TUTORIAL 5. THE HYPERFINE STRUCTURE OF HYDROGEN
The hyperfine structure of hydrogen arises from the interaction of a source's magnetic momentum field (Sp-2) — proton or electron — and the target's Sp-2 circuit. However, the momentum in the field must be changing and this occurs due to the relative motion of the electron and proton. Since the electron Sp-2 circuit needs to retain its action invariance h, any change is compensated by an exchange of action with its Sp-3 circuit. It is essentially Faraday's law of electromagnetic induction on a microscopic level. Four parameters will now be introduced: A and B (magnetic potential); Γ (relative-to-proton speed); and k (hyperfine-shift multiplier). We shall see that the constraints on these parameters suggest some important relationships that argue against curve-fitting. We shall also see how they contribute to the full hyperfine calculation:
νfull = ν3D + Δνshift + Δνhf
5.1 A different target: the midpoint not the centroid
Conventional QED calculations reference the Lamb shift to the hyperfine centroid, the amplitude-weighted mean of the several hyperfine components. The PTV model instead starts with a modified fine structure frequency ν3D and aims for the arithmetic midpoint of the two main components. This choice demonstrates geometric simplicity.
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The PTV splitting is symmetric in construction, the same electromagnetic induction raises and lowers the electron frequency by equal and opposite amounts νhf about the midpoint rendering it a natural reference.
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For the ground state of hydrogen, the PTV Lamb shift is ~ 450 times greater than the traditional Lamb-shift frequency, providing a more demanding test of the correction framework.
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Targeting the midpoint reveals a striking regularity. The magnetic potential νshift that bridges the gap obtains a B ~ -3.000 power law across all angular momentum states.
Reaching the Hyperfine Midpoint
The fine structure formula — Eq. (51) in Paper B — is modified by a D-function (introducing the factor Γ ) that amounts to including the relative-to-proton speed of the electron. This is the result of the proton speed induced by the electron's magnetic momentum field, as represented in Eqs (12)-(13) from Paper C which is a calculation for the modified fine structure frequency ν3D that includes the relativistic reduced mass, see Figure 7 below. Added to this is the magnetic potential shift Δν (that includes the A and B parameters as well as the bound-state distance d2bar, the latter being calculated in Paper B) — see below and Eq. (22) in Paper C. This results from the changing proton Sp-2 field acting on the electron integrated from infinity to the bound-state location. The addition of these two terms targets the hyperfine midpoint. The value of B is optimized against the known data. Focusing on the lowest three states of a set (e.g. nP3/2), the factor Γ is varied until the change in B is constant, see Eq. (23) in Paper C. This determines B, and the parameter A is then selected to obtain the lowest hyperfine midpoint of the set.
Magnetic potential shift: Δνshift = A(d2bar)^B
shift
5.2 The role of gamma (Γ)
The central new parameter of the hyperfine theory — see Paper C — is Γ, an empirically determined multiplier that encodes the relative rotation senses of the proton and electron Sp-2 (poloidal) circuits. It enters the calculation through the D-function and captures a reaction in the proton that manifests in a speed change due to the electron's magnetic momentum field.
Figure 7 The adjustment to the fine structure equation to include the relativistic reduced mass Mr and the proton-speed adjustment represented by the D-function. When Γ > 0, the proton moves towards the electron and when Γ < 0 it recedes. The calculated frequency in Eq. (12) is in MHz, and it must be added to the magnetic potential frequency in Eq. (22) using the optimised A and B parameters together with the bound state distance d2bar. Here, M = mo/mo'. Source: Paper C.
The near side of the electron's Sp-2 circuit receives a stronger proton field than the far side. This characteristic together with the relative proton-electron Sp-2 rotation senses determines whether the targeted Sp-2 circuit increases or reduces in action.
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Same sense Sp-2 (Γ > 0): The changing electron magnetic field acts in opposition and is stronger at the near side of the proton circuit, reducing its Sp-2 action. The proton borrows action from Sp-3 to compensate reducing its Sp-3 energy. This is an attraction, the proton moves towards the electron, and the electron's speed relative to the proton increases, raising ν3D in Eq. (12). The changing proton magnetic field in turn diminishes the electron Sp-2 action and Sp-3 energy is redistributed to compensate. Since A < 0, this red-shifts ν3D through the magnetic potential shift Δνshift — Eq. (22) in Paper C — as well as the hyperfine shift Δνhf — Eqs. (35) in Paper C, see Figure 9 below.
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Opposite sense Sp-2 (Γ < 0): The electron field now reinforces the proton Sp-2 action. The overload in Sp-2 action means it is redistributed into Sp-3 which in turn is overloaded. So the proton moves away from the electron as a repulsion reducing ν3D. The reciprocal effect on the electron where A > 0 simultaneously blue-shifts ν3D due to the magnetic potential term, and blue-shifts the hyperfine shift, see Figure 10 below.
