docs: restructure knowledge base into tree layout + gA extraction module
- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
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@@ -25,19 +25,19 @@ $$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 \;
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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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| $F_0(Z,W)$ | Point charge Fermi function | [[corrections/01-fermi-function]] | ✓ |
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| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[corrections/02-finite-size]] | ✓ |
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| $U(Z,W)$ | Charge distribution correction | [[corrections/02-finite-size]] | ✓ |
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| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[corrections/02-finite-size]] | ✓ |
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| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[corrections/05-radiative]] | ✓ |
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| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[corrections/06-recoil]] | ✗ |
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| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[corrections/06-recoil]] | — |
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| $S(Z,W)$ | Atomic electron screening | [[corrections/03-screening]] | ✓ |
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| $X(Z,W)$ | Atomic exchange effect | [[corrections/04-exchange]] | ✓ |
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| $r(Z,W)$ | Atomic mismatch / overlap | [[corrections/08-atomic-overlap]] | — |
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| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[corrections/08-atomic-overlap]] | — |
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| $C(Z,W)$ | Nuclear structure shape factor | [[nuclear-structure/01-shape-factors]] | — |
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| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[nuclear-structure/01-shape-factors]] | — |
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## Baseline: Phase Space
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@@ -49,7 +49,7 @@ with $p = \sqrt{W^2-1}$, $W$ total energy in units of $m_e$, and $W_0$ the endpo
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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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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]].
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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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@@ -6,7 +6,7 @@ tags:
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status: active
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aliases: [Phase Space, Baseline Spectrum]
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cssclasses: []
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related_components: [[01-fermi-function]], [[07-recoil-effects]]
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related_components: [[corrections/01-fermi-function]], [[corrections/06-recoil]]
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---
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# Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$
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@@ -41,7 +41,7 @@ $$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for
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Deviations from linearity near the endpoint indicate:
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- Non-zero neutrino mass
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- Weak magnetism / recoil effects ([[07-recoil-effects]])
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- Weak magnetism / recoil effects ([[corrections/06-recoil]])
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- New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]])
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## Calculator Implementation
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@@ -62,7 +62,7 @@ The `PhaseSpace` calculator accepts a `transition_type` parameter that determine
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| `F2U`, `F3` | 2, 3 | $p_\nu^4 + \frac{3}{10}p_\nu^2 p_e^2 + p_e^4$ |
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| `F3U`, `F4` | 3, 4 | $p_\nu^6 + 7p_\nu^4 p_e^2 + 7p_\nu^2 p_e^4 + p_e^6$ |
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The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. See `[[10-nuclear-structure]]` for the full shape factor treatment.
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The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. See `[[nuclear-structure/01-shape-factors]]` for the full shape factor treatment.
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## Energy Range
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@@ -1,325 +0,0 @@
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---
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title: "14 – gA-Driven Shape Factor Parameterization"
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date: 2026-05-13
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tags:
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- beta-spectrum/shape-factor
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- beta-spectrum/gA
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- beta-spectrum/forbidden
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status: active
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aliases: [Spectrum Shape Method, SSM, gA quenching, shape factor parameterization]
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related_components: [[10-nuclear-structure]], [[07-recoil-effects]]
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---
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# gA-Driven Shape Factor Parameterization — The Spectrum Shape Method (SSM)
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## Overview
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For **forbidden nonunique β decays**, the shape factor $C(Z,W)$ depends on the weak axial-vector coupling constant $g_A$ in a nontrivial, energy-dependent way. This creates a powerful observable: by fitting the theoretical spectrum shape to high-precision experimental data, one can **extract the effective value of $g_A$** (and $g_V$) for a given nuclear transition. This technique is called the **Spectrum Shape Method (SSM)**, introduced in Kostensalo et al. (2017).
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> [!tip] Why this matters for 99Tc
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> The $^{99}\text{Tc}$ β spectrum is a **second-forbidden nonunique** transition ($9/2_1^+ \to 5/2_1^+$). Its shape factor is **strongly sensitive** to $g_A$ — making it an excellent laboratory for determining the quenched effective coupling constants. Paulsen et al. (2024) measured the spectrum with MMCs down to $<1$ keV and extracted $g_V^{\text{eff}} = 0.376(5)$, $g_A^{\text{eff}} = 0.574(36)$.
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## 1. Theoretical Foundation
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### 1.1 Shape Factor Decomposition
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The shape factor $C(w_e)$ decomposes into vector, axial-vector, and mixed parts:
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$$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
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where $w_e = W$ is the total electron energy in units of $m_e c^2$. The three components $C_V$, $C_A$, $C_{VA}$ are **purely nuclear-structure dependent** — they are computed from nuclear matrix elements (NMEs) and are **independent of the coupling constants**.
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After integration over the spectrum:
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$$\tilde{C} = g_V^2\, \tilde{C}_V + g_A^2\, \tilde{C}_A + g_V g_A\, \tilde{C}_{VA} \tag{2}$$
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The integrated components $\tilde{C}_i$ are energy-independent constants.
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### 1.2 The β Spectrum Equation
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The full differential spectrum (Paulsen et al., 2024, Eq. 5):
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$$N(W)\,dW = \frac{G_F^2 \cos^2\theta_C}{2\pi^3}\, F(Z,W)\, p W (W_0 - W)^2\, X(W)\, C(W)\, \rho(Z,W)\, dW$$
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where:
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- $F(Z,W)$ — Fermi function (Coulomb correction)
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- $X(W)$ — screening and exchange corrections
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- $C(W)$ — **shape factor** (nuclear structure)
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- $\rho(Z,W)$ — atomic overlap correction
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- $G_F \cos\theta_C$ — effective Fermi constant
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### 1.3 Traditional Parametrization (Ad-Hoc)
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Historically, the shape factor for $^{99}\text{Tc}$ has been approximated as **first-forbidden unique** (Reich & Schüpferling, 1974; standard reference [15]):
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$$C_{\text{adhoc}}(W) = q^2 + \lambda p^2 = (W_0 - W)^2 + \lambda (W^2 - 1) \tag{3}$$
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with $\lambda$ as a free fitting parameter. This is **not derived from nuclear structure** — it is an empirical ansatz. Paulsen et al. (2024) extracted $\lambda \approx 0.65$ from fits, but this parametrization:
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- Cannot distinguish $g_V$ from $g_A$ effects
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- Provides no insight into nuclear structure
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- Fails below $\sim$100 keV when compared to high-precision data
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### 1.4 SSM Parametrization (Theory-Driven)
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The SSM replaces Eq. (3) with the **microscopically computed** shape factor:
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$$C_{\text{SSM}}(W; g_A) = g_V^2\, C_V(W) + g_A^2\, C_A(W) + g_V g_A\, C_{VA}(W) \tag{4}$$
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where $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ are computed from shell-model (or MQPM) one-body transition densities (OBTDs).
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## 2. Algorithm for Parameterization
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### 2.1 Input Requirements
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| Quantity | Source | Notes |
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|---|---|---|
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| $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ | Nuclear shell model (NuShellX@MSU) or MQPM | Computed for the specific transition |
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| $W_0$ (endpoint) | Independent measurement or fit | Paulsen et al.: $W_0 = 295.82(16)$ keV |
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| Experimental spectrum $N_{\text{exp}}(W)$ | Calorimetry / semiconductor / MMC | Corrected for detector response, background, energy losses |
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| $F(Z,W)$, $X(W)$, $\rho(Z,W)$ | Calculator modules | Already implemented in project |
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| $g_V$ | Fixed (CVC: $g_V = 1.0$, or treated as free) | Paulsen et al. used free $g_V$ fit |
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### 2.2 Computation of Shape Factor Components
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The shape factor in the **Behrens-Bühring** formalism (truncated lepton current approximation, Paulsen et al. Eq. 10):
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$$C(W) = \sum_{K,k_e,k_\nu} \left[ e_k^2 W k_e\, m_K(k_e,k_\nu) - 2\mu e_k W k_e\, \gamma\, m_K(k_e,k_\nu) \right]$$
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where the sum runs over lepton quantum numbers $k_e$, $k_\nu$ and nuclear multipole order $K$. The key point: **each component** $C_i(W)$ ($i = V, A, VA$) is computed separately from the NMEs.
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**Practical computation via shell model:**
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1. **OBTD calculation** (NuShellX):
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- Compute one-body transition densities for the transition
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- For $^{99}\text{Tc}$: dominant transition $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$
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- Sub-dominant: $1g_{7/2} \to 1g_{9/2}$ with OBTD $\approx 0.010$
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2. **Coulomb displacement energy** $E_C$ (for relativistic vector current):
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- Leading order: $E_C^{(3)} = \frac{\int_0^\infty g_f V(r) g_i (r/R)^K r^2 dr}{\int_0^\infty g_f g_i (r/R)^K r^2 dr}$
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- Energy-dependent: $E_C^{(4)}$ includes full lepton current dependence
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- For $^{99}\text{Tc}$: $E_C^{(4)} \approx 10.5$ MeV (GLEKPN valence space)
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3. **Vector current from CVC**:
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$$V_F^{221} = -\sqrt{\frac{R}{10}}\left[W_0 - (m_n - m_p) + E_C\right] V_F^{220}$$
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For $^{99}\text{Tc}$, $[W_0 - (m_n - m_p)] = -0.487$ MeV — small Q-value makes $E_C$ critical.
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### 2.3 Fitting Procedure
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```
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Step 1: Precompute shape factor components C_i(W) from nuclear model
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→ Store as lookup tables: C_V(W), C_A(W), C_VA(W) for W ∈ [1, W0]
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Step 2: For trial (gA, gV) pair:
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a. Compute C(W; gA, gV) = gV²·C_V(W) + gA²·C_A(W) + gV·gA·C_VA(W)
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b. Compute full spectrum:
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N_theo(W) ∝ p·W·(W0−W)²·F(Z,W)·X(W)·C(W; gA, gV)·ρ(Z,W)
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c. Normalize: ∫ N_theo(W) dW = 1
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Step 3: Compare with experimental spectrum:
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χ²(gA, gV) = Σ [N_exp(W_i) − N_theo(W_i; gA, gV)]² / σ_i²
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Step 4: Minimize χ² over (gA, gV) grid
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→ Extract best-fit values and confidence intervals
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```
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### 2.4 Energy Range Selection
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Paulsen et al. (2024) recommend fitting in the range **20–275 keV** to avoid:
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- **Below ~20 keV**: detector threshold effects, atomic exchange modeling uncertainties
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- **Above ~275 keV**: endpoint region with poor statistics, resolution distortion
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The χ² landscape is approximately quadratic, enabling simple uncertainty estimation at χ² = χ²_min + 1.
