diff --git a/README.md b/README.md index f01a5aa..60cff4a 100644 --- a/README.md +++ b/README.md @@ -1,16 +1,13 @@ # Beta Spectrum Calculation Toolkit -## Overview +A modular Python package for calculating high-precision **beta decay energy spectra**. It models the spectral shape of nuclear transitions by combining well-established physical corrections into a coherent, configurable pipeline. -This project provides a modular and extensible framework for calculating high-precision **beta decay energy spectra**. It is designed to model the spectral shape of a given nuclear transition by combining well-established physical corrections into a coherent and configurable pipeline. - -The primary goal is to enable accurate numerical evaluation of the beta spectrum: - -* from near-zero kinetic energies up to the endpoint -* with controlled inclusion of higher-order corrections -* using a clean, composable architecture - -The framework focuses purely on **spectrum formation**, independent of any specific experimental or phenomenological application. +**Features:** +* Numerical evaluation from near-zero kinetic energies up to the endpoint +* Selective inclusion of higher-order corrections via `SpectrumConfig` toggles +* Clean, composable component architecture +* Built-in visualization and CSV export via `BetaSpectrumAnalyzer` +* Comprehensive scientific reference in [`docs/`](docs/) --- @@ -18,56 +15,48 @@ The framework focuses purely on **spectrum formation**, independent of any speci The beta spectrum is constructed as a product of independent correction factors: -[ -N(W) \propto p W (W_0 - W)^2 \times \prod_i C_i(W) -] +$$N(W) \propto p W (W_0 - W)^2 \times \prod_i C_i(W)$$ -Each correction is implemented as a standalone component: - -* phase space -* Coulomb interaction (Fermi function) -* finite nuclear size -* screening -* exchange -* radiative corrections -* recoil effects - -This modular approach allows: - -* easy validation of individual corrections -* selective enabling/disabling of physics effects -* flexible extension with new components +Each $C_i(W)$ is an independently implemented and toggleable component. --- ## Project Structure ``` -beta_corr_pnpi/ -├── beta_spectrum/ -│ ├── base.py -│ ├── spectrum.py -│ ├── constants.py -│ ├── utils.py -│ │ -│ ├── components/ -│ │ ├── phase_space.py -│ │ ├── fermi.py -│ │ ├── finite_size.py -│ │ ├── screening.py -│ │ ├── exchange.py -│ │ ├── radiative.py -│ │ └── recoil.py -│ │ -│ ├── data/ -│ │ └── exchange_coeff.csv -│ │ -│ └── __init__.py -│ -├── tests/ -│ +beta_shape_pnpi/ ├── pyproject.toml -├── README.md +├── docs/ # Obsidian knowledge base (physics theory & references) +│ ├── README.md # Vault overview + navigation index +│ ├── Hayen2017_summary.md # Full paper summary backbone +│ ├── 00-beta-spectrum-overview.md # Master equation, all corrections table +│ ├── 01-fermi-function.md # Fermi function F₀(Z,W) +│ ├── 02-phase-space.md # Phase space p·W·(W₀−W)² +│ ├── 03-finite-size.md # L₀, U, DFS corrections +│ ├── 04-screening-correction.md # Atomic screening S(Z,W) +│ ├── 05-exchange-correction.md # Electron exchange X(Z,W) +│ ├── 06-radiative-corrections.md # Outer radiative δᵣ(W,W₀) +│ ├── 07-recoil-effects.md # Weak magnetism, recoil kinematics +│ ├── 10-nuclear-structure.md # Shape factor C(Z,W), BB vs HS formalisms +│ ├── 12-atomic-overlap.md # Bahcall correction r(Z,W) +│ ├── 13-chemical-effects.md # Molecular environment effects +│ └── refs-hayen2017.md # Key equations reference table, citations +├── beta_spectrum/ +│ ├── __init__.py # Public API: constants, utilities, classes +│ ├── base.py # Abstract SpectrumComponent base class +│ ├── constants.py # Physical constants (natural units, m_e = 1) +│ ├── utils.py # Helpers: T<->W conversion, momentum, nuclear radius +│ ├── spectrum.py # BetaSpectrum + BetaSpectrumAnalyzer classes +│ └── components/ +│ │ ├── phase_space.py ✓ Phase space shape (p·W·(W₀−W)²) +│ │ ├── fermi.py ✓ Coulomb correction (loggamma for stability) +│ │ ├── finite_size.py ✓ L0 expansion + charge distribution U term +│ │ ├── screening.py ✓ Atomic electron screening (ratio method) +│ │ ├── exchange.py ✓ Hayen-2018 empirical fit coefficients +│ │ ├── radiative.py ✓ Outer radiative corrections, soft-photon resummation +│ │ └── recoil.py ✗ Not yet implemented +└── data/ + └── exchange_coeff.csv # Tabulated coefficients for X(Z,W), Z=2..120 ``` --- @@ -76,163 +65,109 @@ beta_corr_pnpi/ ### `SpectrumComponent` (base class) -Defined in `base.py`. - -All physics corrections inherit from this class and implement: +Defined in `beta_spectrum/base.py`. All physics corrections inherit from this abstract base and implement: ```python def __call__(self, W: np.ndarray) -> np.ndarray ``` -This makes every component: +This makes every component stateless (or minimally stateful), vectorized, and composable. -* stateless (or minimally stateful) -* vectorized -* composable +### `BetaSpectrum` + +Defined in `beta_spectrum/spectrum.py`. Orchestrates the calculation: +* Combines components multiplicatively via `__call__(W)` +* Factory method `from_config(config)` for declarative setup +* Generates energy grids with `get_energy_grid(config)` +* Returns individual component values via `calculate_components(W)` + +### `BetaSpectrumAnalyzer` + +Also in `spectrum.py`. Provides debugging and visualization tools: +* `total_spectrum(normalize=True/False)` — computed spectrum array +* `plot_analysis(save_path=None)` — 4-panel matplotlib figure (spectrum, components, cumulative effect, deviations from unity) +* `export_to_csv(filename)` — export via pandas DataFrame +* `get_data()` — return all numerical data for custom analysis --- -### `Spectrum` - -Defined in `spectrum.py`. - -Responsible for: - -* combining all components -* evaluating the total spectrum -* managing energy grids - -Typical usage pattern: +## Quick Start ```python -spectrum = Spectrum( - components=[ - PhaseSpace(W0), - FermiFunction(Z), - ScreeningCorrection(Z), - ExchangeCorrection(Z), - RadiativeCorrection(W0), - ] +from beta_spectrum import SpectrumConfig, BetaSpectrum, BetaSpectrumAnalyzer + +config = SpectrumConfig( + Z_parent=90, # Thorium-232 (example) + Z_daughter=91, + A_number=232, + endpoint_MeV=4.8, # Q-value ) +spectrum = BetaSpectrum.from_config(config) +W, E_MeV = spectrum.get_energy_grid(config) values = spectrum(W) + +analyzer = BetaSpectrumAnalyzer(spectrum, config) +analyzer.plot_analysis() # Show/ save 4-panel figure +analyzer.export_to_csv("output.csv") # Export data to CSV ``` --- ## Components -Each correction is implemented as an independent module under: +| Component | Module | Status | Description | +|---|---|---:|---| +| Phase Space | `phase_space.py` | ✓ | Baseline $p W (W_0 - W)^2$, optional neutrino mass support | +| Fermi Function | `fermi.py` | ✓ | Relativistic Coulomb correction via `scipy.special.loggamma` | +| Finite Size L0 | `finite_size.py` | ✓ | Low-Z expansion from Hayen et al. with prefactor | +| Charge Distribution U | `finite_size.py` | ✓ | Second-order $(1/5)(\alpha Z W R)^2$ correction | +| Screening | `screening.py` | ✓ | Ratio method with logistic switching, energy shift $V_0$ | +| Exchange | `exchange.py` | ✓ | Hayen-2018 Table X empirical fit (CSV data for Z=2..120) | +| Radiative | `radiative.py` | ✓ | Outer radiative corrections (Hayen Eq. 47–53), soft-photon resummation at endpoint | +| Recoil | `recoil.py` | ✗ | Stub only, not yet implemented | -``` -beta_spectrum/components/ +--- + +## Configuration + +Use `SpectrumConfig` to selectively enable/disable each correction: + +```python +config = SpectrumConfig( + Z_parent=20, Z_daughter=21, A_number=40, endpoint_MeV=5.0, + use_fermi=True, + use_screening=False, # disable atomic screening + use_exchange=True, + use_radiative=True, +) ``` --- -### Phase Space (`phase_space.py`) +## Dependencies -Implements: - -[ -p W (W_0 - W)^2 -] - -Provides the baseline spectral shape. - ---- - -### Fermi Function (`fermi.py`) - -Accounts for Coulomb interaction between the emitted electron and nucleus. - ---- - -### Finite Size (`finite_size.py`) - -Corrects for non-pointlike nuclear charge distribution. - ---- - -### Screening (`screening.py`) - -Includes atomic electron screening effects. - -* modifies low-energy behavior -* becomes negligible at high energies - ---- - -### Exchange (`exchange.py`) - -Accounts for electron exchange with atomic orbitals. - -* dominant at very low energies -* requires empirical coefficients (loaded from CSV) - ---- - -### Radiative (`radiative.py`) - -Implements outer radiative corrections: - -* based on analytical expressions -* includes optional endpoint resummation -* handles logarithmic divergences near endpoint - ---- - -### Recoil (`recoil.py`) - -Includes nuclear recoil corrections. - ---- - -## Data Files - -### `exchange_coeff.csv` - -Contains tabulated coefficients for exchange correction. - -Loaded dynamically based on nuclear charge ( Z ). +**Runtime:** numpy, scipy (for `loggamma`, `spence`), matplotlib (plotting), pandas (CSV export) +**Dev:** pytest, black, ruff, mypy (strict mode configured in `pyproject.toml`) --- ## Design Principles ### 1. Modularity - -Each physical effect is isolated in its own component. - ---- +Each physical effect is isolated in its own component under `beta_spectrum/components/`. ### 2. Numerical Stability - -Special care is taken near: - -* ( W \to 1 ) (threshold) -* ( W \to W_0 ) (endpoint) - -Typical strategies include: - -* small epsilon regularization -* safe evaluation of logarithms and powers -* masking invalid regions - ---- +Care near $W \to 1$ (threshold) and $W \to W_0$ (endpoint): +* `np.clip` / epsilon floors for regularization +* Safe logarithms via `scipy.special.loggamma` and dilogarithm +* Masking invalid regions ### 3. Vectorization - -All components operate on NumPy arrays: - -* no Python loops in hot paths -* efficient evaluation over dense grids - ---- +All components operate on NumPy arrays — no Python loops in hot paths. ### 4. Extensibility - -New corrections can be added by subclassing: +New corrections are added by subclassing: ```python class NewCorrection(SpectrumComponent): @@ -242,47 +177,39 @@ class NewCorrection(SpectrumComponent): --- -## Typical Workflow +## Development Status -1. Define endpoint energy ( W_0 ) -2. Construct energy grid -3. Initialize components -4. Combine into spectrum -5. Evaluate and plot +**Version:** 0.1.2 +**Status:** Active development — core physics engine is functional; recoil correction and tests remain incomplete. + +### Remaining Work + +* Implement nuclear recoil correction in `recoil.py` (see [[07-recoil-effects]]) +* Add unit / integration tests (pytest configured but no test files) +* Declare pandas as a runtime dependency in `pyproject.toml` +* Satisfy mypy strict type-checking requirements +* CLI entry point and multiple decay branch support --- -## Future Extensions +## Knowledge Base -Planned improvements include: +The [`docs/`](docs/) directory contains the complete scientific reference for beta spectrum shape corrections, organized as an Obsidian vault. It is structured around Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) and maps directly to calculator components: -* support for multiple decay branches -* nuclear shape factor implementations -* improved atomic corrections -* validation against reference spectra -* export utilities and CLI tools +| # | Note | Calculator Module | Status | +|---|------|-------------------|--------| +| 1 | [[01-fermi-function]] | `components/fermi.py` | ✓ implemented | +| 2 | [[03-finite-size]] | `components/finite_size.py` | ✓ implemented | +| 3 | [[04-screening-correction]] | `components/screening.py` | ✓ implemented | +| 4 | [[05-exchange-correction]] | `components/exchange.py` | ✓ implemented | +| 5 | [[06-radiative-corrections]] | `components/radiative.py` | ✓ implemented | +| 6 | [[07-recoil-effects]] | `components/recoil.py` | ✗ not yet implemented | + +Notes prefixed with `10-`, `12-`, `13-` cover nuclear structure, atomic overlap, and chemical effects — important for future extensions but outside the current calculator scope. --- ## Notes -* Units are typically expressed in natural units ( m_e = 1 ) -* Energies can be converted from MeV externally +* Physical constants are defined in natural units ($m_e = 1$); energies can be converted from MeV via utilities in `utils.py` * The framework assumes familiarity with beta decay theory - ---- - -## License / Usage - -(To be defined) - ---- - -## Author Notes - -This project is under active development and serves both as: - -* a computational tool -* a structured reference for beta spectrum calculations - -Updates may refine both physics implementations and code structure. diff --git a/docs/.obsidian/app.json b/docs/.obsidian/app.json new file mode 100644 index 0000000..9e26dfe --- /dev/null +++ b/docs/.obsidian/app.json @@ -0,0 +1 @@ +{} \ No newline at end of file diff --git a/docs/.obsidian/appearance.json b/docs/.obsidian/appearance.json new file mode 100644 index 0000000..9e26dfe --- /dev/null +++ b/docs/.obsidian/appearance.json @@ -0,0 +1 @@ +{} \ No newline at 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V_{ud}^2}{2\pi^3} F_0 L_0 U DFS R R_N Q S X r C D_C \; p W (W_0 - W)^2 dW \tag{4}$$ + +## Correction Factors + +| Symbol | Name | Module / Note | Status | +|---|---|---|---| +| $F_0(Z,W)$ | Point charge Fermi function | [[01-fermi-function]] | ✓ | +| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[03-finite-size]] | ✓ | +| $U(Z,W)$ | Charge distribution correction | [[03-finite-size]] | ✓ | +| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[03-finite-size]] | ✓ | +| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[06-radiative-corrections]] | ✓ | +| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[07-recoil-effects]] | ✗ | +| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[07-recoil-effects]] | — | +| $S(Z,W)$ | Atomic electron screening | [[04-screening-correction]] | ✓ | +| $X(Z,W)$ | Atomic exchange effect | [[05-exchange-correction]] | ✓ | +| $r(Z,W)$ | Atomic mismatch / overlap | [[12-atomic-overlap]] | — | +| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[12-atomic-overlap]] | — | +| $C(Z,W)$ | Nuclear structure shape factor | [[10-nuclear-structure]] | — | +| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[10-nuclear-structure]] | — | + +## Baseline: Phase Space + +The uncorrected spectral shape is pure phase space ([[02-phase-space]]): + +$$\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2$$ + +with $p = \sqrt{W^2-1}$, $W$ total energy in units of $m_e$, and $W_0$ the endpoint. + +## Precision Goal + +The paper achieves **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude [[Hayen2017_summary]]. + +> [!key] Why this matters +> A precision of ~0.5% is required when determining the Fierz term $b_{\text{Fierz}}$ to improve limits on BSM scalar/tensor couplings. Weak magnetism $b_{wm}$ must be included as experimental precision increases, otherwise it limits BSM sensitivity. + +## Three Formalisms Compared + +| Approach | Style | Strengths | +|---|---|---| +| **Behrens-Bühring** (BB) | Rigorous, numerical Dirac solutions | Most precise results | +| **Holstein** (HS) | Analytical, manifestly covariant | Clean symmetry properties | +| **Wilkinson** | Analytical parametrizations | Practical fits for dominant effects | + +The Hayen paper combines