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).

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

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Atomic Overlap / Mismatch Correction r(Z,W)