The Casimir force at nuclear separations

Why the library suggests it

Before the argument, the scale. The Casimir force is measured across micrometres; this card is about a gap nine orders of magnitude smaller, and the interesting part is both how steeply the force grows on the way down and what stops being true when it arrives.

The descent to a femtometre

Everyone measures the Casimir force across micrometres. The two papers behind this card follow it down nine decades further, to the femtometre gap between two nucleons, and the first thing to understand is how steeply it grows on the way — and the second is what stops being true when it gets there. Move the separation and watch both.

What the ladder counts

The interaction energy, which grows as the inverse cube of the separationThe force, which grows as the inverse fourth power

The dial

Separation

1E-15 m

Back to the nucleon separation

• The interaction energy, which grows as the inverse cube of the separation
• Measured here

What the ideal law returns here

Interaction energy:

5.44E0 MeV

Force:

2.61E3 N

Orders climbed since a micrometre:

27

Balance temperature:

1.14E12 K

Tabulated temperature:

8.7E11 K

Electron and positron pair density:

2E43 m⁻³

Inside the band the two papers compute. Here the gap is no longer empty: the zero-point energy in it corresponds to a temperature around ten to the eleventh kelvin, which is enough to fill it with electron and positron pairs, and those pairs screen the interaction into the shape nuclear physics writes as the Yukawa potential. The plain inverse-cube law on the ladder is the unscreened starting point, not the answer the papers end with.

Separation: 1E-15 metres. Interaction energy: 5.44E0 MeV. Orders climbed since a micrometre: 27. Inside the band the two papers compute. Here the gap is no longer empty: the zero-point energy in it corresponds to a temperature around ten to the eleventh kelvin, which is enough to fill it with electron and positron pairs, and those pairs screen the interaction into the shape nuclear physics writes as the Yukawa potential. The plain inverse-cube law on the ladder is the unscreened starting point, not the answer the papers end with.

Orders of magnitude climbed since one micrometre

• Measured here (1E-6 m) — Across micrometres the Casimir force is an ordinary laboratory effect, measured many times over. Everything to the right of this mark is calculation.
• Top of the modelled band (3E-15 m) — Three femtometres is the far end of the band the papers tabulate, and their temperature there — about 2.9 times ten to the eleventh kelvin — is already well past the point at which the gap fills with electrons and positrons.
• Casimir equals Coulomb (2.6E-15 m) — Setting the Casimir energy between two protons equal to their Coulomb energy gives a cubic whose solution is 2.6 femtometres. That is the paper's own evidence that the vacuum force is of nuclear size at nuclear size.
• The nucleon separation (1E-15 m) — One femtometre is the surface separation the model takes between two nucleons of proton radius 0.8 femtometres. The ideal law evaluated there returns a binding energy of around 5 million electronvolts, inside the measured range of 1.1 to 8.8 million electronvolts per nucleon.

What stops being true on the way down. An empty gap is the assumption that fails first. The same free energy carries a black-body term, and setting it equal to the attractive zero-point term fixes a temperature that rises as the gap closes — around a hundred billion kelvin at these separations. At that temperature the gap fills with electron and positron pairs, and from there the papers compute a screened interaction whose exponential decay has the same form as the Yukawa potential. This instrument marks where the unscreened law stops being the right picture; it does not draw the screened one, because the screening length depends on the plasma's magnetic permeability and on any applied field, and that is the open calculation the card is about.

What the idealisation assumes, and it assumes a lot: two perfectly conducting flat plates of nucleon cross-section, at zero temperature, with an empty gap between them. Real nucleons are not flat plates, and the authors are explicit that these are simple estimates offered as a possible complementary contribution, not a replacement for established nuclear theory.

At the nucleon separation the review splits its result: minus 0.9 million electronvolts from the zero-frequency term and minus 3.6 million electronvolts from the higher terms, totalling 4.5 million electronvolts per nucleon, against measured binding energies per nucleon running from 1.1 in deuterium to 8.8 in nickel-62.

The forward marker the same identification produces: matching the plasma frequency to the pion rest energy gives a meson mass of 267 electron masses against the experimentally reported 264, and the companion paper carries the magnetic permeability through the same step to see how the screening length moves.

The calculations are in the literature and peer-reviewed. What has not been looked for is the field dependence they predict, and the run that would look for it is a conventional low-energy nuclear physics experiment.

The plate law, the balance temperature and the binding split come from this sheet

The tabulated temperatures, the pair density and the equilibrium separation come from this sheet

Suman Panja, Luiza Inacio, Subhojit Pal and Mathias Boström add the plasma's magnetic permeability and any applied magnetic field to the calculation, and both change the screening length, with the resulting energies landing in the range of measured nuclear binding energies (Casimir forces across magnetic plasmas at nuclear separations, 2025). The companion review works Barry Ninham's original proposal through: two nucleons modelled as proton-sized conducting plates a femtometre apart, the gap filled with electron-positron pairs, returning a binding energy of about 4.5 million electronvolts per nucleon and a meson mass of 267 electron masses against a measured 264 (High-Temperature Plasma in Casimir Physics, 2026). Read those next to the peer-reviewed proposal that conditioning the form of an electromagnetic field, rather than raising its strength, changes the effective barrier two nuclei face (Specially Conditioned EM Fields to Reduce Nuclear Fusion Input Energy Needs, 2012).

The experiment or build

The theoretical step is cheap and is the right first move: the model says an applied magnetic field changes the screening length, so compute the predicted change in a nuclear reaction rate as a function of applied field, for a reaction whose low-energy cross-section is already well measured. Then run it — a low-energy deuteron beam on a target inside a high-field magnet, with the field swept and everything else held. The settling measurement is the fusion reaction rate at fixed beam energy and fixed target density, plotted against applied magnetic field, against a field-free control run in the same session. A rate that moves with the field, and returns when the field is removed, is the cleanest possible version of the vacuum-catalysed fusion claim. Note the discipline the coiner of that term himself applies: the hypothesis is an increase in tunnelling rate, not a change in the bare Coulomb law, and the measurement above tests exactly that.