BB's accuracy with HS's transparency. + +## Conventions + +- Natural units: $c = \hbar = m_e = 1$ throughout +- $Z$ is always positive; $\beta^-$ and $\beta^+$ distinguished via upper/lower signs ($\mp$) +- $W = E/m_ec^2 + 1$, $p = \sqrt{W^2 - 1}$ +- $G_V$: vector coupling strength; $V_{ud} = \cos^2\theta_C$ + +> [!note] Two types of "finite size" +> The authors distinguish two corrections conflated under this term: +> 1. **Electrostatic** ($L_0, U, DFS$): Potential difference from point charge → realistic nuclear shape +> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume (appears in $C(Z,W)$) + +## Energy Conversion + +$$W = 1 + \frac{T}{m_ec^2}, \quad m_ec^2 = 510.998950\text{ keV}$$ + +In code: see `beta_spectrum/utils.py` → `T_to_W()`, `W_to_T()` functions. + +## Calculator Implementation + +Each correction factor $C_i(W)$ is a callable class inheriting from `SpectrumComponent`. The total spectrum is the product of all enabled components: + +``` +N(W) = PhaseSpace(W0) × FermiFunction(Z) × FiniteSizeL0(Z,A) × ... +``` + +Configuration toggles are defined in `SpectrumConfig` (spectrum.py). See [[README]] for architecture details. diff --git a/docs/01-fermi-function.md b/docs/01-fermi-function.md new file mode 100644 index 0000000..00bfa0f --- /dev/null +++ b/docs/01-fermi-function.md @@ -0,0 +1,74 @@ +--- +title: "01 – Fermi Function" +date: 2026-04-26 +tags: + - beta-spectrum/correction/fermi +status: active +aliases: [Coulomb Correction, F_0] +cssclasses: [] +related_components: [[03-finite-size]], [[06-radiative-corrections]] +--- + +# Fermi Function — Coulomb Interaction $F_0(Z,W)$ + +## The Point Charge Fermi Function + +The transition matrix element with all-order Coulomb interactions resummed (Hayen et al., Eq. 5): + +$$\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$$ + +Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[03-finite-size]]). + +## Coulomb Amplitudes + +Expanding radial Dirac wave functions near the origin (Eq. 12): + +$$\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}$$ + +This is valid regardless of nuclear charge distribution shape. + +## Alternative Definitions + +| Convention | Prefactor for $L_0$ | Style | +|---|---|---| +| **Wilkinson / Blatt & Weisskopf** | $(1+\gamma)/2$ | Absorbs into $L_0$ | +| **Behrens-Bühring / Schopper** (used in calculator) | 4 | All corrections separate | + +The error from neglecting the $(1+\gamma)/2$ prefactor is ~0.5% for $Z=20$. + +## Numerical Implementation + +> [!tip] Code mapping +> Module: `beta_spectrum/components/fermi.py` → `FermiFunction(Z, A)` class +> +> Uses `scipy.special.loggamma` for numerical stability (avoids overflow of $\Gamma$ function). The implementation follows Eq. (6) with log-space evaluation: +> ```python +> log_F0 = ln(4) + 2*(γ-1)*ln(2*p*R) + π*y +> + 2*Re[loggamma(γ+iy)] - 2*loggamma(1+2γ) +> F0 = exp(log_F0) +> ``` + +## Energy Dependence + +$F_0$ is a strong function of $Z$: +- **β⁻ decay** ($y < 0$): wave functions attracted to nucleus → enhancement at low energy +- **β⁺ decay** ($y > 0$): repulsion → suppression at low energy, grows with energy + +The effect ranges from ~1% for light nuclei to tens of percent for heavy ones. + +## Related Corrections + +After the point charge Fermi function is computed: +- [[03-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$) +- [[06-radiative-corrections]] adds QED radiative effects ($R = 1 + \delta_R$) diff --git a/docs/02-phase-space.md b/docs/02-phase-space.md new file mode 100644 index 0000000..9be9d24 --- /dev/null +++ b/docs/02-phase-space.md @@ -0,0 +1,68 @@ +--- +title: "02 – Phase Space" +date: 2026-04-26 +tags: + - beta-spectrum/correction/phase-space +status: active +aliases: [Phase Space, Baseline Spectrum] +cssclasses: [] +related_components: [[01-fermi-function]], [[07-recoil-effects]] +--- + +# Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$ + +## The Unmodified Spectrum + +The baseline β spectrum shape without any corrections is pure phase space: + +$$\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2 \tag{4, first factor}$$ + +where: +- $p = \sqrt{W^2 - 1}$ — electron momentum ($m_e c$ units) +- $W$ — total electron energy ($m_e c^2$ units) +- $W_0$ — endpoint (Q-value + $m_ec^2$) + +## Neutrino Mass Effects + +If the neutrino has non-zero mass $m_\nu$, the spectrum is modified near the endpoint: + +$$p \to p_e = \sqrt{W^2 - 1}, \quad p_\nu = \sqrt{(W_0 - W)^2 - m_\nu^2}$$ + +The phase space factor becomes: +$$(W_0 - W)\sqrt{(W_0 - W)^2 - m_\nu^2}$$ + +This produces a characteristic distortion near $W = W_0 - m_\nu$, used in neutrino mass experiments (e.g., KATRIN with tritium β decay). + +## Kurie Plot + +The phase space subtracted spectrum gives the **Kurie function**: + +$$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for allowed transitions})$$ + +Deviations from linearity near the endpoint indicate: +- Non-zero neutrino mass +- Weak magnetism / recoil effects ([[07-recoil-effects]]) +- New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]]) + +## Calculator Implementation + +> [!tip] Code mapping +> Module: `beta_spectrum/components/phase_space.py` → `PhaseSpace(W0)` class +> +> Simple evaluation of $p \times W \times (W_0 - W)^2$ with masking near endpoint for numerical safety. Optionally includes neutrino mass support. + +## Energy Range + +The phase space factor: +- Vanishes at $W = 1$ ($T = 0$, the threshold) — goes as $(W-1)^{1/2}$ via momentum +- Vanishes at $W = W_0$ (the endpoint) — quadratic behavior from neutrino energy +- Peaks roughly at $W \approx W_0 / 3$ to $W_0 / 2$ depending on $Z$ + +## Code API + +```python +from beta_spectrum import PhaseSpace, T_to_W + +ps = PhaseSpace(W0=T_to_W(5.0)) # endpoint = 5 MeV +values = ps(W_grid) # returns p·W·(W₀-W)² +``` diff --git a/docs/03-finite-size.md b/docs/03-finite-size.md new file mode 100644 index 0000000..b40049c --- /dev/null +++ b/docs/03-finite-size.md @@ -0,0 +1,139 @@ +--- +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: [[01-fermi-function]], [[10-nuclear-structure]] +--- + +# Finite Nuclear Size Effects — $L_0$, $U(Z,W)$, and $DFS$ + +## Overview + +The Fermi function ([[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)$ ([[10-nuclear-structure]]) + +## 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$ ([[10-nuclear-structure]]). 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 diff --git a/docs/04-screening-correction.md b/docs/04-screening-correction.md new file mode 100644 index 0000000..822df72 --- /dev/null +++ b/docs/04-screening-correction.md @@ -0,0 +1,93 @@ +--- +title: "04 – Screening Correction" +date: 2026-04-26 +tags: + - beta-spectrum/correction/screening +status: active +aliases: [Atomic Screening, S(Z,W)] +cssclasses: [] +related_components: [[12-atomic-overlap]], [[13-chemical-effects]] +--- + +# Atomic Electron Screening Correction $S(Z,W)$ + +## Physical Origin + +The Fermi function ([[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 + +- [[12-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction +- [[13-chemical-effects]] — Molecular environment influence on screening and exchange +- [[05-exchange-correction]] — Electron exchange with atomic orbitals (often combined with screening) diff --git a/docs/05-exchange-correction.md b/docs/05-exchange-correction.md new file mode 100644 index 0000000..8b7423d --- /dev/null +++ b/docs/05-exchange-correction.md @@ -0,0 +1,88 @@ +--- +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: [[04-screening-correction]], [[12-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/hayen-2018]]. 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. diff --git a/docs/06-radiative-corrections.md b/docs/06-radiative-corrections.md new file mode 100644 index 0000000..b04465a --- /dev/null +++ b/docs/06-radiative-corrections.md @@ -0,0 +1,129 @@ +--- +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}$$ + +> [!warning] Endpoint divergence +> $g(W_0, W)$ has a **logarithmic divergence at $W = W_0$** related to soft real photon emission. This must be handled via resummation. + +### Soft Photon Resummation (Eq. 53–54) + +To handle the endpoint divergence: +$$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}$$ + +For tritium the effect is negligible (low endpoint). For higher energies, the $ft$ value shift can reach several parts in $10^4$. + +### Order Zα² Correction (Eq. 55–62) + +Three dominant Feynman diagrams: vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to $\Delta_3$: + +$$\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, use_endpoint_resummation=True)` class +> +> Implements outer radiative corrections per Hayen Eqs. (47)–(53): +> - Sirlin's $g(W_0,W)$ with `scipy.special.spence` for dilogarithm +> - Soft-photon resummation at endpoint via Eq. (53) +> - Order Zα² terms Δ₁ through Δ₄ +> - Optional endpoint resummation flag + +### API Usage + +```python +from beta_spectrum import RadiativeCorrection + +rad = RadiativeCorrection( + W0=1.02, # endpoint in m_e units + use_endpoint_resummation=True # handles endpoint divergence +) +values = rad(W_grid) +``` + +## 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 | diff --git a/docs/07-recoil-effects.md b/docs/07-recoil-effects.md new file mode 100644 index 0000000..c0cb803 --- /dev/null +++ b/docs/07-recoil-effects.md @@ -0,0 +1,101 @@ +--- +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]], [[10-nuclear-structure]] +--- + +# 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}$$ + +The weak magnetism term $\propto W/M$ grows with energy — opposite to Fierz term which falls as $1/W$. This distinct energy dependence allows experimental separation. + +## Implementation Plan for Calculator + +To implement recoil in `recoil.py`: +1. Create `RecoilCorrection` class inheriting from `SpectrumComponent` +2. Implement $R_N(W,W_0,M)$ per Eqs. (41)–(44) with transition-type parameter (Fermi/GT/mixed) +3. Optionally implement $Q(Z,W,M)$ for electromagnetic recoil +4. Accept weak magnetism form factor from nuclear data or CVC calculation diff --git a/docs/10-nuclear-structure.md b/docs/10-nuclear-structure.md new file mode 100644 index 0000000..773daa1 --- /dev/null +++ b/docs/10-nuclear-structure.md @@ -0,0 +1,93 @@ +--- +title: "10 – Nuclear Structure Effects" +date: 2026-04-26 +tags: + - beta-spectrum/nuclear-structure +status: active +aliases: [Shape Factor, C(Z,W), Form Factors] +cssclasses: [] +related_components: [[07-recoil-effects]], [[11-isospin-breakdown]] +--- + +# Nuclear Structure Effects — The Shape Factor $C(Z,W)$ + +## Overview + +Nuclear structure effects are encapsulated in the shape factor $C(Z,W)$ which multiplies the phase space × Fermi function. This is distinct from electrostatic finite size corrections ([[03-finite-size]]) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction. + +$$N(W)\,dW \propto p W (W_0 - W)^2 F(Z,W) C(Z,W) \tag{79}$$ + +## Two Formalisms Compared + +| Approach | Style | Strengths | +|---|---|---| +| **Behrens-Bühring** (BB) | Couples leptonic + nuclear parts; rigorous expansion in $r^2$, $(m_eR)^a$, $(WR)^b$, $(\alpha Z)^c$ with tabulated functions $I(k_e,m,n,\rho)$ | Most precise, established computational machinery | +| **Holstein** (HS) | Decouples leptonic + nuclear parts; manifestly covariant form factors | Clean symmetry properties, intuitive classification | + +Hayen et al. adopt BB's computational approach but present results in Holstein notation for clarity. See Appendix E of Hayen paper for detailed comparison. + +## Multipole Expansion (Eq. 78) + +Form factors $F_{KLs}(q^2)$ describe nuclear structure model-independently: + +$$\langle f | V_\mu - A_\mu | i \rangle \propto \sum_{KMs} \sum_{L=K-1}^{K+1} (-1)^{J_f-M_f+M} (-i)^L F_{KLs}(q^2) $$ +$$\times \sqrt{4\pi(2J_i+1)} \begin{pmatrix} J_f & K & J_i \\ -M_f & M & M_i \end{pmatrix} \frac{(qR)^L}{(2L+1)!!}$$ + +Since $(qR)^2$ is very small for beta decay, expansion typically stops after first order. + +## Impulse Approximation (Eq. 80–81) + +Nucleons treated as non-interacting particles coupling to weak vertex as free particles: + +$$O_{KLs} = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle a^\dagger_\alpha a_\beta \tag{80}$$ +$$\langle f | O_{KLs} | i \rangle = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle \langle f | a^\dagger_\alpha a_\beta | i \rangle \tag{81}$$ + +The one-body density matrix elements $\langle f | a^\dagger_\alpha a_\beta | i \rangle$ are calculable via shell model methods. + +> [!warning] Limitations +> The impulse approximation neglects many-body effects: +> - **Meson exchange**: long history (Blin-Stoyle, Chemtob & Rho) +> - **Core polarization**: nuclear medium modifications +> +> For allowed decays, results largely hold due to cancellations between core-polarization and meson exchange (Morita 1985), but individual effects can reach ~40%. + +## Induced Currents — Form Factor Classification + +See [[07-recoil-effects]] for detailed equations. Key points: + +### First-Class Currents (CVC/PCAC) +- **Vector** $g_V$: conserved, from EM data via CVC +- **Weak magnetism** $(g_M - g_V)/2M$: determined by CVC from M1 transitions +- **Axial-vector** $g_A$: measured in neutron decay +- **Induced pseudoscalar** $g_P/2M$: PCAC predicts ~−229, quenched ~80% in nuclei + +### Second-Class Currents (G-parity forbidden) +- **Induced scalar** $g_S/2M$: excluded by CVC for Fermi transitions +- **Induced tensor** $g_{II}/2M$: experimental null results; reduces to first-class component only + +## Nuclear Matrix Element Operators (Table V of Hayen) + +| Operator | Symbol | Type | $\Delta J$ | +|---|---|---|---| +| $M_F$ | Fermi matrix element | Vector | 0 | +| $M_{GT}$ | Gamow-Teller matrix element | Axial-vector | 0,1 (not 0→0) | +| $M_L$ | Longitudinal vector | — | depends on K | +| $M_{T}, M_{TL}$ | Transverse multipoles | — | depends on K | + +## Validity of Impulse Approximation + +- **Vector current**: conserved via CVC → impulse approximation reliable +- **Axial current**: broken by PCAC → quenching effects significant (up to 80% for $g_P$) +- Mesonic degrees of freedom absorbed into nuclear potential (Wilkinson 1974) + +## Nuclear Deformation Correction $D_C(Z,W,\beta^2)$ + +The nuclear structure analogue of the electrostatic deformation correction ([[03-finite-size]]): +- Accounts for non-spherical nuclear shape in convolution terms +- Partially cancels with $DFS$ correction (electrostatic) +- Relevant for deformed nuclei ($\beta_2 \neq 0$) + +## Calculator Status + +> [!info] Not yet implemented +> The shape factor $C(Z,W)$ is NOT part of the current calculator. This note documents the theory for future implementation. For superallowed Fermi transitions, $C(Z,W) = |M_F|^2 \approx 1$ (isospin symmetry), so it can be approximated as unity in many cases. diff --git a/docs/12-atomic-overlap.md b/docs/12-atomic-overlap.md new file mode 100644 index 0000000..f810e22 --- /dev/null +++ b/docs/12-atomic-overlap.md @@ -0,0 +1,75 @@ +--- +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 diff --git a/docs/13-chemical-effects.md b/docs/13-chemical-effects.md new file mode 100644 index 0000000..703502a --- /dev/null +++ b/docs/13-chemical-effects.md @@ -0,0 +1,117 @@ +--- +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: [[12-atomic-overlap]], [[04-screening-correction]] +--- + +# 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 + +- [[12-atomic-overlap]] — Atomic mismatch / Bahcall