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## 3. Results from Literature
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### 3.1 $^{99}\text{Tc}$ — Paulsen et al. (2024)
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From the MMC measurement (PTB/LNHB), the extracted values are:
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| Quantity | Value | Method |
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|---|---|---|
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| $Q_\beta$ | 295.82(16) keV | Fit to spectrum shape |
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| $g_V^{\text{eff}}$ | 0.376(5) | From half-life normalization |
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| $g_A^{\text{eff}}$ | 0.574(36) | From spectrum shape fit |
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| $\bar{E}_\beta$ | 98.51(23) keV | From theoretical spectrum |
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| $\log f$ | −0.476 60(22) | From phase space integration |
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| $\log ft$ | 12.3478(23) | Combined |
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**Key finding**: The shape-fit alone gives $g_A^{\text{eff}} = 1.526(92)$ (enhanced, not quenched), but when the half-life is used to normalize the overall rate, the **quenched** values $g_V^{\text{eff}} = 0.376$, $g_A^{\text{eff}} = 0.574$ emerge — consistent with first-forbidden nonunique transitions (Suhonen review: $g_A^{\text{eff}} \approx 0.9$, $g_V^{\text{eff}} \approx 0.3$–0.7).
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### 3.2 $^{87}\text{Rb}$ — Kostensalo et al. (2017)
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Third-forbidden nonunique ($3/2^- \to 9/2^+$):
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- Strong $g_A$ dependence in spectrum shape
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- MQPM and NSM agree on shape evolution
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- $g_A \approx 0.9$ gives best match to available data
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- Best candidate for $g_A$ determination in third-forbidden decays
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### 3.3 $^{113}\text{Cd}$ and $^{115}\text{In}$ — Kostensalo et al. (2017, 2017a)
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Fourth-forbidden nonunique:
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- Most dramatic $g_A$ dependence: bell-shaped spectrum at $g_A \approx g_V$, monotonic otherwise
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- Consistent $g_A \approx 0.9$ extracted across three nuclear models (MQPM, NSM, IBFM-2)
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- Demonstrates **robustness of SSM** against nuclear model details
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### 3.4 Summary of $g_A$ Sensitivity by Forbiddenness
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| Transition | Forbiddenness | $g_A$ Sensitivity | SSM Viability |
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|---|---|---|---|
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| $^{99}\text{Tc}$ | 2nd nonunique | **Strong** | ✅ Excellent |
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| $^{94}\text{Nb}$, $^{98}\text{Tc}$ | 2nd nonunique | **Strong** | ✅ Good |
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| $^{87}\text{Rb}$ | 3rd nonunique | **Strong** | ✅ Good |
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| $^{113}\text{Cd}$, $^{115}\text{In}$ | 4th nonunique | **Very strong** | ✅ Excellent |
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| $^{137}\text{Cs}$ (to $11/2^-$) | 1st unique | Weak (NLO only) | ⚠️ Limited |
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| $^{137}\text{Cs}$ (to $3/2^+$) | 2nd nonunique | **None** (cancellation) | ❌ Not applicable |
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| First-forbidden nonunique | 1st nonunique | **Weak** | ❌ Not useful |
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**Systematic pattern**: Even-forbidden decays show strong $g_A$ dependence; odd-forbidden decays are mostly insensitive.
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## 4. Implementation Plan
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### 4.1 New Module: `components/shape_factor_gA.py`
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```
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ShapeFactorGA
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├── __init__: load C_V(W), C_A(W), C_VA(W) from precomputed tables or nuclear model
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├── evaluate(W, gA, gV=1.0): compute C(W; gA, gV)
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├── evaluate_components(W): return (C_V, C_A, C_VA) separately
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├── chi2_spectrum(gA, gV, W_grid, N_exp, sigma): χ² for spectrum fit
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├── fit_gA(W_grid, N_exp, sigma, W_min, W_max): minimize χ², return (gA_best, gA_err)
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├── fit_gA_gV(W_grid, N_exp, sigma, W_min, W_max): fit both, return (gA, gV, cov)
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└── mean_energy(gA, gV, W0): compute ⟨E⟩ from theoretical spectrum
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```
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### 4.2 Integration with BetaSpectrum
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The shape factor becomes a new `SpectrumComponent`:
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```python
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class ShapeFactorGA(SpectrumComponent):
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"""gA-dependent shape factor for forbidden nonunique decays."""
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def __init__(self, C_V_table, C_A_table, C_VA_table, gA=1.27, gV=1.0):
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# Load precomputed shape factor components
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# C_i tables: W array → C_i(W) values
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def evaluate(self, W):
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return gV**2 * self.C_V(W) + gA**2 * self.C_A(W) + gV*gA * self.C_VA(W)
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def integrate(self, W0):
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"""Compute integrated shape factor for half-life prediction."""
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return integrate(C(W) * p * W * (W0-W)**2 * F(Z,W) dW, 1, W0)
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```
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### 4.3 Precomputed Tables from Nuclear Model
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For $^{99}\text{Tc}$, the shape factor components should be precomputed from NuShellX@MSU (GLEKPN valence space, Mach interaction):
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| W (keV) | $C_V(W)$ | $C_A(W)$ | $C_{VA}(W)$ |
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|---|---|---|---|
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| 1.001 | ... | ... | ... |
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| ... | ... | ... | ... |
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| 1.581 | ... | ... | ... |
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(Endpoint: $W_0 = 1 + 295.82/510.998950 \approx 1.579$)
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### 4.4 Fitting Pipeline
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```python
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def fit_gA_for_Tc95(data_file, W_min=20*keV, W_max=275*keV):
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"""Fit gA to 99Tc beta spectrum data."""
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W, N_exp, sigma = load_spectrum(data_file)
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# Load precomputed shape factor components
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sf = ShapeFactorGA("Tc95_CV_CA_CVA_tables.npy")
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# Grid search + refinement
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best_chi2 = inf
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for gA in np.linspace(0.4, 1.8, 141):
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chi2 = sf.chi2_spectrum(gA=gA, gV=1.0, W=W, N_exp=N_exp, sigma=sigma,
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W_min=W_min, W_max=W_max)
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if chi2 < best_chi2:
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best_chi2 = chi2
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best_gA = gA
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# Refine around minimum
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gA_opt = optimize.minimize_scalar(...)
|
||||
|
||||
# Uncertainty from χ² = χ²_min + 1
|
||||
gA_low, gA_high = find_chi2_one_boundary(gA_opt, sf, ...)
|
||||
|
||||
return gA_opt, gA_low, gA_high
|
||||
```
|
||||
|
||||
## 5. Uncertainty Budget (from Paulsen et al., 2024)
|
||||
|
||||
| Component | Uncertainty on $g_A^{\text{eff}}$ | Relative |
|
||||
|---|---|---|
|
||||
| Fit method (statistics) | 0.0745 | 4.9% |
|
||||
| Coulomb displacement energy | 0.0425 | 2.8% |
|
||||
| Nuclear model | 0.0294 | 1.9% |
|
||||
| Lepton current treatment | 0.0127 | 0.8% |
|
||||
| Fit range | 0.0061 | 0.4% |
|
||||
| Maximum energy | 0.0038 | 0.2% |
|
||||
| Radiative corrections | 0.0033 | 0.2% |
|
||||
| Atomic exchange | 0.0029 | 0.2% |
|
||||
| **Combined** | **0.0919** | **6.0%** |
|
||||
|
||||
Dominant uncertainty: **fit method / statistics**. The nuclear model uncertainty is surprisingly small (1.9%), confirming SSM robustness.
|
||||
|
||||
## 6. Key References
|
||||
|
||||
| Ref | Citation | Relevance |
|
||||
|---|---|---|
|
||||
| [SSM-1] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | Original SSM paper; 26 decays studied |
|
||||
| [SSM-2] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; 16 decays; Table I with integrated shape factors |
|
||||
| [SSM-3] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}\text{Tc}$ high-precision MMC measurement; $g_A$ extraction |
|
||||
| [SSM-4] | Kostensalo et al., *Phys. Rev. C* **95**, 024327 (2017) | $^{113}\text{Cd}$ analysis with three nuclear models; SSM robustness |
|
||||
| [Hayen] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | Comprehensive review; BB formalism; Eq. 79–81 |
|
||||
| [Suhonen] | Suhonen, *From Nucleons to Nucleus* (Springer, 2007) | Nuclear structure background; effective coupling constants |
|
||||
| [Behrens] | Behrens & Bühring, *Electron Radial Wave Functions and Nuclear Beta Decay* (1982) | BB formalism reference; tabulated $I(k_e,m,n,\rho)$ functions |
|
||||
|
||||
## 7. Connection to Existing Modules
|
||||
|
||||
This module integrates with:
|
||||
- **`components/fermi.py`** — Fermi function $F(Z,W)$
|
||||
- **`components/exchange.py`** — Exchange correction $X(W)$ (extended to forbidden transitions)
|
||||
- **`components/finite_size.py`** — Finite-size correction (daughter nucleus radius)
|
||||
- **`components/screening.py`** — Screening correction
|
||||
- **`spectrum.py`** — `BetaSpectrum` multiplicative composition
|
||||
- **`docs/10-nuclear-structure.md`** — Theoretical background on shape factors
|
||||
|
||||
## 8. Practical Considerations
|
||||
|
||||
### 8.1 Which transitions to prioritize?
|
||||
|
||||
For $g_A$ extraction via SSM, the selection criteria are:
|
||||
1. **Strong $g_A$ dependence** in spectrum shape (even-forbidden, nonunique)
|
||||
2. **High branching ratio** (preferably > 90%)
|
||||
3. **Measurable Q-value** with good energy resolution
|
||||
4. **Low background** environment
|
||||
|
||||
For the $^{99}\text{Tc}$ project, this is the primary candidate — it satisfies all criteria.
