Hyperfine structure and isotopes: what the nucleus contributes

1. The nucleus supplies more than positive charge

Hydrogen and deuterium have the same nuclear charge but different nuclear masses and structures. Ordinary hydrogen has proton spin I = 1/2; deuterium has deuteron spin I = 1. NIST’s atomic-data notes identify these isotope and spin distinctions and specify which isotope its hydrogen line tables describe. [NIST hydrogen atomic data]

This separation matters for SMM/DTT because a proposed proton–electron phase relation does not automatically determine each of these quantities. Paper 21 organizes mass, magnetic response, and nuclear structure as distinct inputs or derivation targets. [Paper 21, §§9–10 and 15]

An isotope comparison becomes informative when the electronic rule is held fixed and the changed nuclear inputs are declared. Allowing a new electronic normalization for each isotope would discard that shared theoretical constraint.

2. Worked isotope shift from reduced mass

For a nucleus of mass M, the reduced mass is μ(M) = m_e/(1 + m_e/M). A heavier nucleus makes μ slightly larger and the bound-state energies slightly more negative. Every gross interval increases by the same fractional reduced-mass factor.

With the declared 2022 CODATA electron-to-proton and electron-to-deuteron mass ratios, this ratio exceeds one by about 2.72099 × 10⁻⁴. For the 1S–2S gross interval, the calculated deuterium-minus-hydrogen difference is about 671.0 GHz. This is the reduced-mass contribution alone, before relativistic, radiative, and nuclear isotope terms. [NIST CODATA mass ratios]

A useful symbolic check writes x_H = m_e/m_p and x_D = m_e/m_d. Then μ_D/μ_H − 1 = (x_H − x_D)/(1 + x_D). The sign is positive because the deuteron is heavier.

Use nuclear masses in this formula. A tabulated neutral-atom mass already includes an electron and a binding-energy correction; inserting it directly as M introduces an avoidable inconsistency.

3. Add proton and electron angular momenta

In the 1S ground state, electronic orbital angular momentum vanishes and J = S has j = 1/2. Coupling it to proton spin I gives total F = I + J, with F = 0 or F = 1. The simplest rotationally invariant hyperfine Hamiltonian is

For hydrogen, I = j = 1/2. The F = 1 triplet lies at A/4 and the F = 0 singlet at −3A/4 relative to the hyperfine centroid. Their separation is exactly A within this model. The multiplicities are three and one, reflecting the 2F + 1 magnetic sublevels.

This weighted-centroid identity is a useful check. An unweighted average of the two multiplet energies does not recover the centroid. Comparing a calculated centroid with a selected hyperfine component requires the appropriate correction.

The algebra determines the pattern once A is supplied. It does not derive A. That coefficient comes from magnetic response and the spatial state, which provide a separate test of an atomic theory.

4. Magnetic contact response sets the leading coefficient

For point magnetic moments in an S-state Coulomb envelope, the leading Fermi contact coefficient can be written

Here μ_B = eℏ/(2m_e), μ_N = eℏ/(2m_p), and g_e is the positive magnitude used in the convention μ_electron = −g_eμ_BS/ℏ. Some constants tables instead incorporate the electron’s sign into its listed g factor. The convention must be translated consistently.

Using |ψ_nS(0)|² = 1/(πa_μ³n³) gives the equivalent energy expression

For a worked leading example, set g_e = 2 and use the reference proton magnetic response and masses. The result is A_1S/h ≈ 1.419 GHz, corresponding to a wavelength near 21 cm. This is a contact-model calculation; electron anomalous response, recoil, and proton structure refine it.

The model displays an important dependency: a spin-one-half label alone does not fix a magnetic moment. A temporal derivation of the coefficient must supply both the representation and the response normalization.

5. Deuterium changes the angular pattern as well as the mass

For deuterium’s 1S state, I = 1 and j = 1/2. The possible total values are F = 1/2 and F = 3/2. The same angular formula gives E_3/2 = A_D/2 and E_1/2 = −A_D, so their separation is 3A_D/2. Their multiplicities are four and two, and the weighted centroid again vanishes.