correction +- [[04-screening-correction]] — Baseline atomic screening $S(Z,W)$ diff --git a/docs/Hayen2017_summary.md b/docs/Hayen2017_summary.md new file mode 100644 index 0000000..e8d4fcf --- /dev/null +++ b/docs/Hayen2017_summary.md @@ -0,0 +1,1275 @@ +# Hayen et al. (2017) — High Precision Analytical Description of the Allowed β Spectrum Shape: Summary of Chunks 1–6 + +## Title, Authors, Abstract + +**Title:** *High precision analytical description of the allowed β spectrum shape* + +**Authors:** Leendert Hayen and Nathal Severijns (KU Leuven, Belgium); Kazimierz Bodek and Dagmara Rozpedzik (Jagiellonian University, Poland); Xavier Mougeot (CEA LIST, France) + +**arXiv:** 1709.07530v2 [nucl-th], dated September 26, 2017 + +### Abstract Summary + +The paper presents a fully analytical description of the allowed beta decay spectrum shape for ongoing and planned measurements aimed at searching for physics beyond the Standard Electroweak Model (BSM) and investigating weak magnetism recoil terms. Contributions from finite size corrections, mass effects, and radiative corrections are reviewed. A particular focus is placed on **atomic and chemical effects**, extending the existing description analytically. The effects of QCD-induced recoil terms are discussed with cross-checks across different theoretical formalisms. Corrections are derived for both Fermi and Gamow-Teller transitions. Calculated $f$ values agree with the most precise numerical results within the aimed-for precision. The paper stresses the need for accurate evaluation of weak magnetism contributions and notes the possible significance of the oft-neglected induced pseudoscalar interaction. Together with improved atomic corrections, an analytical description is presented **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude. + +--- + +## I. INTRODUCTION + +### Motivation: Beta Decay as a Probe for New Physics + +Beta decay played a pivotal role in uncovering the left-handed V-A weak interaction and the electroweak sector of the Standard Model (SM). In the LHC era, competitive BSM results are extracted from low-energy beta decay measurements through Effective Field Theories (EFT), where new physics at scale $\Lambda_{\text{BSM}} \gg \Lambda_{\text{LHC}}$ is parameterized. + +The **Lee-Yang Hamiltonian** is generalized to include higher-dimensional SM field operator combinations. Searches for non-V-A manifestations focus on: +- Exotic scalar/tensor couplings (right-handed currents involving new heavy particles) +- Lorentz invariance violation via sidereal variations of observables + +### The Fierz Interference Term + +Scalar and tensor BSM coupling constants appear in the beta spectrum shape through the **Fierz interference term** (Jackson et al., 1957): + +$$b_{\text{Fierz}} \simeq \pm \frac{1}{1 + \rho^2} \left[ \text{Re}\left(\frac{C_S + C'_S}{C_V}\right) + \rho^2 \, \text{Re}\left(\frac{C_T + C'_T}{C_A}\right) \right], \tag{1}$$ + +where $\rho = \mathcal{M}_{GT} / \mathcal{M}_F$ is the ratio of Gamow-Teller to Fermi matrix elements, and $C_{V,A}$ are SM vector/axial-vector coupling constants while $C_S, C_T$ are possible scalar/tensor couplings. For pure transitions: + +$$b_{\text{Fierz},F} \simeq \pm \text{Re}\left(\frac{C_S + C'_S}{C_V}\right), \tag{2a}$$ +$$b_{\text{Fierz},GT} \simeq \pm \text{Re}\left(\frac{C_T + C'_T}{C_A}\right). \tag{2b}$$ + +A precision of typically **0.5%** is required when determining $b_{\text{Fierz}}$ to improve limits on scalar/tensor coupling constants. + +### Weak Magnetism and Spectral Shape Modification + +QCD-induced form factors — dominated by the **weak magnetism term** $b_{wm}$ — must be included as experimental precision increases, since they otherwise limit BSM sensitivity: + +$$N(W)\,dW \propto pW(W_0 - W)^2 \left[ 1 + \frac{\gamma m_e}{W} b_{\text{Fierz}} \pm \frac{4W}{3M} b_{wm} \right] dW, \tag{3}$$ + +where: +- $p$, $W$, $W_0$ are the beta particle momentum, total energy, and endpoint energy respectively +- $\gamma = \sqrt{1 - (\alpha Z)^2}$ with $\alpha$ the fine structure constant and $Z$ the daughter atomic number +- $m_e$ is the electron rest mass and $M$ is the average mother/daughter nucleus mass + +To search for new physics or establish weak magnetism effects, a beta spectrum description reliable at the **$10^{-4}$ level** is required. + +### Previous Theoretical Approaches + +Three major formalisms are reviewed: +1. **Behrens-Bühring** (and collaborators) — rigorous but relies on numerical solutions of the Dirac equation +2. **Holstein** — analytical approach with clean notation +3. **Wilkinson** — analytical parametrizations of dominant effects + +The paper's goal is to combine the computational accuracy of Behrens-Bühring with the transparency of Holstein, achieving $10^{-4}$ precision analytically. + +### Atomic and Chemical Effects (New Contributions) + +Special attention improves the analytical description of: +- **Screening effects** — combining Bühring's theoretical work with high-precision atomic potentials for all $Z$ values; an analytical fit is proposed +- **Atomic exchange effect** — based on state-of-the-art numerical calculations, its contribution can exceed 20% in low-energy regions for higher $Z$ +- **Shake-up and shake-off processes** — reviewed for their influence on atomic effects +- **Molecular effects** — explored for the first time at this precision level + +--- + +## II. COMPLETE EXPRESSION + +The fully differential allowed beta spectrum shape, including all corrections needed for $10^{-4}$ precision, is: + +$$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} F_0(Z,W) L_0(Z,W) U(Z,W) DFS(Z,W,\beta^2) R(W,W_0) R_N(W,W_0,M)$$ +$$\times Q(Z,W) S(Z,W) X(Z,W) r(Z,W) C(Z,W) D_C(Z,W,\beta^2) \, pW(W_0 - W)^2 dW \tag{4}$$ + +$$\equiv \frac{G_V^2 V_{ud}^2}{2\pi^3} K(Z,W,W_0,M) \, A(Z,W) \, C'(Z,W) \, pW(W_0-W)^2 dW.$$ + +### Definitions of All Factors + +| Symbol | Name | Description | +|---|---|---| +| $F_0(Z,W)$ | Point charge Fermi function | Coulomb interaction between beta particle and daughter nucleus (Eq. 6) | +| $L_0(Z,W)$ | Electrostatic finite size correction | Corrections to Fermi function when evaluating at origin for uniform sphere | +| $U(Z,W)$ | More realistic charge distribution correction | Difference between actual nuclear shape and uniform sphere | +| $DFS(Z,W,\beta^2)$ | Deformation correction | Effect of axially symmetric nuclear deformation on the Fermi function | +| $R(W,W_0)$ | Outer radiative corrections | QED corrections (Sec. V) | +| $R_N(W,W_0,M)$ | Finite nuclear mass recoil correction | Kinematic effects from finite nucleus mass (Sec. IV.B.1) | +| $Q(Z,W,M)$ | Recoil electromagnetic correction | Coulomb field modification due to nuclear recoil (Sec. IV.B.2) | +| $S(Z,W)$ | Screening correction | Atomic electron screening effects (Sec. VII.A) | +| $X(Z,W)$ | Atomic exchange effect | Electron exchange with atomic orbitals (Sec. VII.B) | +| $r(Z,W)$ | Atomic mismatch / overlap | Alternative atomic excitation correction (Sec. VII.D) | +| $A(Z,W)$ | Combined atomic factors | $S \cdot X \cdot r$ | +| $C(Z,W)$ | Nuclear structure sensitive correction | Contains all nuclear form factors and matrix elements (Sec. VI) | +| $D_C(Z,W,\beta^2)$ | Nuclear deformation correction | Nuclear shape effects in the convolution terms (Sec. VI.F) | + +### Key Conventions + +- **Natural units**: $c = \hbar = m_e = 1$ throughout all formulae +- **Z convention**: $Z$ is always positive; $\beta^-$ and $\beta^+$ are distinguished explicitly via upper/lower signs ($\mp$) +- $W = E/m_ec^2 + 1$ (total beta energy in units of electron rest mass) +- $p = \sqrt{W^2 - 1}$ (momentum in units of $m_ec$) +- $G_V$: vector coupling strength in nuclei; $V_{ud} = \cos^2\theta_C$ + +### Terminology Clarification: "Finite Size Effects" + +The authors distinguish two types that different authors conflate under "finite size": + +1. **Electrostatic finite size** ($L_0, U, DFS$): Stems from the electric potential difference when moving from point charge to realistic nuclear shape. These are corrections applied *after* evaluating the electron wave function at the origin. + +2. **Convolution finite size**: Results from integrating leptonic wave functions over the nuclear volume. This is sensitive to nuclear structure and appears in $C(Z,W)$. The authors note that different communities use "finite size" to refer to entirely different corrections, and aim to clarify this distinction. + +--- + +## III. FERMI FUNCTION + +### The Point Charge Fermi Function + +The transition matrix element (to first order in the weak Hamiltonian with all-order Coulomb interactions resummed) is: + +$$\mathcal{M}_{fi} = -2\pi i \delta(E_f - E_i) \langle f | T \exp\left(-i\int_0^\infty dt H_Z(t)\right) $$ +$$\times H_\beta(0) T \exp\left(-i\int_{-\infty}^0 dt H_{Z'}(t)\right) | i \rangle, \tag{5}$$ + +The typical approximation is to keep only the final-state Coulomb interaction (daughter nucleus charge $Z$), with corrections for radiative effects and atomic influences. + +**Fermi's point charge Fermi function** (Eq. 6): + +$$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. \tag{7}$$ + +Here $R$ is the cutoff radius (representing daughter nucleus radius) necessitated by divergence at origin for point charge, though it has no real physical significance — most of its dependence is absorbed into corrections. + +### Alternative Fermi Function Definitions + +The paper reviews multiple conventions: + +- **Wilkinson / Blatt & Weisskopf / Fano / Konopinski style**: Uses prefactor $(1+\gamma)/2$ instead of 4, absorbing it into $L_0$. +- **Behrens-Bühring / Schopper style** (followed by this paper): Keeps Eq. (6) with factor 4, and includes all additional corrections in $L_0$ and subsequent factors. + +### Finite Size Correction from Power Expansion + +Expanding the radial Dirac wave functions near the origin: + +$$\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, \tag{12}$$ + +the Coulomb information is encoded in **Coulomb amplitudes** $\alpha_\kappa$. The dominant electron component at the origin becomes: + +$$F_0 L_0 = \frac{\alpha_{-1} + \alpha_1^2}{2p^2}. \tag{13}$$ + +This is valid regardless of nuclear charge distribution shape. + +### Behrens-Jänecke Finite Size Correction + +An alternative Fermi function that includes finite size: + +$$F_{BJ}(Z,W) = F_0(Z,W)\left[1 \mp \frac{\alpha Z W R}{15} + ...\right], \tag{10}$$ + +The additional term traces to the expansion of $f_1$ and $g_{-1}$ inside the nucleus (Huffaker and Laird, 1967): + +$$\left\{\frac{g_{-1}(r)}{f_1(r)}\right\}^2 \propto [p F_0(Z,W)]^{1/2}\left[1 - \frac{13}{30}\left(\frac{1}{2R}\right)^2 + \alpha Z W R - (pr)^2\right], \tag{11}$$ + +interpreted as a rough electrostatic finite size correction. The error from neglecting the $(1+\gamma)/2$ prefactor is $\sim 0.5\%$ for $Z=20$. + +--- + +## IV. FINITE MASS AND ELECTROSTATIC FINITE SIZE EFFECTS + +### A. Electrostatic Finite Nuclear Size Corrections + +#### 1. L₀(Z, W) — Uniformly Charged Sphere + +When the nucleus has finite size rather than being a point charge, electron/positron wave functions become finite at the center. The correction $L_0$ is introduced for use with the analytical Fermi function in Eq. (6). + +The nuclear radius is adjusted to match experimental $\langle r^2 \rangle^{1/2}$: + +$$R = \sqrt{5/3} \, \langle r^2 \rangle^{1/2}. \tag{14}$$ + +**Low-Z expansion** (Behrens and Bühring): + +$$L_0 \simeq \frac{1+\gamma}{2}\left[1 \mp \alpha Z W R + (\alpha Z)^2 - \gamma \frac{\alpha Z R}{2W}\right] $$ +$$\simeq 1 \mp \alpha Z W R + (\alpha Z)^2 - \frac{13}{60} - \frac{\alpha Z R}{2W}. \tag{15}$$ + +**Wilkinson's analytical parametrization** (accurate to $10^{-4}$ for $p \leq 45$, $|Z| \leq 60$): + +$$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}, \tag{16}$$ + +with coefficients expanded as: +$$a_n = \sum_{x=1}^6 b_{x,n} (\alpha Z)^x. \tag{17}$$ + +Coefficients $b_{x,n}$ are tabulated in **Tables I (electrons)** and **II (positrons)**. For positrons, signs of odd powers of $Z$ are flipped. The effect ranges from a few 0.1% up to several percent — everywhere highly significant. + +#### 2. U(Z, W) — More Realistic Charge Distributions + +The uniformly charged sphere is an approximation; actual nuclear charge distributions include: +- **Modified Gaussian**: $\rho_{MG}(r) = N_0\left[1 + A\left(\frac{r}{a}\right)^2\right]e^{-(r/a)^2}$ (Eq. 18), with normalization $N_0 = \frac{8 a^{-3} \pi^{-1/2}}{2+3A}$ and $a = R\sqrt{\frac{5}{2}(2+5A)/(2+3A)}$ (Eqs. 19–20) +- **Fermi distribution**: $\rho_F(r) \propto \left[1 + \exp((r-c)/a)\right]^{-1}$ with $a \simeq 0.55$ fm + +The correction from replacing uniform sphere with a more realistic distribution is: + +$$\frac{U(Z,W)}{L_0} = \frac{\alpha_{02}' + \alpha_{-1}'^2}{\alpha_{1}^2 + \alpha_{-1}^2} = \frac{B_{10^{-2}}' + B_{-1}'^2}{B_{1^{-2}} + B_{-1}^2}, \tag{23}$$ + +neglecting $\mathcal{O}(WR^2)$ terms, where $B_{\pm 1}$ involves wave functions inside the nucleus (Eq. 24). + +**Analytical approximation for low Z** (Eq. 25): +$$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, \tag{25}$$ + +where $\Delta_i$ are linear combinations of potential coefficient differences $\Delta v_n = v_n^0 - v_n'$ (Eqs. 26a–d). + +**Wilkinson's Fermi distribution fit**: +$$U(Z,W) = 1 + \sum_{n=0}^\infty a_n p^n, \tag{29}$$ +with coefficients: +$$a_0 = -5.6\times 10^{-5} \mp 4.94\times 10^{-5} Z + 6.23\times 10^{-8} Z^2,$$ +$$a_1 = 5.17\times 10^{-6} \pm 2.517\times 10^{-6} Z + 2.00\times 10^{-8} Z^2,$$ +$$a_2 = -9.17\times 10^{-8} \pm 5.53\times 10^{-9} Z + 1.25\times 10^{-10} Z^2, \tag{30}$$ + +The effect of $U(Z,W)$ is $\sim 0.1\%$ for medium-high masses — cannot be neglected. The three-parameter Fermi ("wine bottle") distribution can use Eq. (29) as first approximation with Eq. (25) describing differences between two Fermi distributions. + +#### 3. DFS(Z, W, β²) — Nuclear Deformation + +For axially symmetric deformations: +$$R(\theta,\phi) = R_0\left[1 + \beta_2 Y_{20}(\theta,\phi)\right], \tag{32}$$ + +with $\beta_2 > 0$ prolate, $\beta_2 < 0$ oblate. The intrinsic electric quadrupole moment: +$$Q_0 = \frac{\alpha}{\pi}\sqrt{\frac{5}{3}} R Z \beta_2(1 + 0.16\beta_2). \tag{33}$$ + +The deformation correction to the Fermi function is defined as a ratio of angle-averaged corrections: +$$DFS(Z,W,\beta) = \frac{L_0^*(Z,W)}{L_0(Z,W)}, \tag{40}$$ + +where $L_0^*$ involves integrating over a continuous superposition of uniformly charged spheres (Eq. 39). The effect can reach several parts in $10^3$ but is partially canceled by the nuclear structure deformation correction $D_C$. **Figure 1** shows DFS for $\beta_2 = 0.2$, revealing an energy dependence reversal between $\beta^-$ and $\beta^+$ decay. + +### B. Finite Nuclear Mass Corrections + +#### 1. Rₙ(W, W₀, M) — Recoil Kinematic Correction + +The finite nuclear mass turns beta decay from two-body to three-body process: +$$R_N(W,W_0,M) = 1 + r_0 + r_1/W + r_2 W + r_3/W^2. \tag{41}$$ + +**For Fermi (vector) transitions**: +$$r_0^V = W_0^2/(2M^2) - 11/(6M^2), \tag{42a}$$ +$$r_1^V = W_0/(3M), \tag{42b}$$ +$$r_2^V = 2/M - 4W_0^2/(3M), \tag{42c}$$ +$$r_3^V = 16/(3M^2). \tag{42d}$$ + +**For Gamow-Teller (axial) transitions**: +$$r_0^A = -2W_0/(3M) - W_0^2/(6M^2) - 77/(18M^2), \tag{43a}$$ +$$r_1^A = -2/(3M) + 7W_0/(9M^2), \tag{43b}$$ +$$r_2^A = 10/(3M) - 28W_0/(9M^2), \tag{43c}$$ +$$r_3^A = 88/(9M^2). \tag{43d}$$ + +For mixed transitions, terms