TUTORIAL 4. THE FINE STRUCTURE OF HYDROGEN
With the field-free PTV constructed and its internal potential established, the fine structure energy levels of hydrogen can be derived. The calculation proceeds in two stages: first the energy levels of a free PTV are obtained; then the electron is placed in the field of a proton PTV. This has the effect of redistributing electron energy from its Sp-3 rotational mode into translation motion towards the proton. When exactly half has been displaced it is radiated away and a bound state in formed. In contrast to the Sommerfeld-Dirac theory, the Coulomb potential is not forming the energy levels but modifying them.
4.1 The fine structure formula for a free electron
Sommerfeld's original fine structure derivation quantised the radial momentum of an electron in an external Coulomb potential. The PTV derivation uses the same radial action integral, but applies it to the self-potential of the unloaded ring rather than an external field. A crucial innovation is string density N3. Rather than follow Sommerfeld in allowing the Sp-3 radius to depend on nΦ², the model makes the radius r3o an invariant and lets the string density N3 vary as nΦ². As pointed out in Figure 5, Tutorial 3, the string density is a separate spatial division to the n wavelengths and n interlaced components in each of the nΦ² components.
The action integral over the radial momentum yields a relativistic mass formula with a positive half power.
PTV Relativistic Mass Formula
The radial action integral yields the relativistic mass
m = mo √(1 + α² /Y² )
where Y = nr + √(nΦ² - α²), nr and nΦ are the radial and azimuthal quantum numbers. The positive half power contrasts with the negative half power that emerges from Sommerfeld's theory. This sign difference is not an error but is precisely what the model requires when the proton field is introduced, see Eq. (39) in Paper B.
4.2 The Bound State Condition
When an electron PTV and a proton PTV approach on a common toroidal axis, the proton's Sp-3 electric momentum field penetrates the electron's Sp-3 rotation and, being opposite rotations, displaces Sp-3 rotational energy into translational energy of motion towards the proton. A bound state forms at a specific separation distance d2bar determined by the following condition, see Eq. (86) and Table 2 in Paper B.
Bound State Condition
The bound state is reached when the energy of the proton's electric momentum field inhabiting one electron string — n interlaced components in one wavelength, see Figure 5, Tutorial 3 — equals exactly half the energy of that string. At this separation d2bar , the displaced energy is radiated away, the electron comes to rest in the proton's frame, and a stable bound state is established. Numerical solution gives d2bar ~ 2n for each quantum state (nr , nΦ),
At the bound-state separation, the helical trajectory of the poloidal axis (Sp-2 axis) makes an angle π/4 with the shortest distance r2f' joining the proton and electron Sp-2 circuits. This reduces the Sp-3 speed directed towards the proton by √2. Substituting this into the field-free energy formula — Eq. (49), Paper B — produces the fine structure formula Eq. (51). An expansion to fourth order in α reproduces the fine structure formula exactly. Table 1 in Tutorial 1 shows that the error against the Sommerfeld-Dirac scheme (with no reduced mass) is less than 2 parts in 10 billion.
Figure 6 Proton PTV (left) and electron PTV (right) on a common axis. The proton electric momentum field p3f' opposes the electron Sp-3 momentum p3. As the electron Sp-2 circuit B approaches the proton A along r2f', rotational energy is redistributed into translational energy. At the bound-state distance x = d2bar, the translational energy is emitted and the electron PTV comes to rest in the proton's frame.
4.3 Reduced Mass
The calculation for reduced mass is conducted entirely in the electron Sp-3 ring. The idea is to obtain the speed of the proton Sp-3 field, find the electron Sp-3 speed, and then take the electron Sp-3 speed relative to the proton field as if it were stationary. Here, we use the assumption from Figure 5, Tutorial 3, that there are n wavelengths each with n interlaced components. When a wavelength absorbs field momentum it is divided among the n interlacings reducing its effective strength by 1/n. Also, the field strength is proportional to 1/r2f' ~ 1/(2n). These parts conspire to produce the reduced mass result in Eq. (92), Paper B.
Sp-2 Structure — the n wavelengths each with n interlacings
In Figure 5, Tutorial 3, the Sp-2 structure was provided with n wavelengths and n interlacings for each of the nΦ² strings in the N3 density. This choice has three main advantages.
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The first of n wavelengths absorbs the Sp-3 field momentum, which taken over n time periods reduces the average field momentum absorbed by a factor 1/n. This justifies the reduction factor n in the denominator of the Coulomb law, Eqs (81)-(82) in Paper B. Without this n, we would no have the bound-state distance d2bar ~ 2n.
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The reduced-mass derivation requires both of these points: the field effect is reduce by 1/n due to the n wavelengths, and the bound-state distance d2bar ~ 2n.
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In the hyperfine calculation — Eq. (35) in Paper C — the result depends on there being n² Sp-2 circuits receptive to the changing proton Sp-2 field. It also depends on d2bar ~ 2n. This is explored further in Tutorial 5.
An interesting relationship emerges that connects the fine-structure constant α to the ratio of Sp-2 and Sp-3 radii. It turns out that r2o/r3o = α/√(1 - α²).
Paper A Paper B Paper C Paper D Paper E
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).
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.
Tutorial 5
Tutorial 3
Paper E: Barry R. Clarke, A heuristic model of the Bose-Einstein distribution with distinguishable photons