in Table 1. In the fluid phase, R0 is given by the product between the Gamow rate and the enhancement factor (eq. [17]), that is,



Table 1   Nuclear Reaction Parameters

     The BIM may undergo solidification when &angl0;Γ&angr0; ≡ x1Γ1 + x2Γ2 ≥ 172 (Ogata et al. 1991). Pycnonuclear rates due to lattice vibrations in dense Coulomb solids were previously obtained by solving the Schrödinger equations of scattering between nearest-neighbor particles in the bcc lattice potentials. The rates are given by equation (30), where the corresponding expression for R0 is (Ichimaru et al. 1992)



Here F(Yij) denotes the thermal enhancement factor (Kitamura & Ichimaru 1995) over the ground state, which is expressed as



where Yij = Γij/ represents the ratio between the Einstein frequency and average thermal energy of the ions.

     In Figure 2a we compare the normalized reaction rates (32) for OCP at Rs = 5000, calculated with the enhancement factors (25), (28) and (29). The solid pycnonuclear rate (33)is also plotted for the solid regime, Γ > 172. Here we omit the suffix ij and denote R, Γij, τij and Yij as Rs, Γ, τ and Y, respectively. We observe good agreement among the three sets of calculations, despite the differences in the ways the path integrals in equation (12) were evaluated. The reaction rates in fluids decrease as the temperature is lowered and join almost smoothly into those in solids across the solidification line (Γ = 172), where 3Γ/τ = 1.34. The reaction rates in solids also depend slightly on the temperature, since the value of Y is not significantly larger than unity (Y = 2.43) at Γ = 172 and hence thermal excitation of lattice vibrations is not negligible (Kitamura & Ichimaru 1995).





Fig. 2   Normalized reaction rates R0 for OCP at (a) Rs = 5000 and (b) Rs = 1000. The dots are the computed results; the thick lines depict formula (32) with the fluid enhancement factor (eq. [25]) as well as formula (33) for the solid pycnonuclear rates. The dotted and the dashed curves are the corresponding results with the enhancement factors (28) (Ogata et al. 1993) and (29) (Ichimaru & Kitamura 1999), respectively.

     Figure 2b exhibits the corresponding result for higher density, Rs = 1000. Since the value of 3Γ/τ significantly exceeds unity (3Γ/τ = 2.29) at Γ = 172, the fluid reaction rate obtained by the present theory is almost independent of the temperature near solidification. Similarly, the solid pycnonuclear rates exhibit virtually no temperature dependence, since most of the nuclei are in the zero-point vibrational state even at the melting point (Y = 5.44 ≫ 1 at Γ = 172). We again find that the reaction rates in the fluid phase can join smoothly into those in the solid phase across the fluid-solid transition. The previous theories, such as formulae (28) and (29), however, exhibit anomalous behavior at 3Γ/τ > 2 and do not ensure a smooth connection, since these formulae do not take into account the boundary condition (eq. [24]). In the regime 3Γ/τ < 2, good agreement can be seen between the present calculation and the path-integral Monte Carlo results (eq. [29]), which proves numerical accuracy of formula (25).

     The microscopic states of the particles in fluids generally differ considerably from those in solids; in the former, the nuclei are distributed uniformly in space owing to the large contributions of kinetic energy, while in the latter they are confined within the lattice because of the strong Coulomb repulsion. In the vicinity of the fluid-solid transitions in dense matter, however, even the fluid phase exhibits short-range lattice-like order characterized by the ion-sphere model (Ichimaru 1982) and, since the nuclear reaction rates are governed by the short-range behavior of the pair-correlation functions, it is reasonable that the reaction rates in both phases are comparable near the transition. Such intuition applies equally to electron-screened plasmas, which we shall treat in the following subsection.

