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docs/12-atomic-overlap.md

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+---
+title: "12 – Atomic Overlap Correction"
+date: 2026-04-26
+tags:
+  - beta-spectrum/correction/atomic-overlap
+status: active
+aliases: [Atomic Mismatch, Bahcall Correction, r(Z,W)]
+cssclasses: []
+related_components: [[04-screening-correction]], [[13-chemical-effects]]
+---
+
+# Atomic Overlap / Mismatch Correction $r(Z,W)$
+
+## Physical Origin
+
+β decay results in a sudden change of nuclear potential ($Z \to Z\pm 1$). Initial and final atomic orbital wave functions only partially overlap — the atom must "rearrange" from its initial configuration to accommodate the new nuclear charge. This **atomic mismatch** (or shake-up) correction accounts for discrete excitations into higher bound states.
+
+## Bahcall Correction (Eq. 166)
+
+In first approximation, this reduces to a difference in atomic binding energies:
+
+$$r(Z,W) = 1 - \frac{1}{W_0 - W} \frac{\partial^2 B(G)}{\partial Z^2} \tag{166}$$
+
+where $B(G)$ is the total atomic binding energy for a neutral atom with $Z\pm 1$ protons. The second derivative relates to average excitation energy via:
+$$\Delta E_{ex} = -\frac{1}{2}\frac{\partial^2 B(G)}{\partial Z^2}$$
+
+### Parametrization (Eq. 169–170)
+
+Using Desclaux (1973) numerical values:
+$$K(Z) = -0.872 + 1.270 \, Z^{0.097} + 9.062\times 10^{-11} \, Z^{4.5} \tag{169}$$
+
+The correction then becomes:
+$$r(Z,W) = 1 - \frac{1}{W_0 - W^2}\left[B(G)'' + 2(C'_0 + C_1)\right] \tag{170}$$
+
+## Magnitude Near Endpoint
+
+> [!warning] Critical near endpoint for low-energy transitions
+> Influence felt mainly near the endpoint where $\partial^2 B(G)/\partial Z^2$ can reach **a few hundreds of eV**.
+
+| Isotope | Endpoint | Correction at 15 keV before endpoint |
+|---|---:|---|
+| $^{63}$Ni | 67.2 keV | ~**1%** |
+| $^{241}$Pu | 20.8 keV | similarly important |
+| Tritium ($^3$H) | 18.6 keV | relevant for KATRIN analysis |
+
+The correction increases rapidly past the endpoint (where it's not physically meaningful — this signals breakdown of approximation).
+
+## Bound State β Decay (Eq. 171)
+
+Since the electron is created inside an electronic potential well, there exists a possibility for β decay to be captured into a bound atomic state:
+
+$$\frac{\Gamma_b}{\Gamma_c} = \frac{\pi(\alpha Z)^3}{f(Z,W_0)} (W_0 - 1)^2 \Sigma \tag{171}$$
+
+### Branching Ratios
+
+| System | $\Gamma_b/\Gamma_c$ | Relevance for calculator |
+|---|---:|---|
+| Free neutron | $4.2\times 10^{-6}$ | Negligible |
+| Tritium (T⁺) | ~**1%** | Relevant for precision neutrino mass experiments |
+| Tritium (neutral T) | ~**0.5%** | — |
+| Superallowed decays (MeV Q-value) | << $10^{-4}$ | Negligible |
+
+> [!note] ft analysis only
+> Bound state decay does **not** affect the β spectrum shape (it is a separate final state in S-matrix calculation). It enters when considering the **ft analysis**, so it's not directly relevant for `BetaSpectrum` but may matter for future calculator extensions.
+
+## Calculator Implementation
+
+> [!info] Not yet implemented
+> The atomic mismatch correction $r(Z,W)$ is NOT currently part of the calculator. This note documents the theory for future implementation, particularly important for low-energy transitions (tritium, $^{63}$Ni, $^{241}$Pu).
+
+## Related Notes
+
+- [[04-screening-correction]] — Screening modification from shake-up processes
+- [[11-isospin-breakdown]] — Shake-off process correction
+- [[13-chemical-effects]] — Molecular environment influence on atomic overlap