Where the point-nucleus Dirac atom ends, and what it omits

Units are SI, with x⁰ = ct and metric diag(1, −1, −1, −1), though nothing here needs four-vectors. Topic 7.05 found G and F leaving a point nucleus as r^γ with γ = √(κ² − (Zα)²), and topic 7.06's energies, E = mc²[1 + (Zα)²/(nᵣ + γ)²]^(−1/2), are built on it. γ depends on |κ| = j + ½ alone, so it reaches zero at Zα = j + ½: for every j = ½ level at once, at Z = 1/α = 137.04, derived. The 1s energy, mc²γ, falls to zero there with slope −mc²α Zα/γ, which is −3.4 keV per unit Z at uranium and −163 keV at Z = 137: a vertical tangent. The nᵣ = 1 pair, 2s₁/₂ and 2p₁/₂, ends at mc²/√2 = 361 keV. Nothing happens to 2p₃/₂, whose γ = √(4 − (Zα)²) is still 1.73. As Zα → 0, the 1s exponent γ → 1 returns Quantum Mechanics II's r^(l+1) for an s state, the limit check. Unit 3's spinless 1s ended at Zα = l + ½ = ½, Z = 68.5; the spin has moved the edge to j + ½.

Above the edge γ² is negative, so write γ = iσ with σ = √((Zα)² − 1). Both near-origin solutions, r^(±iσ) = cos(σ ln r) ± i sin(σ ln r), have modulus 1: both are bounded, both square-integrable, and the real combinations are G ∝ cos(σ ln(r/r₀)) with r₀ any length, derived. Even the small component stops being small, since topic 7.05's indicial pair gives |F/G| = |iσ − 1|/(Zα) = 1. Near the origin the angular term κ/r and the Coulomb term Zα/r scale alike, so the equation holds no length that could fix r₀; each choice yields a different set of levels, and the energy formula itself turns complex, E(1s) = imc²σ. This is Unit 3's fall to the centre, nodes piling up without end as r → 0, though slowly: at Z = 140, σ = 0.209 and successive nodes lie a factor e^(π/σ) = 3.3 × 10⁶ apart in r. The point-nucleus Hamiltonian no longer defines the atom, so the theory ends near Z = 137, derived. What it lacks is a statement about short distances, and the nucleus is the only one physics supplies.

Spread the charge within a radius R. Inside, the potential stays finite and the solution is regular; for R ≪ ħ/mc = 386 fm its ratio ρ = F/G at r = R is a pure number set by Zα and the charge profile. Outside, G = A r^γ + B r^(−γ), with F/G equal to a₊ = (γ − 1)/(Zα) on the first root and a₋ = −(1 + γ)/(Zα) on the second. Matching F/G at R gives B/A = R^(2γ)(a₊ − ρ)/(ρ − a₋), derived. For real γ this vanishes as R → 0, so the point limit picks r^γ for every Zα below 1, including the band from 0.866 to 1 where normalisability could not: topic 7.05's choice, settled. The approach slows near the edge, since the irregular share at radius r goes as (R/r)^(2γ), and 2γ is 1.48 for uranium but 0.63 at Z = 130. For imaginary γ, R^(2iσ) = cos(2σ ln R) + i sin(2σ ln R) has modulus 1 and no limit, so the levels depend on R however small it is made. With R finite, nothing singular happens at Z = 137 at all.

With a finite nucleus the 1s level keeps falling through Z = 137. Calculations with R ≈ 1.2 A¹⁄³ fm, quoted, bring it to −mc², the top of the negative-energy continuum, near Z = 172, and bring 2p₁/₂ there near Z = 185. Below −mc², topic 7.05's decay constant ħcλ = √(m²c⁴ − E²) turns imaginary: the state no longer decays at infinity and dissolves into a resonance in the negative-energy continuum, which a one-particle equation cannot interpret. Unit 8's hole theory reads it as a vacuum that, offered an empty 1s, keeps an electron and emits a positron: a charged vacuum, quoted. No such nucleus exists, the heaviest yet made having Z = 118, and uranium-uranium collisions, which assemble Z = 184 for about 10⁻²¹ s, have not shown the effect cleanly, quoted. The point nucleus could not even approach this: its 1s ends at E = 0, half-way down the gap, so the dive is a finite-size phenomenon from start to finish.

Dirac's own correction to Bohr's 1s is −α⁴mc²/8, 43.8 GHz in frequency, and four omissions sit around it. Nuclear motion is the largest: the reduced mass lowers the binding by 13.6059 eV × mₑ/(mₑ + mₚ) = 7.41 meV, 1.79 THz, derived, 41 times Dirac's correction; what the reduced mass misses, of order (mₑ/mₚ)α⁴mc², 0.19 GHz in scale, is 24 MHz for 1s, quoted. The Lamb shift, from the electron's coupling to the quantised field, raises 1s by 8.17 GHz, quoted here; Unit 13 estimates it as Bethe did, without the full QED corrections. Coupling to the proton's magnetic moment splits 1s by 1.42 GHz, F = 1 above F = 0, without moving the centroid, quoted. The proton's size, at first order with |ψ(0)|² = 1/(πa³), raises 1s by (2/3)α⁴mc²(mcrₚ/ħ)² = 1.1 MHz for rₚ = 0.841 fm, derived; mcrₚ/ħ is a pure number, so the dimensions check, and the shift vanishes for a point proton. As fractions of the binding the list runs from 5 × 10⁻⁴ down to 3 × 10⁻¹⁰.

Dirac's degeneracy of 2S₁/₂ and 2P₁/₂ survives most of the list. The reduced mass rescales every level by mᵣ/m, and the leading recoil term beyond it depends, like the Dirac energy, on n and j alone, so neither separates the pair. Hyperfine coupling splits each into F = 0 and F = 1 without moving either centroid. The proton's size acts on s states only, but at n = 2 it is 1.1 MHz/8 = 0.14 MHz. That leaves the radiative shift, measured at 1057.8 MHz with 2S₁/₂ above 2P₁/₂: the one omission that breaks the degeneracy, which Unit 13 estimates as Bethe did, at about 1040 MHz. The ranking is not universal, because the omissions grow differently with Z: the Lamb and size shifts roughly as Z⁴, hyperfine as Z³, the reduced-mass shift as Z²mₑ/M. In hydrogen-like uranium nuclear motion is worth 0.3 eV of the 132.3 keV binding, ²³⁸U has spin zero and so no hyperfine structure, and nuclear size and the radiative shift, about 0.20 keV and 0.27 keV, quoted, lead the list.