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docs: restructure knowledge base into tree layout + gA extraction module

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- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/
- Move all correction notes to corrections/ (01-09)
- Move nuclear structure to nuclear-structure/ with index
- Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO
- Reorganize refs by topic (electromagnetic/, forbidden-decay/)
- Update all wikilinks to new paths
- Remove radiative_spec.md (code algorithm, not reference)
- Lint: all wikilinks verified valid
- Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
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docs/corrections/01-fermi-function.md
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---
title: "01 – Fermi Function"
date: 2026-04-26
tags:
- beta-spectrum/correction/fermi
status: active
aliases: [Coulomb Correction, F_0]
cssclasses: []
related_components: [[corrections/02-finite-size]], [[corrections/05-radiative]]
---
# Fermi Function — Coulomb Interaction $F_0(Z,W)$
## The Point Charge Fermi Function
The transition matrix element with all-order Coulomb interactions resummed (Hayen et al., Eq. 5):
$$\mathcal{M}_{fi} = -2\pi i \delta(E_f - E_i) \langle f | T e^{-i\int_0^\infty dt H_Z(t)} H_\beta(0) T e^{-i\int_{-\infty}^0 dt H_{Z'}(t)} | i \rangle$$
Keeping only the final-state Coulomb interaction (daughter nucleus charge $Z$):
$$F_0(Z,W) = 4(2pR)^{2(\gamma-1)} e^{\pi y} \frac{|\Gamma(\gamma + iy)|^2}{(\Gamma(1+2\gamma))^2} \tag{6}$$
with:
$$\gamma = \sqrt{1 - (\alpha Z)^2}, \quad y = \pm \alpha Z W / p$$
Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[corrections/02-finite-size]]).
## Coulomb Amplitudes
Expanding radial Dirac wave functions near the origin (Eq. 12):
$$\left\{\frac{f_\kappa(r)}{g_\kappa(r)}\right\} = \alpha_\kappa \{(2|\kappa|-1)!!\}^{-1}(pr)^{|\kappa|-1} \sum_{n=0}^\infty \frac{a_{\kappa n}}{b_{\kappa n}} r^n$$
The Coulomb information is encoded in **Coulomb amplitudes** $\alpha_\kappa$. The dominant electron component at the origin:
$$F_0 L_0 = \frac{\alpha_{-1} + \alpha_1^2}{2p^2} \tag{13}$$
This is valid regardless of nuclear charge distribution shape.
## Alternative Definitions
| Convention | Prefactor for $L_0$ | Style |
|---|---|---|
| **Wilkinson / Blatt & Weisskopf** | $(1+\gamma)/2$ | Absorbs into $L_0$ |
| **Behrens-Bühring / Schopper** (used in calculator) | 4 | All corrections separate |
The error from neglecting the $(1+\gamma)/2$ prefactor is ~0.5% for $Z=20$.
## Numerical Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/fermi.py` → `FermiFunction(Z, A)` class
>
> Uses `scipy.special.loggamma` for numerical stability (avoids overflow of $\Gamma$ function). The implementation follows Eq. (6) with log-space evaluation:
> ```python
> log_F0 = ln(4) + 2*(γ-1)*ln(2*p*R) + π*y
> + 2*Re[loggamma(γ+iy)] - 2*loggamma(1+2γ)
> F0 = exp(log_F0)
> ```
## Energy Dependence
$F_0$ is a strong function of $Z$:
- **β⁻ decay** ($y < 0$): wave functions attracted to nucleus → enhancement at low energy
- **β⁺ decay** ($y > 0$): repulsion → suppression at low energy, grows with energy
The effect ranges from ~1% for light nuclei to tens of percent for heavy ones.
## Related Corrections
After the point charge Fermi function is computed:
- [[corrections/02-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$)
- [[corrections/05-radiative]] adds QED radiative effects ($R = 1 + \delta_R$)
docs/corrections/02-finite-size.md
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---
title: "03 – Finite Size Effects"
date: 2026-04-26
tags:
- beta-spectrum/correction/finite-size
status: active
aliases: [Finite Nuclear Size, L_0, U, DFS]
cssclasses: []
related_components: [[corrections/01-fermi-function]], [[nuclear-structure/01-shape-factors]]
---
# Finite Nuclear Size Effects — $L_0$, $U(Z,W)$, and $DFS$
## Overview
The Fermi function ([[corrections/01-fermi-function]]) assumes a point charge nucleus. Real nuclei have finite extent and possibly deformation — these corrections account for the difference in electric potential inside the nuclear volume.
> [!note] Two types of "finite size"
> 1. **Electrostatic** ($L_0$, $U$, $DFS$): Potential difference when moving from point charge to realistic nuclear shape — applied *after* evaluating wave function at origin
> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume — appears in shape factor $C(Z,W)$ ([[nuclear-structure/01-shape-factors]])
## Nuclear Radius
$$R = \sqrt{5/3} \, \langle r^2 \rangle^{1/2} \tag{14}$$
where $\langle r^2 \rangle^{1/2}$ is the root-mean-square charge radius from electron scattering experiments.
---
## $L_0(Z,W)$ — Uniformly Charged Sphere
### Low-Z Expansion (Behrens & Bühring)
$$L_0 \simeq 1 \mp \alpha Z W R + (\alpha Z)^2 - \frac{13}{60} - \frac{\alpha Z R}{2W} \tag{15}$$
### Wilkinson's Analytical Parametrization (Eq. 16)
Accurate to $10^{-4}$ for $p \leq 45$, $|Z| \leq 60$:
$$L_0(Z,W) = 1 + \frac{13}{60}(\alpha Z)^2 \mp \frac{\alpha Z W R(41 - 26\gamma)}{[15(2\gamma-1)]} $$
$$\mp \frac{\alpha Z R \gamma(17 - 2\gamma)}{[30W(2\gamma-1)]} + \frac{R}{W}\sum_{n=0}^5 a_n (WR)^n + 0.41(R-0.0164)(\alpha Z)^{4.5}$$
with $a_n = \sum_{x=1}^6 b_{x,n} (\alpha Z)^x$ (Eq. 17). Coefficients in **Tables I** (electrons) and **II** (positrons); signs of odd powers flipped for positrons.
