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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: corrections/03-screening, corrections/09-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^2can 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
BetaSpectrumbut 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
- corrections/03-screening — Screening modification from shake-up processes
- — Shake-off process correction
- corrections/09-chemical-effects — Molecular environment influence on atomic overlap