3.2. Electron-screened BIM

     Enhancement factors for electron-screened BIM can be obtained by following a procedure analogous to that described in the previous subsection, where the binary potential (eq. [3]) and the screening potentials (eqs. [10] and [11]) are additionally taken into account. The classical turning radius rTP of the binary reaction is expressed in a parametrized form as



where



is the critical temperature of screening (Ichimaru 1993) at which the energy of the Gamow peak τijkBT/3 equals the screening energy Es ≡ ZiZje2/Ds. At high temperatures T ≫ Ts, where electron screening can be regarded as a weak perturbation, equation (35) yields rTP/aij = 3Γij/τij , the turning radius for the Gamow rates. In the strong-screening regime T ≪ Ts, rTP is proportional to Ds; that is, rTP/Ds = (0.095 + 0.055rs)-2/3. The fitting error in formula (35) is confined within 2% for T/Ts > 0.2. We remark here that the earlier theory (Ichimaru 1993) predicted rTP/Ds = 1 in the low-temperature limit, since an approximate truncated Coulomb potential Wij(r) = ZiZje2/r - Es was adopted instead of equation (3). The values of S so obtained can be expressed in an analytic form as



where



     In the thermonuclear regime 3Γij/τij ≪ 1, we may regard h1 as a small perturbation; Smin can be obtained by substituting the unperturbed path, bij = rTP of equation (35), into the integral (16a); hence



with



Fitting formula that reproduces the integral (40) may be expressed as



The thermonuclear enhancement factor is thus obtained as



where rTP is given by equation (35).

     In the pycnonuclear regime, 3Γij/τij ≫ 1, Smin approaches the asymptotic value obtained through the condition analogous to equation (22),



The pycnonuclear enhancement factor in the zero-energy limit is thus obtained as



with Xij ≡ (bij/aij)1/2I. This quantity is computed by numerical integration of equation (16a) with wij given by equation (3) and bij obtained through solutions to equation (43). The result is reproduced by a fitting formula,



The second and the third terms on the right-hand side describe the increment of the pycnonuclear rates due to electron screening.

     The enhancement factors in the intermediate temperature range have been evaluated by minimizing equation (15a) with the aid of the screening potentials (eqs. [10] and [11]). The results are expressed by an analytic formula analogous to equation (25),



where







and ξ is given by equation (44). Formula (46) is applicable to the entire temperature-density regime of BIM fluids. In Figure 3 we compare the computed values of ξij with the predictions of formula (46); numerical accuracy of fitting formula (46) is clearly indicated. It is important to note here that the enhancement factor (46) approaches a constant value in the limit of T → 0; this is a significant improvement over the previous theories (Ichimaru 1993; Ichimaru & Kitamura 1999), which did not take into account the asymptotic behavior (eq. [44]) and the resultant enhancement factors exhibited unphysical divergence in the low-temperature limit.


Fig. 3   Enhancement factors in electron-screened OCP for a/Ds = 0.313 and 1.172. The dots refer to the computed results; the solid curves represent the fitting formula (eq. [46]).

     In the Monte Carlo screening potential (eq. [11]), electron screening was treated through the RPA dielectric function at T = 0. Since the electrons are strongly coupled in the regime rs > 1, we must separately take into account strong electron correlation beyond RPA. For this purpose, we use the screening length Ds for dense electron liquids, which takes into account the Ichimaru-Utsumi (1981) local-field correction (Kitamura & Ichimaru 1998):



This formula also takes into account finite-temperature effect through the electron degeneracy parameter &thetas; ≡ 2mekBT/[ℏ2(3π2ne)2/3], and



In the high-temperature limit (&thetas; ≫ 1), equation (49) reproduces the Debye-Hückel screening length (e.g., Ichimaru 1993).

     For relativistically degenerate electrons (rs < 0.1 and &thetas; → 0), the screening length can be expressed as (Ichimaru & Utsumi 1983)



Equation (49) or equation (51) should be adopted as an expression of Ds.

     In the derivation of the enhancement factor (eq. [46]), we have used the screening potentials (eqs. [10] and [11]) for the entire temperature regime at fixed densities. Such a procedure is partially justified since the screening potentials in the intermediate range of r depend very little on the temperatures (Totsuji & Tokami 1984); their short-range values, however, exhibit temperature dependence (Ogata et al. 1991). The temperature dependence of Hij(0) can be appropriately incorporated into the enhancement factors through interaction free energies of dense matter, as we shall describe in the next section.