> [!warning] Magnitude
> The effect ranges from a few 0.1% up to several percent — everywhere highly significant. Cannot be neglected for precision work at $10^{-4}$ level.
### Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/finite_size.py` → `FiniteSizeL0(Z, A)` class
>
> Uses Wilkinson parametrization (Eq. 16) with coefficients from Hayen Tables I-II. The nuclear radius is computed via `utils.nuclear_radius(A)`.
---
## $U(Z,W)$ — More Realistic Charge Distributions
The uniform sphere approximation is insufficient for high precision. Two common models:
### Modified Gaussian (MG)
$$\rho_{MG}(r) = N_0\left[1 + A\left(\frac{r}{a}\right)^2\right]e^{-(r/a)^2} \tag{18}$$
with $N_0$ and $a$ determined by normalization and RMS radius (Eqs. 19–20).
### Fermi Distribution
$$\rho_F(r) \propto [1 + \exp((r-c)/a)]^{-1}, \quad a \simeq 0.55\text{ fm}$$
The correction is defined as ratio:
$$\frac{U(Z,W)}{L_0} = \frac{\alpha_{02}' + \alpha_{-1}'^2}{\alpha_{1}^2 + \alpha_{-1}^2} \tag{23}$$
### Analytical Approximation for Low Z (Eq. 25)
$$U(Z,W) \approx 1 + \alpha Z W R \Delta_1 + \frac{\gamma}{W}\alpha Z R \Delta_2 + (\alpha Z)^2 \Delta_3 - (WR)^2 \Delta_4$$
### Wilkinson's Fermi Distribution Fit (Eqs. 29–30)
$$U(Z,W) = 1 + \sum_{n=0}^\infty a_n p^n$$
with $a_0, a_1, a_2$ given as quadratic polynomials in $Z$.
> [!tip] Magnitude
> The effect of $U(Z,W)$ is ~0.1% for medium-high masses — cannot be neglected at the $10^{-4}$ precision level.
### Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/finite_size.py` → `ChargeDistributionU(Z, A)` class
>
> Implements Eq. (25) with $\Delta_i$ coefficients computed from nuclear model parameters.
---
## $DFS(Z,W,\beta^2)$ — Nuclear Deformation
For axially symmetric deformations:
$$R(\theta,\phi) = R_0\left[1 + \beta_2 Y_{20}(\theta,\phi)\right] \tag{32}$$
with $\beta_2 > 0$ prolate, $\beta_2 < 0$ oblate. The intrinsic quadrupole moment:
$$Q_0 = \frac{\alpha}{\pi}\sqrt{\frac{5}{3}} R Z \beta_2(1 + 0.16\beta_2) \tag{33}$$
The correction is a ratio of angle-averaged potentials (Eq. 40):
$$DFS(Z,W,\beta) = \frac{L_0^*(Z,W)}{L_0(Z,W)} \tag{40}$$
where $L_0^*$ integrates over a continuous superposition of uniformly charged spheres (Eq. 39).
> [!tip] Magnitude
> Effect can reach several parts in $10^3$. Partially canceled by nuclear structure deformation correction $D_C$ ([[nuclear-structure/01-shape-factors]]). Figure 1 of Hayen shows energy dependence reversal between β⁻ and β⁺ for $\beta_2 = 0.2$.
### Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/finite_size.py` → includes DFS in the finite size module (combined with L₀)
>
> The deformation parameter $\beta_2$ is typically provided by nuclear data tables or shell model calculations.
---
## Summary of Corrections Flow
```
Fermi function F₀(Z,W) [point charge]
↓
× L₀(Z,W) [uniform sphere correction]
↓
× U(Z,W) [realistic charge distribution]
↓
× DFS(Z,W,β²) [nuclear deformation]
```
## References in Hayen Paper
- Section IV.A.1 (Eq. 15–17): $L_0$ uniform sphere
- Section IV.A.2 (Eq. 18–30): $U$ charge distributions
- Section IV.A.3 (Eq. 32–40): $DFS$ deformation
docs/corrections/03-screening.md
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---
title: "04 – Screening Correction"
date: 2026-04-26
tags:
- beta-spectrum/correction/screening
status: active
aliases: [Atomic Screening, S(Z,W)]
cssclasses: []
related_components: [[corrections/08-atomic-overlap]], [[corrections/09-chemical-effects]]
---
# Atomic Electron Screening Correction $S(Z,W)$
## Physical Origin
The Fermi function ([[corrections/01-fermi-function]]) assumes the β particle moves in the Coulomb field of a bare nucleus. In reality, atomic electrons partially screen this charge at large distances from the nucleus, modifying the effective potential experienced by the emitted electron/positron. This is particularly important at low kinetic energies where the β particle spends more time far from the nucleus.
## The Screening Correction
Screening is implemented as a ratio method:
$$S(Z,W) = \frac{F_{\text{screened}}(Z, W)}{F_0(Z, W - V_0/m_ec^2)} \tag{144}$$
where $V_0$ is the screening potential (binding energy of the outermost atomic electrons). The denominator evaluates the Fermi function at a shifted energy representing the unscreened case.
### Screening Potential
The screening potential $V_0$ depends on:
- Atomic number $Z$ (parent atom)
- Chemical/physical state (neutral atom, ionized molecule)
- Electronic configuration
For neutral atoms, $V_0$ is typically a few eV to tens of eV — small compared to nuclear potentials but significant at β energies below ~10 keV.
### Molecular Screening Correction
In molecular environments, the screening potential changes:
$$\Delta S_{\text{Mol}} \quad (\text{Eq. 176})$$
Since screening is typically at sub-percent level maximum, molecular deviations are expected at the **few $10^{-4}$** level — relevant for precision work down to 1 keV.
