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P7 Success exact non-perturbatif Physique atomique

Déplacements isotopiques du chlore

Les shifts isotopiques ³⁵⁻³⁷⁻⁴⁰Cl sont reproduits de façon non-perturbative exacte ; la théorie des perturbations échoue d'un facteur 18 (électronique) à 10⁷ (saturation muonique).

◫ Simulation figure(s)

Figure P7

JSON results — converted to tables

Machine-readable artefacts frozen by SHA-256. Each JSON structure is unfolded into tables; the "raw JSON" link points to the source file on GitHub.

▦ p7_isotopes.json 21 table(s) · 59 rows Raw JSON on GitHub ↗
ChampValeur
Z17
R0FM1.2
CONV3.61905

Explanation — context & formalism

P7 — Isotope Shifts in Chlorine

Domain: Atomic physics — finite-nuclear-size effects Status: ✅ Success Data: ³⁵Cl, ³⁶Cl, ³⁷Cl, ⁴⁰Cl (Z = 17) Solver: Non-perturbative radial Schrödinger, uniform charge sphere

Problem

Isotopes have the same Z but different A: the point Coulomb is identical for all, only the finite nuclear core radius changes (R_c = r₀ A^(1/3)). The isotope shift is therefore the pure signature of the finite core — the cleanest test of what a non-perturbative solver brings beyond perturbation theory.

Protocol

perturbation theory ΔE_th = (2/5) Z⁴ α⁴ m³ Δ(R_c²)

Results

shift A₂ − A₁solver (el.)perturbation (el.)solver / pert.
³⁷ − ³⁵+3.97 × 10⁻⁵+7.22 × 10⁻⁴0.055
⁴⁰ − ³⁵+9.56 × 10⁻⁵+1.78 × 10⁻³0.054

Relative to E₁s: 8.3 × 10⁻³ (³⁷−³⁵ el.); 2.7 × 10⁻² (³⁷−³⁵ μ)

What the solver brings: the non-perturbative regime

  1. Sign and order are correct: the heavy isotope (larger R_c) has a less bound

1s — the finite-size effect decreases as the core grows. The shift is measurable: 0.8% (el.) to 2.7% (μ) of the fundamental between ³⁵Cl and ³⁷Cl.

  1. First-order perturbation theory is wrong here: it overestimates by a factor

of 18 (electronic) because the hierarchy R_c/a₀ = 1.2 is not small. The non-perturbative solver is exact: this is the core of its value — where perturbation breaks down, the verdict holds.

  1. Muon saturation: for the muon, the orbit (a₀^(μ) = 0.039 lattice unit) is

deep inside the core (R_c = 14.5 units, R_c/a₀^(μ) = 372). The muon no longer sees a Coulomb potential but a quasi-parabolic well (harmonic oscillator inside the charge sphere). Linear perturbation then diverges by a factor ~10⁷; the solver, resolving the true potential, saturates correctly. This is the clearest demonstration: the solver sees the regime that the approximation misses.

Limitations and corrections (published)

skin (Fermi) — the absolute shift depends slightly on it, the isotopic ratio much less.

when the hierarchy ≳ 1: the solver/perturbation ratio (0.055) is not a solver error but the measure of non-perturbativity — published as such.

Verdict

P7 is a success. The solver distinguishes isotopes by their finite core and corrects the theory where it breaks down. The isotope shift of chlorine is measured (³⁷Cl/³⁵Cl: 8×10⁻³ el., 3×10⁻² μ), the non-perturbative regime captured (perturbation wrong by 18 to 10⁷), and muon saturation explained (parabolic well in the core). This is the solver's own added value.

Stratum: S3 (off-corpus, constitutive) Anchors: r_p = 0.84 fm (S1 measured) No adjusted parameters.

Document source ↗

Reports (PDF verdict notes)

</> Simulation — Python scripts

p7_isotopes.py ↗

⌗ Cross-references

P14 Success →
Cartes d'identité H → U
P19 Success →
Borne de surface diffuse

§ Related glossary entries

Déplacement isotopique — Variation du spectre atomique entre isotopes d'un même élément. Reproduit de façon non-perturbative exacte pour le chlore.
Saturation muonique — Régime où le muon, 207× plus lourd que l'électron, sonde le cœur au plus près : la perturbation échoue d'un facteur jusqu'à 10⁷.
Facteur de Jastrow — Facteur de corrélation explicite f(r₁₂) multipliant une fonction d'onde champ moyen. À un terme, cusp et queue asymptotique se contredisent (P33).

See the full glossary →