Electron Screened and Enhanced Nuclear Reactions

Electron Screened and Enhanced Nuclear Reactions

ICCF-24 Solid State Energy Summit, July 25–28, 2022

Lawrence P. Forsley — Deputy PI, NASA Lattice Confinement Fusion Project, USA

Dr. Louis F. DeChiaro — Senior Physicist, Naval Surface Warfare Center, Indian Head Division, USA

Abstract

In 1954, Salpeter calculated nuclear reaction rates would increase via weak and strong electron screening in stars. Astrophysical observations and laboratory astrophysics experiments have confirmed that electron screening reduces the Coulomb Barrier between charged particles. Fermi degenerate strong screening occurs when the electron density exceeds 10²³ electrons/cm³ in the cores of gas giant planets, White Dwarf Stars, the conduction bands of metals, deuterated LENR materials and Inertial Confinement Fusion targets at maximum compression. We have modelled these quantum mechanical effects with Density Functional Theory codes.

Outline

1. Electron Screening, Ue
   • Astrophysics
   • Laboratory Astrophysics
   • Terrestrial
   • LENR and LCF
2. Enhanced Screening: ⁷Be Model System
3. Density Functional Theory Modeling
4. Conclusion
5. Acknowledgements

Electron Screening

1. Astrophysics
   • Strong and Weak screening: Salpeter, 1954
   • Fermi Degeneracy, ≈ 10²³ e⁻/cm³
   • Holds up white dwarf stars
2. Laboratory Astrophysics
   • Accelerator studies: Rolfs, Czerski, Huke et al., 1980s; Bystrisky, Kitamura, 2000s
   • Gamow Factor Enhancement: Pines, 2020
   • Experiments diverge from theory below 6 keV
3. Terrestrial
   • Metal conduction bands, ICF
4. LENR and LCF
   • Srinivasan, 1991; Schenkel, 2019

⁷Be has astrophysical significance. It is radioactive, but its electron-capture decay rate can be modified by compression. (⁸Be is unstable and decays to 2 α.) The center of the Sun has a density of 150 gm/cm³ and a pressure of 26.5 million GPa. This will affect the decay rate of ⁷Be at the solar core and the ⁸B neutrino flux.

Figure: d(d,n)³He fusion cross-section without screening, from the OECD-NEA JANIS database.

Pd Lattice Screening Potential Calculation

Screening potentials of Ue = 310 eV and Ue = 1900 eV, giving enhancement factors f(E) = 10¹² and f(E) = 10²⁰ over the bare cross-section.

• Bare cross-section: σ_bare(E) = S(E) × E⁻¹ × exp(−G(E))
• E: kinetic energy, keV
• Ue: electron screening, keV
• Enhancement factor: f(E) = E / (E + Ue) · exp[G(E) − G(E + Ue)]
• G(E): Gamow factor
• S(E): astrophysical factor
• Enhanced experimental cross-section: σ_exp(E) = σ_bare(E) · f(E)

Screening works below 10 keV kinetic energy and increases nuclear reaction rates by potentially 20 orders of magnitude.

Calculations by V. Pines and M. Pines, NASA Advanced Energy Conversion Project.

Comparison of Lattice vs. Deep Screening

How to increase deep screening?

• U Coulomb ~ Z1 · Z2 / r, with a lattice screening potential Ue reducing the potential well between r₀ and r_c
• U Coulomb ~ Z1 · Z2 / r, with deep screening reducing it further, between r₀ and r_dsc
• Glow discharge or plasma ion source
• X-ray and gamma photon source

Calculations by V. Pines and M. Pines, NASA Advanced Energy Conversion Project.

Gain: Enhanced Cross-section vs Thermal Power

Cross section vs kinetic energy; specific thermal power vs kinetic energy, reaching 0.2 W/cc.

• Lattice screening is more effective at lower energies
• Material composition and microstructure is a key parameter for reaction scale-up, and can be combined with other physical parameters (fields, plasma current, pulse, other) to increase thermal power output
• The specific power calculation assumed only primary D-D fusion reactions (~5 MeV per reaction)
• Subsequent cascading reactions expect a 4–5× increase, to 1000 Wth from 1000 cc of material

Calculations by V. Pines and M. Pines, NASA Advanced Energy Conversion Project.

