94 lines
3.9 KiB
Markdown
94 lines
3.9 KiB
Markdown
---
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title: "00 – Beta Spectrum Overview"
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date: 2026-04-26
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tags:
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- beta-spectrum/overview
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status: active
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aliases:
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- Master Equation
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cssclasses: []
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---
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# Beta Decay Spectrum — Full Expression
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## The Master Equation
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The fully differential allowed β spectrum shape, including all corrections needed for $10^{-4}$ precision (Hayen et al., Eq. 4):
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$$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} \; K(Z,W,W_0,M) \; A(Z,W) \; C'(Z,W) \; p W (W_0 - W)^2 dW$$
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or expanded:
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$$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} F_0 L_0 U DFS R R_N Q S X r C D_C \; p W (W_0 - W)^2 dW \tag{4}$$
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## Correction Factors
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| Symbol | Name | Module / Note | Status |
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|---|---|---|---|
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| $F_0(Z,W)$ | Point charge Fermi function | [[01-fermi-function]] | ✓ |
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| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[03-finite-size]] | ✓ |
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| $U(Z,W)$ | Charge distribution correction | [[03-finite-size]] | ✓ |
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| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[03-finite-size]] | ✓ |
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| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[06-radiative-corrections]] | ✓ |
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| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[07-recoil-effects]] | ✗ |
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| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[07-recoil-effects]] | — |
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| $S(Z,W)$ | Atomic electron screening | [[04-screening-correction]] | ✓ |
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| $X(Z,W)$ | Atomic exchange effect | [[05-exchange-correction]] | ✓ |
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| $r(Z,W)$ | Atomic mismatch / overlap | [[12-atomic-overlap]] | — |
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| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[12-atomic-overlap]] | — |
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| $C(Z,W)$ | Nuclear structure shape factor | [[10-nuclear-structure]] | — |
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| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[10-nuclear-structure]] | — |
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## Baseline: Phase Space
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The uncorrected spectral shape is pure phase space ([[02-phase-space]]):
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$$\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2$$
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with $p = \sqrt{W^2-1}$, $W$ total energy in units of $m_e$, and $W_0$ the endpoint.
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## Precision Goal
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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]].
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> [!key] Why this matters
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> 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.
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## Three Formalisms Compared
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| Approach | Style | Strengths |
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|---|---|---|
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| **Behrens-Bühring** (BB) | Rigorous, numerical Dirac solutions | Most precise results |
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| **Holstein** (HS) | Analytical, manifestly covariant | Clean symmetry properties |
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| **Wilkinson** | Analytical parametrizations | Practical fits for dominant effects |
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The Hayen paper combines BB's accuracy with HS's transparency.
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## Conventions
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- Natural units: $c = \hbar = m_e = 1$ throughout
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- $Z$ is always positive; $\beta^-$ and $\beta^+$ distinguished via upper/lower signs ($\mp$)
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- $W = E/m_ec^2 + 1$, $p = \sqrt{W^2 - 1}$
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- $G_V$: vector coupling strength; $V_{ud} = \cos^2\theta_C$
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> [!note] Two types of "finite size"
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> The authors distinguish two corrections conflated under this term:
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> 1. **Electrostatic** ($L_0, U, DFS$): Potential difference from point charge → realistic nuclear shape
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> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume (appears in $C(Z,W)$)
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## Energy Conversion
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$$W = 1 + \frac{T}{m_ec^2}, \quad m_ec^2 = 510.998950\text{ keV}$$
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In code: see `beta_spectrum/utils.py` → `T_to_W()`, `W_to_T()` functions.
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## Calculator Implementation
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Each correction factor $C_i(W)$ is a callable class inheriting from `SpectrumComponent`. The total spectrum is the product of all enabled components:
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```
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N(W) = PhaseSpace(W0) × FermiFunction(Z) × FiniteSizeL0(Z,A) × ...
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```
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Configuration toggles are defined in `SpectrumConfig` (spectrum.py). See [[README]] for architecture details.
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