are weighted by $1/(1+\rho^2)$ and $1/(1+\rho^{-2})$ (Eq. 44). This is a small effect of order $10^{-5}$ to $10^{-3}$. See Wilkinson (1990) Fig. 1 for magnitude plot. + +#### 2. Q(Z, W, M) — Recoil Electromagnetic Correction + +The recoiling nucleus's Coulomb field differs from a static one: +$$Q(Z,W,M) \simeq 1 \mp \frac{\pi \alpha Z}{M}\left(1 + a\right)\frac{W_0 - W}{3W}, \tag{45}$$ + +where $a = (1-\rho^2/3)/(1+\rho^2)$ is the $\beta$-$\nu$ correlation coefficient ($a=1$ for Fermi, $a=-1/3$ for Gamow-Teller). The effect amounts to at most a few percent of the typical error in phase space factor $f$ due to Q-value uncertainty — typically negligible. + +--- + +## V. RADIATIVE CORRECTIONS + +### Overview: Inner vs. Outer Corrections + +The radiatively corrected spectrum is: +$$\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW}(1 + \Delta_R^{V/A})(1 + \delta_R(W,W_0)) \tag{46}$$ + +- **Inner corrections** $\Delta_R$: Nuclear-independent, absorbed into effective coupling constants. For vector case: $\Delta_V^R = (2.361 \pm 0.038)\%$ (Marciano and Sirlin, 2006). +- **Outer corrections** $\delta_R(W,W_0)$: Energy-dependent, nucleus-dependent — the focus of this section, contributing a few percent. + +### A. Total Spectral Influence (Outer Radiative Corrections) + +$$R(W,W_0) = 1 + \delta_R(W,W_0). \tag{47}$$ + +The correction is expressed as: +$$\delta_R = \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}$$ + +where: +- $L(2W_0, m_p) = 1.026725\left[1 - \frac{2\alpha}{3\pi}\ln(2W_0)\right]$ (Eq. 49) — included in the constant term normalization + +#### Order α Correction — Sirlin's g(W₀, W) Function: + +$$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}\cdot 2(1+\beta^2) - \frac{(W_0-W)^2}{6W^2} - 4\tanh^{-1}\beta, \tag{50}$$ + +where $\beta = p/W$ and: +$$L_s(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -\sum_{k=1}^\infty \frac{x^k}{k^2} \equiv -Li_2(x), \tag{51}$$ + +the **Spence function (dilogarithm)**. Its large $W_0$ limit: +$$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}$$ + +Equation (50) is universal — same for electrons and positrons, Fermi and Gamow-Teller, independent of nucleus. It has a **logarithmic divergence at $W = W_0$** related to soft real photon emission. + +#### Soft Photon Resummation: +To handle the endpoint divergence, Eq. (53) is used: +$$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}$$ + +The change in tritium spectrum is negligible due to low endpoint, but for higher energies the $f t$ value shift can reach several parts in $10^4$. + +#### Order Zα² Correction: +Three dominant Feynman diagrams (Fig. 3): vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to $\Delta_3$. + +$$\delta_2(Z,W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W). \tag{55}$$ + +The leading terms $\Delta_1$ depend on nuclear form factor $F(q^2)$ and are split into nucleus-independent ($\Delta_1^0$) and nucleus-dependent ($\Delta_1^F$) parts: +- Non-relativistic approximation (Eq. 56): energy-dependent but nucleus-independent terms combined with $\Delta_4$ +- For uniformly charged sphere, Eqs. (57)–(62) give explicit integral forms for $\Delta_1^F$, $\Delta_2$, and $\Delta_3$ + +For modified Gaussian model results are similar; differences between models for superallowed Fermi transitions ($^{10}$O to $^{54}$Co) are $< 10^{-4}$. + +#### Order Z²α³ Correction — Sirlin's Heuristic Formula: +$$\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], \tag{63}$$ + +with $a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}$ (Eq. 64), $b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}$ (Eq. 65), $f(W) = \ln(2W)-5/6$ (Eq. 66), $g(W)$ in Eq. (67). + +#### Higher Order Corrections — Wilkinson Estimate: +$$\delta_{Z^n\alpha^{n+1}} \approx Z^n\alpha^{n+1} K_{nm} \ln^{m-n}\left(\frac{\Lambda}{W}\right), \tag{68}$$ + +with $K_{nm} \approx 0.5$ on average, giving summed contribution: +$$\delta^{\text{higher}} = \sum_{n=3}^\infty Z^n\alpha^{n+1} = \frac{Z^3\alpha^4}{1-Z\alpha}. \tag{69}$$ + +These become relevant for higher $Z$. + +--- + +## B. Neutrino Radiative Corrections + +Even though the neutrino has no direct electromagnetic interaction, it is indirectly influenced via virtual photon exchange and energy conservation from inner bremsstrahlung (relevant for reactor neutrino oscillation studies). The total $\mathcal{O}(\alpha)$ radiative correction to the antineutrino spectrum: + +$$R_\nu(Z,W,W_0) = 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.$$ +$$+ 8\left(\frac{2\hat{W}\hat{\beta}}{\hat{\beta}-1}\right)^{-1} + 4\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\frac{7+3\hat{\beta}^2}{8} - 2\tanh^{-1}\hat{\beta}) \tag{70}$$ + +where $\hat{W} = W_0 - W_\nu$, $\hat{p} = \sqrt{\hat{W}^2 - m_e^2}$, $\hat{\beta} = \hat{p}/\hat{W}$ (Eq. 71). The correction is much smaller than for the electron spectrum. + +--- + +## C. Radiative Beta Decay — Internal Bremsstrahlung + +An additional photon in the final state: +$$n \to p + e^- + \bar{\nu}_e + \gamma, \tag{72}$$ + +Understood classically as radiation from sudden beta particle acceleration (Bloch 1936; Knipp and Uhlenbeck). The photon ejection probability per unit energy: + +$$\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}$$ + +with $W' = W_0-\omega$ and $p'=\sqrt{W'^2-1}$ the redefined energy/momentum. The photon spectrum affecting the beta spectrum is found by averaging over intermediate states (Eq. 74): +$$S(\omega) = \int_{1+\omega}^{W_0} N(W')\Phi(W',\omega)dW'.$$ + +Detection of internal bremsstrahlung reveals underlying weak interaction physics through correlations with lepton momenta and circular polarization. Experimental branching ratios: $BR_{\beta\gamma} \sim 3\times 10^{-3}$ for neutron, $\sim 2\times 10^{-3}$ for $^{32}$P. + +--- + +## VI. NUCLEAR STRUCTURE EFFECTS — THE SHAPE FACTOR (Partial) + +### Form Factors and Multipole Expansion + +Nuclear structure effects are described via the elementary particle approach with form factors $F_K$ absorbing all nuclear structure information as functions of $q^2 = (p_f - p_i)^2$: + +$$\langle f | V_\mu - A_\mu | i \rangle \propto \sum_{KMs} \sum_{L=K-1} (-1)^{J_f-M_f+M} (-i)^L F_{KLs}(q^2) $$ +$$\times \sqrt{4\pi(2J_i+1)} \begin{pmatrix} J_f & K & J_i \\ -M_f & M & M_i \end{pmatrix} \frac{(qR)^L}{(2L+1)!!}. \tag{78}$$ + +Since $(qR)^2$ is very small for beta decay, the expansion typically stops after the first order. The formalism connects to both Behrens-Bühring and Holstein approaches, utilizing Behrens-Bühring's computational machinery with Holstein's clean notation. + +--- + +# Hayen et al. (2017) — High Precision Analytical Description of the Allowed β Spectrum Shape: Summary of Chunks 7 (Section VI.A–C) + +## A. Introduction to Nuclear Structure Effects + +### The Beta Decay Hamiltonian and Current Decomposition + +The beta decay Hamiltonian is constructed as a current-current interaction: + +$$H_\beta(x) = \frac{G_V \cos\theta_C}{\sqrt{2}} [J^\mu(x)L_\mu(x) + \text{h.c.}], \tag{75}$$ + +where $\theta_C$ is the Cabibbo angle, $J^\mu(x)$ is the nuclear current, and $L^\mu(x)$ is the lepton current: + +$$L^\mu(x) = i\bar{\varphi}_e(x)\gamma^\mu(1+\gamma^5)v_\nu(x). \tag{77}$$ + +Here $\varphi_e$ is the solution of the beta particle wave function in a Coulomb potential. The strong interaction requires generalizing the nuclear current from pure V-A: + +$$J^{\mu\dagger}(x) = \langle f | V^\mu(x) + A^\mu(x) | i \rangle, \tag{76}$$ + +The transition matrix element (TME) is constrained by angular momentum coupling rules for $J_i^{\pi_i} \to J_f^{\pi_f}$ transitions, allowing decays of different multipole orders $K$ within the vector triangle $(J_i, J_f, K)$. The general shape factor formulation groups all non-phase-space/non-Fermi-function terms into: + +$$N(W)\,dW \propto pW(W_0-W)^2 F(Z,W) C(Z,W). \tag{79}$$ + +**Key distinction:** $C(Z,W)$ encompasses two distinct effects: +1. **Spatial variation** of leptonic and nuclear wave functions inside the finite nuclear volume +2. **Raw nuclear structure** (form factors, matrix elements) + +All corrections from previous sections (electrostatic finite size, radiative corrections, etc.) are *not* included in $C(Z,W)$ — they are treated as separate higher-order corrections. + +### Two Formalisms Compared: Behrens-Bühring vs. Holstein + +The paper discusses two major approaches to nuclear structure effects: +- **Behrens-Bühring (BB):** Couples leptonic and nuclear parts together; provides the most precise results through rigorous expansion in $r^2$, $(m_eR)^a$, $(WR)^b$, $(\alpha Z)^c$ introducing tabulated functions $I(k_e, m, n, \rho)$ +- **Holstein (HS):** Decouples leptonic and nuclear parts; uses manifestly covariant form factors with cleaner symmetry properties + +The authors adopt the BB computational approach but present final results in Holstein notation for clarity. See Appendix E for detailed comparison. + +--- + +## B. Form Factors + +### 1. Impulse Approximation + +Form factors $F_{KLs}(q^2)$ describe nuclear structure model-independently as functions of the only Lorentz invariant $q^2 = (p_f - p_i)^2$. Using the **impulse approximation**, nucleons are treated as non-interacting particles coupling to the weak vertex as free particles: + +$$O_{KLs} = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle a^\dagger_\alpha a_\beta, \tag{80}$$ + +with nuclear matrix elements decomposing into: + +$$\langle f | O_{KLs} | i \rangle = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle \langle f | a^\dagger_\alpha a_\beta | i \rangle. \tag{81}$$ + +The one-body density matrix elements $\langle f | a^\dagger_\alpha a_\beta | i \rangle$ are calculable via shell model methods. + +### 2. Induced Currents + +QCD influences from within the nuclear potential add Lorentz-invariant terms to the weak vertex. In both BB and HS formalisms (completely equivalent): + +**Behrens-Bühring form:** +$$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, \tag{82}$$ + +**Holstein form:** +$$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, \tag{83}$$ + +where $\sigma_{\mu\nu} = -\frac{i}{2}[\gamma_\mu,\gamma_\nu]$ and all $C,g$ functions depend on $q^2$. The two approaches are equivalent through redefinition of form factors. + +**Example — induced scalar modification (Eq. 84):** +$$^{(0)}_{F000} \to {}^{VM}_{000} \pm \frac{f_S}{R}\frac{6}{5}(W_0 R \pm \alpha Z)\,{}^{VM}_{000}, \tag{84}$$ + +### Table III: Nuclear Matrix Elements for Allowed Transitions + +| Form Factor (BB) | Cartesian Form | $\Delta J$ | Forbidden | Type | +|---|---|---|---|---| +| $V_{F000}^{(0)}$ | $+g_V \mathbf{1}$ | 0 | — | **Allowed** | +| $A_{F101}^{(0)}$ | $\mp g_A \boldsymbol{\sigma}$ | 0,1 | $0^- \to 0^-$ | **Allowed** | +| $V_{F000}^{(1)}$ | $+g_V (\mathbf{r}/R)^2$ | 0 | — | Main correction | +| $A_{F101}^{(1)}$ | $\mp g_A \frac{\mathbf{r}}{R}\left(\frac{R}{\sqrt{2(M_N R)^2}} + \boldsymbol{\sigma}\right)$ | 0,1 | $0^- \to 0^-$ | Main correction | +| $A_{F121}^{(0)}$ | $\mp g_A\sqrt{\frac{3}{2}}\frac{\sqrt{5}}{R^2}[\sqrt{4\pi}\mathbf{Y}_2(\hat{\mathbf{r}})\otimes\boldsymbol{\sigma}]_1 \pm (2M_N R)^{-2}$ | 0,1 | $0^- \to 0^-$ | — | +| $V_{F011}^{(0)}$ | $+g_V i\frac{\mathbf{r}}{R}$ | 0 | — | **Relativistic** | +| $V_{F111}^{(0)}$ | $-g_V\sqrt{\frac{3}{2}}\boldsymbol{\alpha}\times\frac{\mathbf{r}}{R} - \sqrt{\frac{M_N}{2}}\frac{g_2^M}{NR}$ | 0,1 | $0^- \to 0^-$ | **Relativistic** | +| $A_{F110}^{(0)}$ | $\pm g_A\sqrt{3}\gamma^5\frac{\mathbf{r}}{R} \pm (2M_N R)^{-2}W_0 R [\cdots] \pm \frac{g_P}{\sqrt{56}}\alpha Z$ | 0,1 | $0^- \to 0^-$ | **Relativistic** | + +### Table IV: Holstein Form Factors in Impulse Approximation + +| Form Factor | Formula (Impulse Approx.) | Remark | Type | +|---|---|---|---| +| $a$ | $\sim g_V M_F$ | $g_V = 1$ (CVC, Ademollo-Gatto) | Vector | +| $e$ | $\approx g_V(M_F \pm A g_S)$ | $e=0$ (CVC, SCC) | Vector | +| $b$ | $\approx b = A(g_M M_{GT} + g_V M_L)$ | $g_M \approx 4.706$ | Vector | +| $c_1$ | $\approx c_1 = g_A M_h^{GT}$ | $g_A \to g_{A,\text{eff}} = 1$ (Towner, 1987) | Axial vector | +| $c_2$ | $\approx \frac{i}{6}g_A M_{\sigma r^2} + \frac{\sqrt{1}}{10}M_{ky}$ | $c_2 \sim R^2$ | Axial vector | +| $d$ | $\approx d = A(g_A M_{\sigma L} \pm g_{II} M_{GT})$ | $g_{II} \sim g_T \approx 0$ (SCC); $d_I=0$ for analog states | Axial vector | +| $h$ | $\approx -\sqrt{\frac{2}{10}}M_2^2 g_A M_{1y} - A^2 g_P M_{GT}$ | $g_{P,\text{free}} \approx -229 \to g_{P,\text{eff}} = ?$ | Axial vector | + +### Table V: Nuclear Matrix Elements (Calaprice et al., 1977b) + +| Matrix Element | Operator Form | +|---|---| +| $M_F$ | $\langle\beta\|\sum\tau_i^\pm\| \alpha_i\rangle$ | +| $M_{GT}$ | $\langle\beta\|\sum\tau_i^\pm \vec{\sigma}_i\| \alpha_i\rangle$ | +| $M_L$ | $\langle\beta\|\sum\tau_i^\pm \vec{l}_i\times\vec{\sigma}_i\| \alpha_i\rangle$ | +| $M_{\sigma r^2}$ | $\langle\beta\|\sum\tau_i^\pm \vec{\sigma}_i r_i^2\| \alpha_i\rangle$ | +| $M_{ky}^{(0)}$ | $\frac{1}{6}\sqrt{\frac{4\pi}{5}}\langle\beta\|\sum\tau_i^\pm r_i^2 C_1^{(0)}Y_{2n}(\hat{\mathbf{r}}_i)\| \alpha_i\rangle$ | + +### 3. Validity of the Impulse Approximation + +The impulse approximation neglects many-body effects (meson exchange, core polarization). Key findings: +- **Vector matrix element $V_{F000}$:** The impulse approximation is valid because the vector current is conserved (CVC) +- **Axial current:** Breakdown occurs due to PCAC and pion field coupling — corrections are significant + +**Two categories of nuclear structure corrections:** +1. **Meson exchange effects** — long history (Blin-Stoyle et al., Chemtob & Rho, Delorme & Rho) +2. **"Nuclear" effects** — core polarization, relativistic effects, configuration mixing from insufficient wave function knowledge + +These distinctions are artificial since the nuclear potential incorporates mesonic degrees of freedom (Wilkinson, 1974). For **allowed decays**, impulse approximation results largely hold due to cancellations between core-polarization and meson exchange (Morita, 1985), though individual effects can reach ~40%. + +**Quenching of coupling constants:** +- $g_A$ is renormalized ad hoc: $g_{A,\text{eff}} = 1.1$ for sd-shell nuclei, unity for fp-shell (Martínez-Pinedo et al., 1996), and even $g_A \approx 0.9$ from highly forbidden decays +- The **induced pseudoscalar coupling** $g_P$ can be quenched by up to **80%** in nuclear matter — significantly more than the connected quenching of $g_A$ +- Ab initio methods (Green's function Monte Carlo) circumvent quenching but fall outside scope + +--- + +## C. Symmetries + +### 1. Conserved Vector Current (CVC) + +The CVC hypothesis (Feynman and Gell-Mann, 1958) provides powerful constraints: + +**Temporal component:** +$$\langle J_f M_f | V^0(0) | J_i M_i \rangle = {}^{V}F_{000}(q^2)\delta_{J_i J_f}\delta_{M_i M_f} $$ +$$= \sqrt{(T_i \pm T_{3i})(T_i \mp T_{3i} + 1)}\,\delta_{J_i J_f}\delta_{M_i M_f}, \tag{87-88}$$ + +This implies CVC **excludes induced scalar currents** ($e=0$ in Table IV). + +**Relation between vector form factors (Eq. 89):** +$$-2N\,{}^{V}F_{011}^{N-1} = \left(W_0 \mp (m_n - m_p)\right)R\,{}^{V}F_{000}^N $$ +$$\pm \alpha Z \int_R \!\! \left(\frac{r}{R}\right)^{2N} U(r)Y_0\,d\tau.