The deuteron magnetic moment has its own coefficient. Replacing the proton mass in a hydrogen formula while retaining the proton magnetic moment would not produce a deuterium hyperfine calculation. Higher nuclear response is also different because the deuteron is a composite proton–neutron bound state.

At nonzero magnetic field, the Zeeman interaction can mix zero-field F sectors with the same magnetic projection. F then ceases to be an exact label in the intermediate-field regime. A full field-dependent calculation diagonalizes hyperfine and Zeeman terms together.

These distinctions offer a practical theoretical check of universality: keep the form of the electronic coupling fixed, change only the independently specified nuclear properties, and calculate the resulting multiplet. A successful source theory should explain which quantities transfer unchanged and which are outputs of a different nuclear state.

6. Distinguish universal local response from nuclear size

A universal contact insertion δH = λδ³(r) gives a level shift λ/(πa_μ³n³) for nS. If λ is genuinely the same physical coefficient in hydrogen and deuterium, the contact shift scales as μ³. That is a more specific statement than merely saying that both atoms contain the same electron.

A nuclear finite-charge-radius term instead contains an additional radius factor. Its leading contribution scales as μ³r_N²/n³. The hydrogen–deuterium difference consequently depends on μ_D³r_d² − μ_H³r_p². Nuclear polarizability and recoil add further terms.

The n⁻³ dependence alone cannot identify the physical origin, because both examples have it. One needs the coefficient relation across isotopes or additional angular and state dependence. Paper 14 makes this separation a central part of differential comparisons. [Paper 14, isotope and nuisance analysis]

The ordinary nuclear contribution must therefore be included before an isotope residual is attributed to electron holonomy or temporal completion. This is a theoretical identifiability requirement, not simply a matter of adding more decimal places to the same calculation.

7. Related atoms require a clear transfer rule

Muonium contains an electron and a positive muon. Muonic hydrogen contains a negative muon and a proton. Their names are similar, but the replacement occurs on opposite sides of the atom and changes the relevant mass and nuclear sensitivities substantially.

Positronium contains an electron and a positron with equal masses. Reduced mass changes, both particles move relativistically at comparable orders, and annihilation channels contribute. A hydrogen formula with a substituted nuclear mass is therefore only the beginning of its theory.

Antihydrogen contains an antiproton and a positron. A temporal orientation proposal needs a defined transformation law before it predicts any sign change. Paper 21 treats a distinct orientation-sensitive correction as conditional on such a derivation; ordinary charge reversal alone is not a calculation of a hydrogen–antihydrogen energy difference. [Paper 21, §9]

These systems are useful future extensions because they separate electronic, nuclear, two-body, and orientation assumptions. Each deserves its own effective Hamiltonian and uncertainty account rather than being grouped into a single unconstrained residual fit.

8. A focused SMM/DTT programme for nuclear interfaces

The immediate theoretical task is to specify the proton and electron source sectors sufficiently to calculate inertia, magnetic response, and the interaction between them. Neutrality and a stable composite sector are important preliminary results, but they do not by themselves give a mass ratio, charge radius, or hyperfine coefficient.

One productive milestone would derive a dimensionless mass or magnetic-response ratio independently and propagate it through the reduced-mass or contact formulas above. A second would derive a shared operator whose isotope dependence is fixed by the nuclear state, with no new normalization chosen for each atom.

Paper 21’s input hierarchy supports this order of work: constants and response parameters are supplied first, atomic consequences follow, and reserved observables assess the resulting linked calculation. [Paper 21, §§10 and 27]

This makes hyperfine and isotope theory a natural collaboration area. It connects representation theory, two-body dynamics, nuclear structure, and the proposed temporal source while keeping each mathematical dependency visible.

Continue reading

• Coupling, reduced mass, and atomic scales
• Finite nuclear size and radiative corrections
• Distinguish shared and isotope-dependent effects
• Explore the consequences of changing nuclear mass