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The optimal injection energy for beam-driven proton–boron fusion lies above the 675 keV resonance

This paper demonstrates that for beam-driven proton–boron fusion, injecting protons at an optimal energy of 0.85 MeV—slightly above the 675 keV resonance but well below the 3.7 MeV rate coefficient peak—increases the engineering gain per unit of injected energy by 25% without requiring new physics or confinement schemes.

Original authors: Oleg Agamalov

Published 2026-10-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Oleg Agamalov

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to harness the power of the stars, scientists have long looked to a specific recipe involving hydrogen and boron. Unlike the more common fusion fuel mix of hydrogen isotopes, this combination offers a distinct advantage: it produces almost no neutrons, which means less radiation damage to the surrounding machinery and a cleaner path to electricity. However, making this reaction happen is notoriously difficult. It requires heating the fuel to temperatures far higher than those needed for other fusion types, and until recently, the energy required to create those conditions seemed to outweigh the energy the reaction could ever produce. The prevailing strategy to solve this has been to focus on a specific energy level where the atoms are most likely to collide and fuse, a point known as a resonance.

A new analysis by independent researcher Oleg Agamalov challenges the long-held assumption that aiming for this peak collision point is the most efficient way to run a machine. While the probability of two atoms fusing is indeed highest at a specific energy, the overall efficiency of a machine depends on more than just that single moment of impact. It depends on the entire journey of the fuel particles as they slow down and lose energy. Agamalov's work suggests that by slightly adjusting the starting speed of the fuel, engineers can extract significantly more energy from the same amount of input power, without needing to invent new physics or build entirely new types of reactors.

For decades, researchers designing machines to drive proton-boron fusion have targeted a specific energy level of 675 kiloelectronvolts. This number corresponds to a sharp peak in the likelihood of a proton hitting a boron nucleus and splitting it into three helium particles. The logic was straightforward: if the reaction happens most often at this energy, then injecting the fuel at exactly this speed should yield the best results. This approach has become the standard for every non-thermal fusion scheme currently being pursued, with the belief that placing the ions at this resonance maximizes the power balance.

However, Agamalov points out that this view misses a crucial detail about how these machines actually work. In a driven device, the fuel is not sitting still in a hot soup; it is a stream of fast particles injected into a target. Once injected, these particles do not fuse instantly and stop. Instead, they travel through the fuel, colliding with electrons and gradually losing speed, a process known as slowing down. During this entire journey, from the moment they are injected until they come to a halt, they are constantly trying to fuse. The total energy produced is not just about the moment of injection, but the sum of all the fusion attempts made while the particle is slowing down.

When you calculate the total energy gained by the machine relative to the energy spent to create the beam, the math changes. The analysis shows that the most efficient starting point is not the 675 kiloelectronvolt resonance, but a slightly higher energy of 0.85 megaelectronvolts. At this higher speed, the particles spend a bit more time in the energy range where fusion is still quite effective, and the cost of accelerating them to this higher speed is outweighed by the extra fusion reactions they trigger as they slow down. The study calculates that shifting the injection energy from the traditional 675 kiloelectronvolts to 0.85 megaelectronvolts would increase the energy gain of the system by 25 percent.

This finding is significant because it does not require a breakthrough in materials, a new way to confine the plasma, or a change in the magnetic fields used to hold the fuel. It is a simple adjustment to the voltage of the injector. The paper explicitly rules out the idea that one should aim for the absolute highest possible energy where the fusion rate is theoretically fastest, which occurs near 3.7 megaelectronvolts. While the reaction rate is indeed highest there, the energy cost to get the particles to that speed is so immense, and the time they spend in less productive energy ranges is so long, that the overall efficiency of the machine drops by 40 percent compared to the 0.85 megaelectronvolt optimum.

The research relies on integrating the known behavior of how protons lose energy in an electron gas with the measured probabilities of fusion at different speeds. By running these calculations, the author demonstrates that the "sweet spot" for engineering gain is a plateau rather than a sharp peak, sitting comfortably between the traditional resonance and the theoretical maximum. The result is a clear, actionable recommendation for anyone building a proton-boron fusion device: turn up the voltage slightly. This adjustment offers a substantial, immediately accessible improvement in performance, turning a marginal power balance into a more viable one simply by changing the injection energy.