|
||||
|
||||
### 8.2 Quenching vs. Enhancement
|
||||
|
||||
The extracted $g_A^{\text{eff}}$ from shape fits alone can be **enhanced** ($>1.27$) because the shape factor normalization absorbs the overall rate. The **quenched** values emerge only when the half-life is used for absolute normalization:
|
||||
|
||||
$$g_i^{\text{eff, quenched}} = g_i^{\text{eff, shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
|
||||
|
||||
For $^{99}\text{Tc}$: $t_{1/2}^{\text{theo}} \approx 30 \times 10^3$ a vs. $t_{1/2}^{\text{exp}} \approx 212 \times 10^3$ a → quenching factor $\approx 0.38$.
|
||||
|
||||
### 8.3 Nuclear Model Dependencies
|
||||
|
||||
While SSM is robust against nuclear model details (MQPM vs. NSM vs. IBFM-2 agree to $\sim$10%), the **absolute normalization** of shape factor components varies:
|
||||
- For $^{99}\text{Tc}$: MQPM gives $\sim$2× larger $\tilde{C}$ than NSM
|
||||
- This affects half-life predictions but **not** the shape evolution
|
||||
- The $g_A$ extraction from shape is thus model-independent to $\sim$2% (see uncertainty budget)
|
||||
|
||||
## 9. Future Extensions
|
||||
|
||||
1. **First-forbidden unique decays**: Include NLO terms for $g_A$ sensitivity
|
||||
2. **Highly forbidden decays** (4th–6th order): $^{96}\text{Zr}$, $^{48}\text{Ca}$ — strong $g_A$ dependence in even-forbidden channels
|
||||
3. **Two-body currents**: Recent ab initio calculations suggest quenching may be unnecessary when 2BC are included (Carbone et al.)
|
||||
4. **Multi-parameter fits**: Simultaneous fit of $g_A$, $g_V$, $Q_\beta$, and nuclear structure parameters
|
||||
5. **Machine learning surrogate**: Train emulator on shape factor components for rapid χ² evaluation
|
||||
+65
-39
@@ -1,5 +1,5 @@
|
||||
---
|
||||
title: Beta Spectrum Wiki
|
||||
title: Beta Spectrum Calculator — Knowledge Base
|
||||
date: 2026-04-26
|
||||
tags:
|
||||
- vault/overview
|
||||
@@ -12,61 +12,83 @@ This Obsidian vault serves as the scientific reference and development workspace
|
||||
|
||||
## 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.
|
||||
Organize the physics theory behind β-decay spectrum calculations into structured, interconnected notes that 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]] |
|
||||
| Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | Review article | [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]] |
|
||||
|
||||
## Notes Structure
|
||||
## Directory 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)
|
||||
```
|
||||
docs/
|
||||
├── README.md ← You are here
|
||||
├── 00-beta-spectrum-overview.md ← Master equation & full correction list
|
||||
├── 02-phase-space.md ← Baseline spectral shape
|
||||
├── corrections/ ← All correction components
|
||||
│ ├── 01-fermi-function.md ← Fermi function F₀(Z,W)
|
||||
│ ├── 02-finite-size.md ← Nuclear size: L₀, U, DFS
|
||||
│ ├── 03-screening.md ← Atomic screening S(Z,W)
|
||||
│ ├── 04-exchange.md ← Atomic exchange X(Z,W)
|
||||
│ ├── 05-radiative.md ← Outer radiative R(W,W₀)
|
||||
│ ├── 06-recoil.md ← Recoil: R_N, Q, weak magnetism
|
||||
│ ├── 07-detector-response.md ← Detector convolution
|
||||
│ ├── 08-atomic-overlap.md ← Atomic mismatch r(Z,W)
|
||||
│ └── 09-chemical-effects.md ← Molecular environment
|
||||
├── nuclear-structure/ ← Nuclear theory
|
||||
│ ├── index.md ← Overview
|
||||
│ └── 01-shape-factors.md ← C(Z,W), NMEs, impulse approximation
|
||||
├── gA-extraction/ ← gA extraction (new, in development)
|
||||
│ ├── 01-ssm-algorithm.md ← SSM step-by-step algorithm
|
||||
│ ├── 02-ssnme.md ← sNME and ESSM formalism
|
||||
│ ├── 03-tc99-case-study.md ← 99Tc literature results
|
||||
│ └── 04-implementation-todo.md ← Implementation TODO list
|
||||
└── refs/ ← Paper summaries
|
||||
├── electromagnetic/ ← CVC, radiative corrections
|
||||
├── forbidden-decay/ ← Forbidden decay studies
|
||||
└── (organized by topic)
|
||||
```
|
||||
|
||||
## 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
|
||||
├── PhaseSpace → 02-phase-space.md
|
||||
├── FermiFunction → corrections/01-fermi-function.md
|
||||
├── FiniteSizeL0 → corrections/02-finite-size.md
|
||||
├── ChargeDistributionU → corrections/02-finite-size.md
|
||||
├── ScreeningCorrection → corrections/03-screening.md
|
||||
├── ExchangeCorrection → corrections/04-exchange.md
|
||||
├── RadiativeCorrection → corrections/05-radiative.md
|
||||
└── (recoil.py) → corrections/06-recoil.md (not yet implemented)
|
||||
|
||||
BetaSpectrum (spectrum.py) — multiplicative composition + from_config()
|
||||
BetaSpectrumAnalyzer (spectrum.py) — plotting, CSV export
|
||||
```
|
||||
|
||||
## Correction Implementation Status
|
||||
|
||||
| # | Correction | Module | Note | Status |
|
||||
|---|---|---|---|---|
|
||||
| 1 | Phase space | `phase_space.py` | Baseline | ✓ implemented |
|
||||
| 2 | Fermi function | `fermi.py` | Coulomb interaction | ✓ implemented |
|
||||
| 3 | Finite size | `finite_size.py` | L₀, U, DFS | ✓ implemented |
|
||||
| 4 | Screening | `screening.py` | Atomic screening | ✓ implemented |
|
||||
| 5 | Exchange | `exchange.py` | Atomic exchange | ✓ implemented |
|
||||
| 6 | Radiative | `radiative.py` | Outer RC | ✓ implemented |
|
||||
| 7 | Recoil | `recoil.py` | R_N, Q, weak magnetism | ✗ not yet implemented |
|
||||
| 8 | Shape factor | `shape_factor_gA.py` | C(Z,W) for forbidden | — planned |
|
||||
| 9 | Detector | `detector_response.py` | Convolution | — planned |
|
||||
|
||||
## gA Extraction (New)
|
||||
|
||||
For forbidden nonunique decays (e.g., $^{99}\text{Tc}$), the shape factor depends on $g_A$ in a measurable way. See:
|
||||
|
||||
- [[gA-extraction/01-ssm-algorithm]] — The SSM algorithm for extracting $g_A$ from spectral shapes
|
||||
- [[gA-extraction/04-implementation-todo]] — Step-by-step implementation plan
|
||||
|
||||
## Key Variables and Units
|
||||
|
||||
- $W$ — total electron energy in units of $m_e c^2$ (natural units, $m_e = 1$)
|
||||
@@ -77,3 +99,7 @@ BetaSpectrumAnalyzer (spectrum.py) — plotting, CSV export
|
||||
- $\gamma = \sqrt{1 - (\alpha Z)^2}$
|
||||
|
||||
Conversion: $W = 1 + T / m_e c^2$, with $m_e c^2 = 510.998950$ keV.
|
||||
|
||||
## Precision Goal
|
||||
|
||||
The calculator aims for **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei, following the approach of Hayen et al. (2017).
|
||||
|
||||
@@ -6,7 +6,7 @@ tags:
|
||||
status: active
|
||||
aliases: [Coulomb Correction, F_0]
|
||||
cssclasses: []
|
||||
related_components: [[03-finite-size]], [[06-radiative-corrections]]
|
||||
related_components: [[corrections/02-finite-size]], [[corrections/05-radiative]]
|
||||
---
|
||||
|
||||
# Fermi Function — Coulomb Interaction $F_0(Z,W)$
|
||||
@@ -24,7 +24,7 @@ $$F_0(Z,W) = 4(2pR)^{2(\gamma-1)} e^{\pi y} \frac{|\Gamma(\gamma + iy)|^2}{(\Gam
|
||||
with:
|
||||
$$\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]]).
|
||||
Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[corrections/02-finite-size]]).
|
||||
|
||||
## Coulomb Amplitudes
|
||||
|
||||
@@ -70,5 +70,5 @@ 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$)
|
||||
- [[corrections/02-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$)
|
||||
- [[corrections/05-radiative]] adds QED radiative effects ($R = 1 + \delta_R$)
|
||||
@@ -6,18 +6,18 @@ tags:
|
||||
status: active
|
||||
aliases: [Finite Nuclear Size, L_0, U, DFS]
|
||||
cssclasses: []
|
||||
related_components: [[01-fermi-function]], [[10-nuclear-structure]]
|
||||
related_components: [[corrections/01-fermi-function]], [[nuclear-structure/01-shape-factors]]
|
||||
---
|
||||
|
||||
# Finite Nuclear Size Effects — $L_0$, $U(Z,W)$, and $DFS$
|
||||
|
||||
## Overview
|
||||
|
||||
The Fermi function ([[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.
|
||||
The Fermi function ([[corrections/01-fermi-function]]) assumes a point charge nucleus. Real nuclei have finite extent and possibly deformation — these corrections account for the difference in electric potential inside the nuclear volume.
|
||||
|
||||
> [!note] Two types of "finite size"
|
||||
> 1. **Electrostatic** ($L_0$, $U$, $DFS$): Potential difference when moving from point charge to realistic nuclear shape — applied *after* evaluating wave function at origin
|
||||
> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume — appears in shape factor $C(Z,W)$ ([[10-nuclear-structure]])
|
||||
> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume — appears in shape factor $C(Z,W)$ ([[nuclear-structure/01-shape-factors]])
|
||||
|
||||
## Nuclear Radius
|
||||
|
||||
@@ -109,7 +109,7 @@ $$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$.