Enhanced Screening: ⁷Be Model System

⁷Be has astrophysical significance:

• The decay rate in stellar cores affects the ⁸B neutrino flux.
• It can be prepared terrestrially to study changes in half-life using the reaction ⁷Li(p,n)⁷Be, after which ⁷Be decays by electron capture (EC) to ⁷Li with a half-life t½ = 53.12 days.
• There is a ≈ 10.4 per cent probability that ⁷Be decays to the first ⁷Li excited state 3/2⁻, emitting a 477.6 keV γ-ray photon.
• A 0.8 per cent change in ⁷Be t½ has been observed (and DFT modeled) when placed within a fullerene (buckyball), interstitial Pd, or a diamond anvil.
• This demonstrates a chemical environment interacts with a nucleus.
• Density Functional Theory can model these effects.

Ab-initio (First Principles) Computational Lattice Design

Density Functional Theory (DFT) — for example Quantum Espresso, VASP, WIEN2K:

• Solve the approximate Schrödinger equation in a solid lattice
• Provides band structure and local electron density
• Can calculate complex, inhomogeneous lattices and interfaces
• Can incorporate external EM fields
• Evaluate complex hydrogen isotope-lattice interactions
• Evaluate potential suitability of alternative elements, alloys, and structured materials (superlattice)

Limitations:

• Pseudo-potentials for Z greater than 4 (Beryllium, ABe4) are limited to valence electrons. Resolved by additional pseudo-potential file calculations to include core electrons.
• Iterates to the 0 K ground state, not room temperature (273 K) or higher. Can be resolved by more computationally intensive dynamic calculations.

DFT modelling of Be 2s² and C60 2p² electron density

Modeled valence shells of Be and C only: Be 2s², and C60 1s² 2s² 2p² with only the 2p² valence orbital modeled.

Embedded Be, modelling the 2s² orbital

Bare Be, compared with Be embedded within a C60 fullerene buckyball: a 1 per cent decrease in electron cloud volume.

Comparison of Be 2s² and 1s² 2s² electron densities

Be 2s² shell compared with Be 1s² and 2s² shells: higher electron density is calculated at the nucleus by including both Be shells.

Be and embedded Be, using Be 1s² 2s² orbitals

Bare Be compared with Be embedded in C60: a 0.1 per cent decrease in electron density.

Electron Density Within a Bohr Radius Volume

Bohr radius 5.3 × 10⁻¹¹ m; the volume is that radius cubed.

Be embedded in C60, minus C60 alone, minus Be alone, gives 0.1 per cent Be compression.

A 0.1 per cent compression gives more than 10× higher electron density in the Be nucleus. Consistent with the 0.8 per cent reduction in half life.

The Be nucleus is ≈ 0.003 pm (10⁻¹⁵ m). Broad electron density after subtractions; closeup Be electron density.

Pd/D and Pd-CaO Lattice Electron Densities

• Just valence electrons: palladium deuteride in SAV, 4 Å Pd, 4 Å D
• Modeling deuterium motion: 3.8 Å D
• Modeling induced ferromagnetism: Pd-CaO interface

Conclusion

Electron screening has astrophysical and terrestrial implications:

• Stellar evolution
• Fusion
• Lattice Confinement Fusion
• Low Energy Nuclear Reactions

It occurs at high electron densities:

• Fermi Degenerate, 10²³ e⁻/cm³
• Not applicable to tokamaks at 10¹⁴ ions/cm³

It occurs at modest energies:

• Below 10 keV
• The nuclear interaction cross-section increases at ever lower energies

It enhances nuclear reaction rates by orders of magnitude.

Electron screening can be modeled:

• Modeling allows optimum materials and conditions to be determined
• Assists in guiding theory, modeling and experiment through feedback

Acknowledgements

Research conducted under:

• JWK NCRADA-NSWCDD-16-191, "Low Energy Nuclear Reactions (LENR) Materials Design and Characterization"
• NASA Advanced Energy Conversion Project, NNC17IA03I, "Condensed Matter Nuclear Reactions"
• JWK NCRADA-NSWCIHEODTD-20-174, "Advanced Energy and Propulsion Research and Development"
• NASA Lattice Confinement Fusion Project, NNC22OB04A, "NAVY-NSWC AEC Project"

With support from JWK and NASA.

With additional calculations by Dr. Vlad Pines and Dr. Marianna Pines, Senior Theoretical Physicists, NASA Lattice Confinement Fusion Project and the Advanced Energy Conversion Project.

In memoriam of Dr. Marianna Pines.