## Energy Dependence
- **Dominant at low energies** (T < ~50 keV): where the β particle's wave function is most affected by atomic electron cloud
- **Negligible at high energies**: $S(Z,W) \to 1$ as $W \gg V_0$
- The effect reverses sign between β⁻ and β⁺ (electrons attracted to/repelled from electron cloud differently)
## Shake-Up Effects on Screening
Atomic excitations during β decay modify screening:
> [!warning] Tritium case
> In tritium β⁻ decay, the screening potential can change by as much as **20%** because only a single electron contributes to atomic Coulomb interaction. For heavier systems, relative changes decrease but scale with $Z$. Neglecting shake-up introduces errors of order $\sim 1 \times 10^{-4}$.
Shake-off (continuum ejection) has smaller effect:
- With $\Delta Z_{\text{eff}} \approx 0.3$–$0.4$, yields ~**0.1%** relative change in screening potential → few $10^{-5}$ level in spectral shape
- **Negligible at target precision** for most nuclei
## Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/screening.py` → `ScreeningCorrection(FermiFunction)` class
>
> Uses ratio method with smooth logistic switching function. The screening potential $V_0$ is passed as a parameter and depends on the parent atom's atomic number.
### API Usage
```python
from beta_spectrum import ScreeningCorrection, FermiFunction
screen = ScreeningCorrection(
FermiFunction(Z_parent=90), # bare nucleus reference
V0=0.015 # screening potential in m_e units (~7.6 keV)
)
values = screen(W_grid)
```
## Size Comparison with Other Corrections
From Hayen et al. Table (Section VIII overview):
| Correction | Magnitude | Energy Range Most Relevant |
|---|---|---|
| Screening $S$ | $10^{-3}$ – $10^{-4}$ | T < 50 keV |
| Exchange $X$ | up to ~20% at lowest E | T < 10 keV |
| Molecular screening $\Delta S_{\text{Mol}}$ | few $10^{-4}$ | T < 5 keV (molecular) |
## Related Notes
- [[corrections/08-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction
- [[corrections/09-chemical-effects]] — Molecular environment influence on screening and exchange
- [[corrections/04-exchange]] — Electron exchange with atomic orbitals (often combined with screening)
docs/corrections/04-exchange.md
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---
title: "05 – Exchange Correction"
date: 2026-04-26
tags:
- beta-spectrum/correction/exchange
status: active
aliases: [Atomic Exchange, X(Z,W), Hayen-2018]
cssclasses: []
related_components: [[corrections/03-screening]], [[corrections/08-atomic-overlap]]
---
# Atomic Electron Exchange Correction $X(Z,W)$
## Physical Origin
When a β electron is emitted from the nucleus, it can exchange with bound atomic electrons due to indistinguishability (Pauli exclusion principle). This **electron exchange** effect modifies the β spectrum at low energies where the continuum electron's wave function overlaps significantly with atomic orbitals.
> [!key] Magnitude
> The exchange contribution can exceed **20% in low-energy regions for higher Z** nuclei — it is one of the largest atomic corrections and absolutely critical for precision work below ~10 keV.
## Theory: Full Potential Approach
Exchange requires radial wave functions of both continuum and bound states integrated over **all space** (not just near origin). This is computationally demanding for analytical description because:
- Exchange needs integration over entire spatial domain
- Hydrogenic approach with simple screening is insufficient for high precision
- Must use realistic atomic potentials (Dirac-Hartree-Fock-Slater)
### Three Potential Models Compared (Hayen Fig. 9)
| Model | Description | Accuracy |
|---|---|---|
| **Unoptimized** | Basic Hartree-Fock-Slater, no exchange optimization | Lower bound |
| **Salvat** | Parametrized atomic potential | Intermediate |
| **Optimized** | Full potential with exchange optimization included | Target ($10^{-4}$) |
Screening strength is adjusted for best agreement with bound state energies (decreasing importance for higher $n$).
## Analytical Parametrisation — Hayen et al. 2018
The exchange correction $X(W)$ as a function of kinetic energy uses empirical fit coefficients from Table X of [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]]. Coefficients are tabulated for all atoms with $Z = 2$ through $z = 120$.
### Shell Structure Effects
> [!tip] Figure 9 observation
> Clear shell closure effects observed in orbital exchange contributions: at noble gas Z values, the exchange correction shows characteristic kinks. These must be captured by interpolation between tabulated Z values.
## Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/exchange.py` → `ExchangeCorrection(Z)` class
>
> Loads empirical fit coefficients from `data/exchange_coeff.csv` (119 rows, $Z = 2$–$120$). Uses interpolation between tabulated values.
### Data File
```
beta_spectrum/data/exchange_coeff.csv
├── Row count: 119 (Z=2 to Z=120)
├── Columns: Z, fit coefficients a₀...aₙ for energy parametrisation
└── Source: Hayen et al., Phys. Rev. D 97, 013006 (2018), Table X
```
### API Usage
```python
from beta_spectrum import ExchangeCorrection
exch = ExchangeCorrection(Z=90) # Z of parent atom
values = exch(W_grid) # returns correction factor at each W
```
## Energy Dependence
- **Dominant at very low energies** (T < ~10 keV): where continuum electron overlaps with atomic orbitals
- Decreases rapidly above ~50 keV (continuum wave function too short-ranged for significant overlap)
- For medium/high Z nuclei, shake-off probability in $s_{1/2}$ or $p_{1/2}$ state is < 0.1% — negligible
## Shake-Up Influence on Exchange
In low-Z nuclei (e.g., tritium):
- Shake-up probabilities can be significant (~25%)
- Exchange with higher-lying ns orbitals must be included even if unoccupied in initial state
For most practical cases, inclusion of shake-up changes the exchange correction at the few $10^{-4}$ level down to 0.5 keV for tritium only — otherwise negligible.
## Code Status: ✓ Implemented
The `ExchangeCorrection` class is fully implemented and tested. The CSV data file covers the full range from light (He, Z=2) to transuranic (U, Z=92+) elements.