|
||||
> Effect can reach several parts in $10^3$. Partially canceled by nuclear structure deformation correction $D_C$ ([[nuclear-structure/01-shape-factors]]). Figure 1 of Hayen shows energy dependence reversal between β⁻ and β⁺ for $\beta_2 = 0.2$.
|
||||
|
||||
### Calculator Implementation
|
||||
|
||||
@@ -6,14 +6,14 @@ tags:
|
||||
status: active
|
||||
aliases: [Atomic Screening, S(Z,W)]
|
||||
cssclasses: []
|
||||
related_components: [[12-atomic-overlap]], [[13-chemical-effects]]
|
||||
related_components: [[corrections/08-atomic-overlap]], [[corrections/09-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 Fermi function ([[corrections/01-fermi-function]]) assumes the β particle moves in the Coulomb field of a bare nucleus. In reality, atomic electrons partially screen this charge at large distances from the nucleus, modifying the effective potential experienced by the emitted electron/positron. This is particularly important at low kinetic energies where the β particle spends more time far from the nucleus.
|
||||
|
||||
## The Screening Correction
|
||||
|
||||
@@ -88,6 +88,6 @@ From Hayen et al. Table (Section VIII overview):
|
||||
|
||||
## 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)
|
||||
- [[corrections/08-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction
|
||||
- [[corrections/09-chemical-effects]] — Molecular environment influence on screening and exchange
|
||||
- [[corrections/04-exchange]] — Electron exchange with atomic orbitals (often combined with screening)
|
||||
@@ -6,7 +6,7 @@ tags:
|
||||
status: active
|
||||
aliases: [Atomic Exchange, X(Z,W), Hayen-2018]
|
||||
cssclasses: []
|
||||
related_components: [[04-screening-correction]], [[12-atomic-overlap]]
|
||||
related_components: [[corrections/03-screening]], [[corrections/08-atomic-overlap]]
|
||||
---
|
||||
|
||||
# Atomic Electron Exchange Correction $X(Z,W)$
|
||||
@@ -37,7 +37,7 @@ Screening strength is adjusted for best agreement with bound state energies (dec
|
||||
|
||||
## 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$.
|
||||
The exchange correction $X(W)$ as a function of kinetic energy uses empirical fit coefficients from Table X of [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]]. Coefficients are tabulated for all atoms with $Z = 2$ through $z = 120$.
|
||||
|
||||
### Shell Structure Effects
|
||||
|
||||
@@ -6,7 +6,7 @@ tags:
|
||||
status: todo
|
||||
aliases: [Nuclear Recoil, R_N, Weak Magnetism]
|
||||
cssclasses: []
|
||||
related_components: [[02-phase-space]], [[10-nuclear-structure]]
|
||||
related_components: [[02-phase-space]], [[nuclear-structure/01-shape-factors]]
|
||||
---
|
||||
|
||||
# Nuclear Recoil Effects — $R_N$, $Q$, and Induced Currents
|
||||
@@ -6,7 +6,7 @@ tags:
|
||||
status: active
|
||||
aliases: [Atomic Mismatch, Bahcall Correction, r(Z,W)]
|
||||
cssclasses: []
|
||||
related_components: [[04-screening-correction]], [[13-chemical-effects]]
|
||||
related_components: [[corrections/03-screening]], [[corrections/09-chemical-effects]]
|
||||
---
|
||||
|
||||
# Atomic Overlap / Mismatch Correction $r(Z,W)$
|
||||
@@ -70,6 +70,6 @@ $$\frac{\Gamma_b}{\Gamma_c} = \frac{\pi(\alpha Z)^3}{f(Z,W_0)} (W_0 - 1)^2 \Sigm
|
||||
|
||||
## 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
|
||||
- [[corrections/03-screening]] — Screening modification from shake-up processes
|
||||
- — Shake-off process correction
|
||||
- [[corrections/09-chemical-effects]] — Molecular environment influence on atomic overlap
|
||||
@@ -6,7 +6,7 @@ tags:
|
||||
status: active
|
||||
aliases: [Molecular Screening, Molecular Exchange, Saenz-Froelich]
|
||||
cssclasses: []
|
||||
related_components: [[12-atomic-overlap]], [[04-screening-correction]]
|
||||
related_components: [[corrections/08-atomic-overlap]], [[corrections/03-screening]]
|
||||
---
|
||||
|
||||
# Chemical / Molecular Effects on β Spectrum
|
||||
@@ -113,5 +113,5 @@ For molecular experiments requiring these corrections:
|
||||
|
||||
## Related Notes
|
||||
|
||||
- [[12-atomic-overlap]] — Atomic mismatch / Bahcall correction
|
||||
- [[04-screening-correction]] — Baseline atomic screening $S(Z,W)$
|
||||
- [[corrections/08-atomic-overlap]] — Atomic mismatch / Bahcall correction
|
||||
- [[corrections/03-screening]] — Baseline atomic screening $S(Z,W)$
|
||||
@@ -0,0 +1,172 @@
|
||||
---
|
||||
title: "SSM — Spectrum Shape Method for gA Extraction"
|
||||
date: 2026-05-14
|
||||
tags:
|
||||
- gA/extraction/algorithm
|
||||
- gA/ssm
|
||||
- beta-spectrum/forbidden
|
||||
status: active
|
||||
aliases: [Spectrum Shape Method, gA fitting, shape factor fit]
|
||||
related_notes:
|
||||
- gA-extraction/02-ssnme
|
||||
- gA-extraction/03-tc99-case-study
|
||||
- gA-extraction/04-implementation-todo
|
||||
- nuclear-structure/01-shape-factors
|
||||
- corrections/05-radiative
|
||||
---
|
||||
|
||||
# Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction
|
||||
|
||||
## 1. Physical Basis
|
||||
|
||||
For **forbidden nonunique β decays**, the shape factor $C(w_e)$ decomposes into three energy-dependent components with distinct coupling-constant dependence:
|
||||
|
||||
$$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
|
||||
|
||||
where $w_e$ is the total electron energy in units of $m_e c^2$. The components $C_V$, $C_A$, $C_{VA}$ are computed from nuclear many-body theory and are **independent of the coupling constants**. The spectrum shape is sensitive to $g_A$ through the interference between vector, axial-vector, and mixed terms — a purely nuclear-structure effect unique to nonunique forbidden decays.
|
||||
|
||||
> [!note] Why nonunique decays?
|
||||
> Unique forbidden decays have a single dominant NME → universal spectrum shape independent of nuclear structure. Nonunique decays involve multiple NMEs → the relative weight of vector vs. axial contributions depends on $g_A$, creating observable spectral shape differences.
|
||||
|
||||
## 2. The Full Spectrum Model
|
||||
|
||||
The differential spectrum (Paulsen et al., 2024, Eq. 5):
|
||||
|
||||
$$N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)$$
|
||||
|
||||
where:
|
||||
- $\mathcal{N}$ — overall normalization (absorbs $G_F^2 \cos^2\theta_C / 2\pi^3$)
|
||||
- $F(Z,W)$ — Fermi function (Coulomb correction) — [[corrections/01-fermi-function]]
|
||||
- $p W (W_0 - W)^2$ — phase space — [[02-phase-space]]
|
||||
- $X(W)$ — atomic exchange correction — [[corrections/04-exchange]]
|
||||
- $S(Z,W)$ — atomic screening — [[corrections/03-screening]]
|
||||
- $R(W)$ — outer radiative correction — [[corrections/05-radiative]]
|
||||
- $C(W; g_A, g_V)$ — **shape factor** — [[nuclear-structure/01-shape-factors]]
|
||||
- $\rho(Z,W)$ — atomic overlap correction — [[corrections/08-atomic-overlap]]
|
||||
|
||||
## 3. The Algorithm
|
||||
|
||||
### Step 1: Compute shape factor components from nuclear model
|
||||
|
||||
**Input:** Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.
|
||||
|
||||
**Output:** Three energy-dependent lookup tables: $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $W \in [1, W_0]$.
|
||||
|
||||
**Method:**
|
||||
1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
|
||||
2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
|
||||
3. Sum over all contributing multipole orders $K$ and lepton quantum numbers $k_e, k_\nu$
|
||||
|
||||
**CVC constraint for vector current:**
|
||||
The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:
|
||||
|
||||
$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}$$
|
||||
|
||||
where $E_C$ is the Coulomb displacement energy. The sNME is critical for small-$Q$ decays like $^{99}\text{Tc}$ ($[W_0 - (m_n - m_p)] = -0.487$ MeV).
|
||||
|
||||
**Practical note:** For $^{99}\text{Tc}$, the dominant transition is $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$. The sub-dominant $1g_{7/2} \to 1g_{9/2}$ transition contributes OBTD $\approx 0.010$. See Kostensalo & Suhonen (2017), Table VII.
|
||||
|
||||
### Step 2: Build theoretical spectrum for trial $g_A$
|
||||
|
||||
For each trial value of $g_A$ (with $g_V = 1.0$ fixed by CVC):
|
||||
|
||||
1. **Compute shape factor:**
|
||||
$$C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)$$
|
||||
(using $g_V = 1.0$)
|
||||
|
||||
2. **Compute full spectrum:**
|
||||
$$N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)$$
|
||||
|
||||
3. **Normalize:** Choose $\mathcal{N}$ so that $\int_1^{W_0} N_{\text{theo}}(W; g_A)\, dW = 1$ (or match the total counts in the experimental spectrum)
|
||||
|
||||
### Step 3: Compare with experimental spectrum
|
||||
|
||||
**Input:** Experimental spectrum data $(W_i, N_i, \sigma_i)$ for $i = 1, \dots, N_{\text{bins}}$.
|
||||
|
||||
**χ² function:**
|
||||
$$\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}$$
|
||||
|
||||
**Fit range selection (Paulsen et al., 2024):**
|
||||
- **Lower bound:** ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
|
||||
- **Upper bound:** ~275 keV for $^{99}\text{Tc}$ — above this, endpoint region with poor statistics and resolution distortion
|
||||
- Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range
|
||||
|
||||
### Step 4: Minimize χ² and extract $g_A$
|
||||
|
||||
**Grid search:** Evaluate $\chi^2(g_A)$ on a grid (e.g., $g_A \in [0.4, 1.8]$ with step 0.01).
|
||||
|
||||
**Refinement:** Use `scipy.optimize.minimize_scalar` (Brent's method) around the grid minimum.
|
||||
|
||||
**Uncertainty:** Find $g_A$ values where $\chi^2 = \chi^2_{\text{min}} + 1$. These define the 68% confidence interval.
|
||||
|
||||
**Output:** $(g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})$
|
||||
|
||||
### Step 5 (optional): Simultaneous $g_A$ and $g_V$ fit
|
||||
|
||||
If $g_V$ is not fixed to 1.0 (e.g., testing CVC violation):
|
||||
|
||||
$$\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}$$
|
||||
|
||||
Fit both parameters simultaneously. The correlation between $g_A$ and $g_V$ is typically strong.