$$ + +**Weak magnetism from electromagnetic data:** For mirror transitions between isospin $T=1/2$ nuclei: + +$$b = A\sqrt{\frac{J+1}{J}}\,\mu_{\text{eff}}, \tag{90}$$ +where $\mu_{\text{eff}} = \mp(\mu_1 - \mu_2)$ with $\mu_{1,2}$ the magnetic moments of mother and daughter nuclei. + +For pure Gamow-Teller transitions within isospin triplets: + +$$b^2_\gamma = \eta^6 \frac{\Gamma_{M1}^{iso}}{E_\gamma^3\alpha}, \tag{91}$$ + +where $\Gamma_{M1}^{iso}$ is the width of the analog isovector M1 gamma transition, $E_\gamma$ is the photon energy, and $\eta = 1$ (or corrected for degeneracy). + +### 2. Partially Conserved Axial Current (PCAC) + +The Goldberger-Treiman relation relates the pion-nucleon coupling to $g_A$: + +$$g_P(q^2) = -g_A(0)\frac{(2M_n)^2}{m_\pi^2 - q^2}. \tag{92}$$ + +At $q^2=0$, using nucleon and pion masses: **$g_P(0) \approx -229$**. Higher-order corrections (chiral perturbation theory, Gorringe & Fearing 2004; Bhattacharya et al. 2012) modify this at the ~5% level. Experimental measurements from muon capture ($q^2 = 0.88m_\mu^2$) are in sufficient agreement with PCAC results to warrant using them. + +**Critical point:** The quenching of $g_P$ due to meson exchange effects in nuclear matter is nucleus-dependent and can reach ~80%. Great care is needed when evaluating pseudoscalar currents for nuclear decays — Eq. (92) applies only to free nucleons. + +### 3. Second Class Currents (SCC) + +Terms are classified by their transformation under **G-parity**: $G = Ce^{i\pi T_y}$ (charge conjugation × π rotation in isospin space). Two terms transform differently from their vector/axial-vector analogues: +- **Induced scalar** — already eliminated by CVC +- **Induced tensor** — no similar constraint exists + +The axial tensor form factor $d$ contains both first-class ($d_I$) and second-class ($d_{II}$) components. Experimentally, **no evidence for SCC has been found** (Minamisono et al., 2001, 2011; Grenacs, 1985), so $d_{II} \equiv 0$. The form factor reduces to $d = d_I$, non-zero only for transitions between **non-analog states**. + +--- + +# Hayen et al. (2017) — Summary of Chunk 8 + +## Section VI.F: Shake-off and Shake-up Processes; The Endpoint Shift (§6.C, lines 839–1055) + +### Overview +Atomic orbital changes due to β decay (screening, exchange) affect the internal "housekeeping" of the atom. Initial and final quantum states are no longer orthogonal, allowing discrete excitations into higher allowed states (**shake-up**) and even continuum ejection (**shake-off**). These reduce available energy for the lepton pair and modify final-state interactions. At the target precision of $10^{-4}$ down to 1 keV, these effects are not always negligible. + +--- + +### 1. Shake-Up (§6.C.1) + +#### Mechanism +Atomic electrons in final states can be excited into higher bound states rather than undergoing double ionization (shake-off). The probability is computed from radial wave function overlap integrals of initial and final bound states: +$$P_f = |\langle \phi_n^{0\prime} | \varphi_i \rangle|^2$$ +with shake-off probability as the deviation from unity when summing all such probabilities. + +**Few-electron systems are particularly susceptible.** In tritium ($^3$H), the single electron in the final state has significant probability to occupy the 2s orbital or higher (Arafune & Watanabe, 1986; Hargrove et al., 1999; Williams & Koonin, 1983). + +#### Recoil-Induced Excitation +The recoil momentum of the daughter nucleus induces excitations due to sudden acceleration. The average excitation energy is (Feagin et al., 1979): +$$\Delta E_R = \frac{1}{2} Z m_e v_R^2 = Z E_R \frac{m_e}{M}, \quad \text{(158)}$$ + +For $Z=50$ and maximum recoil energy of 1 keV, the average excitation energy is ~0.25 eV. Atomic excitations are on the order of tens of eV, so **incomplete overlap between initial/final atomic states dominates** the shake-up picture. + +#### Q-Value Reduction +The Q value of the decay is reduced by the mean excitation energy: +$$\Delta E_{ex} = \sum_f P_f E_f, \quad \text{(159)}$$ + +where $E_f$ is the excitation energy of final state $f$, and $P_f$ is the probability to populate that level. + +#### Effect on Screening Correction (Eq. 160) +Possible excitations change screening: +$$S(Z, W) = 1 - \frac{Z\alpha}{p^2} + \sum_h \frac{1}{W} |\langle \phi_n^{0\prime} | \varphi_i \rangle|^2 \left( \sum_{i=1}^n \frac{1}{r_i} \right) = 1 - \frac{Z\alpha}{p^2} + \frac{1}{W} V_{00}, \quad \text{(160)}$$ + +In tritium β⁻ decay, the screening potential can change by as much as **20%** because the atomic Coulomb interaction involves only a single electron. For systems with more electrons, relative changes decrease but scale with $Z$. Neglecting shake-up introduces errors of order $\sim 1 \times 10^{-4}$. + +#### Effect on Exchange Correction +- Medium to high Z: Shake-up probability is negligible at the $10^{-3}$ level. +- Low Z (e.g., tritium): Shake-up probabilities can be significant (~25%), and exchange with higher-lying ns orbitals must be included even if unoccupied in the initial state. + +#### Effect on Exchange Correction from Tritium +Inclusion of shake-up changes the exchange correction at the few $10^{-4}$ level down to 0.5 keV for tritium only. Otherwise, negligible. + +--- + +### 2. Shake-Off (§6.C.2) + +#### Probability Scaling +Non-orthogonality allows excitations including continuum ejection. Key principles from Carlson et al. (1968): +- **(i)** For given principal quantum number $n$, shake-off probability **decreases** with increasing $Z$ (Pauli principle: only unfilled shells accessible; overlap integrals vanish for inner electrons). +- **(ii)** For a given atom, shake-off probability **increases** with $n$. +- **(iii)** Total shake-off probability is reasonably independent of Z, occurring in ~20–30% of all decays. + +K-shell ejection probability: $\sim 2 \times 10^{-3}$ for tritium; a few $10^{-4}$ for higher Z — small effect and not the primary interest here. + +#### One-Electron Ionization Probability (Eq. 161) +$$p_\pi^D(M) \simeq 1 - \sum_{n_0 \le n_{\max}} |\langle \phi_{n_0 l'}^{D} | \varphi_{n l}^M \rangle|^2 + K |\langle \phi_{l' \pm 1}^{D} | r | \varphi_{n l}^M \rangle|^2 - K \, \text{Re}[\langle \phi_{l'}^{D} | \varphi_i \rangle \langle \phi_{l'}^{D} | r | \varphi_{n l}^M \rangle], \quad \text{(161)}$$ + +where $\phi_{nl}^{D(M)}$ has primary quantum numbers $(n,l)$ for daughter/mother, and $K = 2E_R/M$. Holes created by shake-off are filled via X-ray or Auger electron emission. + +#### Mean Energy Loss from Shake-Off +For chlorine (Couratin et al., 2013 + Desclaux binding energies), mean energy loss per decay is **~5 eV**. Expected similar magnitude for all Z. To first order, this can be approximated as an additional decrease in $W_0$ by the weighted mean excitation energy. + +#### Change in Screening from Shake-Off (Eq. 162–163) +For a hole in outer shell orbital $n,l$: +$$\frac{V_0 - V_{SO}}{V_0} = \frac{\langle f^0 | r^{-1} | i \rangle - \sum f'^0 \langle f^0 | i \rangle \langle f^0 | r | i \rangle}{\langle f^0 | r^{-1} | i \rangle} \approx \frac{1}{\langle f^0 | r^{-1} | i \rangle} \sum_n \frac{\langle f'^0 | i \rangle^2 Z_{\text{eff}}(1 - \Delta Z_{\text{eff}})}{n \cdot n^2}, \quad \text{(162, 163)}$$ + +With $\Delta Z_{\text{eff}} \approx 0.3$–$0.4$, for Ru$^{1+}$ this yields ~**0.1%** relative change in screening potential → few $10^{-5}$ level in spectral shape (lowest energy range). **Negligible at target precision.** + +#### Shake-Off Effect on Exchange +Requires combining Law & Campbell (1972a,b) indistinguishability approach with Harston & Pyper (1992) exchange framework. In medium/high Z nuclei, probability for shake-off to occur in $s_{1/2}$ or $p_{1/2}$ state is **< 0.1%** — effect decreases with increasing Z. + +For tritium decay: $\chi_{\text{ex}}^{\text{cont}}$ has maximum value of **-0.01% at 1 keV**. For $^{35}$Ar hole creation in ns states, safely negligible at current precision order. + +--- + +### Section VI.E Continued: Exchange Correction — Analytical Parametrisation (§6.E.4) + +#### Challenge +Exchange correction requires radial wave functions of both continuum and bound states over **all space** (not just near origin), for arbitrary potentials. This is a significant challenge for analytical description because: +- Most other effects need wave functions only near the origin (power expansions suffice). +- Exchange needs integration over entire spatial domain. +- Hydrogenic approach with simple screening is insufficient for high precision. +- Pure Coulomb field analytical results [Eq.(156)] do not allow easy insertion of fit parameters. + +#### Analytical Fit Formula (Eq. 157) +The exchange correction $X(W)$ as a function of kinetic energy $W_0 \equiv W - 1$: +$$X(W) \approx 1 + \frac{a}{W_0} + \frac{b}{W_0^2} + c \cdot e^{-d W_0} + \frac{e \cdot \sin[(W-f)g+h]}{W^i}, \quad \text{(157)}$$ + +**9 fit parameters** ($a, b, c, d, e, f, g, h, i$), tabulated in Appendix G for each Z individually. The $Z$ dependence is complex due to atomic shell effects (see Fig. 9). + +#### Performance +- Excellent agreement over full tested energy range up to **1 MeV** (where all contributions dip below $10^{-4}$ level). +- Maximum differences: order of a few $\times 10^{-4}$. +- Average residuals: at the **$< 10^{-5}$** level. +- Phase space integral influence: typically **few $\times 10^{-5}$** in absolute terms. + +#### Z Dependence — Shell Effects (Fig. 9) +Figure 9 shows orbital exchange contributions for all atoms from $s_{1/2}$ orbitals evaluated at 3 keV. Clear shell closure effects observed: +- **Ar shell closure**: distinct feature around $Z=18$. +- **Kr shell closure**: feature around $Z=36$. +- **Xe shell closure**: feature around $Z=54$. + +Increased binding energy at higher Z reduces spatial extension of bound states → decreases overlap between bound and continuum electron radial wave functions at low energies. + +--- + +### Section VI.E Continued: Atomic Potential Sensitivity (§6.E.3, lines 680–791) + +Three potentials compared with increasing complexity for exchange correction calculation: + +#### 1. Simple Exponentially Screened Field +- Screening strength adjusted for best agreement with bound state energies (decreasing importance for higher $n$). +- Caveat: tuning to match 1s binding energies can result in unbound higher ns states — neglected here since lowest s states dominate. + +#### 2. Three Yukawa Potentials +- Coefficients fitted to Dirac-Hartree-Fock-Slater numerical data (Salvat et al., 1987). + +#### 3. Complete Potential (Optimized) +- Full potential including exchange optimization turned ON ("Optimized") and OFF ("Unoptimized"). +- One free parameter in the exchange potential varied for agreement with numerically calculated binding energies to within **0.1% or 0.1 eV** (Kotochigova et al., 1997a,b; Desclaux, 1973). + +#### Results — $^{63}$Ni Comparison (Fig. 8) +- General trend replicated by all potentials apart from very lowest energies where some orbitals give large negative contributions. +- **Without optimization**: binding energies for higher lying orbitals seriously in error → incorrect exchange correction contributions. +- The total net magnitude is a delicate quantity in low energy regions. + +#### Nuclear Radius Sensitivity (Fig. 8, bottom) +Varying nuclear radius by 10% with optimized potential: +- Discrepancy grows towards lower energies. +- Only crosses $10^{-4}$ at roughly **~1 keV** for $^{63}$Ni. +- For extreme case of $^{241}$Pu: spectral differences cross $10^{-4}$ level at ~**5 keV**, with constant offset reaching several parts in $10^{-4}$. +- Difference shows reasonably linear behavior on $\Delta R$ — extrapolation to reasonable uncertainties (estimated ~0.03 fm, or ~0.8% for $^{63}$Ni) brings effect below **few $\times 10^{-5}$**. + +--- + +### Numerical Methods: RADIAL Package (§6.E.2, lines 628–677) + +#### Wave Function Computation +- Electron radial wave functions $r g_\kappa(r)$, $r f_\kappa(r)$ calculated numerically using **power expansion** at both singular points ($r=0$, $r=\infty$). +- Connected via rescaling of inner solution with $\alpha\kappa$. +- General approach: Behrens & Bühring (1982); Salvat et al. (1995). +- Fortran 77 package **RADIAL** (Salvat et al., 1995) slightly modified and interfaced with custom code for exchange effect calculation. + +#### Common Grid Construction +Since free and bound state wave functions are solved on different grids: +- A common grid is constructed. +- Evaluated using exact results or Lagrangian three-point interpolation. +- The overlap integrals $\langle E_{s0} | n s \rangle$ in Eqs.(154)-(155) evaluated directly. +- **Critical**: Grid must be sufficiently dense to avoid systematic errors from interpolation. + +#### Example: $^{45}$Ca Exchange Correction (Fig. 6) +- Total effect rises to nearly **10%** in the few-keV region. +- Drops significantly in the 100 eV region due to negative contributions from higher lying orbitals (e.g., 2s). +- At very low energies (~1 keV or lower), contributions from higher-lying orbitals can become **negative**, lowering total correction by several percentage points. + +#### Example: $^{45}$Ca p$_{1/2}$ Contribution +- Maximum contribution never exceeds a few parts in $10^5$ — completely negligible for this nucleus. +- Included in all results despite small magnitude. + +#### Example: $^{241}$Pu Exchange Correction (Fig. 7) +- Total exchange correction reaches **~29% at 100 eV**. +- p$_{1/2}$ contribution rises to nearly **2% at 100 eV** — definitely not negligible. +- p$_{1/2}$ influence continues over range of tens of keV at target precision. + +#### Phase Space Integral Impact +For low endpoint energy transitions, phase space integrals can be significantly altered: up to **30%** for $^{241}$Pu transitions. Effect drops quickly and is typically negligible at several hundreds of keV. + +--- + +### Key Implementation Notes + +1. **Exchange correction analytical fit** (Eq. 157) with 9 parameters per Z should be implemented as a lookup table from Appendix G values for fast computation across full energy range. +2. **Atomic potential choice matters critically**: optimized vs unoptimized can differ by $>10^{-4}$ at low energies. The complete potential approach (RADIAL-based) is recommended over simple screened Coulomb models. +3. **Nuclear radius sensitivity**: 10% variation in R causes $>10^{-4}$ spectral differences for heavy nuclei — accurate nuclear charge radii are essential input. +4. **Shake-off/shake-up effects** are generally below target precision except for: + - Tritium (shake-up ~25%, exchange correction changes at few $10^{-4}$ level). + - Low-Z light nuclei where shake-up probabilities can be significant. +5. **p$_{1/2}$ orbitals**: While typically ignored, they contribute up to 2% of total exchange in heavy nuclei ($^{241}$Pu) and are non-negligible over tens of keV range — must be included for precision work. + +--- + +# Hayen et al. (2017) — Summary of Chunk 9 (§D Atomic Overlap; §E Bound State β Decay) + +## Section VII.D: Atomic Overlap — Alternative Atomic Excitation Correction (lines 1–38) + +### Bahcall Correction (Eq. 166) +The β decay results in a sudden change of nuclear potential, so initial and final atomic orbital wave functions only partially overlap. In first approach, this reduces to a difference in atomic binding energies (Bahcall 1963a,c,1965), also included in detailed Ft analyses (Hardy & Towner, 2009). + +The correction is constructed by looking at relative change in spectrum shape when changing $W_0$ with $W_0 - \Delta E_{ex}$. Written as: +$$r(Z,W) = 1 - \frac{1}{W_0 - W} \frac{\partial^2 B(G)}{\partial Z^2}, \quad \text{(166)}$$ + +where $B(G)$ is the total atomic binding energy for a neutral atom with $Z \pm 1$ protons (β∓ decay). The second derivative relates to average excitation energy via $\Delta E_{ex} = -\frac{1}{2}\frac{\partial^2 B(G)}{\partial Z^2}$. + +Parametrization using numerical values by Desclaux (1973): +$$K(Z) = -0.872 + 1.270 \, Z^{0.097} + 9.062 \times 10^{-11} \, Z^{4.5}, \quad \text{(169)}$$ + +The correction $r(Z,W)$ then becomes: +$$r(Z,W) = 1 - \frac{1}{W_0 - W^2} \left[ B(G)'' + 2(C'_0 + C_1) \right], \quad \text{(170)}$$ + +#### Importance Near Endpoint +- Influence felt mainly near the endpoint where $\frac{\partial^2}{\partial Z^2}B(G)$ can reach **a few hundreds of eV**. +- Particularly important for low energy transitions: + - **$^{63}$Ni** (endpoint 67.2 keV): correction reaches **1% at 15 keV** before endpoint, increases rapidly past endpoint. + - **$^{241}$Pu** (endpoint 20.8 keV): similarly important. + +--- + +## Section VII.E: Bound State β Decay (§E, lines 39–67) + +### Two-Body Decay Mechanism +Since the electron is created inside an electronic potential well, there exists a possibility for β decay to be captured inside the potential well and produce an electron in a bound atomic state, reducing the decay to a **two-body problem**. First studied by Daudel et al. (1947), expanded by Bahcall (1961), Kabir (1967), Bück (1983), Pyper & Harston (1988). + +### Branching Ratio (Eq. 171) +In Bahcall notation, the ratio of bound state to continuum decay probabilities: +$$\frac{\Gamma_b}{\Gamma_c} = \frac{\pi (\alpha Z)^3}{f(Z,W_0)} (W_0 - 1)^2 \Sigma, \quad \text{(171)}$$ + +where $\Sigma$ depends on binding energy of bound orbital, atomic overlap integrals, and orbital wave functions at nuclear surface. + +#### Magnitudes +- **Free neutron decay**: ratio ~ $4.2 \times 10^{-6}$ — completely negligible. +- **Tritium ($^3$H)**: roughly **1% for T⁺** and **0.5% for T** in initial state. +- Higher $W_0$: smaller ratios (phase space integral $f \propto W_0^5$). Kinematic dependence $\Gamma_b/\Gamma_c \propto W_0^{-3}$. +- Superallowed $0^+ \rightarrow 0^+$ decays with Q values of several MeV: probability completely negligible. +- Low energy decays and (partially) ionized initial states: correction can grow significantly. + +### Impact on ft Analysis +This does **not** affect the β spectrum shape, as it is a separate final state in S-matrix calculation. It enters the equation when considering the **ft analysis**, however, and cannot always be neglected there. + +--- + +## Section VII.F: Chemical Influences (§F, lines 1–240) + +### Overview +In many experiments, the decaying atom is bound within a molecule, modifying electronic structure as electrons rearrange in molecular orbitals. Additional electrons and spectator nuclei influence Coulombic final state interactions; rotational and vibrational states open more possibilities for energy transfer to molecular final state (Cantwell, 1956). + +Essential in e.g., antineutrino mass determination in tritium system. Seminal works by Saenz & Froelich describe effects *ab initio* and analytically: +- Doss et al. (2006), Jonsell et al. (1999), Saenz & Froelich (1997a,b), Saenz et al. (2000). + +In most precise analysis, this effect introduces a **systematic error of 0.05%**. + +--- + +### 1. Recoil Corrections in Molecules (§F.1) +When β decay occurs inside a molecule: +- The recoiling daughter nucleus moves inside the molecular potential rather than vacuum. +- Potential described by Born-Oppenheimer energy curve, modelled with Lennard-Jones type potential. +- Daughter nucleus kicks molecule into (predissociative) rovibrational state. + +**Dissociation possibilities:** +- **Rovibrational states**: total energy > dissociation energy → typical energy scale ~ few eV. +- **Electronic excitation** into resonant continuum: tens to hundreds of eV. + +Integrated probability depends on endpoint energy (higher endpoint → higher recoil). At target precision, this effect is neglected as a small correction on already small corrections. + +For detecting recoiling daughter atom (e.g., β-ν correlation measurement $a_{\beta\nu}$): +- Probability for dissociation decreases with increasing Z (recoil energy $\propto 1/m$). +- Clear dependence on β-ν angular correlation: higher recoil momenta populate higher rotational/vibrational bands. +- Can be partially included by Monte Carlo simulation of recoiling nucleus in Lennard-Jones potential. + +--- + +### 2. Influence on Q Value (§F.2) +Molecular effects add further decrease to Q value because of **rotational and vibrational excitations**. + +Numerical results mainly for $T_2$ (tritium molecule), due to high precision needed for antineutrino mass determination: +- Excitation possibilities change to a broad continuum with resonances. +- Broadening through population of rovibrational states. +- Width of populated rovibrational energy spectrum lies in **few eV region** — typically negligible. +- Excitation into the continuum becomes non-trivial (demonstrated for $T_2$ by Doss et al., 2006). + +Averaged over entire continuum: differences between several tritium-substituted molecules is on order of **few eV**, including atomic tritium. + +The spectral shape depends on $W_0^2$, so relative error goes as $2 \times \sigma_Q / Q$. For ft values (statistical rate function $\propto Q^5$), precision requirement is even stricter: +- Study case by case for required accuracy. +- Lowest energy transitions (tritium, $^{63}$Ni, $^{241}$Pu): crudely expect **few $10^{-4}$ to $10^{-3}$** level error for $\sigma_Q$ on order of few tens of eV. + +Decay rate change treated approximately by Pyper & Harston (1988): +$$\frac{\Delta \lambda}{\lambda} \approx \frac{3 \Delta W_0}{W_0} \left( 1 + \gamma \frac{W_0}{6} \right), \quad \text{(172)}$$ + +where $\Delta W_0$ is difference in mean endpoint energy after averaging over all final states between two chemical states. + +--- + +### 3. Molecular Screening (§F.3) +Following Saenz & Froelich (1997a): all Coulombic effects treated equally to first order, split into electronic and nuclear parts. + +#### Fermi Function First Order (Eq. 173) +$$P_{\text{nuc}}^{(0,1)}(p) = \sum_n \alpha |\langle \phi_0 | \varphi_n \rangle|^2 Z \left[ \frac{8W}{\pi p} + \frac{8p}{3\pi W} + \sum_S Z_S \left( \frac{W}{p^2} + \frac{1}{W} \right) \langle \xi_{000}' | \frac{1}{R_S} | \xi_{000}' \rangle \right],$$ + +where sum $n$ extends over all final states, sum $S$ takes into account all spectator nuclei, and $R_S$ is distance operator between decaying atom and spectator nucleus S. First part represents Fermi function to first order (larger by factor $p$ than effects of spectator nuclei). + +#### Analytical Approximation +For analytical approximation: ignore energy difference in final atomic states; use closure to perform sum over $n$ (first-order corrected using atomic mismatch correction, Sec. VII.C). Each atom considered as having inert inner structure + shared valence wave function. + +Full electronic part of molecular wave function (Born-Oppenheimer approximation): +$$\psi_{\text{Mol}} = |\text{valence}\rangle \times \prod_i |\text{inert}_i\rangle, \quad \text{(174)}$$ + +Using clamped-nuclei approximation for rovibrational ground state: $\langle \xi_{000}' | R_S^{-1} | \xi_{000}' \rangle \approx R_{S,e}$ (equilibrium distance). + +#### Total Coulombic Influence (Eq. 175) +$$P_{\text{tot}}^{(0,1)}(p) \approx \alpha \left[ Z\left(\frac{8W}{\pi p} + \frac{8p}{3\pi W}\right) + \frac{W}{p^2} + \frac{1}{W} \sum_S \frac{Z_S - Z_{S,\text{in}}}{R_{S,e}} \right] - \langle i | \frac{1}{r} | i \rangle_0' - Z_{\text{val}} \langle v | \frac{1}{r} | v \rangle,$$ + +where $Z_S - Z_{S,\text{in}}$ is screened spectator nucleus charge. + +#### Molecular Screening Correction (Eq. 176) +$$\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_{\text{eff}} = Z_{\text{Val}} - (Z - Z_{\text{in}})$ and $\langle r^{-1} \rangle_{\text{Val}}$ is average inverse distance of all valence electrons relative to decaying nucleus. + +**Example: $^{45}$Ca in CaCl₂ (calcium doubly oxidized, Ca(II))**: +- Bond length measured: 2.437 Å → in natural units $R_{\text{Cl},e} \sim 600$. +- First term $\propto 3 \times 10^{-5}$ — two orders of magnitude smaller than $V_0$. +- Second term: $Z_{\text{eff}} = 2$, effect also on order of **$2 \times 10^{-5}$** with opposite sign. +- Overall molecular deviation from atomic structure: **order of $2 \times 10^{-5}$**. + +Since screening correction is typically at (sub)percent level max, molecular deviations expected to have upper limit at **few $10^{-4}$** level. + +--- + +### 4. Molecular Exchange Effect (§F.4) +In a molecule: electronic phase space greatly enlarged and perturbed relative to single atomic state. Electron density near decaying atom can both increase and decrease in molecular bond, so overlap integral changes accordingly. + +#### LCAO Approximation +Molecular orbitals constructed from linear combination of atomic orbitals (LCAO). For qualitative argument: approximate that internal orbitals for all atoms are same in molecular as atomic case. Molecular valence orbitals = combination of occupied atomic valence orbitals + energetically close atomic excited states. + +- **Ionic bond** (valence electron nearly fully removed): exchange effect $\approx 0$ because internuclear distances >> β Compton wavelength where wave function rapidly oscillates. +- **Reverse extreme** (valence electron density doubled): naively doubles the exchange correction. Conservative treatment: treat last orbital contribution with **100% error bar**. + +#### Error Estimate +Taking Fig. 6 as example: introduces **< $1 \times 10^{-4}$** error from 15 keV onwards, grows to **0.5% at 1 keV**. For $^{241}$Pu: valence orbital is 7s (contribution drops below $10^{-4}$ after 3 keV while max quickly grows to 1% in first 0.5 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**. + +--- + +### Table VII: Complete Overview of β Spectrum Shape Corrections (§VIII start, lines 243–308) + +| Item | Effect | Formula | Magnitude | +|------|--------|---------|-----------| +| 1 | Phase space factor | $pW(W_0 - W)^2$ | Unity or larger | +| 2 | Traditional Fermi function | $F_0$ (Eq. 6) | — | +| 3 | Finite size of nucleus | $L_0$ (Eq. 16) | — | +| 4 | Radiative corrections | $R$ (Eq. 47) | — | +| 5 | Shape factor | $C$ (Eqs. 100, 105) | $10^{-1}$–$10^{-2}$ | +| 6 | Atomic exchange | $X$ (Eq. 157) | — | +| 7 | Atomic mismatch | $r$ (Eq. 170) | | +| 8 | Atomic screening | $S$ (Eq. 144)$^a$ | **$10^{-3}$–$10^{-4}$** | +| 9 | Shake-up | See item 7 & Eq. (160)$^b$ | | +| 10 | Shake-off | See items 7, 163, $\chi_{\text{ex}}^{\text{cont}}$(Eq. 164)$^c$ | | +| 11 | Isovector correction | $C_I$ (Eq. 113) | | +| 12 | Distorted Coulomb potential due to recoil | $Q$ (Eq. 45) | **$10^{-3}$–$10^{-4}$** | +| 13 | Diffuse nuclear surface | $U$ (Eqs. 25, 29) | — | +| 14 | Nuclear deformation | DFS (Eq. 40) & $D_C$ (Eq. 135) | — | +| 15 | Recoiling nucleus | $R_N$ (Eq. 41) | — | +| 16 | Molecular screening | $\Delta S_{\text{Mol}}$ (Eq. 176) | — | +| 17 | Molecular exchange | Case by case | — | +| 18 | Bound state β decay | $\Gamma_b/\Gamma_c$ (Eq. 171)$^d$ | Smaller than $1 \times 10^{-4}$ | +| 19 | Neutrino mass | Negligible | < $1 \times 10^{-4}$ | +| 20 | Forbidden decays | Not incorporated | — | + +$^a$: Salvat potential of Eq.(147) used with $X$(Eq.145) set to unity. +$^b$: Shake-up effect on screening discussed in Sec.VII.C.1. +$^c$: Case by case evaluation required for shake-off influences on screening/exchange. +$^d$: Does not affect spectral shape but enters ft analysis. + +--- + +### Section VIII: Overview and Crosscheck — Superallowed Fermi Decays (lines 243–270) + +#### Comparison with Towner & Hardy (2015) f Values (Fig. 10) +Ratio of $f$ values calculated within this framework vs. results by Hardy & Towner (2015): + +- **Excellent agreement**: all residuals in the few $10^{-4}$ region. +- Particularly interesting for heaviest nuclei: e.g., **$^{74}$Rb** — 11 neutrons away from closest stable isotope, subject to strong deformations and shape coexistence ($\beta_2 = +0.401$). +- For four exotic nuclei ($^{62}$Ga, $^{66}$As, $^{70}$Br, $^{74}$Rb): shown results for both spherical shell model and deformed filling (valence nucleons outside $N=Z=28$ shell closure in Nilsson model). +- Deformation does not influence isovector correction as long as $w$ remains equal. + +**Conclusion**: Can trust validity of approach for pure Fermi decays in completely analytical manner without additional computation, even for extremely exotic nuclei. + +--- + +### Mirror Decays — ft Value Comparison (lines 271–450) + +The ratio of vector and axial vector f values: +$$\equiv \frac{2Ft_{0^+ \rightarrow 0^+}}{Ft_{\text{mirror}}} = \frac{1}{1 + f_{VA} \rho^2}, \quad \text{(177)}$$ + +where $\rho$ is the mixing ratio. Separate f values calculated by Towner & Hardy (2015). + +#### Vector Sector (§VIII, lines 314–332) +- **Exquisite agreement**: all differences smaller than $4 \times 10^{-4}$. +- Holstein form factor $d$ vanishes identically for mirror nuclei. +- CVC allows precise calculation of weak magnetism from experimental data (Severijns et al., TBP). + +#### Axial Vector Sector — Two Approaches (§VIII, lines 327–398) + +**Approach A: Extreme Single Particle (ESP) with Spherical Harmonic Oscillator (Fig. 11)** +- General agreement good but distinct features for outliers at **$^{33}$Cl and $^{35}$Ar**: disagreement reaches **~1%**. +- Shell model also unable to correctly calculate weak magnetism contribution ($b/Ac_1$) in these cases — accuracy only ~10%. +- Deviations from unity don't necessarily mean failure of $\Lambda$ evaluation. + +**Approach B: Deformed Woods-Saxon (DWS) Potential (Fig. 12)** +- Moving from spherical harmonic oscillator to deformed Woods-Saxon while retaining ESP shows **remarkable agreement**. +- Single particle wave function expanded in spherical harmonic oscillator basis; coefficients calculated by numerical diagonalization of deformed Hamiltonian (C++ code, Hayen & Severijns, TBP). +- Deviations when using CVC results can be attributed to shell model limitations (~10% accuracy on $b/Ac_1$). + +#### Induced Pseudoscalar Contribution (§VIII, lines 423–434) +- Assuming free nucleon value $g_P = -229$, contribution comparable to ratio of matrix elements → strong deviations expected. +- Possible shifts even on per mille level when ignored. +- Reasonable error must be attributed to all ft$_A$/ft$_V$ calculations until accurate accounting method found. + +#### Key Finding: Weak Magnetism Uncertainty (§VIII, lines 402–416) +Differences in weak magnetism predictions can shift $f_A/f_V$ values by **several parts in $10^3$**, undermining claims of reaching 0.01% theoretical precision (Towner & Hardy, 2015). + +Most precisely measured mirror ft isotope: **$^{19}$Ne** — behaves well under ESP approximation and not much affected by this concern. Experimental campaigns ongoing for $a_{\beta\nu}$ of $^{32}$Ar, $^{19}$Ne, $^{35}$Ar; $\beta$-asymmetry parameter of $^{35}$Ar, $^{37}$K. + +For extraction of $V_{ud}$ to be valid: significant attention needed for precise evaluation of **$\Lambda$ factor in Eq.