Technical Summary: Optimal Injection Energy for Beam-Driven Proton–Boron Fusion

Problem Statement
Current non-thermal schemes for proton–boron-11 (p–¹¹B) fusion universally target the 675 keV laboratory energy, where the nuclear cross-section ( $\sigma$ ) for the $p + ^{11}\text{B} \to 3\alpha + 8.7 \text{ MeV}$ reaction exhibits a pronounced resonance. The prevailing assumption is that concentrating the ion distribution at this resonance maximizes fusion performance. However, this approach overlooks the fact that the fusion rate depends on the rate coefficient, $\langle \sigma v \rangle$ , rather than the cross-section alone. Because the relative velocity ( $v$ ) increases with the square root of energy, the global maximum of $\langle \sigma v \rangle$ occurs at approximately 3.7 MeV. While 3.7 MeV is the kinematic ceiling for ignition, it is practically inaccessible for beam-driven devices due to the immense drive power required. Consequently, the field lacks an optimized injection energy that balances the high-energy cost of injection against the integrated fusion yield over the ion's slowing-down history.

Methodology
The paper evaluates the engineering gain ( $Q$ ) for a beam-driven device, defined as the total fusion energy produced per unit of energy invested in the injected particle. Rather than focusing on the instantaneous rate at the injection energy, the analysis integrates the reaction rate over the entire slowing-down history of the fast ion.

The methodology employs the following analytical framework:

Slowing-Down Dynamics: In the regime where the fast proton velocity is much smaller than the electron thermal velocity ( $v_f \ll v_{te}$ ), the energy loss rate is governed by electron drag, described by the Spitzer slowing-down time ( $\tau_{se}$ ).
Gain Formulation: The gain per unit injected energy is proportional to the integral of the reaction rate coefficient weighted by the inverse of the injection energy ( $1/E_0$ ). This weighting accounts for two factors: the higher cost of creating high-energy ions and the time spent in energy regions where the cross-section is low (specifically the "valley" between 1 and 3 MeV).
Cross-Section Data: The calculation utilizes an analytic fit by Tentori and Belloni for center-of-mass energies above 0.668 MeV and a low-energy parameterization by Becker et al. below that threshold, joined by a linear bridge.
Sensitivity Analysis: The study examines the impact of the Coulomb logarithm ( $\ln \Lambda$ ) drifting across the slowing-down range to ensure the robustness of the optimum.

Key Contributions and Results
The analysis yields a specific, non-intuitive optimum for beam-driven systems:

Optimal Injection Energy: The engineering gain is maximized at an injection energy of 0.85 MeV. This value is slightly above the 675 keV resonance but significantly below the 3.7 MeV theoretical maximum of $\langle \sigma v \rangle$ .
Performance Gain: Injecting protons at 0.85 MeV rather than at the widely targeted 675 keV resonance increases the beam-driven gain per unit energy invested by 25% (normalized gain of 1.00 vs. 0.80).
Plateau Effect: The optimum is not a sharp peak but a plateau. The gain remains within 3% of its maximum for injection energies between 0.77 and 0.96 MeV.
Inefficiency of High Energies: Targeting the 3.6–3.7 MeV range is shown to be highly inefficient for driven devices. Despite a higher instantaneous rate coefficient, the massive initial energy investment and the long transit time through the low-cross-section region result in a gain that is 40% worse than the 0.85 MeV optimum.

Significance and Claims
The paper asserts that this finding represents a substantial, practically accessible improvement for the development of proton–boron fusion as a driven system. The significance of the result lies in its simplicity and immediate applicability:

No New Physics or Hardware: Achieving this 25% gain requires no new physics, no new confinement schemes, and no changes to magnetic configurations.
Implementation: The improvement is achieved solely by increasing the injector voltage from the standard 675 keV to 0.85 MeV.
Paradigm Shift: The work challenges the field's uniform focus on the 675 keV resonance, demonstrating that the "true" figure of merit for a beam-driven device is the integrated yield over the slowing-down history, not the peak cross-section.

In conclusion, the paper posits that shifting the injection energy to 0.85 MeV is an immediate, low-risk optimization that enhances the efficiency of any existing or planned beam-driven p–¹¹B system.

Technical Summary: Optimal Injection Energy for Beam-Driven Proton–Boron Fusion

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