|
||||
|
||||
### Step 6 (optional): Half-life normalization — quenched $g_A$
|
||||
|
||||
The $g_A$ extracted from **shape alone** can be enhanced ($>1.27$) because the normalization constant $\mathcal{N}$ absorbs the overall rate. To obtain the **quenched** values consistent with half-life:
|
||||
|
||||
1. Compute the theoretical partial half-life for the best-fit $(g_A, g_V)$:
|
||||
$$t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW$$
|
||||
|
||||
2. Compare with experimental partial half-life:
|
||||
$$t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}$$
|
||||
|
||||
3. Extract quenched values:
|
||||
$$g_i^{\text{quenched}} = g_i^{\text{shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
|
||||
|
||||
For $^{99}\text{Tc}$: $t_{1/2}^{\text{theo}} \approx 30 \times 10^3$ a vs. $t_{1/2}^{\text{exp}} \approx 212 \times 10^3$ a → quenching factor $\approx 0.38$.
|
||||
|
||||
## 4. The sNME Problem and ESSM
|
||||
|
||||
### The Problem
|
||||
|
||||
The sNME (small relativistic vector NME) is extremely difficult to compute in nuclear shell model because it involves contributions outside the valence shell. The CVC prediction gives a "canonical" value, but this assumes a perfect many-body theory — which finite valence spaces are not.
|
||||
|
||||
### The Solution: Enhanced SSM (ESSM)
|
||||
|
||||
Kumar et al. (2021) proposed treating the sNME as a **fitting parameter** alongside $g_A$. This is the **Enhanced Spectrum Shape Method (ESSM)**:
|
||||
|
||||
1. Fix $g_A$ at a trial value
|
||||
2. Vary the sNME to simultaneously fit the **experimental half-life** (not just the shape)
|
||||
3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each $g_A$ value:
|
||||
- **sNME(c):** closer to the CVC value
|
||||
- **sNME(f):** further from the CVC value
|
||||
4. Compare both solutions' spectral shapes with the data
|
||||
|
||||
The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See [[gA-extraction/02-ssnme]] for details.
|
||||
|
||||
## 5. Uncertainty Budget
|
||||
|
||||
| Component | Uncertainty on $g_A$ | Relative | Dominant for |
|
||||
|---|---|---|---|
|
||||
| Fit method (statistics) | 0.0745 | 4.9% | All SSM analyses |
|
||||
| Coulomb displacement energy | 0.0425 | 2.8% | Small-$Q$ decays |
|
||||
| Nuclear model | 0.0294 | 1.9% | Model comparison |
|
||||
| Lepton current treatment | 0.0127 | 0.8% | High-$Z$ nuclei |
|
||||
| Fit range | 0.0061 | 0.4% | All |
|
||||
| Endpoint energy | 0.0038 | 0.2% | Precision $Q$-value needed |
|
||||
| Radiative corrections | 0.0033 | 0.2% | All |
|
||||
| Atomic exchange | 0.0029 | 0.2% | Low-energy region |
|
||||
| **Combined** | **0.0919** | **6.0%** | — |
|
||||
|
||||
Source: Paulsen et al. (2024), Table VIII.
|
||||
|
||||
## 6. Key References
|
||||
|
||||
| Ref | Citation | Use |
|
||||
|---|---|---|
|
||||
| [SSM-orig] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | SSM introduction; 26 decays |
|
||||
| [SSM-Tc94-98] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc |
|
||||
| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}$Tc MMC measurement; $g_A$ extraction (enhanced value) |
|
||||
| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc MMC measurement; quenched $g_A$ via ESSM |
|
||||
| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | Enhanced SSM formalism |
|
||||
| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME analysis; five second-forbidden decays |
|
||||
| [Hayen2017] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | BB formalism; shape factor theory |
|
||||
@@ -0,0 +1,106 @@
|
||||
---
|
||||
title: "sNME — Small Relativistic Vector NME"
|
||||
date: 2026-05-14
|
||||
tags:
|
||||
- gA/extraction/ssnme
|
||||
- beta-spectrum/forbidden
|
||||
- nuclear-structure/NME
|
||||
status: active
|
||||
aliases: [small relativistic vector nuclear matrix element, sNME, CVC vector current]
|
||||
related_notes:
|
||||
- gA-extraction/01-ssm-algorithm
|
||||
- gA-extraction/03-tc99-case-study
|
||||
- nuclear-structure/01-shape-factors
|
||||
---
|
||||
|
||||
# The Small Relativistic Vector NME (sNME) — Why It Matters
|
||||
|
||||
## What is the sNME?
|
||||
|
||||
In forbidden β decay, the relativistic vector current has two components:
|
||||
- **Large NME (l-NME):** Computed from the large components of nucleon wave functions. Reliably calculable in NSM.
|
||||
- **Small NME (sNME):** Couples the large and small components. **Extremely difficult** to compute because it involves contributions outside the valence shell.
|
||||
|
||||
In a perfect many-body theory with infinite valence space, the sNME is related to the l-NME by the **CVC hypothesis**:
|
||||
|
||||
$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME} \tag{1}$$
|
||||
|
||||
where:
|
||||
- $W_0$ — endpoint energy in MeV
|
||||
- $m_n - m_p \approx 1.293$ MeV — neutron-proton mass difference
|
||||
- $E_C$ — Coulomb displacement energy (depends on nuclear charge distribution)
|
||||
- l-NME — large vector NME (computable in NSM)
|
||||
|
||||
### Why is sNME important?
|
||||
|
||||
Despite being "small" (typically $< 0.1$ in absolute units), the sNME can influence β-spectral shapes and half-lives **quite strongly** because:
|
||||
|
||||
1. **Quadratic dependence:** The half-life $t_{1/2} \propto 1/(\text{l-NME} + \text{sNME})^2$, so even a small sNME shifts the total matrix element significantly.
|
||||
2. **Small Q-value amplification:** For $^{99}\text{Tc}$, $[W_0 - (m_n - m_p)] = -0.487$ MeV — the negative mass difference nearly cancels the endpoint energy, making the sNME contribution relatively large.
|
||||
3. **Shape dependence:** Different sNME values produce measurably different spectral shapes, especially below 100 keV.
|
||||
|
||||
## The CVC Value — An Idealization
|
||||
|
||||
The CVC prediction (Eq. 1) is an **idealization** — it pertains to a perfect many-body theory. In practice:
|
||||
- NSM calculations with finite valence spaces give sNME $\approx 0$ (the valence space truncation kills it)
|
||||
- The CVC value serves as a **reference**, not a physical prediction
|
||||
|
||||
## ESSM: sNME as a Fitting Parameter
|
||||
|
||||
### The Enhanced Spectrum Shape Method (ESSM)
|
||||
|
||||
Kumar et al. (2021) proposed using the sNME as an additional fitting parameter:
|
||||
|
||||
1. **Fix** $g_A$ at a trial value
|
||||
2. **Vary** the sNME to match the **experimental partial half-life**
|
||||
3. Because $t_{1/2} \propto (\text{l-NME} + \text{sNME})^{-2}$, the equation $t_{1/2}^{\text{theo}}(g_A, \text{sNME}) = t_{1/2}^{\text{exp}}$ typically yields **two solutions** for the sNME:
|
||||
- **sNME(c):** closer to the CVC value
|
||||
- **sNME(f):** further from the CVC value
|
||||
|
||||
### Two-Solution Structure
|
||||
|
||||
For each $g_A$ value, there are two sNME solutions (or none, if the experimental half-life cannot be reproduced). This creates two distinct spectral shape predictions:
|
||||
|
||||
- The **sNME(c)** solution is theoretically preferred (closer to CVC)
|
||||
- The **sNME(f)** solution may also be physical — only experiment can decide
|
||||
|
||||
### Application to $^{99}\text{Tc}$
|
||||
|
||||
Song et al. (2025) applied ESSM to $^{99}\text{Tc}$ with three nuclear models:
|
||||
|
||||
| Model | $g_A^{\text{eff}}$ | sNME (c) [fm³] | sNME (f) [fm³] | $\chi^2_\nu$ | p-value |
|
||||
|---|---|---|---|---|---|
|
||||
| **CloseCVC (CVC fixed)** | 1.1 | 0.0674 | — | 1.106 | 0.12 |
|
||||
| | 1.0 | 0.0681 | — | 1.244 | 0.004 |
|
||||
| | 1.0 | 0.0651 | — | 1.644 | <0.0001 |
|
||||
| **FarCVC (sNME fitted)** | 1.2 | −0.0698 | — | 1.066 | 0.22 |
|
||||
| | 1.0 | −0.0669 | — | 1.097 | 0.14 |
|
||||
| | 1.0 | −0.0683 | — | 1.095 | 0.14 |
|
||||
|
||||
Key findings:
|
||||
- **Paulsen et al. (CloseCVC):** $g_A^{\text{eff}} = 1.526(92)$ — enhanced value
|
||||
- **Song et al. (FarCVC/ESSM):** $g_A^{\text{eff}} \approx 1.0$–1.2 — quenched value
|
||||
- The discrepancy arises from the **sNME treatment**, not from the data
|
||||
|
||||
## sNME in Other Transitions
|
||||
|
||||
Ramalho & Suhonen (2024) studied five second-forbidden nonunique decays:
|
||||
|
||||
| Transition | $g_A$ Sensitivity | sNME Sensitivity |
|
||||
|---|---|---|
|
||||
| $^{60}\text{Fe} \to {}^{60}\text{Co}$ | Strong | Weak |
|
||||
| $^{94}\text{Nb} \to {}^{94}\text{Mo}$ | Strong (jj45pnb) | Weak |
|
||||
| $^{98}\text{Tc} \to {}^{98}\text{Ru}$ | Strong | Moderate (below 100 keV) |
|
||||
| $^{126}\text{Sn} \to {}^{126}\text{Sb}$ | Weak | **Strong** |
|
||||
| $^{129}\text{I} \to {}^{129}\text{Xe}$ | Weak | **Strong** |
|
||||
|
||||
For $^{99}\text{Tc}$, the $g_A$ sensitivity is **strong** and the sNME sensitivity is **moderate** — making it an excellent candidate for combined $g_A$ + sNME fitting.