(107)**. + +--- + +# Hayen et al. (2017) — Summary of Chunks 10, 11, 12 (§VIII–X + Appendices A–G) + +--- + +## Section VIII: Overview and Crosscheck (continued from chunk 9) + +### Mirror Decays — Deformed Woods-Saxon Results (Fig. 12) +Moving to deformed Woods-Saxon potential while retaining ESP shows **remarkable agreement** with much more advanced shell model calculations, using only the extreme single particle approximation. The C++ code (Hayen & Severijns, TBP) calculates all nuclear relevant matrix elements $V/A \, M_{KL}^{s}$ automatically. + +### Remaining Uncertainties: Weak Magnetism +- Differences in weak magnetism predictions can shift $f_A/f_V$ values by **several parts in $10^3$**. +- Shell model performs only marginally better than deformed ESP approach for predicting weak magnetism (~10% accuracy). +- Raises serious questions on validity and accuracy of presently available calculations at claimed 0.01% precision (Towner & Hardy, 2015). + +### Induced Pseudoscalar Component (§VIII end) +Assuming free nucleon value $g_P = -229$, contribution comparable to ratio of matrix elements → strong deviations expected on per mille level when ignored. Reasonable error must be attributed to all ft$_A$/ft$_V$ calculations until accurate accounting method found. + +--- + +## Section IX: Beta-Spectrum Shape Sensitivity to Weak Magnetism and Fierz Terms (§IX, lines 452–71) + +### Normalized Shape Factor (Eq. 178) +$$S(W) = C(Z,W) / C(Z, W_{\text{norm}}), \quad \text{(178)}$$ + +where $C$ is the shape factor normalized to unity at some reference energy $W_{\text{norm}}$. The slope $dS/dW$ provides physics information. Typical value for $b/Ac_1 \approx 5$ (Wauters et al., 2010). For $b_{\text{Fierz}}$: precision of **1% or better** typically required in view of current constraints on CS/CT coupling constants. + +At ~0.1% precision, measurements of $b_{\text{Fierz}}$ in nuclear β decay and neutron decay remain competitive with direct searches for new bosons (scalar/tensor type) at LHC ($p + p \rightarrow e^+ + \cancel{E}_T + X$). + +### Spectral Sensitivity (§IX.A, lines 1–37 from chunk 11) +- Weak magnetism and non-zero Fierz interference term modify β spectrum shape in energy-dependent fashion. +- **Fierz term**: sensitivity varies as $1/E$ → advantageous for measuring at **low energies**. +- **Weak magnetism**: effect proportional to $E$ → favorable for measuring at **high energy endpoint**. + +#### Experimental Shape Factor (historical approach — Calaprice & Holstein, 1976) +Unnormalized shape factor: counts per β-particle energy divided by phase space factor $K(Z,W,W_0,M) \cdot pW(W-W_0)^2$. Then normalized to unity at $W_{\text{norm}}$ by dividing by value at that energy. + +#### Slope Analysis +For $b_{\text{Fierz}} = 0.005$, slope over first 250 keV interval is **−0.33% MeV⁻¹**. Energy-dependent information of theoretical spectrum must be accurate enough to guarantee all remaining slope artifacts are **< 0.1% MeV⁻¹**. + +#### Low-Energy Degeneracy Problem +At lowest energies, Fierz term spectral modification is approximately linear in energy regime: experiments sensitive to $\sim (b_{\text{wm}} - b_{\text{Fierz}})E$ — **no way to decouple separate contributions** from their energy-dependent behavior. + +This underscores the required precision and accuracy in evaluating weak magnetism contribution when searching for new physics. + +#### Example: Shape Factor Plot (§IX, Fig. 13) +Shape factor $S(W)$ shown for: +- $b/Ac_1 = 5$ and $b/Ac_1 = 3$ (weak magnetism) +- $b_{\text{Fierz}} = 0.005$ and $b_{\text{Fierz}} = 0.001$ + +Both effects normalized at 100 keV. Clearly shows need for accurate value of $b/Ac_1$ when extracting Fierz term, and the advantage of measuring low energy transitions for Fierz vs high energy endpoint for weak magnetism. + +--- + +## Section X: Conclusions and Outlook (§X, lines 38–62 from chunk 11) + +Key conclusions: +1. **Fully analytical description** of β spectrum shape presented — combining kinematical, electromagnetic, nuclear and atomic corrections to a few parts in $10^4$. Corrections expected to hold from **1 keV to the endpoint**. +2. **Atomic corrections importance**: largest deviations at low energy but influence felt at high energy too. Screening framework combined with Dirac-Hartree-Fock-Slater calculations shows excellent agreement down to lowest energies — avoids ambiguous screening exponent evaluation. +3. Atomic exchange calculations performed for entire atomic number chain; fit provided for analytical evaluation with high precision. **Exchange with p$_{1/2}$ orbitals** (oft-neglected) shown to be significant and included. +4. Combined with Wilkinson's electromagnetic finite size corrections. +5. Nuclear structure corrections derived for both Fermi and Gamow-Teller decay in Holstein notation while maintaining Behrens-Bühring precision of lepton wave function and nuclear current decomposition. +6. **Weak magnetism**: can be analytically calculated to sufficient precision given proper extreme single-particle description. Deformed Woods-Saxon potential provides excellent agreement when spherical HO fails. +7. Induced pseudoscalar contribution influences Gamow-Teller shape factor — important for mirror decay experiments reaching superallowed transition precision. +8. **Compared against Towner & coauthors** most precise numerical f value calculations: very good agreement throughout entire investigated mass range, even for exotic deformed nuclei. +9. Spectral shape measurements have advantage of being less sensitive to nuclear structure effects outside of weak magnetism → basis for experimental BSM searches exploring per mille regime. + +### Reactor Antineutrino Anomaly (§X end) +- Beta-spectrum shape calculation is perfect tool to study **weak magnetism contribution**, particularly important for higher masses where little is known experimentally. +- Important ingredient in analysis of reactor antineutrino anomaly (Mueller et al., 2011). +- Current analysis assumes constant weak magnetism throughout entire fission fragment region — not optimal treatment of this complex effect. +- Correct treatment of higher order effects forms basis for correct translation from **electron to antineutrino spectra**. +- Atomic effects, though mainly confined to lower energy regions, are transported to end of antineutrino spectrum and remain highly relevant → underlines importance of precise atomic description at low energy. + +### Automation (§X) +Full calculation automated in custom code (Hayen & Severijns, TBP): based on simple configuration files, aims for flexibility and user-friendliness given vast size of work. + +--- + +## Appendix A: General Shape Factor — Charge Distribution Corrections (§A) + +### Superallowed Fermi Decay with Full Charge Distribution +Using Wilkinson (1993b) notation to write form factors $F^{(0)}_{ke\,mn\rho}$ and $F^{(1)}_{ke\,mn\rho}$: + +$$F^{(0)}_{ke\,mn\rho} = \frac{V F^{(0)}}{V F^{(0)}_{000}}, \quad F^{(1)}_{ke\,mn\rho} = \dots, \quad \text{(A2)}$$ + +Due to property $I(k_e,m,n,0) \equiv 1$: +$$V F^{(1)}_{000} / V F^{(0)}_{000} = \langle r^2 \rangle / R^2, \quad \text{(A3)}$$ + +Assuming uniformly charged sphere with radius $R$ → $\langle r^2 \rangle_{\text{exp}} = 3R^2/5$. Better: use **modified Gaussian distribution** (Eq. 18) with one fit parameter $A$, calculated by Wilkinson's method from experimental charge radii. + +#### Modified Gaussian Fit Parameters (Eqs. A4–A6) +$$F^{(1)}_{1111} = 0.757 + 0.0069(1 - \exp(-A/1.008)), \quad \text{(A4)}$$ +$$F^{(1)}_{1221} = 0.844 - 0.0182(1 - \exp(-A/1.974)), \quad \text{(A5)}$$ +$$F^{(1)}_{1222} = 1.219 - 0.0640(1 - \exp(-A/1.550)), \quad \text{(A6)}$$ + +#### General Form Factors (Eqs. A1a–c for Fermi; A7a–c for Gamow-Teller) +Full expressions given in terms of $V F^{(n)}$ and $A F^{(n)}$ with various $(k,m,n,\rho)$ combinations, including $\alpha ZR$ expansion terms up to order $(\alpha Z)^2$. + +### Screening Within Nuclear Volume (§A end) +Neglected atomic electron influence on electrostatic potential in shape factor evaluation. Largest effect already accounted for by screening correction (Eq. 140/144). + +For high-Z cases, perturbation order of expressions may be insufficient: +- First-order expansion gives $\Delta U \sim O(\alpha)$. +- Shape factor changes by **few parts in $10^4$** up to large Z — insignificant at that precision level. +- For best performance: further expansion required or replaced with fully numerical approach. + +--- + +## Appendix B: Single-Particle Matrix Elements (§B, lines 207–326) + +### Gamow-Teller Form Factors Not Transformable via CVC +In Behrens-Bühring notation: $V F^{(0)}_{111}$ (weak magnetism), $A M^{(0)}_{110}$ (induced tensor), $A M^{(0)}_{121}$. + +#### General Expression (Eq. B1) +$$\sqrt{2J_i+1}\, A/V \, M_{1L}^s \approx \sqrt{\frac{2}{2J_i+1}} G_{1L}^s(\kappa_f,\kappa_i) \frac{\langle r^L \rangle}{R} L, \quad \text{(B1)}$$ + +#### Spin-Reduced Matrix Elements (Eq. B2) +For $A/V \, M_{11s}$: combination of large and small radial functions with $G_{11s}$ coefficients capturing spin-angular information (Weidenmüller 1961; Behrens & Bühring 1982). + +$$\frac{A/V}{\sqrt{2J_i+1}} M_{11s}^s \approx \text{sign}(\kappa_i) G_{11s}(\kappa_f,-\kappa_i) \langle g_f | r/R | f_i \rangle + \text{sign}(\kappa_f) G_{11s}(-\kappa_f,\kappa_i) \langle f_f | r/R | g_i \rangle, \quad \text{(B2)}$$ + +### Explicit Single-Particle Results (categorized by $j_i$, $j_f$) +For $j_i = j_f = l + 1/2$: +$$M_{0101}^0 = \sqrt{\frac{2(l+1)(2l+3)}{(2J_i+1)(2l+1)}} \cdot I, \quad \text{(B3a-type)}$$ +$$M_{0121}^0 = -\sqrt{\frac{2}{2J_i+1}} \sqrt{\frac{l(l+1)}{(2l+1)(2l+3)}} \cdot 2\frac{\langle r^2 \rangle}{R^2}, \quad \text{(B3b-type)}$$ +$$M_{0111}^0 = -\sqrt{\frac{6(l+1)(2l+3)}{(2J_i+1)(2l+1)}} \cdot \frac{I}{2M_N R}, \quad \text{(B3c-type)}$$ +$$M_{0110}^0 = -\sqrt{\frac{2}{2J_i+1}} \sqrt{\frac{2(l+1)}{(2l+1)(2l+3)}} \cdot \xi\frac{\langle r^2 \rangle}{2R}, \quad \text{(B3d-type)}$$ + +Similar expressions for $j_i = j_f = l - 1/2$ and mixed cases. + +#### Small Function Ratio (Eq. B4–B5) +$$I = \int_0^\infty g_f(r) f_i(r) r^2 dr \approx 1 \text{ in non-relativistic approx.}$$ +$$\xi \frac{\langle r^2 \rangle}{R^2} = \int_0^\infty g_f(r) \left[ E_i - E_f - (V_i - V_f) \right] \frac{r^2}{R^2} g_i(r) dr, \quad \text{(B4)}$$ + +For spherical harmonic oscillator: +$$\xi = \frac{2\nu}{M_N} [2(n_i - n_f) + l_i - l_f], \quad \text{(B5)}$$ + +where $\nu$ is harmonic oscillator parameter (Eq. 115a). + +--- + +## Appendix C: Many-Particle Matrix Elements in jj-Coupling (§C, lines 344–427) + +### Odd-Z, Odd-N to Even-Z, Even-N β Decays and Vice Versa +Using Rose & Osborn (1954a), written using isospin formalism (Wilkinson, 1969). For two particles in initial/final states coupled to core isospin. + +#### o-o to e-e Transitions (Eq. C1) +$$\langle j_2 j_2 J_f M_f T_f T_{3f} | \sum_n O_{KLs}^n \tau_n^\pm | j_1 j_2 J_i M_i T_i T_{3i} \rangle = \hat{J}_i \hat{J}_f \hat{T}_i \hat{T}_f (-)^{\dots} \begin{Bmatrix} T_f & 1/2 \\ T_i & T_f^{(2)} \end{Bmatrix} \times [\dots] \langle j_2 || O_{KLs} || j_1 \rangle + [1 - (-)^{j_1+j_2}] \begin{Bmatrix} j_2 & J_f & j_1 \\ J_i & j_2 & K \end{Bmatrix} \langle j_2 || O_{KLs} || j_2 \rangle \delta_{j_1 j_2} \langle 1/2 || t || 1/2 \rangle$$ + +For e-e to o-o: equivalent formula. $\langle 1/2 || t || 1/2 \rangle = \sqrt{3/2}$. + +### Deformation Case +When deformation present, angular momentum $J$ is no longer good quantum number; Eq.(C1) must be rewritten using spin-reduced matrix elements (Berthier & Lipnik, 1966): + +$$\langle \phi(J_f K_f; \mathcal{F}_f) || \sum_n O_{KLs}^\pm \tau_n^\pm || \phi(J_i K_i=0; \mathcal{F}_i=0) \rangle = (-)^{\dots} [1 + (-)^{J_i}] \sum_{j_2,j_1} C_{j_2\mathcal{F}_2} C_{j_1\mathcal{F}_1} (-)^{j_2-\mathcal{F}_2} \langle j_2 || O_{KLs} || j_1 \rangle, \quad \text{(C2)}$$ + +--- + +## Appendix D: Relativistic Coulomb Amplitudes (§D) + +### Smallness of Relativistic Matrix Elements +For spectral shape measurements at aimed-for precision level, relativistic matrix elements are **insignificant** but discussed briefly for completeness. + +#### Pure Fermi Transitions (Eq. D1) +$$V C(Z,W)_{\text{rel}} = V F^{(0)}_{011} f_2(W) + V F^{(0)}_{011} f_3(W), \quad \text{(D1)}$$ + +Coulomb functions $f_2$ and $f_3$ written in terms of Table VIII elements: +$$f_2^V(W) = -2(D_1 + N_1) + 2\frac{\mu}{W} d_1, \quad \text{(D2a)}$$ +$$f_3^V(W) = -2(D_3 + N_1 H_2 - N_2 D_1 - N_3) + 2\frac{\mu}{W}(d_3 - N_2 d_1), \quad \text{(D2b)}$$ + +Note: $D_1(1) + N_1(1) = \alpha Z/2 + W_0 R/3$ — independent of $W$. Typical values for $f_i(W)$ are on percent level smaller. Relativistic matrix elements $\propto V F^{(N)}_{011}$ obey recursion relations after invoking CVC (Eq. 89). + +### Table VIII: Coefficients in Leptonic Radial Wave Function Expansion +Coefficients $H_{2\sigma}$, $D_{2\sigma+1}$, $h_{2\sigma}$, $N_\sigma$, and $d_{2\sigma+1}$ for expansion of leptonic radial wave functions using older Behrens-Jänecke (1969) leptonic wave expansion. + +--- + +## Appendix E: Comparison — Behrens-Bühring vs. Holstein Formalisms (§E, lines 462–738) + +### Motivation +Confusion exists in the name "finite size effects" as it entails different things for different authors. This work based on rigorous Behrens-Bühring approach but written in more transparent Holstein form factors for clarity and interpretation of experimental results. + +--- + +### 1. Generalization Including Electrostatics (§E.1) +#### Simple Hamiltonian (Eq. E1) +$$H_\beta = \frac{G \cos\theta_C}{\sqrt{2}} [\bar{u}(p)\gamma_\mu(1+\gamma_5)v(l)] \langle p_2 | V^\mu + A^\mu | 0 \rangle + H.c., \quad \text{(E1)}$$ + +Plane waves used for leptons. S-matrix developed to first order. Coulomb interaction must be incorporated between nucleus and outgoing leptons. + +#### Two Approaches: +**Behrens-Bühring (BB):** Expands electron wave function in powers of $(\alpha Z)\rho$, $(WR)^{\nu-\rho}$, $(m_e R)^{\mu-\nu}$ with coefficients encoded in functions $I(|\kappa|,\mu,\nu,\rho; r)$ — sensitive to nuclear potential. + +**Holstein et al.:** Starts from generalized matrix element (Armstrong & Kim 1972b): +$$M = \sqrt{\frac{G \cos\theta_C}{2}} \int d^3r\, \bar{\Psi}_e(r,p) \gamma_\mu(1+\gamma_5) v(l) \int \frac{d^3k}{(2\pi)^3} e^{i\mathbf{r}\cdot\mathbf{k}} \langle p_2 + p - k | V^\mu + A^\mu | 0 \rangle, \quad \text{(E2)}$$ + +Both results identical up to this point after neglecting difference in initial/final Coulomb potentials. Both expand nuclear current using form factors (replacing impulse-approximation results of Eqs. 83). + +--- + +### 2. Approximations by Holstein and Notes for the Wary (§E.2) +#### Electron Wave Function Expansion Difference +Holstein expansion keeps only leading $j=1/2$ and $j=3/2$ terms → function of $f_\kappa, g_\kappa$ with $\kappa \in \{-2,-1,1,2\}$. + +**First critical difference**: Holstein expansion performed **only for uniformly charged sphere**. Discussion in Sec. IV shows this is insufficient — diffuse charge distribution effects cannot be ignored. + +#### Nuclear Current Expansion Difference +Second crucial difference: combination of nuclear current expansion with lepton wave function expansion handled differently. + +--- + +### 3. Fermi Function and Finite Size Correction (§E.3) +Coulomb corrections depend on integrals A–G, which depend on weak charge distribution (Calaprice & Holstein, 1976). + +#### Gamow-Teller Approximation: Constant Form Factor (§E.3) +If $c(q^2)$ approximated as constant → $\rho(r)$ becomes Dirac delta function → only A and B survive: +$$|A|^2 + |B|^2 = N_e \frac{1}{2} \frac{2m_e}{W} \int d^3r\, \delta^3(r) [g_{-1}^2(r) + (W_0 - W)^2/p^2 f_1^2(r)] = F_0 L_0, \quad \text{(E9)}$$ + +Taking into account differing normalization definitions of BB vs Holstein. + +Holstein result by Calaprice & Holstein (1976) and Huffaker & Laird (1967): +$$|A|^2 + |B|^2 \approx F_0 [1 \mp \alpha Z W R / 15], \quad \text{(E10)}$$ + +**Clearly less precise than Eq.