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______________________________________________________________________
title: "06 – Radiative Corrections"
date: 2026-04-26
tags:
- beta-spectrum/correction/radiative
status: active
aliases: [Outer Radiative, δ_R, Sirlin Function]
cssclasses: []
related_components: \[[10-nuclear-structure]\], \[[07-recoil-effects]\]
______________________________________________________________________
# Outer Radiative Corrections — $R(W,W_0) = 1 + \\delta_R(W,W_0)$
## Overview: Inner vs. Outer Corrections
The radiatively corrected spectrum (Hayen et al., Eq. 46):
$$\\frac{d\\Gamma}{dW} = \\frac{d\\Gamma_0}{dW}(1 + \\Delta_R^{V/A})(1 + \\delta_R(W,W_0))$$
| Type | Symbol | Dependence | Treatment |
| --------- | ------------------ | --------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------- |
| **Inner** | $\\Delta_R$ | Nuclear-independent, absorbed into effective coupling constants | $\\Delta_V^R = (2.361 \\pm 0.038)%$ (Marciano & Sirlin 2006) — NOT part of spectrum shape calculation |
| **Outer** | $\\delta_R(W,W_0)$ | Energy-dependent, nucleus-dependent | This is the correction implemented in code (\[[06-radiative-corrections]\]) |
## Total Spectral Influence (Eq. 47–48)
$$R(W,W_0) = 1 + \\frac{\\alpha}{2\\pi}\\left[g(W_0,W) - 3\\ln\\left(\\frac{m_p}{2W_0}\\right)\\right] + Z\\alpha^2\\sum_i \\Delta_i(W) + Z^2\\alpha^3 \\delta\_{3h} + ... \\tag{48}$$
### Order α — Sirlin's $g(W_0, W)$ Function (Eq. 50–52)
The universal correction (same for electrons and positrons, Fermi and GT):
$$g(W_0,W) = 3\\ln(m_p) - \\frac{3}{4} + \\frac{4\\beta}{1+\\beta}\\left[4\\tanh^{-1}\\beta - 1\\right] $$
$$+ \\frac{W_0-W}{\\beta}\\left(-\\frac{3}{3W} + \\ln[2(W_0-W)]\\right) $$
$$- \\frac{(W_0-W)^2}{6W^2} + 4\\frac{\\tanh^{-1}\\beta}{\\beta}(1+\\beta^2) - 4\\tanh^{-1}\\beta$$
where $\\beta = p/W$.
The Spence function (dilogarithm):
$$L_s(x) = \\int_0^x \\frac{\\ln(1-t)}{t} dt = -Li_2(x) = -\\sum\_{k=1}^\\infty \\frac{x^k}{k^2} \\tag{51}$$
Large $W_0$ limit:
$$g(W_0 \\to \\infty, W) = 3\\ln\\left(\\frac{M_N}{2W_0}\\right) + \\frac{81}{10} - \\frac{4\\pi^2}{3} \\tag{52}$$
> [!warning] Endpoint divergence
> $g(W_0, W)$ has a **logarithmic divergence at $W = W_0$** related to soft real photon emission. This must be handled via resummation.
### Soft Photon Resummation (Eq. 53–54)
To handle the endpoint divergence, the leading logarithmic term $C(\\beta)\\ln(W_0 - W)$ is replaced with an exponentiated form:
$$t(\\beta)\\ln(W_0 - W) \\to (W_0-W)^{t(\\beta)} - 1 \\tag{53}$$
where:
$$t(\\beta) = \\frac{2\\alpha}{\\pi}\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right] \\tag{54}$$
$$C(\\beta) = 4\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right]$$
**Implementation note:** The resummation is only applied when $W_0 - W < \\delta\_{\\text{cut}}$ (default $\\delta\_{\\text{cut}} = 10^{-3}$ in $m_e$ units, ≈ 0.5 keV). For $W_0 - W \\gg \\delta\_{\\text{cut}}$, the standard logarithmic form is used. This ensures resummed and standard modes are identical far from the endpoint.
For tritium the effect is negligible (low endpoint). For higher energies, the $ft$ value shift can reach several parts in $10^4$.
### Order Zα² Correction (Eq. 55–62)
Three dominant Feynman diagrams: vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to $\\Delta_3$:
$$\\delta_2(Z,W) = Z\\alpha^2 \\sum\_{i=1}^4 \\Delta_i(W) \\tag{55}$$
The leading term $\\Delta_1$ depends on nuclear form factor $F(q^2)$ and is split into:
- **Nucleus-independent** part $\\Delta_1^0$: energy-dependent terms combined with $\\Delta_4$
- **Nuclear-dependent** part $\\Delta_1^F$: explicit integral forms for uniformly charged sphere
For modified Gaussian model, differences between models for superallowed Fermi transitions ($^{10}$O to $^{54}$Co) are < $10^{-4}$.
### Order Z²α³ — Sirlin's Heuristic Formula (Eq. 63–67)
$$\\delta\_{3h} = Z^2\\alpha^3 \\left[a\\ln\\left(\\frac{\\Lambda}{W}\\right) + bf(W) + \\frac{4\\pi}{3}g(W) - 0.649\\ln(2W_0)\\right]$$
with $a = \\frac{\\pi}{3} - \\frac{\\sqrt{3}}{2}$, $b = \\frac{4}{3\\pi^4}(\\pi^2 - \\gamma_E) - \\frac{\\pi^2}{18}$.
### Higher Order (Eq. 68–69)
$$\\delta\_{Z^n\\alpha^{n+1}} \\approx Z^n\\alpha^{n+1} K\_{nm} \\ln^{m-n}\\left(\\frac{\\Lambda}{W}\\right), \\quad K\_{nm} \\approx 0.5 \\tag{68}$$
$$\\sum\_{n=3}^\\infty Z^n\\alpha^{n+1} = \\frac{Z^3\\alpha^4}{1-Z\\alpha} \\tag{69}$$
Relevant for higher $Z$ nuclei only.
## Neutrino Radiative Corrections (Eq. 70–71)
Even though neutrinos have no direct electromagnetic interaction, they receive indirect corrections via virtual photon exchange and energy conservation from inner bremsstrahlung:
$$R\_\\nu = 1 + \\frac{\\alpha}{2\\pi}\\left[3\\ln\\left(\\frac{m_p}{m_e}\\right) + \\frac{23}{4} - Li_2\\left(-\\frac{8}{\\hat{\\beta}}\\right) + ...\\right]$$
where $\\hat{W} = W_0 - W\_\\nu$. Much smaller than electron spectrum correction — relevant for reactor neutrino oscillation studies.