|
||||
|
||||
## References
|
||||
|
||||
| Ref | Citation | Use |
|
||||
|---|---|---|
|
||||
| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | ESSM formalism |
|
||||
| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc with ESSM |
|
||||
| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME in five second-forbidden decays |
|
||||
| [Behrens] | Behrens & Bühring, *Electron Radial Wave Functions and Nuclear Beta Decay* (1982) | sNME formalism |
|
||||
@@ -0,0 +1,105 @@
|
||||
---
|
||||
title: "$^{99}$Tc Case Study — gA Extraction Results"
|
||||
date: 2026-05-14
|
||||
tags:
|
||||
- gA/extraction/tc99
|
||||
- beta-spectrum/forbidden/second
|
||||
status: active
|
||||
aliases: [Tc99 gA, 99Tc beta spectrum, SSM Tc99]
|
||||
related_notes:
|
||||
- gA-extraction/01-ssm-algorithm
|
||||
- gA-extraction/02-ssnme
|
||||
- refs/2024_Paulsen_T99_beta_spectrum/summary
|
||||
- refs/2025_Song_tc99_measurement_for_g_v/summary
|
||||
---
|
||||
|
||||
# $^{99}$Tc — The Premier Laboratory for gA Extraction
|
||||
|
||||
## Decay Properties
|
||||
|
||||
| Property | Value |
|
||||
|---|---|
|
||||
| Transition | $^{99}\text{Tc}(9/2_1^+) \to {}^{99}\text{Ru}(5/2_1^+)$ |
|
||||
| Forbiddenness | Second-forbidden nonunique |
|
||||
| Branching ratio | 99.99855(30)% |
|
||||
| Q-value | 295.82(16) keV (Paulsen et al.) |
|
||||
| Half-life | 211.5(1.1) $\times 10^3$ a (ENSDF) |
|
||||
| Dominant transition | $2d_{5/2} \to 1g_{9/2}$ (OBTD $\approx 0.478$) |
|
||||
| Sub-dominant | $1g_{7/2} \to 1g_{9/2}$ (OBTD $\approx 0.010$) |
|
||||
|
||||
## Two Contradictory Results
|
||||
|
||||
### Result 1: Paulsen et al. (2024) — Enhanced $g_A$
|
||||
|
||||
**Measurement:** MMC at PTB Berlin + LNHB Paris. Two independent setups, consistent results.
|
||||
|
||||
**Method:** Fixed sNME to CVC value (CloseCVC).
|
||||
|
||||
**Result:**
|
||||
$$g_A^{\text{eff}} = 1.526(92)$$
|
||||
|
||||
This is an **enhanced** value — significantly above the free-nucleon value of 1.27.
|
||||
|
||||
**Follow-on values:**
|
||||
- $g_V^{\text{eff}} = 0.376(5)$ — quenched (via half-life normalization)
|
||||
- $g_A^{\text{eff, quenched}} = 0.574(36)$ — quenched
|
||||
- $\bar{E}_\beta = 98.51(23)$ keV
|
||||
- $\log ft = 12.3478(23)$
|
||||
|
||||
**Key claim:** This resolves an inconsistency — first-forbidden nonunique transitions show $g_A^{\text{eff}} \approx 0.9$, while fourth-forbidden show $g_A^{\text{eff}} \approx 0.9$. The $^{99}\text{Tc}$ result suggests $g_A$ is **enhanced** in second-forbidden decays.
|
||||
|
||||
### Result 2: Song et al. (2025) — Quenched $g_A$
|
||||
|
||||
**Measurement:** MMC at LLNL (KRISS sensor). Independent of Paulsen et al.
|
||||
|
||||
**Method:** ESSM — sNME fitted as free parameter (FarCVC).
|
||||
|
||||
**Result:**
|
||||
$$g_A^{\text{eff}} \approx 1.0\text{--}1.2$$
|
||||
|
||||
This is a **quenched** value, consistent with first-forbidden and fourth-forbidden nonunique decays.
|
||||
|
||||
**Best fit:** glekpn Hamiltonian, $g_A^{\text{eff}} = 1.2$, sNME = −0.0698 fm³, $\chi^2_\nu = 1.066$, p-value = 0.22
|
||||
|
||||
**Key claim:** The Paulsen result is an artifact of fixing sNME to CVC. When sNME is fitted, the data prefer quenched $g_A$, consistent with the broader pattern.
|
||||
|
||||
## The Crucial Difference: sNME Treatment
|
||||
|
||||
| Aspect | Paulsen et al. | Song et al. |
|
||||
|---|---|---|
|
||||
| sNME treatment | Fixed to CVC value | Fitted as free parameter (ESSM) |
|
||||
| $g_A^{\text{eff}}$ | 1.526(92) (enhanced) | 1.0–1.2 (quenched) |
|
||||
| Nuclear models | NSM (GLEKPN) only | NSM (glekpn, jj45pnb) + MQPM |
|
||||
| Half-life constraint | Separate normalization | Simultaneous shape + half-life fit |
|
||||
| Consistency with 113Cd, 115In | Inconsistent | Consistent |
|
||||
|
||||
## What This Means for Your Experiment
|
||||
|
||||
The $^{99}\text{Tc}$ case demonstrates that **the treatment of sNME is the dominant theoretical systematic** in gA extraction. The two experiments (Paulsen and Song) measured essentially the same spectrum (Paulsen's data was used as input for Song's analysis) but obtained **contradictory results** because of different theoretical assumptions.
|
||||
|
||||
**For your data analysis:**
|
||||
1. **Do not fix sNME to CVC** — treat it as a fitting parameter
|
||||
2. **Use ESSM** — simultaneous fit to spectral shape + half-life
|
||||
3. **Try multiple nuclear models** — glekpn, jj45pnb, and MQPM (if available)
|
||||
4. **Report both sNME solutions** (c and f) for each model
|
||||
5. **The "optimal" choice** (closest to CVC) is theoretically motivated but requires experimental validation
|
||||
|
||||
## Your Polynomial Fit: Connecting to SSM
|
||||
|
||||
You've fitted $C(W) = A \cdot W + B \cdot W^{-1} + C \cdot W^2$. This empirical parametrization can be **cross-checked** against the SSM prediction:
|
||||
|
||||
1. Compute $C_{\text{SSM}}(W; g_A)$ for a range of $g_A$ values
|
||||
2. Fit each SSM curve to the polynomial form $aW + bW^{-1} + cW^2$ over your fit range
|
||||
3. Extract the effective $(A, B, C)$ as a function of $g_A$
|
||||
4. Compare your fitted $(A, B, C)$ to the $g_A$-dependent curve to identify which $g_A$ range is consistent
|
||||
|
||||
This provides a **model-independent bridge** between your empirical fit and the physical $g_A$ extraction.
|
||||
|
||||
## References
|
||||
|
||||
| Ref | Citation |
|
||||
|---|---|
|
||||
| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) |
|
||||
| [Song2025] | Song et al., arXiv:2506.17544 (2025) |
|
||||
| [Kostensalo2017] | Kostensalo & Suhonen, *Phys. Rev. C* **96**, 024317 (2017) |
|
||||
| [ESSM] | Kumar et al., *Eur. Phys. J. A* **57**, 225 (2021) |
|
||||
@@ -0,0 +1,175 @@
|
||||
---
|
||||
title: "Implementation TODO — gA Extraction Pipeline"
|
||||
date: 2026-05-14
|
||||
tags:
|
||||
- gA/extraction/todo
|
||||
- implementation-plan
|
||||
status: active
|
||||
aliases: [gA pipeline, SSM implementation, shape factor module]
|
||||
related_notes:
|
||||
- gA-extraction/01-ssm-algorithm
|
||||
- gA-extraction/02-ssnme
|
||||
- gA-extraction/03-tc99-case-study
|
||||
---
|
||||
|
||||
# Implementation TODO — gA Extraction Pipeline
|
||||
|
||||
> [!info] Status
|
||||
> **Research phase.** Experimental data to be added by user. Implementation to follow after documentation is complete.
