NASA/TP-20205001617 Nuclear Fusion Reactions in Deuterated Metals Vladimir Pines and Marianna Pines PineSci Consulting, Avon Lake, Ohio Arnon Chait and Bruce M. Steinetz Glenn Research Center, Cleveland, Ohio Lawrence Forsley JWK Corporation, Annandale, Virginia Robert C. Hendricks, Gustave C. Fralick, and Theresa L. Benyo Glenn Research Center, Cleveland, Ohio Bayarbadrakh Baramsai, Philip B. Ugorowski, and Michael D. Becks Vantage Partners, LLC, Brook Park, Ohio Richard E. Martin Cleveland State University, Cleveland, Ohio Nicholas Penney Ohio Aerospace Institute, Brook Park, Ohio Carl E. Sandifer II Glenn Research Center, Cleveland, Ohio June 2020 NASA STI Program . . . in Profile Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA Scientific and Technical Information (STI) Program plays a key part in helping NASA maintain this important role. 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Contents Summary ........................................................................................................................................................................1 1.0 Introduction ............................................................................................................................................................2 2.0 Nuclear Fusion Cross Section of Bare Nucleus Ions..............................................................................................2 3.0 Nuclear Fusion with Electron Screening ................................................................................................................3 3.1 Coulomb Barrier Screening by Lattice Electrons...........................................................................................3 3.2 Coulomb Barrier Screening by Plasma Particles ...........................................................................................5 3.3 Coulomb Barrier Screening by Conduction Electrons in Metal Lattice.........................................................8 3.4 Screening of Reacting Hydrogen Isotope Nuclei by Atomic Shell (Bound) Electrons in Deuterated Metals..........................................................................................................................................8 4.0 General Screening Case For Reacting Hydrogen Isotope Nuclei...........................................................................9 5.0 Coulomb Scattering on Target Nuclei ....................................................................................................................9 5.1 Light Particles Elastic Coulomb Scattering (e–, e+)........................................................................................9 5.2 Heavy Particle Elastic Coulomb Scattering (p, d, α) ...................................................................................11 5.3 Compton Scattering on Free Deuteron.........................................................................................................12 6.0 Neutron Elastic Scattering on Deuteron Nuclei....................................................................................................14 7.0 Neutron Energy Loss in Elastic Collisions With Deuteron Nuclei.......................................................................15 8.0 Summary of Results .............................................................................................................................................16 References ...................................................................................................................................................................16 Summary Nuclear fusion reactions of D-D are examined in an environment comprised of high density cold fuel embedded in metal lattices in which a small fuel portion is activated by hot neutrons. Such an environment provides for enhanced screening of the Coulomb barrier due to conduction and shell electrons of the metal lattice, or by plasma induced by ionizing radiation (γ quanta). We show that neutrons are far more efficient than energetic charged particles, such as light particles (e−, e+) or heavy particles (p, d, α) in transferring kinetic energy to fuel nuclei (D) to initiate fusion processes. It is well-known that screening increases the probability of tunneling through the Coulomb barrier. Electron screening also significantly increases the probability of large- versus small-angle Coulomb scattering of the reacting nuclei to enable subsequent nuclear reactions via tunneling. This probability is incorporated into the astrophysical factor S(E). Aspects of screening effects to enable calculation of nuclear reaction rates are also evaluated, including Coulomb scattering and localized heating of the cold fuel, primary D-D reactions, and subsequent reactions with both the fuel and the lattice nuclei. The effect of screening for enhancement of the total nuclear reaction rate is a function of multiple parameters including fuel temperature and the relative