4. EVALUATION OF Hij(0)

     It can be proven (Ichimaru & Kitamura 1996) for classical charged fluids that the value of Hij(0) is related to the increment of the interaction free energy before and after a reaction. In this section we construct a free-energy formula for dense electron-screened BIM fluids in order to evaluate h0 in equation (47).

     Assuming the Born-Oppenheimer approximation, the interaction free energy fint of dense electron-screened BIM fluid per ion in units of kBT may be expressed as (Kitamura & Ichimaru 1998)



where fxc refers to the exchange-correlation free energy of a quantum electron liquid per electron in units of kBT with Γe ≡ e2/aekBT, fex denotes the excess free energy formula of a classical OCP,



with



denoting the weak-coupling (Γ ≪ 1) expression derived by Abe (1959), γ = 0.57721... is Euler's constant, and



represents the Monte Carlo result for the strong-coupling regime 1 ≤ Γ < 180 (Slattery, Doolen &DeWitt 1982).

     In equation (52), we have introduced an effective Coulomb coupling parameter for the screened ion system through



where κi is an adjustable parameter. In order to determine a precise expression of κi valid for 0 < rs ≤ 1, let us consider a balance between the total free energy of OCP consisting of ionic species "i" with and without electron screening:



which should be negative. This quantity was calculated for the parameter ranges 0.001 ≤ rs ≤ 0.1 and 10 ≤ Γ ≤ 160 by treating the screening due to relativistically degenerate electrons as a weak perturbation (Ichimaru & Utsumi 1984). The OCP variational calculations were carried out in the strong-screening regime 0.1 ≤ rs ≤ 1 (Galam & Hansen 1976). The expression for κi has been determined so as to reproduce these numerical data for hydrogen plasmas (Zi = 1) as



This expression is more accurate than the previous evaluation, κi = 0.85 (Kitamura & Ichimaru 1998), which is appropriate at moderate densities rs ≈ 0.1–2 but leads to unphysical results Δf > 0 when rs < 0.1.

     In terms of the excess free energy formula (eq. [53]), the screening potential h0 at r = 0 can finally be expressed as (Ichimaru & Kitamura 1996)



where











These parameters should be evaluated with equations (49) or (51). In Figure 1, the values of h0 predicted by formula (59) are compared with those obtained in § 2 through extrapolation of the Monte Carlo data. We observe reasonable agreement between the two sets of estimations. Consequently, we use equation (59) for the evaluation of the enhancement factor (eq. [47]).

5. TOTAL REACTION RATES

     The total reaction rate (per unit volume) is given by the product between the binary rate and the enhancement factor Aij, that is,



(Ichimaru & Kitamura 1999). Here σij is the cross section of binary reaction with relative kinetic energy E(= μijv2/2), and the angle brackets denote the thermal average over the Boltzmann distribution at temperature T.

     At short distances, the binary scattering potential (eq. [3]) can be approximated by a truncated Coulomb potential



for which σij can be obtained analytically; the thermal average can then be expressed as (Ichimaru & Kitamura 1999)



We carry out the numerical integration and obtain an analytic expression for the reaction rate (eq. [63]) as



with



     For electron-screened pycnonuclear reactions at low temperatures T < Ts, however, rTP may become comparable to Ds, as we can see in equation (35); the truncated Coulomb potential (eq. [64]) is not valid, and expression (3) should be used instead. Consequently, we replace the exponential terms in equation (67) by exp[-αijπ(Ds/r)1/2], the corresponding WKB reaction rate derived from equation (37), to obtain



The two expressions on the right-hand side coincide at T = Ts. The fluid reaction rates are obtained by substituting equation (68) into R0 in formula (66).