## Internal Bremsstrahlung (Eq. 72–74)
An additional photon in the final state: $n \\to p + e^- + \\bar{\\nu}\_e + \\gamma$
Photon ejection probability per unit energy:
$$\\Phi(W,\\omega) = \\frac{\\alpha p}{\\pi\\omega p'}\\left[\\frac{W+W'^2}{p^2}\\ln\\left(\\frac{W+p}{W'-p}\\right) - 2\\right] \\tag{73}$$
Experimental branching ratios: $BR\_{\\beta\\gamma} \\sim 3\\times 10^{-3}$ (neutron), $\\sim 2\\times 10^{-3}$ ($^{32}$P).
## Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/radiative.py` → `RadiativeCorrection(W0, Z, A, use_endpoint_resummation, delta_cut)` class
>
> Implements outer radiative corrections per Sirlin (1967) and Sirlin (1987):
>
> - **O(α)**: Universal Sirlin function $g(W,W_0)$ with `scipy.special.spence` for dilogarithm
> - **O(Zα²)**: Finite nuclear size correction $\\delta_2(Z,W)$ using modified Gaussian model
> - Soft-photon resummation near endpoint via Sirlin (1987) prescription (only when $W_0 - W < \\delta\_{\\text{cut}}$)
> - Small-beta Taylor expansion for numerical stability at threshold
> - Optional endpoint resummation flag
### API Usage
```python
from beta_spectrum import RadiativeCorrection
# Light nucleus (tritium-like) — resummation has negligible effect at low W0
rad = RadiativeCorrection(
W0=1.02,
Z=1,
use_endpoint_resummation=True
)
# Heavy nucleus (uranium) with nuclear model correction
rad = RadiativeCorrection(
W0=5.0,
Z=92,
A=238, # affects Delta_F nuclear model term
use_endpoint_resummation=True,
delta_cut=1e-3 # only apply resummation within 1 keV of endpoint
)
# Standard (non-resummed) mode for comparison
rad_standard = RadiativeCorrection(
W0=5.0,
Z=92,
use_endpoint_resummation=False
)
values = rad(W_grid)
```
> [!note] Resummation scope
> The soft-photon resummation (Eq. 53) is only applied when $W_0 - W < \\delta\_{\\text{cut}}$ (default $10^{-3}$ in $m_e$ units ≈ 0.5 keV). Far from the endpoint, both resummed and standard modes produce identical results. This matches the Sirlin (1987, 2011) prescription where resummation replaces only the leading logarithmic divergence near $W \\to W_0$.
## Magnitude Summary
| Term | Typical Size | Notes |
| ---------------------------- | ------------ | -------------------------------- |
| $\\alpha/2\\pi \\cdot g$ | ~1–3% | Universal, dominant contribution |
| $Z\\alpha^2 \\Delta_i$ | ~0.01–0.1% | Nuclear-form-factor dependent |
| $Z^2\\alpha^3 \\delta\_{3h}$ | < 0.01% | Only for heavy nuclei |
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---
title: "07 – Recoil Effects"
date: 2026-04-26
tags:
- beta-spectrum/correction/recoil
status: todo
aliases: [Nuclear Recoil, R_N, Weak Magnetism]
cssclasses: []
related_components: [[02-phase-space]], [[nuclear-structure/01-shape-factors]]
---
# Nuclear Recoil Effects — $R_N$, $Q$, and Induced Currents
## Status in Calculator
> [!todo] Not yet implemented
> Module: `beta_spectrum/components/recoil.py` exists but is **empty** (stub only). This correction needs to be added.
## Why Recoil Matters
The finite nuclear mass turns beta decay from a two-body into a three-body process, introducing recoil corrections that become significant at the $10^{-4}$ precision level. Additionally:
- **Weak magnetism** $b_{wm}$ must be included as experimental precision increases — otherwise it limits BSM sensitivity
- The oft-neglected **induced pseudoscalar interaction** may be significant
## Kinematic Recoil Correction $R_N(W,W_0,M)$ (Eq. 41–43)
$$R_N = 1 + \frac{r_0}{W} + \frac{r_1}{W^2} + r_2 W + \frac{r_3}{W^2} \tag{41}$$
### For Fermi (vector) transitions:
$$r_0^V = W_0^2/(2M^2) - 11/(6M^2), \quad r_1^V = W_0/(3M)$$
$$r_2^V = 2/M - 4W_0^2/(3M), \quad r_3^V = 16/(3M^2)$$
### For Gamow-Teller (axial) transitions:
$$r_0^A = -2W_0/(3M) - W_0^2/(6M^2) - 77/(18M^2), \quad r_1^A = -2/(3M) + 7W_0/(9M^2)$$
$$r_2^A = 10/(3M) - 28W_0/(9M^2), \quad r_3^A = 88/(9M^2)$$
### Mixed transitions:
Terms are weighted by $1/(1+\rho^2)$ and $1/(1+\rho^{-2})$ where $\rho = \mathcal{M}_{GT}/\mathcal{M}_F$.
> [!tip] Magnitude
> Order $10^{-5}$ to $10^{-3}$. See Wilkinson (1990) Fig. 1 for magnitude plot.
## Electromagnetic Recoil Correction $Q(Z,W,M)$ (Eq. 45)
The recoiling nucleus's Coulomb field differs from a static one:
$$Q \simeq 1 \mp \frac{\pi\alpha Z}{M}\left(1 + a\right)\frac{W_0 - W}{3W} \tag{45}$$
where $a = (1-\rho^2/3)/(1+\rho^2)$ is the β-ν correlation coefficient ($a=1$ for Fermi, $a=-1/3$ for GT).
> [!note] Typically negligible
> The effect amounts to at most a few percent of the typical error in phase space factor $f$ due to Q-value uncertainty — usually not limiting.