scattering probability between the fuel and lattice metal nuclei. Screening also significantly increases the probability of interaction between hot fuel and lattice nuclei increasing the likelihood of Oppenheimer-Phillips processes opening a potential route to reaction multiplication. We demonstrate that the screened Coulomb potential of the target ion is determined by the nonlinear Vlasov potential and not by the Debye potential. In general, the effect of screening becomes important at low kinetic energy of the projectile. We examine the range of applicability of both the analytical and asymptotic expressions for the well-known electron screening lattice potential energy Ue, which is valid only for E >> Ue (E is the energy in the center of mass reference frame). We demonstrate that for E ≤ Ue, a direct calculation of Gamow factor for screened Coulomb potential is required to avoid unreasonably high values of the enhancement factor f(E) by the analytical—and more so by the asymptotic—formulas. 1.0 Introduction Electron screening is essential for efficient nuclear fusion reactions to occur. Screening effects on fusion reaction rates as measured in deuterated materials have been demonstrated to be important. The nuclear reaction rate includes two primary factors: the Coulomb scattering of the projectile nuclei on the target nuclei as well as nuclei tunneling through the Coulomb barrier. During elastic scattering of charged projectiles on a target nucleus, such as a deuteron, some of the energy of the projectile particle is transferred to the target nucleus, hence heating it. Depending on the projectile particle energy and the efficiency of kinetic energy transfer during the scattering event, the target deuteron may become energetic enough to enable subsequent nuclear fusion reactions via tunneling through the Coulomb barrier. Electron screening may play a significant role in this process because of hot fuel interacting with lattice nuclei in the highly screened environment, as has been demonstrated in the companion experimental work reported in Steinetz et al. (Ref. 1). In the current work we analyze the electron screening effect on Coulomb scattering and the tunneling process involving charged projectiles. We then demonstrated the superior efficiency of the kinetic energy transfer by energetic neutrons on the target deuteron nuclei resulting in subsequent nuclear reactions. Such a process is a key ingredient in achieving and sustaining nuclear reactions. 2.0 Nuclear Fusion Cross Section of Bare Nucleus Ions In the standard case of subbarrier quantum tunneling through the Coulomb barrier between positive nucleus ions, the nuclear fusion cross section of bare nucleus ions σbare(E) can be written (Ref. 2) as: σbare(E) = [S(E)/E] exp[-G(E)] (1) where E is the energy in the CM (center of mass) reference frame, G(E) is the Gamow factor, and S(E) is the astrophysical S-factor containing the details of nuclear interactions. It is noted that the theoretical development proceeds in Gauss units (not SI). In the nonrelativistic case, the relation between energy E in the CM frame and the kinetic energy K1∞ of the projectile nucleus ion in the laboratory (lab) frame takes the simple form: K1∞ = (1/2) m1 v1∞^2 = (1 + m1/m2) E (2) In the lab frame, the target nucleus ion with mass m2 is at rest (v2 = 0), and the projectile nucleus ion with mass m1 has velocity v1∞ at infinity. In the Wentzel-Kramers-Brilloin (WKB) approximation, G(E) involves the evaluation of the following integral (Ref. 2): G(E) = (2/ħ) ∫_{r0}^{rctp} {2µ [UC(r) - E]}^(1/2) dr (3) Here, UC(r) is the Coulomb potential energy (or the Coulomb barrier), UC(r) = Z1e Z2e / r of a projectile nucleus with charge Z1e in the Coulomb field Z2e/r of the target nucleus; µ = m1m2/(m1 + m2) is the reduced mass of projectile and target nuclei; r0 = (R1 + R2) is the classical distance of closest approach with nuclei radii R1 and R2; and rctp is the classical turning point, determined from: E = UC(rctp) → rctp = Z1e Z2e / E (4). Evaluation of the integral gives the standard expression for the Gamow factor: GC(E) = (EG/E)^(1/2) (2/π) [ arccos((E/VC)^(1/2)) - (E/VC)^(1/2) (1 - E/VC)^(1/2) ] (5) where VC = Z1e Z2e / r0 is the full height of the Coulomb barrier, EG = 2µc^2 (πα Z1 Z2)^2 is the Gamow energy, and α = e^2 / ħc. In the limit of (E/VC)^(1/2) << 1, the Gamow factor reduces to the Sommerfeld expression: GC,asymptotic(E) = (EG/E)^(1/2) [ 1 - (4/π) (E/VC)^(1/2) + ... ] → (EG/E)^(1/2) (6) 3.0 Nuclear Fusion with Electron Screening 3.1 Coulomb Barrier Screening by Lattice Electrons In experiments with deuteron beams and deuterated targets in insulators/semiconductors, relatively small enhancements were found for D(d, p)T due to screening by localized atomic shell electrons. In host metals, much larger enhancements occur due to dynamic screening by free-moving conduction electrons ('lattice screening'). Screening is approximated by a uniform negative shift -Ue of