     At high temperatures T ≫ Ts, the exponential factor exp(-αijπ) in equation (68) approaches exp(-τij + Es/kBT), which coincides with the Gamow rate multiplied by the correction factor due to weak electron screening (Ichimaru 1993). At low temperatures T ≪ Ts, both αij and Ds depend predominantly on the density of the degenerate electrons, and hence equation (68) yields electron-screened pycnonuclear rates that are virtually independent of the temperature. Since αij > 1 in the limit of T → 0, the low-temperature reaction rates take on values significantly smaller than those predicted by the previous theories (Ichimaru 1993; Ichimaru & Kitamura 1999), which adopted αij = 1.

     In the weak-screening regime T > Ts, the reaction rates are dominated by scattering with relative bombarding energy equal to the "effective" Gamow peak, τijkBT/3 - Es, while in the strong-screening regime T < Ts, the corresponding energy is the Boltzmann peak, kBT/2 (Ichimaru 1993). In equation (66), the cross section factor may therefore be evaluated at an effective energy defined by



     The criterion for solidification of electron-screened BIM may approximately be given by &angl0;Γs&angr0; ≡ x1Γ + x2Γ ≥ 172. Pycnonuclear rates in electron-screened BIM solids may be obtained by modifying equation (33) so as to additionally take into account the effects of electron screening in two ways: First, electron screening may soften the short-range lattice potentials, leading to an increase of the tunneling probabilities for the reacting pairs. The resultant increment of the contact probabilities may be approximately incorporated through the second and the third terms on the right-hand side of equation (45). Secondly, electron screening reduces the frequencies of lattice vibrations, since the interionic Coulomb repulsion is weakened. The reduction of the Einstein frequency may be taken into account through the electron-screened parameter (Ichimaru & Kitamura 1999)



introduced in a way similar to equation (56), with



Solid pycnonuclear rates may thus be given by equation (66), where the corresponding expression for R0 is



6. 12C-12C REACTION IN WHITE DWARFS

     Explosive nuclear burning in the interiors of accreting white dwarfs has been considered as a possible mechanism of Type I supernovae (Nomoto 1982). Nuclear runaway is expected to take place when the rate of nuclear energy generation exceeds the energy loss rate by neutrinos and radiation. As a white dwarf progenitor of a Type I supernova, we may typically consider a white dwarf with an interior consisting of carbon-oxygen BIM (i = "C", j = "O"), at a central mass density of 108–1010 g  cm-3 and a temperature of 107–109 K (Nomoto 1982; Hernanz et al. 1988). Since evolution lines of cool white dwarfs may follow close to the fluid-solid transitions, ignition may be greatly affected by the assessment of pycnonuclear rates in dense carbon-oxygen matter near solidification. In addition, the electrons are relativistic (rs < 0.01) so that the screening length (eq. [51]) should be appropriately taken into account.

     In Figure 4a, the contour of PCC/ρm = 10-5 W g-1 for 12C-12C reaction, an approximate condition for ignition (Nomoto 1982), is plotted on the phase diagram of dense carbon matter. Here, Pij denotes the power production rate obtained from equation (66) in accordance with Pij = RijQij, where Qij denotes the energy released per reaction. The values of Qij are listed in Table 1 for various reactions. At high densities near 1010 g  cm-3, the reaction rates are virtually independent of the temperature; the pycnonuclear rates expressed by equations (24) and (44) are manifested. No discontinuity of the ignition curve is observed at the fluid-solid transition, ΓCC = 172 or Γ = 172. We find that the net effect of electron screening on the total reaction rates is relatively small; ignition curve shifts only slightly toward a low-density side due to electron screening. This does not mean that electron polarization itself is negligibly small, however; electron screening enhances the binary reaction rates but reduces the enhancement factors due to interionic correlation, since Coulomb repulsion between ions is weakened as a result of electron screening. As a consequence of such a cancellation, the net effect of electron screening turns out relatively small.





Fig. 4   Contour of PCC/ρm = 10-5 W g-1 in dense carbon-oxygen BIM calculated for (a) xO = 0 and (b) xO = 0.9. The solid lines account for electron screening; the dotted lines neglect electron screening (Ds = ∞).