|
||||
|
||||
## Phase 1: Nuclear Model Input (P0 — prerequisite)
|
||||
|
||||
- [ ] **1.1** Obtain precomputed shape factor components $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $^{99}\text{Tc}$
|
||||
- [ ] Option A: Use Kostensalo & Suhonen (2017) MQPM results (published in PRC 96, 024317)
|
||||
- [ ] Option B: Use Ramalho & Suhonen (2024) NSM results (Front. Phys. 12, 1455778)
|
||||
- [ ] Option C: Run shell model with KShell or another available code
|
||||
- [ ] **Decision:** Which source to use? (See [[gA-extraction/03-tc99-case-study]] for comparison)
|
||||
- [ ] **1.2** Store components as lookup tables (CSV or NumPy):
|
||||
- `data/Tc95_CV.csv` — $C_V(W)$ vs. $W$
|
||||
- `data/Tc95_CA.csv` — $C_A(W)$ vs. $W$
|
||||
- `data/Tc95_CVA.csv` — $C_{VA}(W)$ vs. $W$
|
||||
- Columns: `W (unitless)`, `C_V`, `C_A`, `C_VA`
|
||||
- $W$ range: $[1.001, 1.580]$ (corresponds to $T \in [0.5, 295]$ keV)
|
||||
- [ ] **1.3** Decide on sNME handling:
|
||||
- [ ] Fix to CVC value (Paulsen et al. approach)
|
||||
- [ ] Fit as free parameter (ESSM — Song et al. approach) **RECOMMENDED**
|
||||
- If ESSM: also need l-NME from nuclear model
|
||||
|
||||
## Phase 2: Shape Factor Module (P1 — depends on 1.1)
|
||||
|
||||
Create `beta_spectrum/components/shape_factor_gA.py`:
|
||||
|
||||
- [ ] **2.1** `ShapeFactorGA` class:
|
||||
```python
|
||||
class ShapeFactorGA:
|
||||
def __init__(self, C_V_table, C_A_table, C_VA_table, sNME=None):
|
||||
# Load tables, interpolate to arbitrary W
|
||||
# sNME: if None, use CVC value; if float, use fitted value
|
||||
|
||||
def evaluate(self, W, gA, gV=1.0):
|
||||
"""Compute C(W; gA, gV) = gV²·CV(W) + gA²·CA(W) + gV·gA·CVA(W)"""
|
||||
|
||||
def evaluate_components(self, W):
|
||||
"""Return (C_V(W), C_A(W), C_VA(W))"""
|
||||
```
|
||||
- [ ] **2.2** Interpolation: Use `scipy.interpolate.interp1d` with `kind='cubic'` for smooth evaluation
|
||||
- [ ] **2.3** Validation: Check that $C(W)$ is monotonically reasonable for $g_A \in [0.4, 1.8]$
|
||||
- [ ] **2.4** Unit test: `test_shape_factor_gA_consistency` — verify decomposition matches Eq. (1)
|
||||
|
||||
## Phase 3: Fitting Pipeline (P2 — depends on 1.2 + 2.1)
|
||||
|
||||
Create `beta_spectrum/gA_fit.py`:
|
||||
|
||||
- [ ] **3.1** `fit_gA_spectrum(W_exp, N_exp, sigma_exp, shape_factor, W_min, W_max)`:
|
||||
```python
|
||||
def fit_gA_spectrum(W_exp, N_exp, sigma_exp, sf, W_min=0.020, W_max=0.275):
|
||||
"""
|
||||
Fit gA to experimental spectrum using χ² minimization.
|
||||
|
||||
Parameters
|
||||
----------
|
||||
W_exp : array_like
|
||||
Electron total energy (unitless, W = 1 + T/m_ec²)
|
||||
N_exp : array_like
|
||||
Measured counts (or normalized spectrum)
|
||||
sigma_exp : array_like
|
||||
Uncertainties on N_exp
|
||||
sf : ShapeFactorGA
|
||||
Precomputed shape factor
|
||||
W_min, W_max : float
|
||||
Fit range in unitless W
|
||||
|
||||
Returns
|
||||
-------
|
||||
gA_best, gA_low, gA_high, chi2_min, chi2_curve
|
||||
"""
|
||||
```
|
||||
- [ ] **3.2** Grid search: $g_A \in [0.4, 1.8]$ with 0.01 step
|
||||
- [ ] **3.3** Brent's method refinement around grid minimum
|
||||
- [ ] **3.4** Confidence interval: find $g_A$ where $\chi^2 = \chi^2_{\text{min}} + 1$
|
||||
- [ ] **3.5** χ² curve output for visualization
|
||||
- [ ] **3.6** Unit test: `test_fit_gA_reproduces_input` — inject known $g_A$, verify recovery
|
||||
|
||||
## Phase 4: ESSM Extension (P2 — optional but recommended)
|
||||
|
||||
- [ ] **4.1** `fit_gA_sNME(W_exp, N_exp, sigma_exp, sf, l_NME, t_half_exp)`:
|
||||
- Simultaneous fit of $g_A$ and sNME
|
||||
- Match both spectral shape AND experimental half-life
|
||||
- Return both (sNME_c, sNME_f) solutions
|
||||
- [ ] **4.2** Half-life computation:
|
||||
```python
|
||||
def compute_half_life(sf, gA, gV, sNME, W0):
|
||||
"""Compute t_1/2 from integrated shape factor"""
|
||||
C_integral = integrate(C(W; gA, gV, sNME) * phase_space * Fermi * ... dW)
|
||||
return kappa / C_integral
|
||||
```
|
||||
- [ ] **4.3** Validation against Song et al. (2025) results for $^{99}\text{Tc}$
|
||||
|
||||
## Phase 5: Validation (P3 — depends on 3.1 + 4.1)
|
||||
|
||||
- [ ] **5.1** Reproduce Paulsen et al. (2024) result: $g_A^{\text{eff}} = 1.526(92)$ with CVC-fixed sNME
|
||||
- [ ] **5.2** Reproduce Song et al. (2025) result: $g_A^{\text{eff}} \approx 1.0\text{--}1.2$ with ESSM
|
||||
- [ ] **5.3** Cross-check: polynomial fit $A \cdot W + B \cdot W^{-1} + C \cdot W^2$ vs. SSM for same $g_A$
|
||||
- [ ] **5.4** Sensitivity analysis: vary $W_{\text{min}}$, $W_{\text{max}}$, nuclear model, sNME
|
||||
|
||||
## Phase 6: Experimental Data Analysis (P4 — depends on user providing data)
|
||||
|
||||
- [ ] **6.1** Import raw spectrum data (format TBD)
|
||||
- [ ] **6.2** Apply detector response correction (if needed) — [[corrections/07-detector-response]]
|
||||
- [ ] **6.3** Run `fit_gA_spectrum` with appropriate fit range
|
||||
- [ ] **6.4** Run `fit_gA_sNME` (ESSM) if nuclear model input available
|
||||
- [ ] **6.5** Generate publication-quality plots:
|
||||
- Measured spectrum + best-fit theoretical curve
|
||||
- Residuals plot
|
||||
- χ² vs. $g_A$ curve
|
||||
- Comparison with polynomial fit
|
||||
- [ ] **6.6** Uncertainty budget: propagate detector, model, and systematic uncertainties
|
||||
|
||||
## Phase 7: Paper Draft Support
|
||||
|
||||
- [ ] **7.1** Generate tables: fitted $g_A$, sNME, $\chi^2$, confidence intervals
|
||||
- [ ] **7.2** Generate figures for paper
|
||||
- [ ] **7.3** Cross-reference with literature values
|
||||
- [ ] **7.4** Draft methodology section using [[gA-extraction/01-ssm-algorithm]] as reference
|
||||
|
||||
## File Structure (target)
|
||||
|
||||
```
|
||||
beta_shape_pnpi/
|
||||
├── beta_spectrum/
|
||||
│ ├── components/
|
||||
│ │ └── shape_factor_gA.py ← Phase 2
|
||||
│ └── gA_fit.py ← Phase 3 + 4
|
||||
├── data/
|
||||
│ ├── Tc95_CV.csv ← Phase 1
|
||||
│ ├── Tc95_CA.csv
|
||||
│ ├── Tc95_CVA.csv
|
||||
│ └── experimental/ ← Phase 6 (user-provided)
|
||||
├── tests/
|
||||
│ └── test_shape_factor_gA.py ← Phase 2 + 3
|
||||
│ └── test_gA_fit.py
|
||||
└── docs/
|
||||
└── gA-extraction/
|
||||
├── 01-ssm-algorithm.md
|
||||
├── 02-ssnme.md
|
||||
├── 03-tc99-case-study.md
|
||||
└── 04-implementation-todo.md
|
||||
```
|
||||
|
||||
## Decision Points
|
||||
|
||||
1. **Nuclear model source:** Which precomputed $C_i(W)$ to use? (MQPM vs. NSM)
|
||||
2. **sNME strategy:** CVC-fixed or ESSM? (ESSM recommended)
|
||||
3. **Fit range:** What $W_{\text{min}}$, $W_{\text{max}}$ for your detector?
|
||||
4. **Detector corrections:** Do you need response deconvolution before fitting?
|
||||
|
||||
## References for Implementation
|
||||
|
||||
| Item | Source |
|
||||
|---|---|
|
||||
| Shape factor formalism | Kostensalo & Suhonen, PRC 96, 024317 (2017) |
|
||||
| ESSM formalism | Kumar et al., EPJA 57, 225 (2021) |
|
||||
| sNME analysis | Ramalho & Suhonen, Front. Phys. 12, 1455778 (2024) |
|
||||
| $^{99}\text{Tc}$ MMC data | Paulsen et al., PRC 110, 055503 (2024) |
|
||||
| $^{99}\text{Tc}$ ESSM analysis | Song et al., arXiv:2506.17544 (2025) |
|
||||
| BB formalism | Behrens & Bühring (1982) |
|
||||
@@ -6,14 +6,14 @@ tags:
|
||||
status: active
|
||||
aliases: [Shape Factor, C(Z,W), Form Factors]
|
||||
cssclasses: []
|
||||
related_components: [[07-recoil-effects]], [[11-isospin-breakdown]]
|
||||
related_components: [[corrections/06-recoil]],
|
||||
---
|
||||
|
||||
# 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.
|
||||
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 ([[corrections/02-finite-size]]) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction.
|
||||
|
||||
$$N(W)\,dW \propto p W (W_0 - W)^2 F(Z,W) C(Z,W) \tag{79}$$
|
||||
|
||||
@@ -53,7 +53,7 @@ The one-body density matrix elements $\langle f | a^\dagger_\alpha a_\beta | i \
|
||||
|
||||
## Induced Currents — Form Factor Classification
|
||||
|
||||
See [[07-recoil-effects]] for detailed equations. Key points:
|
||||
See [[corrections/06-recoil]] for detailed equations. Key points:
|
||||
|
||||
### First-Class Currents (CVC/PCAC)
|
||||
- **Vector** $g_V$: conserved, from EM data via CVC
|
||||
@@ -82,7 +82,7 @@ See [[07-recoil-effects]] for detailed equations. Key points:
|
||||
|
||||
## Nuclear Deformation Correction $D_C(Z,W,\beta^2)$
|
||||
|
||||
The nuclear structure analogue of the electrostatic deformation correction ([[03-finite-size]]):
|
||||
The nuclear structure analogue of the electrostatic deformation correction ([[corrections/02-finite-size]]):
|
||||
- Accounts for non-spherical nuclear shape in convolution terms
|
||||
- Partially cancels with $DFS$ correction (electrostatic)
|
||||
- Relevant for deformed nuclei ($\beta_2 \neq 0$)
|
||||
@@ -0,0 +1,29 @@
|
||||
---
|
||||
title: "Nuclear Structure — Overview"
|
||||
date: 2026-05-14
|
||||
tags:
|
||||
- beta-spectrum/nuclear-structure
|
||||
status: active
|
||||
aliases: [Nuclear Theory, Shape Factor, NME]
|
||||
---
|
||||
|
||||
# Nuclear Structure Effects in β Decay
|
||||
|
||||
## Overview
|
||||
|
||||
Nuclear structure enters the β spectrum through the **shape factor** $C(Z,W)$, which encodes how the nuclear wave functions and transition matrix elements modify the spectrum beyond the point-nucleus Coulomb interaction.