## Weak Magnetism and Induced Currents (Section VI)
The weak vertex receives QCD-induced form factors beyond pure V-A:
### Behrens-Bühring Form (Eq. 82):
$$J_\mu^{BB} = i\langle \bar{u}_p | C_V\gamma_\mu - f_M\sigma_{\mu\nu}q^\nu + if_S q_\mu $$
$$- \frac{C_A}{C_V}\left(\gamma_\mu\gamma^5 - f_T\sigma_{\mu\nu}\gamma^5 q^\nu\right) + i\frac{f_P}{C_V}\gamma^5 q_\mu | u_n \rangle$$
### Holstein Form (Eq. 83):
$$J_\mu^{HS} = i\langle \bar{u}_p | g_V\gamma_\mu - \frac{g_M-g_V}{2M}\sigma_{\mu\nu}q^\nu + i\frac{g_S}{2M}q_\mu $$
$$+ g_A\gamma^5\gamma_\mu - \frac{g_{II}}{2M}\sigma_{\mu\nu}\gamma^5 q^\nu + i\frac{g_P}{2M}\gamma^5 q_\mu | u_n \rangle$$
| Form Factor | Name | CVC/PCAC Status |
|---|---|---|
| $C_V = g_V$ | Vector (Fermi) | Conserved — from EM data |
| $f_M/C_V$ | Weak magnetism | **Determined by CVC** from M1 transitions |
| $f_S/g_S$ | Induced scalar | Excluded by CVC for Fermi; Second-class current |
| $C_A = g_A$ | Axial-vector (GT) | Measured in neutron decay ($g_A \approx 1.276$) |
| $f_T$ | Tensor | First-class if exists |
| $g_{II}$ | Induced tensor | Second-class current → experimental null results |
| $f_P/g_P$ | **Induced pseudoscalar** | PCAC prediction: $g_P(0) \approx -229$, but quenched ~80% in nuclei |
### CVC Constraint on Weak Magnetism (Eq. 87–91)
For mirror transitions and isovector M1 decays, weak magnetism is determined from electromagnetic data — no free parameters:
$$\mu_p - \mu_n = g_M(0)/\mu_N \quad (\text{from CVC})$$
### PCAC / Goldberger-Treiman (Eq. 92)
$$g_P(0) \approx -229 \quad (\text{free nucleon})$$
With ~5% higher-order corrections. In nuclear medium, $g_P$ is **quenched by up to 80%** — great care needed when evaluating pseudoscalar currents for nuclear decays.
## Spectral Shape Modification (Eq. 3)
Weak magnetism and Fierz term modify the spectrum shape:
$$N(W)\,dW \propto p W (W_0 - W)^2 \left[1 + \frac{\gamma m_e}{W} b_{\text{Fierz}} \pm \frac{4W}{3M} b_{wm}\right] dW \tag{3}$$
The weak magnetism term $\propto W/M$ grows with energy — opposite to Fierz term which falls as $1/W$. This distinct energy dependence allows experimental separation.
## Implementation Plan for Calculator
To implement recoil in `recoil.py`:
1. Create `RecoilCorrection` class inheriting from `SpectrumComponent`
2. Implement $R_N(W,W_0,M)$ per Eqs. (41)–(44) with transition-type parameter (Fermi/GT/mixed)
3. Optionally implement $Q(Z,W,M)$ for electromagnetic recoil
4. Accept weak magnetism form factor from nuclear data or CVC calculation
docs/corrections/07-detector-response.md
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______________________________________________________________________
title: "08 – Detector Response Convolution"
date: 2026-05-01
tags:
- beta-spectrum/detector-response
status: active
aliases: [Convolution, Response Matrix, Smearing]
cssclasses: []
related_components: \[[00-beta-spectrum-overview]\], \[[06-radiative-corrections]\]
______________________________________________________________________
# Detector Response Convolution — $M(E_j) = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i$
## Overview
Detector response convolution bridges the gap between theoretical beta spectra and experimentally measured data. A theoretical spectrum $\\Phi(W)$ assumes perfect energy resolution, while real detectors smear the true energy distribution due to finite resolution, charge collection inefficiencies, and electronic noise.
The convolution operation:
$$M_j = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i$$
where:
- $\\Phi(W_i)$ — theoretical differential spectrum at true energy $W_i$
- $R(E_j; W_i)$ — detector response probability: probability that an electron of true energy $W_i$ deposits energy $E_j$ in channel $j$
- $\\Delta W_i$ — energy bin width at $W_i$
- $M_j$ — predicted measured counts in detector channel $j$
## Two Modes of Operation
### Tabulated Response
Receive a pre-computed response matrix from Monte Carlo simulations (GEANT4, EGSNRC) or dedicated calibration measurements. The matrix has shape $(N\_{\\text{calib}}, N\_{\\text{channels}})$, where each row $i$ is the detector response to mono-energetic electrons at calibration energy $E_i$.
```python
from beta_spectrum import DetectorResponse
# Load from MC simulation output
detector = DetectorResponse.from_mc_simulation(
channel_energies=channel_grid,
response_matrix=mc_response,
calibration_energies=mc_energies,
)
```
### Analytical Model
Parameterized model with Gaussian core + exponential tail, suitable for quick estimates when no tabulated response is available.
```python
detector = DetectorResponse.from_gaussian_params(
channel_energy_range=(1.0, 1.6), # in m_e units
n_channels=4096,
sigma_a=0.0, # energy-independent resolution (m_e)
sigma_b=0.0, # sqrt(E) contribution (m_e^{-1/2})
tail_fraction=0.0, # fraction of events in exponential tail
tau=0.01, # decay constant of tail (m_e)
model="gaussian", # "gaussian" | "gaussian_tail" | "tikhonov"
fano_factor=0.0, # Fano factor (0.12 for Si, ~0 for MMC)
)
```
## Analytical Response Models
### Gaussian Core
Pure Gaussian peak with energy-dependent resolution:
$$G(E; E_0, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left(-\\frac{(E - E_0)^2}{2\\sigma^2}\\right)$$
### Gaussian + Exponential Tail
Models low-energy tailing from charge collection inefficiencies in semiconductor detectors (standard for Si-PIPS detectors):
$$f(E) = (1 - f\_{\\text{tail}}) \\cdot G(E; E_0, \\sigma) + f\_{\\text{tail}} \\cdot T(E; E_0, \\sigma, \\tau)$$
where $T$ is the Tikhonov function:
$$T(E) = \\frac{1}{2\\tau} \\exp\\left(\\frac{\\tau(E - E_0)}{2} + \\frac{\\sigma^2\\tau^2}{8}\\right) \\cdot \\text{erfc}\\left(\\frac{E - E_0}{\\sqrt{2}\\sigma} + \\frac{\\sigma\\tau}{\\sqrt{2}}\\right)$$
### Energy-Dependent Resolution
$$\\sigma(E) = \\sqrt{\\sigma_a^2 + (\\sigma_b \\sqrt{E})^2 + F \\cdot E \\cdot w}$$
- $\\sigma_a$ — energy-independent contribution (electronic noise, etc.)