     Sahrling & Chabrier (1998) likewise calculated enhancement factors in dense electron-screened carbon-oxygen plasmas on the basis of their Monte Carlo screening potentials and reached the conclusion that the effect of electron screening on the carbon ignition curve is quite small. We remark that their fitting formulae for the enhancement factors were constructed in numerical terms so as to reproduce the computed data for Γ < 60 in the carbon-oxygen plasmas, while the present formulae are based on explicit considerations of the boundary conditions (42) and (44) in physical terms and can be applied to dense BIM consisting of any ionic species.

     Other calculations (Ichimaru & Ogata 1991; Ichimaru & Kitamura 1999), on the other hand, predicted rather large effect of electron screening on the reaction rates, leading to a significant discontinuity in the ignition curve across the fluid-solid transition. As we have explained in conjunction with Figure 2b, since these calculations do not explicitly take into account the characteristics of the fluid pycnonuclear processes (eqs. [24] and [44]), they are not applicable to the vicinity of solidification.

     For a comparison, we present in Figure 4b a corresponding plot for carbon-oxygen mixture with the molar fraction of oxygen xO = 0.9.

7. DEUTERON BURNING IN GIANT PLANETS AND BROWN DWARFS

     The interiors of giant planets (Jupiter, Saturn, and extrasolar giant planets) and brown dwarfs are composed mainly of dense fluid metallic hydrogen (Burrows & Liebert 1993; Hubbard et al. 1997). For objects with masses larger than about 13.5MJ (with MJ = 1.9 × 1030 g being the Jupiter mass), deuteron burning is expected to take place through the reaction D(p,γ)3He in the early stages of evolution (Saumon et al. 1996; Hubbard et al. 1997).

     Jupiter is emitting infrared radiation approximately twice as intense as the total amount of radiation that it receives from the Sun (Hubbard 1980). The common explanation to the origin of such an excess infrared luminosity is adiabatic (convective) cooling or sedimentation of helium through phase separation (Hubbard 1980). Possibility of deuteron burning in giant planets is connected to a theoretical account for the internal power sources (Horowitz 1991; Ichimaru 1993; Ichimaru & Kitamura 1999).

     The basic features of D(p,γ)3He reactions in dense proton-deuteron BIM (i = "p", j = "d") appropriate to the interiors of giant planets and brown dwarfs are summarized in Figure 5. Here, the mass fraction of deuteron has been assumed to be 6 × 10-5 (Anders & Grevesse 1989). The line 3Γpd/τpd = 1 provides the boundary between the thermonuclear and the pycnonuclear regimes. We observe that the temperatures near the center of Jupiter ("J") and brown dwarfs ("BD") are high enough to satisfy the thermonuclear conditions, 3Γpd/τpd < 1 and T > Ts. The Jovian interior (ρm = 5 g cm-3, T = 2 × 104 K) is, however, relatively close to the pycnonuclear regime; both the protons and deuterons are strongly coupled (Γ = Γ = 7.7 > 1), hence we predict a substantial enhancement Apd = 110 due to internuclear correlation. Electrons are degenerate and their density is relatively small (rs = 0.812) so that electron polarization is significant (ae/Ds = 1.13). Nevertheless, total power production rate remains minuscule, Ppd/ρm = 1.41 × 10-34 W g-1, which is far smaller than the average Jovian luminosity of 2.4 × 10-13 W g-1 (Hubbard 1980). Consequently, p-d reactions cannot account for the Jovian excess infrared luminosity. For comparison, at the same mass density and temperature, the rate of d-d reaction, which includes the two branches D(d,p)T and D(d,n)3He, amounts to Pdd/ρm = 2.27 × 10-42 W g-1.


Fig. 5   Physical parameters for p-d reaction in proton-deuteron BIM with Xd = 6 × 10-5. The dots indicate the conditions at the center of Jupiter ("J") as well as the brown dwarfs ("BD") at the age of 10 Gyr modeled by Burrows & Liebert (1993) with the numbers denoting masses in units of the solar mass. The contours of Ppd/ρm (W g-1) = 10-30, 10-10, and 1 are depicted by the solid curves. The line corresponding to Ppp/ρm = 10-15 W g-1 for the truncated p-p chain is illustrated by the chain curve.