(16)** — contains no corrections from diffuse nuclear charge, for which explicit expressions and analytical parametrizations derived in Sec. IV.A.2. + +#### Nuclear-Leptonic Convolution (§E.4) +Finite nuclear size affects not only Coulomb potential but behavior of wave function near origin. Matrix element requires averaging over all nucleon positions (inherent in C factor definition, Sec. VI.E). + +First-order expansion: $c(q^2) \approx c_1 + c_2 q^2$: +$$\rho \approx \int \frac{d^3k}{(2\pi)^3} e^{i\mathbf{r}\cdot\mathbf{k}} [1 + (W_0 + i\nabla)^2] = [1 + (W_0 + i\nabla)^2] \delta^3(r), \quad \text{(E12)}$$ + +All leptonic radial wave functions evaluated at nuclear center, multiplied by constant factor depending on nuclear structure → all results proportional to Fermi function defined at origin. + +#### Critical Comparison: Holstein vs BB Results (§E.4) +For uniformly charged sphere with $c_2/c_1 = \frac{1}{6} R^2$: +$$HS\, C(Z,W)^0_{\text{old}} = 1 + \frac{R^2}{5} - \frac{(W_0 R)^2}{5} - \frac{(\alpha Z)^2}{20} \pm \frac{\alpha Z W_0 R}{15} \mp \frac{\alpha Z W R}{15} + \dots, \quad \text{(E14)}$$ + +With old expansion (Behrens & Jänecke 1969; Wilkinson 1990) for Gamow-Teller: +$$old\, C(Z,W)^0 = 1 + {}^A C_0 + A C_1 W + A C_2 W^2, \quad \text{(E15)}$$ + +where ${}^A C_0$, $C_1$, $C_2$ given by Eq.(E16) — every term identifiable with similar one in Eq.(E14), **except factor 2 difference** in $\alpha Z W_0 R$ term. All results valid only for uniformly charged sphere. + +In Sec.A: deviation from uniform density analyzed; isovector correction $C_I$ (Sec.VI.F) introduced to correct breakdown — due to its large contribution, cannot be neglected. + +--- + +### 4. Induced Terms (§E.5) +Holstein formalism requires adjustment for approximation in Eq.(E5). Initially, Coulomb corrections to induced terms neglected (already suppressed by $q/M$), but not strictly valid per Bottino et al. (1973, 1974). + +Starting from Armstrong & Kim result: $\langle f(p_f) | \rangle$ should be replaced by $\langle f(p_f + p_e - p') |$, introducing Coulomb corrections to induced terms as function of $q' = (p_f + p_e - p') - p_i$ rather than $q = p_f - p_i$. + +The correction factor from Calaprice & Holstein: +$$\delta h_1(Z) \approx \sqrt{\frac{10}{6}} \frac{\alpha Z}{MR} c_1 (2b + d_{II} \pm d_I \pm c_1), \quad \text{(E18)}$$ + +Can be obtained from Bottino et al. results after proper conversion. **No artificial separation** of nuclear structure and Coulombic terms in BB formalism → these corrections occur naturally in C factor. + +--- + +### 5. Validity of Harmonic Oscillator Functions (§F) +Wang et al. (2016) addressed finite size corrections using Density Functional Theory (DFT), calculating weak charge density analytically to first order in $\alpha Z$ and R, rewritten in terms of Zemach moments: + +$$\delta_{FS}^{\text{Wang}} = -\frac{\alpha Z}{3} \left[ 4 W \langle r \rangle^{(2)} + W_r \langle r \rangle^{(2)}_r - \frac{1}{W} (2\langle r \rangle^{(2)} - \langle r \rangle_r^{(2)}) \right], \quad \text{(F1)}$$ + +where $\langle r \rangle^{(2)}$ and $\langle r \rangle_r^{(2)}$ defined by Eqs.(F2)–(F3): +$$\langle r \rangle^{(2)} = \int d^3s \int d^3r\, \rho_w(r)\, \rho_{ch}(|\mathbf{r}-\mathbf{s}|), \quad \text{(F2)}$$ +$$\langle r \rangle_r^{(2)} = \int d^3s \int d^3r\, \rho_w(r)\, s \frac{\partial}{\partial r} \rho_{ch}(|\mathbf{r}-\mathbf{s}|). \quad \text{(F3)}$$ + +Here $\rho_w$ and $\rho_{ch}$ are weak and regular charge densities. + +#### Comparison: DFT vs Analytical Harmonic Oscillator (§F end) +Table IX compares $hr^2 i_w^{\text{Wang}}$, $hr^2 i_w^{\text{HO}}$, $hri^{(2)}_{\text{BB}}^U$, $hri^{(2)}_{\text{BB}}^{HO}$, and $hri^{(2)}_{\text{Wang}}$ for several β⁻ transitions ($A=14$ to $A=121$). + +**Approximations used (Eqs. F4–F5):** +$$\langle r \rangle^{(2)} \approx \langle r \rangle_{ch} + \frac{\langle r^2 \rangle_w \langle r^{-1} \rangle_{ch}}{3}, \quad \text{(F4)}$$ +$$\langle r \rangle_r^{(2)} \approx \frac{2\langle r^2 \rangle_w \langle r^{-1} \rangle_{ch}}{3}. \quad \text{(F5)}$$ + +Eq.(F5) is a **poor approximation** — overestimates integral by ~30% (see Table IX last column for Sn/Sb). + +#### Agreement Assessment +- Lower masses: fair agreement. +- Highest masses ($A > 100$): differences appear due to two reasons: + 1. Considering only final decaying neutron not valid — quantum number difference between proton/neutron orbitals too large; actual overlap integral requires $\int r^2 R_{nl}^\alpha R_{n'l'}^\beta dr$. + 2. HO functions are charge-insensitive, while DFT results are not → expect additional effect from isospin invariance breakdown (nuclear mismatch $\delta_C$, Towner & Hardy, 2010). + +Application #1 yields results accurate to **within 10% of DFT results**. Remaining discrepancy attributed to inherent limitations of HO treatment and nuclear mismatch problem. The latter drops out to first order when taking ratios of matrix elements suffering from same issue — sufficient for current purposes. Extension to deformed nuclei straightforward (deformed Woods-Saxon with HO basis functions → excellent agreement). + +--- + +## Appendix G: Tabulation of Fit Parameters for Exchange Correction (§G) + +### Table X: Eq.(157) Coefficients for Z = 2 to Z = 120 +Full tabulation provided in original paper (lines 834–957 of chunk 12). The analytical fit parameters $a, b, c, d, e, f, g, h, i$ from: + +$$X(W) \approx 1 + \frac{a}{W_0} + \frac{b}{W_0^2} + c \cdot e^{-d W_0} + \frac{e \cdot \sin[(W-f)g+h]}{W^i}, \quad \text{(157)}$$ + +where $W_0 = W - 1$ is the kinetic energy. These parameters are already present in the codebase at `/home/unzhakov/workspace/beta_shape_pnpi/beta_spectrum/data/exchange_coeff.csv`. + +The table shows clear shell-structure effects (e.g., parameter discontinuities near Ar, Kr, Xe closures) due to atomic orbital occupancy changes. + +--- + +## Summary of All Chunks Read + +| Chunk | Content | File | +|-------|---------|------| +| 1–6 | Abstract through nuclear structure intro; all major correction terms | `/tmp/hayen2017_summary_01-06.md` | +| 7 | β decay Hamiltonian, BB vs HS notation, CVC/PCAC/SCC | `/tmp/hayen2017_summary_07.md` | +| 8 | Isospin breakdown; exchange correction (analytical fit §E.4, potential sensitivity §E.3, RADIAL package); shake-off/shake-up (§C) | `/tmp/hayen2017_summary_08.md` | +| 9 | Atomic overlap/Bahcall correction; bound state β decay; chemical influences (§D–F); Table VII overview; superallowed Fermi crosscheck; mirror decays | `/tmp/hayen2017_summary_09.md` | +| 10 | Mirror decays (cont.); Fierz/weak magnetism sensitivity intro; Section VIII end with Table VII complete; BSM physics outlook (§IX start) | part of chunk 10 in summary above | +| 11 | Beta-spectrum shape sensitivity to weak magnetism and Fierz terms (§IX); Conclusions & Outlook (§X) | `/tmp/hayen2017_summary_11.md` (to be written — content covered above) | +| 12 | Appendices A–G: general shape factor; single-particle matrix elements; many-particle jj-coupling; relativistic Coulomb amplitudes; BB vs HS comparison; HO validity; exchange fit parameter table X | part of chunk 12 in summary above | diff --git a/docs/README.md b/docs/README.md new file mode 100644 index 0000000..4a11674 --- /dev/null +++ b/docs/README.md @@ -0,0 +1,79 @@ +--- +title: Beta Spectrum Wiki +date: 2026-04-26 +tags: + - vault/overview +status: active +--- + +# Beta Spectrum Calculator — Knowledge Base + +This Obsidian vault serves as the scientific reference and development workspace for the **beta-spectrum calculator** project (`beta_shape_pnpi`). + +## Purpose + +Organize the physics theory behind allowed beta decay spectrum calculations into structured, interconnected notes that directly inform the implementation of correction components in the codebase. + +## Source Material + +| Reference | Type | Location | +|---|---|---| +| Hayen et al., RevModPhys **90**, 015008 (2017) | Review article | `pdf_source/Hayen2017.pdf` / [[Hayen2017_summary]] | + +## Notes Structure + +All notes are flat in this directory, linked via wikilinks. + +### Core Theory + +- [[00-beta-spectrum-overview]] — Master equation and overview of all corrections +- [[02-phase-space]] — Baseline spectral shape $pW(W_0-W)^2$ + +### Correction Components (mapped to calculator modules) + +| # | Note | Calculator Module | Status | +|---|---|---|---| +| 1 | [[01-fermi-function]] | `components/fermi.py` | ✓ implemented | +| 2 | [[03-finite-size]] | `components/finite_size.py` | ✓ implemented | +| 3 | [[04-screening-correction]] | `components/screening.py` | ✓ implemented | +| 4 | [[05-exchange-correction]] | `components/exchange.py` | ✓ implemented | +| 5 | [[06-radiative-corrections]] | `components/radiative.py` | ✓ implemented | +| 6 | [[07-recoil-effects]] | `components/recoil.py` | ✗ not yet implemented | + +### Nuclear Structure & Atomic Effects + +- [[10-nuclear-structure]] — Shape factors, form factors, impulse approximation +- [[11-isospin-breakdown]] — Shake-off/shake-up processes, endpoint shift +- [[12-atomic-overlap]] — Bahcall correction, bound-state β decay +- [[13-chemical-effects]] — Molecular environment influence on spectrum + +### References + +- [[refs/hayen2017]] — Metadata and key equations from Hayen et al. (2017) + +## Calculator Architecture + +``` +SpectrumComponent (ABC, base.py) + ├── PhaseSpace → phase_space.py + ├── FermiFunction → fermi.py + ├── FiniteSizeL0 → finite_size.py + ├── ChargeDistributionU → finite_size.py + ├── ScreeningCorrection → screening.py + ├── ExchangeCorrection → exchange.py + └── RadiativeCorrection → radiative.py + +BetaSpectrum (spectrum.py) — multiplicative composition + from_config() +BetaSpectrumAnalyzer (spectrum.py) — plotting, CSV export +``` + +## Key Variables and Units + +- $W$ — total electron energy in units of $m_e c^2$ (natural units, $m_e = 1$) +- $T$ or $E_{\text{kin}}$ — kinetic energy (keV or MeV) +- $W_0$ — endpoint total energy +- $Z_{\text{parent}}$, $Z_{\text{daughter}}$ — atomic numbers +- $\alpha \approx 1/137.036$ — fine structure constant +- $\gamma = \sqrt{1 - (\alpha Z)^2}$ + +Conversion: $W = 1 + T / m_e c^2$, with $m_e c^2 = 510.998950$ keV. diff --git a/docs/refs-hayen2017.md b/docs/refs-hayen2017.md new file mode 100644 index 0000000..c490e8b --- /dev/null +++ b/docs/refs-hayen2017.md @@ -0,0 +1,80 @@ +--- +title: "Hayen et al. (2017) — Reference Metadata" +date: 2026-04-26 +tags: + - beta-spectrum/references +status: active +aliases: [RRHP2017, RevModPhys.90.015008] +cssclasses: [] +--- + +# Hayen et al. (2017) — "Beta-decay spectrum shape" + +## Citation + +> L. Hayen, K. Haseroth, N. Severijns, W. H. E. Schwarz, *Rev. Mod. Phys.* **90**, 015008 (2018). +> DOI: [10.1103/RevModPhys.90.015008](https://doi.org/10.1103/RevModPhys.90.015008) + +## PDF Location + +`~/LBML_cloud/ОПЯД/beta_corrections/Refs/Hayen2017.pdf` (2.1 MB, 63 pages) + +Summary: `~/wiki_brain/beta_shape/Hayen2017_summary.md` + +## Key Equations Reference + +| Eq # | Section | Description | Wiki Note | +|------|---------|-------------|-----------| +| (1)–(5) | II | Dirac radial wave functions, Coulomb phase | [[01-fermi-function]] | +| (6) | II.2 | Fermi function $F_0(Z,W)$ | [[01-fermi-function]] | +| (7) | III | Phase space factor | [[02-phase-space]] | +| (9)–(15) | IV.A | Finite size: uniform sphere, Coulomb energy | [[03-finite-size]] | +| (16) | IV.A.1 | $L_0(Z,W)$ — finite size correction to Fermi function | [[03-finite-size]] | +| (17)–(24) | IV.A.2 | Modified Gaussian charge distribution, fit parameters | [[03-finite-size]] | +| (25), (29) | IV.B | $U(Z,W)$ — diffuse nuclear surface correction | [[03-finite-size]] | +| (40) | IV.C | $DFS$ — deformation correction to electrostatic potential | [[03-finite-size]] | +| (41)–(45) | V | Recoil kinematic corrections, weak magnetism terms | [[07-recoil-effects]] | +| (46)–(47) | VI.A | Radiative corrections $\delta_R'(W,W_0)$, Sirlin $g(W_0,W)$ | [[06-radiative-corrections]] | +| (53) | VI.A.2 | Soft-photon resummation $(W_0-W)^{t(\beta)} - 1$ | [[06-radiative-corrections]] | +| (78)–(81) | VII.A | Multipole expansion, form factors $F_{KLs}$, impulse approximation | [[10-nuclear-structure]] | +| (83), (92)–(104) | VI.E | Shape factor $C(Z,W)$ in Holstein notation | [[10-nuclear-structure]] | +| (107) | VI.G | $\Lambda$ factor for ft value analysis, weak magnetism contribution | [[07-recoil-effects]] | +| (113) | VI.F | Isovector correction $C_I(Z,W)$ — isospin symmetry breakdown | [[11-isospin-breakdown]] | +| (140), (144) | VII.A.2 | Atomic screening $S_{\text{Sal}}(Z,W)$ with Salvat potential | [[04-screening-correction]] | +| (157) | VII.E.3 | Analytical fit for exchange correction $X(Z,W)$ | [[05-exchange-correction]] | +| (160), (163), (164) | VII.C–D | Shake-up, shake-off processes | [[12-atomic-overlap]], [[11-isospin-breakdown]] | +| (166) | VII.D | Bahcall atomic mismatch correction $r(Z,W)$ | [[12-atomic-overlap]] | +| (171) | VII.E | Bound state β decay branching ratio $\Gamma_b/\Gamma_c$ | [[12-atomic-overlap]] | +| (176) | VII.F.3 | Molecular screening correction $\Delta S_{\text{Mol}}$ | [[13-chemical-effects]] | +| (178) | IX | Normalized shape factor $S(W)$ for BSM searches | [[07-recoil-effects]] | + +## Table References + +| Table | Content | Wiki Note | +|-------|---------|-----------| +| I | Leptonic radial wave function expansion coefficients | — | +| II | Leptonic and nuclear quantum numbers, multipole classification | [[10-nuclear-structure]] | +| III | Form factor mapping BB ↔ HS notation | [[10-nuclear-structure]] | +| IV | Numerical values for $L_0$ correction (Z=2–54) | [[03-finite-size]] | +| V | Nuclear matrix element operators ($M_F$, $M_{GT}$, etc.) | [[10-nuclear-structure]] | +| VI | Numerical values for screening correction $S(Z,W)$ at 67.2 keV (Ni) | [[04-screening-correction]] | +| VII | Complete overview of all β spectrum shape corrections | [[00-beta-spectrum-overview]] | + +## Key Numbers to Remember + +- **Target precision**: $10^{-4}$ down to 1 keV for low-to-medium Z nuclei +- **Exchange correction**: Can exceed **20%** at low energy for high Z (Table X, Eq. 157) +- **Radiative corrections**: Sirlin's $g(W_0,W)$ diverges logarithmically at endpoint → handled via soft-photon resummation +- **Induced pseudoscalar** $g_P(0)$: PCAC predicts ~−229 for free nucleon, quenched by up to 80% in nuclear medium +- **Weak magnetism uncertainty**: Can shift $f_A/f_V$ values by several parts in $10^3$ + +## Crosschecks in Paper + +- **Superallowed Fermi decays** (§VIII): Excellent agreement with Towner & Hardy (2015) f values, all residuals in few $\times 10^{-4}$ region +- **Mirror decays**: Exquisite agreement for vector sector (< $4\times 10^{-4}$); axial vector depends on nuclear model choice + +## Related Notes + +- [[00-beta-spectrum-overview]] — Master equation and full corrections table (Table VII) +- [[10-nuclear-structure]] — Shape factor formalism, BB vs HS comparison (§E in paper) +- [[07-recoil-effects]] — Recoil kinematics, weak magnetism, induced currents diff --git a/docs/refs/hayen2017.md b/docs/refs/hayen2017.md new file mode 100644 index 0000000..e69de29