the Coulomb barrier: Ue = Z1e Z2e / λsc (7) Screened potential: UC,sc(r) = (Z1e Z2e / r) exp(-r/λsc) (8) For r0 ≤ r ≤ rctp << λsc, expanding the exponential gives UC,sc(r) ≈ UC(r) - Ue (10). This approximation is justified for E >> Ue (13). For E ≤ Ue, direct numerical evaluation of G(E) is required. The enhancement factor is defined as: f(E) = σexp(E) / σbare(E) (15) Under the lattice potential approximation: fUe(E) = [E / (E + Ue)] exp[GC(E) - GC(E + Ue)] (17) Asymptotic formula: fUe,asymptotic(E) = [E / (E + Ue)] exp[ (Ue / 2E) (EG/E)^(1/2) ] (18) Direct numerical evaluation: Gdirect(E) = (2/ħ) ∫_{r0}^{r*ctp} {2µ [UC,sc(r) - E]}^(1/2) dr (19) fdirect(E) = exp[GC(E) - Gdirect(E)] (21) Table I comparison for ErD3 (Ue = 347 eV): - E = (1/2)Ue: fdirect = 1.09×10^12, fUe = 2.36×10^13, fUe,asymptotic = 1.9×10^32 - E = Ue: fdirect = 9.89×10^5, fUe = 3.09×10^6, fUe,asymptotic = 1.96×10^11 - E = 2Ue: fdirect = 539, fUe = 676, fUe,asymptotic = 8286 - E = 3Ue: fdirect = 45.7, fUe = 46.4, fUe,asymptotic = 127 3.2 Coulomb Barrier Screening by Plasma Particles In dense non-equilibrium two-temperature plasma channels produced by ionizing radiation, hot electrons (Te >> Ti) cannot form bound states. The Debye length is determined by ion Debye length λDi: λD ≈ λDi = [kTi / (4π ni0 e^2)]^(1/2). For ErD3 at room temperature, λD = 4.15×10^-10 cm, yielding Ue = 347 eV. 3.3 Coulomb Barrier Screening by Conduction Electrons in Metal Lattice Due to quantum degeneracy, conduction electrons form a Fermi gas. In a classical one-component model, screening length λDe,c gives Ue = 347 eV for ErD3. 3.4 Screening by Atomic Shell Electrons Screening by bound shell electrons is modeled using the Thomas-Fermi model with screening length λTF = 1.4 a0 / Z^(1/3). 4.0 General Screening Case For Reacting Hydrogen Isotope Nuclei When combining screening mechanisms (shell electrons + conduction electrons, or shell electrons + plasma electrons), the total screening potential is Ue,sc = e^2 / λsc (57), where: 1/λsc^2 = 1/λTF^2 + 1/λDe,c^2 or 1/λsc^2 = 1/λTF^2 + 1/λD^2 (58). 5.0 Coulomb Scattering on Target Nuclei 5.1 Light Particles Elastic Coulomb Scattering (e–, e+) Relativistic electron scattering differential cross section is given by Mott-like scattering with atomic screening parameter θmin = ħ / (pe λTF). For 2 MeV electrons on deuteron targets, the total cross section σe-d is 38.41 kb, but the mean deuteron recoil energy transferred per collision is only Ed = 24.75 meV. 5.2 Heavy Particle Elastic Coulomb Scattering (p, d, α) For heavy charged particles (protons, deuterons, alphas), Rutherford-type scattering dominates. Total cross sections are large (e.g., σp-D = 5.76 Mb for 3 MeV protons, σd-D = 11.51 Mb for 3 MeV deuterons, σα-D = 91.48 Mb for 3 MeV alphas), but mean recoil energies remain very small (41.4 meV to 44 meV) because forward small-angle scattering heavily dominates. Conduction electron screening significantly increases the probability of large-angle back-scattering. 5.3 Compton Scattering on Free Deuteron Calculated using the Klein-Nishina cross section. For 2 MeV gamma photons, the Compton cross section on deuterons is σC = 49.43 nb, transferring a mean recoil energy of ED = 2.13 keV. Photons transfer kinetic energy much more efficiently per collision than charged particles. 6.0 Neutron Elastic Scattering on Deuteron Nuclei Below the deuteron disintegration threshold (Kn_th = 3.4 MeV), neutron scattering on deuterons is purely elastic and isotropic in the CM frame (σsc(θCM) = σsc / 4π). Kinematic analysis shows that the energy distribution of scattered neutrons is uniformly distributed over the interval [αn Kn, Kn], where αn = [(md - mn)/(md + mn)]^2. 7.0 Neutron Energy Loss in Elastic Collisions With Deuteron Nuclei The average kinetic energy transferred from a neutron to a deuteron in an elastic collision is: K'd = (1/2)(1 - αn) Kn = [2 mn md / (mn + md)^2] Kn = (4/9) Kn (121) For 2.45 MeV neutrons (from D-D fusion), the total cross section is σsc ≈ 3 b, and the mean deuteron recoil energy is 1.09 MeV. Table II Summary (Mean Deuteron Recoil Energies): - Light particles (e-, e+) Ee = 2 MeV: σ = 38.41 kb, Recoil Energy = 24.75 meV - Heavy particles Ep = 3 MeV: σ = 5.76 Mb, Recoil Energy = 41.4 meV - Heavy particles Ed = 3 MeV: σ = 11.51 Mb, Recoil Energy = 42.7 meV - Heavy particles Eα = 3 MeV: σ = 91.48 Mb, Recoil Energy = 44 meV - Compton γ Eγ = 2 MeV: σ = 49.43 nb, Recoil Energy = 2.13 keV - Neutron n En = 2.45 MeV: σ = 3 b, Recoil Energy = 1.09 MeV Thus, energetic neutrons are orders of magnitude more effective in transferring kinetic energy to fuel nuclei to initiate fusion reactions. 8.0 Summary of Results This study indicates the crucial role of electron screening on the overall efficiency of nuclear fusion events between charged particles. Neutrons are far more efficient than energetic charged particles in transferring kinetic energy to initiate fusion. 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