     Nuclear reaction rates for metallic hydrogen in the Jovian interior were previously studied by Horowitz (1991), where the Thomas-Fermi approximation was used for the electron screening lengths and the screening potentials were evaluated with the hypernetted-chain (HNC) equations. Since electrons are strongly coupled, Thomas-Fermi approximation underestimates the screening length; it is essential to take into account the effect of electron correlation in the screening length, as we have considered in conjunction with equation (49). The HNC scheme is insufficient for accurate description of the short-range correlation in strongly coupled plasmas (Ichimaru 1993); the Monte Carlo screening potentials (eqs. [10] and [11]) are therefore more accurate. Moreover, Horowitz extrapolated the screening potential to r = 0 by assuming Hij(r) to be a linear function of r instead of a quadratic function as indicated by Widom (1963); this may result in an overestimation of Hij(0) and hence the enhancement factors.

     Also plotted in Figure 5 are the estimated central mass densities and temperatures of hydrogen in model brown dwarfs with several different masses, calculated at the age of 10 Gyr (Burrows & Liebert 1993). In these objects, primordial deuterons are considered to burn away in the early epoch, so that the relevant nuclear process is the "truncated" p-p chain, H(p,e+νe)D(p,γ)3He, with the effective Q-value of 6.084 MeV (Burrows & Liebert 1993). We find that the nuclear reactions under these conditions are basically thermonuclear (3Γpp/τpp ≪ 1); the enhancement factor is largely dominated by the leading term on the right-hand side of equation (47), that is, ξij ≈ βHij(0). For the p-p reaction at ρm = 1.0 × 103 g cm-3 and T = 2.0 × 106 K, we have rs = 0.139, ae/Ds = 0.525, 3Γpp/τpp = 0.13, App = 1.80 and Ppp/ρm = 5.76 × 10-10 W g-1.

8. PYCNONUCLEAR REACTIONS IN METALLIC HYDROGEN

     As we can see in Figure 5, pycnonuclear reaction of metallic hydrogen (3Γij/τij ≫ 1) may be expected at temperatures below about 104 K. Though the central part of any stellar object may be too hot to satisfy this condition, it is an interesting issue from the point of view of condensed matter physics (Kitamura & Ichimaru 1998; Nellis, Louis, & Ashcroft 1999). Metallization of hydrogen was recently achieved at pressure of 1.4 Mbar and estimated temperature of (2–3) × 103 K through the shock-compression experiment (Weir, Mitchell, & Nellis 1996). Much higher pressures and lower temperatures may be realized through adiabatic compression of low-temperature fluid/solid molecular hydrogen (Aoki & Meyer-ter-Vehn 1994). In this connection, it is significant to investigate the possibility for detecting the pycnonuclear p-d reactions in laboratories (Ichimaru & Kitamura 1999).

     As Figure 5 illustrates, the basic characteristics of the pycnonuclear reactions in metallic hydrogen are similar to those in dense carbon matter as seen in Figure 4. In metallic hydrogen, however, the polarization of electrons is so conspicuous that the condition T < Ts is satisfied near solidification. We thus expect an electron-screened fluid pycnonuclear reaction, which is described by formula (68) for T < Ts; such a fluid pycnonuclear process was not considered in the earlier calculation by Horowitz (1991).

     When &angl0;Γs&angr0; > 172, metallic hydrogen may undergo solidification and forms a quantum solid. We again find that the reaction rates in the fluid phase join almost continuously into those in the solid phase, though the reaction rates in both phases differ by 1–2 orders of magnitude at solidification. These discrepancies may arise in part from the crudeness of the solid pycnonuclear rate (eq. [72]), recalling that it has been constructed by modifying expression (33) in an approximate way. In order to evaluate the solid pycnonuclear rates under strong electron screening more precisely, we must construct lattice potentials for Coulomb solids interacting via the binary potential (eq. [3]) and solve the relevant scattering Schrödinger equations between the nearest-neighbor particles.