5.3 Relationships between parameters
The form of the magnetic potential shift has already been examined in section 5.2. After the fine structure frequency has been adjusted with the D-function, this shift is the term that bridges the remaining gap to the hyperfine midpoint. The empirically determined parameters A and B have been chosen to match the midpoint frequencies, but some noteworthy discoveries have been made in the process. It turns out the B = -3.000 across all five angular momentum families: nS1/2, nP1/2, nP3/2, nP5/2, and nD5/2. With both signs of Γ, that amounts to ten independent fits in total — see Tables II and III in Paper C. This power on the bound state distance d2bar in the magnetic potential shift is the expected scaling for a magnetic dipole-dipole interaction, and its near-integer value to several decimal places suggests a genuine physical mechanism rather than coincidental fitting. This value even carries over to the deuterium hyperfine midpoints — see Tables XI and XII in Paper C. Tritium data is scarce but on the three states tested in nS1/2 B = -3.00 for both signs of Γ — see Table XV in paper C. This is also the case for the same three states of ³He+ — see Table XVI in Paper C.
Even more striking is the ratio A/Γ which follows a nuclear scaling law. Across all ten sets of states for hydrogen, this ratio stays close to the mean value -28,651,713 MHz — see Tables II and III in Paper C. Extending the investigation to deuterium, tritium, ³He+, and 7Li2+ we find the following law, see Figure 8 below.
Figure 8 Extract from paper C showing the nuclear scaling law that applies across five elements. Here, An is the nuclear mass number, Zn is the nuclear charge, and Ze is a posited number of units of angular momentum in the electron's internal potential.
5.4 Hyperfine splitting mechanism
The final step is to calculate Δνhf , the equal and opposite frequency displacement of the two hyperfine levels about the midpoint. This is a fundamentally different mechanism from the magnetic potential shift Δνshift.
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The magnetic potential shift reflects the cumulative interaction energy of a momentum field on a Sp-2 circuit that changes with position, integrated from infinity to the bound state location. The differential and integral cancel out leaving the boundary value at d2bar. Through energy redistribution between Sp-2 and Sp-3 modes, this reduces or increases the energy of translation at the bound-state location.
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The hyperfine splitting arises from the effect of a changing momentum field taken over the electron's Sp-2 time period at the bound-state location. Again this redistributes action between Sp-2 and Sp-3 modes.
The derivation begins with the change in the proton Sp-2 field momentum Δp2f cut by the electron's Sp-2 circuit as the electron moves through distance Δx = vxT2 and Δy = vyT2 . This results in Eq. (35) in Paper C.
The n² factor in the hyperfine formula
There are n² Sp-2 circuits that are receptive to the proton field in each of the nΦ² strings, see Figure 5 in Tutorial 3. Each of the n² circuits receives the same momentum change and their energies are additive, since they belong to a single string. However, the nΦ² strings are parts of an intensity so their frequencies are not additive.
The result in Eq. (35) of Paper C scales as 1/n³, which follows naturally from Y ~ n and d2bar ~ 2n. This is consistent with the well-known quantum mechanical prediction for hyperfine splitting, but derived here from the geometry of the field interaction rather from matrix elements of the Fermi contact interaction. Figures 9 and 10 below show the full frequency diagrams for the 2S1/2 state of hydrogen when Γ > 0 (same sense Sp-2) and Γ < 0 (opposing Sp-2). This is calculated in full detail in Appendix B of Paper C.
A fourth parameter k serves as a multiplier of the hyperfine shift Eq. (35). It turns out that for the nS1/2 states k = 1 which means no adjustment is necessary. This serves as validation for there being parallel Sp-2 field lines cutting at 45° degrees to the line joining the Sp-2 centres as depicted in Figure 3 from Tutorial 2. However, in Table VII from Paper C, lower values of k are required for other sets of states. This could easily be accounted for by varying the eccentricity of the electron Sp-2 circuits keeping the perimeter the same. This would appropriately reduce the receptivity of the circuit to the proton field.
Figure 9 The 2S1/2 state of hydrogen calculated from the PTV model with Γ > 0. The unadjusted fine structure value has no reduced mass, the adjusted value ν3D includes the reduced mass and the proton-speed correction, and the magnetic potential shift is νshift.
The stealth positron
When we examine the Lorentz force in Tutorial 6 we shall see why one of the Sp-2 rotations for the electron is a stealth positron that doesn't get ionized.
Figure 10 The 2S1/2 state of hydrogen calculated from the PTV model with Γ < 0. The unadjusted fine structure value has no reduced mass, the adjusted value ν3D includes the reduced mass and the proton-speed correction, and the magnetic potential shift is νshift.
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 6
Tutorial 4
Paper E: Barry R. Clarke, A heuristic model of the Bose-Einstein distribution with distinguishable photons
DATA AVAILABILITY
Click here for the calculation for Eq. (35) in Paper C, the BASIC programs, and their data output.