Introduction to The Vortex Atom
2. The Fundamental Unit
The invariant
The program identifies the fundamental irreducible element as a finite string that follows a helical trajectory where linear translation and intrinsic rotation are fluidly interchangeable.
Mechanical refraction
This provides a purely mechanical basis for the refractive index n. In dispersive media, linear action is redistributed into rotational action (Sp-1) allowing the photon to slow down while maintaining frequency and action conservation.
Figure 1. A finite string that follows a helical path with a rake angle π/4 at speed √2c on a notional tube. This produces a Sp-1 rotation at speed c and a linear motion at speed c.
See also, Barry R. Clarke, 'Reinterpretation of the Grangier experiment using a multiple-triggering single-photon model'. Modern Physics Letters B, 37(15), 2023.
3. Particle Morphology and Momentum Fields
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Sp-1 (String): Optical SAM. The intrinsic rotation of the string as it takes a helical path.
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Sp-2 (Poloidal): Optical OAM. The rotation of the Sp-1 axis around a notional tube. This defines rest mass and is constrained to an angular momentum ħ. This rotation generates a magnetic momentum field.
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Sp-3: (Toroidal): Rotation of the notional tube axis around the toroidal centre. The electron or proton is the Sp-3 structure. This rotation generates the electric momentum field (charge) and (when aligned with a conductor) the macroscopic magnetic field surrounding the wire. The electron is thus a stationary resonant state of toroidal momentum rather than a point-particle in orbit (for example, in a hydrogen atom).
Matter is modelled as a captured helical string closed into a double-loop vortex. The structure possesses three nested degrees of freedom.
Figure 2. (a) Sp-3 toroidal ring with Sp-2 tube passing around it. The momentum fields p are shown. (b) Sp-1 axis at speed c rotating around notional Sp-2 tube.
Barry R. Clarke, The Quantum Puzzle: Critique of Quantum Theory and Electrodynamics, World Scientific Publishing, 2017.
See also, Barry R. Clarke, A photonic toroidal vortex model of the hydrogen atom fine structure, Quantum Studies Mathematics and Foundations, 12, 19 (2025).
4. Fundamental Departures from Maxwell
It should be kept in mind that Maxwell's equations are only macroscopic laws of association. For example, when a current of a certain magnitude flows in a conductor, it is associated through an equation with a magnetic field of a certain strength circulating around it. Maxwell's equations provide no mechanism for what might be happening on a microscopic level. The PTV substructure renders several Maxwellian devices obsolete.
Intrinsic magnetism
Although the PTV model generates a magnetic momentum field from Sp-2 rotation, the observed magnetic field around a current-carrying conductor arises from the electron's Sp-3 field (electric momentum flow). The field does not require the motion of charge to exist; it only requires common toroidal alignment.
Momentum flow v rotation
Magnetic field lines represent a flow of momentum. There is no intrinsic rotational momentum (curl) around a magnetic field line.
Obsolescence of displacement current
Maxwell's ∂D/∂t represented fictitious ether charges and its only utility was to obtain a wave equation. In PTV theory, the rotating light-string provides the wave-nature inherently.
Local field generation
Induction is an internal redistribution of action between Sp-2 and Sp-3 to preserve the Sp-2 action h. A changing magnetic momentum field around a target Sp-2 circuit results in a decrease or increase in Sp-3 (electric) momentum. No electric field is generated ex nihilo in the vacuum
See also, Barry R. Clarke, The Lorentz force and the nature of charge from a photonic toroidal vortex model, https://www.preprints.org/manuscript/202603.2233
Geometrical interpretation of the hydrogen atom hyperfine structure
For a preprint version of
the paper click below
Barry R. Clarke, The Vortex Atom: A New Paradigm, World Scientific Publishing, 2021.
RUN THE DATA PROGRAM
Download the Python program (right), copy the code, click on the button 'COMPILER', paste the code into the compiler (CTRL + V), and click on Run. Alternatively, check the data output directly.
Python code
data output
Paper A: Barry R. Clarke, Reinterpretation of the Grangier experiment using a multiple-triggering single-photon model, Modern Physics Letters B , 15, 2350042 (2023).
Paper B: Barry R. Clarke, A photonic toroidal vortex model of the hydrogen atom fine structure, Quantum Studies Mathematics and Foundations, 12, 19 (2025).
The Photonic Toroidal Vortex (PTV) Program
A deterministic substructure for electrodynamics
Barry R. Clarke - Quantum Studies: Mathematics and Foundations12: 19 (2025)
The six tutorials
Tutorial 2: The building blocks
Tutorial 3: The self-potential
Tutorial 4: The fine structure of hydrogen
Construction
The Photonic toroidal vortex model is based on a string that has been given three rotational modes.