- $\\sigma_b$ — $\\sqrt{E}$ term (statistical fluctuations in charge creation)
- $F$ — Fano factor ($\\approx 0.12$ for Si, $\\approx 0$ for MMCs)
- $w$ — mean ionization energy ($\\approx 3.6$ eV for Si, converted to $m_e$ units)
## Calculator Implementation
> [!tip] Code mapping
> Module: `beta_spectrum/components/detector_response.py` → `DetectorResponse` class
>
> - **Analytical models**: Gaussian core, Gaussian + exponential tail (Tikhonov)
> - **Tabulated mode**: Linear interpolation from MC-calculated response matrix
> - **Convolution**: `DetectorResponse.convolve(W, spectrum, normalize=True)`
> - **Integration with BetaSpectrum**: `BetaSpectrum.convolve_with_detector(detector, W, config)`
> - **Integration with BetaSpectrumAnalyzer**: `BetaSpectrumAnalyzer.convolved_spectrum(detector, normalize)`
### API Usage
```python
from beta_spectrum import (
BetaSpectrum, SpectrumConfig, DetectorResponse,
BetaSpectrumAnalyzer, T_to_W
)
# 1. Create theoretical spectrum for 99Tc
config = SpectrumConfig(
Z_parent=43, Z_daughter=44, A_number=99,
endpoint_MeV=0.294 # 99Tc Q-value
)
spectrum = BetaSpectrum.from_config(config)
W, kinetic_MeV = spectrum.get_energy_grid(config)
# 2. Define detector response (sigma = 1.0 keV in m_e units)
sigma_1keV_me = 1.0 / 511.0 # convert keV to m_e units
W0 = T_to_W(0.294)
detector = DetectorResponse.from_gaussian_params(
channel_energy_range=(1.0, W0 + 0.05),
n_channels=512,
sigma_a=sigma_1keV_me,
sigma_b=0.0,
tail_fraction=0.0,
model="gaussian",
fano_factor=0.0,
)
# 3. Convolve
convolved = spectrum.convolve_with_detector(detector, W=W, config=config)
# 4. Or use the Analyzer
analyzer = BetaSpectrumAnalyzer(spectrum, config)
convolved_normalized = analyzer.convolved_spectrum(detector, normalize=True)
```
## Convolution Algorithm
The convolution proceeds in three steps:
1. **Build response matrix**: For each energy bin $W_i$, compute the detector response $R(E_j; W_i)$ across all channels $j$. Result: matrix of shape $(N\_{\\text{bins}}, N\_{\\text{channels}})$.
1. **Weight by spectrum**: Multiply each response row by the theoretical spectrum value $\\Phi(W_i)$ and bin width $\\Delta W_i$.
1. **Sum over energies**: For each channel $j$, sum contributions from all energy bins:
$$M_j = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i = (R^T \\cdot (\\Phi \\cdot \\Delta W))\_j$$
The bin width is computed as the half-bin average: $\\Delta W_i = \\frac{1}{2}(\\Delta W\_{i-1/2} + \\Delta W\_{i+1/2})$.
> [!note] Normalization
> Each response function $R(E; W_i)$ is normalized to unit area before convolution. This ensures that convolving a mono-energetic line produces a response with the same total counts. The final convolved spectrum preserves the integral of the theoretical spectrum.
## Energy Units
All detector response computations use **total energy in units of $m_e c^2$** (natural units, $m_e = 1$):
- $m_e c^2 = 510.998950$ keV
- Conversion: $W = 1 + T / (m_e c^2)$ where $T$ is kinetic energy
- A resolution of $\\sigma = 1.0$ keV corresponds to $\\sigma = 1.0 / 511.0 \\approx 0.00196$ in $m_e$ units
> [!warning] Unit consistency
> The detector channel energy range and the spectrum energy grid must be in the same units (total energy in $m_e$ units). Use `T_to_W()` from `beta_spectrum.utils` to convert kinetic energy in MeV to total energy in $m_e$ units.
## Physical Interpretation
The convolution operation smears the sharp theoretical spectrum into a realistic measured spectrum:
- **Peak broadening**: Finite energy resolution widens spectral features
- **Endpoint smearing**: The sharp cutoff at $W_0$ becomes a gradual fall-off
- **Tail effects**: Charge collection inefficiencies create low-energy tails (modeled by `gaussian_tail`)
- **Count conservation**: Total number of events is preserved (integral of convolved spectrum = integral of theoretical spectrum)
## References
- Hayen et al., RevModPhys **90**, 015008 (2017) — Comprehensive review of beta-decay theory
- Tikhonov, Soviet Physics Doklady **8**, 591 (1964) — Original formulation of the response function
- Sigmund, Phys. Rev. **130**, 1497 (1963) — Low-energy tailing in semiconductor detectors
docs/corrections/08-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: [[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 $\partial^2 B(G)/\partial Z^2$ 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
- [[corrections/03-screening]] — Screening modification from shake-up processes
- — Shake-off process correction
- [[corrections/09-chemical-effects]] — Molecular environment influence on atomic overlap
docs/corrections/09-chemical-effects.md
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---
title: "13 – Chemical Effects"
date: 2026-04-26
tags:
- beta-spectrum/correction/molecular
status: active
aliases: [Molecular Screening, Molecular Exchange, Saenz-Froelich]
cssclasses: []
related_components: [[corrections/08-atomic-overlap]], [[corrections/03-screening]]
---
# Chemical / Molecular Effects on β Spectrum
## Physical Origin
In many experiments (especially precision neutrino mass measurements), the decaying atom is bound within a molecule. This modifies:
- Electronic structure via molecular orbitals
- Coulombic final state interactions from additional electrons and spectator nuclei
- Energy transfer to molecular rovibrational states during decay
> [!warning] Systematic error impact
> In most precise analyses, this introduces a **systematic error of 0.05%**. Primarily relevant for tritium β-decay experiments (KATRIN).