     In Table 2 we list the pycnonuclear rates of p-d reaction in proton-deuteron–mixture fluids with equal molar fractions xp = xd = 1/2, calculated at the solidification condition &angl0;Γs&angr0; = 172. Enhancement by as large as 15 to 21 orders of magnitude is predicted. We find, however, that even in the case of highest density, ρm = 200 g cm-3, the total power production rate takes on a minuscule value on the order of 10-19 W g-1. Judging from the smallness of the reaction rates and extremely high estimated pressures in Table 2, it seems unlikely to detect pycnonuclear reactions at a considerable level in laboratories. The earlier theory by Ichimaru & Kitamura (1999) seriously overestimated these reaction rates by as large as 40 orders of magnitude, because these authors extended equation (29) to electron-screened pycnonuclear processes in an ad hoc fashion without rigorous justifications, and because the truncated Coulomb potential (64) was inappropriately used to evaluate the binary reaction rates at low temperatures.

Table 2   Rates of p-d Reaction and Related Physical Parameters in Proton-Deuteron Fluids with Equal Molar Fraction, Evaluated at the Approximate Solidification Line &angl0;Γs&angr0; = 172

9. CONCLUDING REMARKS

     We have calculated the enhancement of nuclear reactions due to electron screening and internuclear many-body correlation by using the dielectric functions of electron liquids and the existing Monte Carlo data of the screening potentials. Analytic expressions for the reaction rates in dense electron-screened BIM have been presented. Fluid pycnonuclear reactions have been predicted at low temperatures under the conditions of strong ion screening (3Γij/τij ≫ 1) and/or strong electron screening (T ≪ Ts); the former condition is expected to be satisfied in the white-dwarf progenitors of Type I supernovae, and both of the two conditions may be satisfied in ultrahigh-pressure liquid metallic hydrogen near solidification. As the temperature is further lowered, the reaction rates in fluids join smoothly into those in solids across the fluid-solid transitions, &angl0;Γs&angr0; = 172.

     The enhancement factors can be rigorously formulated in terms of quantum-mechanical pair-correlation functions between the reacting pairs, which involves the statistical average over the coordinates of all the nuclei in the plasma (Alastuey & Jancovici 1978; Ichimaru 1993; Ogata 1997). The screening potential, which is related to the classical potential of mean force, is an approximate way of reducing the many-body problem to an effective two-body problem. Alastuey & Jancovici (1978) developed a theory in which corrections due to higher-order correlation were successively estimated through perturbation expansion. They thereby found that the bulk of the contribution to the enhancement factor arises from the leading term, equation (18), and that the first-order correction evaluated through approximate three-body classical correlation functions amounts to only a few percent of the leading term as long as 3Γij/τij < 1.6.

     Recently, first-principles calculations of the enhancement factors were carried out by Ogata (1997) for quantum OCP fluids through direct evaluations of the quantum pair-correlation functions with the path-integral Monte Carlo method. It was thereby shown that quantum fluctuations of the surrounding nuclei may enhance the contact probabilities between the reacting pairs by 1 order of magnitude in the pycnonuclear regime near solidification. Extension of such a calculation to electron-screened plasmas, especially dense metallic hydrogen, is an interesting outstanding issue, since a relatively large quantum effect may be expected owing to the smallness of the mass of hydrogen nuclei. The aim of the present paper has been to construct analytic formulae of nuclear reaction rates within the semiclassical approximation, through which we can understand the overall features of nuclear fusion processes over the entire parameter regime of electron-screened plasmas, from thermonuclear reactions at high temperatures to pycnonuclear reactions near solidification.

     The author thanks S. Ichimaru and H. Totsuji for pertinent discussions on these and related subjects, J. C. Weisheit for his hospitality at the Department of Space Physics and Astronomy of Rice University during an early stage of this work, and acknowledges receipt of a Research Fellowship for Young Scientists from Japan Society for the Promotion of Science.

REFERENCES