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Sp-1. Optical spin angular momentum
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Sp-2. Optical orbital angular momentum
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Sp-3. Toroidal rotation
In Figure (a), a string is wound round a guide tube B and given speed along a helical trajectory of √2c at rake π/4. Both its rotational speed (Sp-1) and its translational speed are c. This is usually called circular polarisation. If we now give the axis of B rotation around another guide tube A we obtain Sp-2 rotation. This is known as optical OAM. Here, the tube has radius r2o. Finally, the tube A is given rotation around a toroidal axis T, see Figure (b). The toroid has radius r3o. This Sp-3 rotation has momentum p3 while the Sp-2 rotation has momentum p2.
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Sp-1 and Sp-2 rotational action h tends to preserve its invariance.
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A changing electric or magnetic field cutting a Sp-2 or Sp-3 circuit increases or diminishes the circuit's rotational action.
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If an invariant rotational action changes, it is compensated by an exchange with translational action.
What's new?
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Definition of mass. Mass is defined in terms of the Sp-2 radius.
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Nature of charge. There are four charge modes — two positive and two negative — resulting from the four possible chiralities of the Sp-2 and Sp-3 rotations.
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The momentum fields. The Sp-2 and Sp-3 rotations of the trapped photon results in spiral momentum fields. The Sp-2 rotation generates a magnetic momentum field and the Sp-3 rotation produces an electric momentum field. The Sp-2 magnetic momentum in the field is ħ/r2f, where r2f is the distance from the Sp-2 centre.
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Quantisation without external potential. Experiments on electron vortices confirm that quantised angular momentum states can exist in the absence of an external potential. In PTV theory, instead of working with an external potential, quantised energy levels can be obtained with an internal potential defined in terms of the toroidal rotation. An external potential simply modifies these states, it is not required for their existence.
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The hyperfine midpoint. Instead of the hyperfine centroid, the PTV model aims for the hyperfine midpoint frequency which allows geometrical symmetry in the hyperfine shift.
Results
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The Sommerfeld-Dirac fine structure energies can be reproduced without an external potential to 2 parts in 10¹ᵒ.
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The hyperfine midpoint can be up to 450 times greater than the hyperfine centroid. Nevertheless, the PTV model obtains less than 0.1 MHz errors across all states.
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Taken over 30 states, the hyperfine shift — which is equal in magnitude each side of the midpoint — has a mean absolute error of 0.0082 MHz against the QED value of 0.0003 MHz.
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The deviation of the hyperfine frequency from experiment is less than 1 part in 10¹ᵒ across all states.
The calculations
There is a single equation from which all hyperfine states can be calculated using four parameters.
νfull = ν3D + Δνshift + Δνhf
where ν3D is the fine structure formula with a proton-speed adjustment, Δνshift is the magnetic potential shift, and Δνhf is the hyperfine shift. The Δν3D includes a parameter Γ, Δνshift includes parameters A and B, and Δνhf includes a parameter k. The calculation can be performed on an electronic calculator and is the same across all states. For example, see the complete calculation for 2S1/2 in Appendix B, Paper C. The Python code (see below) also shows the detailed calculations.
Parameter constraints
(1) For the hyperfine shift, the parameter k = 1 for the nS1/2 states and represents no adjustment. For all other sets of states, Sp-2 eccentricity can account for the k values.
(2) The parameter B is a power in the magnetic potential shift and B = –3.000 across all states, not only for hydrogen but also deuterium and tritium. This suggests a dipole-dipole interaction law.
(3) The A and Γ parameters, which are associated with a cumulative magnetic interaction, are fixed for each set of states (e.g. nP3/2) and, despite their variation across these sets, and across different elements, they obey a universal scaling law:
A/Γ = –28,651,713/(AnZn)
where An is the nuclear number and Zn is the nuclear charge. This holds for hydrogen, deuterium, tritium, 3He+ and 7Li2+.
What Makes This Unusual
Most alternative models of the hydrogen atom achieve basic agreement - say, the Bohr levels - while failing at finer detail. The PTV model tightens as more phenomena are added: the same building block (a photonic helical string Sp-1) and the same three-tier rotation hierarchy generate mass, charge, the fine structure, and the hyperfine structure without any new assumptions at each step. The internal consistency of the framework is itself part of the argument for taking it seriously.
Main page
Tutorial 1
Paper A Paper B Paper C Paper D Paper E
AI BRIEFING DOCUMENT
The quickest way to appreciate the internal consistency of the model is to use Gemini or Claude (Gemini allows more uploads). Papers B, C, and D can be uploaded and an assessment requested. However, AI is inclined not to recognise the full architecture unless guided towards it. Of course, if this is inconsistent, AI will then point it out. It is a matter of focus. For this reason, an AI briefing document is recommended to be uploaded with these papers and this can be downloaded from the link to the right.
AI BRIEFING
Paper C: Barry R. Clarke, Geometrical interpretation of the hydrogen atom hyperfine structure, under peer review.
Paper D: Barry R. Clarke, The Lorentz force and the nature of charge from a Photonic Toroidal Vortex Model, under peer review.
Paper E: Barry R. Clarke, A heuristic model of the Bose-Einstein distribution with distinguishable photons