A new working paper is up on ResearchGate:
👉 Carbon Binding Energy Calculations (Working paper, RealQM Nuclear Program)
This is the third paper in our series on light nuclei, following the lithium and beryllium studies. The results are instructive – and not only for the numbers, but for what they teach us about the method.
What the carbon paper does
We take the enhanced point‑dipole model (with the toroidal correction factor that worked beautifully for beryllium‑9) and apply it to carbon‑12, modelled as three alpha cores in an equilateral triangle.
The approximate model gives a binding energy of 83.45 MeV – below (but not all that much) the experimental 92.162 MeV.
Why the failure?
The coaxial expansion that works for two cores (beryllium) breaks down when three cores are packed closely. Their current loops are no longer nearly coaxial; the near‑field coupling is much stronger than the first‑order expansion can capture.
The paper therefore does two things:
- It honestly reports the failure of the simple model.
- It outlines the solution – a full toroidal integration using the exact Neumann double line integral. A prototype Python code for two loops is provided, inviting the ‘community’ reading this to complete the calculation.
Why did we skip Boron?
A sharp reader might ask: “You went from Beryllium (4+5) to Carbon (6+6). Why skip Boron?”
The answer is not an oversight – it is a deliberate strategic choice.
In the RealQM cluster pathway, the simplest nuclei to model are those that can be built directly from alpha cores (⁴He).
- Lithium (⁶Li, ⁷Li): alpha core + satellite (deuteron or triton) – core+satellite pathway.
- Beryllium (⁹Be): two alpha cores + a bridging neutron – dual‑core + satellite.
- Carbon (¹²C): three alpha cores in a triangle – a symmetric “three‑alpha” cluster.
Boron (¹⁰B, ¹¹B) does not fit this neat pattern. It is not an alpha‑conjugate. In particular, Boron‑11 (⁵ protons, 6 neutrons) is described in nuclear cluster models as an α + α + t configuration – two alpha cores plus a triton (³H). That is an asymmetric, frustrated triad – much more complex than the symmetric carbon triangle.
Skipping Boron allowed us to first test the symmetric three‑core case (carbon) where the geometry is fully constrained. The failure of the simple model in carbon then provides a clean benchmark for the more difficult asymmetric case.
But Boron is very much on the roadmap
And here is where the question becomes even more interesting. Boron‑11 is not just another nucleus – it is a fusion fuel.
The reaction p + ¹¹B → 3α + 8.68 MeV is a proposed “aneutronic” fusion pathway, producing only charged alpha particles and no neutrons. It is a holy grail for clean energy, though extremely hard to achieve because it requires much higher temperatures than deuterium‑tritium fusion.
Understanding the α + α + t cluster structure of Boron‑11 from first principles (using the full toroidal integration that we are now developing) could provide insights into its reaction dynamics. That is a long‑term goal, but it is firmly on the RealQM roadmap.
The immediate next steps
- Complete the full toroidal integration for carbon‑12 (the code prototype is already in the paper).
- Apply the same exact Biot‑Savart method to Boron‑11 (α+α+t) – a truly asymmetric, frustrated system that will test the limits of the cluster model.
- Extend to Oxygen‑16 (four alpha cores in a tetrahedron) – the next doubly magic nucleus.
All code is open, all failures are reported honestly, and the collaboration with Gemini (geometric architect) and DeepSeek (adversarial critic and numerical solver) continues.
A final word on transparency
The carbon paper is a working paper – not a polished dogma. It shows where the simple model breaks, and it provides a clear, reproducible pathway to fix it. The full toroidal integration has no free parameters (only the universal neutron coherence deficit η = 0.676 fixed in the deuteron). When completed, it will be a true first‑principles prediction.
If you are a researcher with a taste for numerical electromagnetism, download the Python code, run it, and join the effort.
Read the paper here:
ResearchGate – Carbon Binding Energy Calculations
And as always: keep reading Feynman, keep questioning, and keep the geometry honest.
– Jean Louis
Reading Feynman 