## 1. Recoil Corrections in Molecules (§F.1)
When β decay occurs inside a molecule:
- The recoiling daughter nucleus moves through the molecular potential, not vacuum
- Potential described by Born-Oppenheimer energy curve, modelled with Lennard-Jones type potential
- Daughter nucleus kicks molecule into (predissociative) rovibrational state
**Dissociation channels:**
| Channel | Energy scale | Relevance |
|---|---:|---|
| Rovibrational states | ~few eV | Negligible at target precision |
| Electronic excitation → resonant continuum | tens–hundreds of eV | — |
Integrated probability depends on endpoint energy (higher endpoint → higher recoil). At $10^{-4}$ precision, this effect is neglected as a small correction on already small corrections.
For β-ν correlation measurements ($a_{\beta\nu}$):
- Dissociation probability decreases with increasing Z (recoil energy $\propto 1/m$)
- Can be partially included via Monte Carlo simulation in Lennard-Jones potential
## 2. Q Value Shift (§F.2)
Molecular effects add further decrease to Q value from **rotational and vibrational excitations**.
For $T_2$ (tritium molecule):
- Excitation possibilities change to a broad continuum with resonances
- Width of populated rovibrational spectrum: **few eV** — typically negligible
- Differences between tritium-substituted molecules: **few eV**, including atomic tritium
### Decay Rate Change (Eq. 172)
$$\frac{\Delta \lambda}{\lambda} \approx \frac{3 \Delta W_0}{W_0} \left( 1 + \gamma \frac{W_0}{6} \right) \tag{172}$$
where $\Delta W_0$ is difference in mean endpoint energy after averaging over all final states between two chemical states.
### Expected Error Level
Spectral shape depends on $W_0^2$, so relative error goes as $2 \times \sigma_Q / Q$. For ft values ($\propto Q^5$):
- Lowest energy transitions (tritium, $^{63}$Ni, $^{241}$Pu): **few $\times 10^{-4}$ to $10^{-3}$** level error for $\sigma_Q \sim$ tens of eV
## 3. Molecular Screening (§F.3) — Saenz & Froelich (1997a)
All Coulombic effects treated equally to first order, split into electronic and nuclear parts.
### Molecular Screening Correction (Eq. 176)
$$\Delta S_{\text{Mol}} = \alpha \left[ \frac{W}{p^2} + \frac{1}{W} \sum_S \frac{Z_S - Z_{S,\text{in}}}{R_{S,e}} - Z_{\text{eff}} \langle r^{-1} \rangle_{\text{Val}} \right]$$
where:
- $Z_S - Z_{S,\text{in}}$ = screened spectator nucleus charge
- $Z_{\text{eff}} = Z_{\text{Val}} - (Z - Z_{\text{in}})$
- $\langle r^{-1} \rangle_{\text{Val}}$ = average inverse distance of all valence electrons relative to decaying nucleus
- Sum over S runs through spectator nuclei, $R_{S,e}$ is equilibrium distance
### Example: $^{45}$Ca in CaCl₂ (Ca(II), doubly oxidized)
- Bond length: 2.437 Å → natural units $R_{\text{Cl},e} \sim 600$
- First term $\propto 3\times 10^{-5}$ — two orders of magnitude smaller than bare $V_0$
- Second term ($Z_{\text{eff}} = 2$): also ~**$2\times 10^{-5}$**, opposite sign
- Overall molecular deviation from atomic structure: **~$2\times 10^{-5}$**
Since screening is typically at (sub)percent level max, molecular deviations have upper limit of **few $\times 10^{-4}$**.
## 4. Molecular Exchange Effect (§F.4)
In a molecule, electronic phase space is greatly enlarged and perturbed relative to single atomic state. Electron density near decaying atom can both increase (bond region) or decrease, modifying the overlap integral.
### LCAO Approximation
Molecular orbitals constructed from linear combination of atomic orbitals. Key cases:
- **Ionic bond** (valence electron nearly fully removed): exchange effect ≈ 0 (internuclear distances >> β Compton wavelength where wave function oscillates rapidly)
- **Reverse extreme** (valence electron density doubled): naively doubles the exchange correction; conservative treatment uses **100% error bar on last orbital contribution**
### Error Estimate
| Isotope | Low-energy effect | High-energy behavior |
|---|---:|---|
| Generic case (Fig. 6) | < $1\times 10^{-4}$ from 15 keV; **0.5% at 1 keV** | — |
| $^{241}$Pu (7s valence) | Max ~1% in first 0.5 keV | Drops below $10^{-4}$ after 3 keV |
Effect of wave function change in lattice vs gas studied by Kolos et al. (1988) for tritium — no significant change found. Due to conservative error bar, **completely absorbed and can be neglected**.
## Calculator Implementation
> [!info] Not planned
> Chemical effects are NOT part of the calculator scope. They represent corrections on top of atomic calculations that only matter in specific experimental contexts (molecular tritium sources). The baseline calculator assumes isolated atoms.
For molecular experiments requiring these corrections:
- Use Saenz & Froelich formalism (§F.3) for screening modification
- Apply conservative error bars for exchange effects (~$10^{-4}$ from 15 keV onward)
- Q value shifts should be handled externally (experiment-specific input)
## Related Notes
- [[corrections/08-atomic-overlap]] — Atomic mismatch / Bahcall correction
- [[corrections/03-screening]] — Baseline atomic screening $S(Z,W)$