unzhakov
030ae3e4b9
- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
94 lines
4.0 KiB
Markdown
94 lines
4.0 KiB
Markdown
---
|
||
title: "00 – Beta Spectrum Overview"
|
||
date: 2026-04-26
|
||
tags:
|
||
- beta-spectrum/overview
|
||
status: active
|
||
aliases:
|
||
- Master Equation
|
||
cssclasses: []
|
||
---
|
||
|
||
# Beta Decay Spectrum — Full Expression
|
||
|
||
## The Master Equation
|
||
|
||
The fully differential allowed β spectrum shape, including all corrections needed for $10^{-4}$ precision (Hayen et al., Eq. 4):
|
||
|
||
$$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} \; K(Z,W,W_0,M) \; A(Z,W) \; C'(Z,W) \; p W (W_0 - W)^2 dW$$
|
||
|
||
or expanded:
|
||
|
||
$$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} F_0 L_0 U DFS R R_N Q S X r C D_C \; p W (W_0 - W)^2 dW \tag{4}$$
|
||
|
||
## Correction Factors
|
||
|
||
| Symbol | Name | Module / Note | Status |
|
||
|---|---|---|---|
|
||
| $F_0(Z,W)$ | Point charge Fermi function | [[corrections/01-fermi-function]] | ✓ |
|
||
| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[corrections/02-finite-size]] | ✓ |
|
||
| $U(Z,W)$ | Charge distribution correction | [[corrections/02-finite-size]] | ✓ |
|
||
| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[corrections/02-finite-size]] | ✓ |
|
||
| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[corrections/05-radiative]] | ✓ |
|
||
| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[corrections/06-recoil]] | ✗ |
|
||
| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[corrections/06-recoil]] | — |
|
||
| $S(Z,W)$ | Atomic electron screening | [[corrections/03-screening]] | ✓ |
|
||
| $X(Z,W)$ | Atomic exchange effect | [[corrections/04-exchange]] | ✓ |
|
||
| $r(Z,W)$ | Atomic mismatch / overlap | [[corrections/08-atomic-overlap]] | — |
|
||
| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[corrections/08-atomic-overlap]] | — |
|
||
| $C(Z,W)$ | Nuclear structure shape factor | [[nuclear-structure/01-shape-factors]] | — |
|
||
| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[nuclear-structure/01-shape-factors]] | — |
|
||
|
||
## 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 Hayen 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 [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/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.
|