|
||||
|
||||
## Contents
|
||||
|
||||
- [[nuclear-structure/01-shape-factors]] — Shape factor $C(Z,W)$: decomposition, multipole expansion, impulse approximation, induced currents
|
||||
|
||||
## Key Concepts
|
||||
|
||||
- **Shape factor decomposition:** $C(W) = g_V^2 C_V(W) + g_A^2 C_A(W) + g_V g_A C_{VA}(W)$
|
||||
- **Forbidden decays:** Nonunique transitions are sensitive to $g_A$ → basis for SSM
|
||||
- **sNME:** Small relativistic vector NME — critical for precision but hard to compute
|
||||
|
||||
## Related
|
||||
|
||||
- [[gA-extraction/01-ssm-algorithm]] — How to extract $g_A$ from shape factor
|
||||
- [[corrections/06-recoil]] — Weak magnetism, induced currents, recoil effects
|
||||
@@ -1,363 +0,0 @@
|
||||
# Algorithm for Radiative Corrections to Beta-Decay Spectra
|
||||
|
||||
## 1. Physical Overview
|
||||
|
||||
The radiatively corrected beta-decay spectrum takes the form:
|
||||
|
||||
\[\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW} \cdot R(W, W_0)\]
|
||||
|
||||
where \(d\Gamma_0/dW\) is the uncorrected spectrum and \(R(W, W_0)\) is a multiplicative correction factor. The correction is organized by powers of the fine-structure constant \(\alpha\) and the nuclear charge \(Z\):
|
||||
|
||||
\[R(W, W_0) = 1 + \delta^{(1)}(W, W_0) + \delta^{(2)}(Z, W) + \delta^{(3)}(Z, W) + \cdots\]
|
||||
|
||||
**Important distinction:** Inner radiative corrections (\(\Delta_R^V\), \(\Delta_R^A\)) are energy-independent constants absorbed into effective coupling constants. They are **not** part of the spectrum shape calculation. Only the outer (energy-dependent) correction \(\delta_R(W, W_0)\) modulates the spectral shape.
|
||||
|
||||
---
|
||||
|
||||
## 2. Input Parameters
|
||||
|
||||
| Symbol | Description | Units |
|
||||
|--------|-------------|-------|
|
||||
| \(W\) | Total electron energy | \(m_e c^2\) |
|
||||
| \(W_0\) | Endpoint energy | \(m_e c^2\) |
|
||||
| \(Z\) | Nuclear charge of daughter | dimensionless |
|
||||
| \(m_p\) | Proton mass | \(m_e c^2\) (\(\approx 1836.15\)) |
|
||||
| \(\alpha\) | Fine-structure constant | \(\approx 1/137.036\) |
|
||||
|
||||
Derived quantities:
|
||||
- Momentum: \(p = \sqrt{W^2 - 1}\)
|
||||
- Velocity: \(\beta = p / W\)
|
||||
- Antineutrino energy (for neutrino spectrum corrections): \(\hat{W} = W_0 - W\)
|
||||
|
||||
---
|
||||
|
||||
## 3. Order \(\alpha\): The Universal Sirlin Function \(g(W, W_0)\)
|
||||
|
||||
### 3.1 Mathematical Form
|
||||
|
||||
The dominant correction, derived in Sirlin (1967) and refined in Sirlin & Zucchini (1986) and Sirlin (1987), is universal—identical for Fermi and Gamow-Teller transitions, electrons and positrons:
|
||||
|
||||
\[g(W, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) - \frac{3}{4} + 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\left[\frac{W_0 - W}{3W} - \frac{3}{2} + \ln\left(\frac{2(W_0 - W)}{m_e}\right)\right] + \frac{4}{\beta}L\left(\frac{2\beta}{1+\beta}\right) + \frac{\tanh^{-1}\beta}{\beta}\left[2(1+\beta^2) + \frac{(W_0-W)^2}{6W^2} - 4\tanh^{-1}\beta\right]\]
|
||||
|
||||
The correction to the spectrum is:
|
||||
|
||||
\[\delta^{(1)}(W, W_0) = \frac{\alpha}{2\pi} \, g(W, W_0)\]
|
||||
|
||||
### 3.2 The Dilogarithm (Spence Function)
|
||||
|
||||
The function \(L(x)\) appearing above is the Spence function (dilogarithm):
|
||||
|
||||
\[L(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -\text{Li}_2(-x/(1-x)) \quad \text{or equivalently} \quad L(x) = -\text{Li}_2(x)\]
|
||||
|
||||
Using the standard dilogarithm \(\text{Li}_2(x) = -\int_0^x \frac{\ln(1-t)}{t} dt = \sum_{k=1}^\infty \frac{x^k}{k^2}\):
|
||||
|
||||
\[L\left(\frac{2\beta}{1+\beta}\right) = -\text{Li}_2\left(\frac{2\beta}{1+\beta}\right)\]
|
||||
|
||||
### 3.3 Numerical Stability Considerations
|
||||
|
||||
**Inverse hyperbolic tangent:**
|
||||
|
||||
\[\tanh^{-1}\beta = \frac{1}{2}\ln\left(\frac{1+\beta}{1-\beta}\right)\]
|
||||
|
||||
When \(\beta \to 1\) (ultra-relativistic limit), compute using the identity:
|
||||
|
||||
\[\tanh^{-1}\beta = \ln\left(\frac{W+p}{m_e}\right) = \ln(W + p) \quad (\text{in } m_e \text{ units where } m_e = 1)\]
|
||||
|
||||
For \(\beta \to 0\) (non-relativistic limit), use the Taylor expansion:
|
||||
|
||||
\[\frac{\tanh^{-1}\beta}{\beta} = 1 + \frac{\beta^2}{3} + \frac{\beta^4}{5} + \cdots\]
|
||||
|
||||
Avoid computing \(\tanh^{-1}\beta\) directly when \(\beta\) is very small.
|
||||
|
||||
**Terms containing \(1/\beta\):** When \(\beta < \epsilon\) (e.g., \(\epsilon = 10^{-6}\)), evaluate the expression using a small-\(\beta\) series expansion to avoid division by zero.
|
||||
|
||||
**Dilogarithm evaluation:** Use a robust numerical implementation. For \(x > 1\), apply the identity:
|
||||
|
||||
\[\text{Li}_2(x) = \frac{\pi^2}{3} - \frac{1}{2}\ln^2(x) - \text{Li}_2(1/x) - i\pi\ln(x) \quad (x > 1)\]
|
||||
|
||||
The imaginary part cancels in the real-valued total correction.
|
||||
|
||||
### 3.4 Endpoint Behavior and Soft-Photon Resummation
|
||||
|
||||
**Critical observation:** The function \(g(W, W_0)\) contains a term proportional to \(\ln(W_0 - W)\), which diverges as \(W \to W_0\). This divergence is physical—arising from the emission of arbitrarily soft real photons—and the total decay rate integrated over all photon energies remains finite.
|
||||
|
||||
For the differential spectrum shape, the leading logarithmic behavior near the endpoint is:
|
||||
|
||||
\[g(W, W_0) \sim 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right) \ln(W_0 - W) + \text{regular terms}\]
|
||||
|
||||
**Resummation prescription (Sirlin 1987, 2011):** Replace the logarithmically divergent term with an exponentiated form:
|
||||
|
||||
\[\ln(W_0 - W) \cdot t(\beta) \quad \longrightarrow \quad (W_0 - W)^{t(\beta)} - 1\]
|
||||
|
||||
where the exponent is:
|
||||
|
||||
\[t(\beta) = \frac{2\alpha}{\pi}\left[\frac{\tanh^{-1}\beta}{\beta} - 1\right]\]
|
||||
|
||||
**Implementation strategy:**
|
||||
|
||||
1. Compute the full \(g(W, W_0)\) including the \(\ln(W_0 - W)\) term.
|
||||
2. For \(W_0 - W < \Delta_{\text{cut}}\) (suggested \(\Delta_{\text{cut}} \sim 10^{-3}\) in units of \(m_e c^2\)), extract the coefficient of \(\ln(W_0 - W)\):
|
||||
|
||||
\[C(\beta) = 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\]
|
||||
|
||||
3. Replace \(C(\beta) \cdot \ln(W_0 - W)\) with \(C(\beta) \cdot \frac{(W_0-W)^{t(\beta)} - 1}{t(\beta)}\) using:
|
||||
|
||||
\[\lim_{t \to 0} \frac{x^t - 1}{t} = \ln x\]
|
||||
|
||||
4. For energies well below the endpoint (\(W_0 - W \gg \Delta_{\text{cut}}\)), no resummation is needed and the original logarithmic form suffices.
|
||||
|
||||
**For low endpoint energies** (e.g., tritium with \(W_0 \approx 1.02\)), the endpoint resummation has negligible effect and can be omitted for simplicity.
|
||||
|
||||
---
|
||||
|
||||
## 4. Order \(Z\alpha^2\): Finite Nuclear Size Corrections
|
||||
|
||||
### 4.1 Structure
|
||||
|
||||
The \(O(Z\alpha^2)\) correction, derived in Sirlin (1987), depends on the nuclear charge distribution:
|
||||
|
||||
\[\delta^{(2)}(Z, W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W)\]
|
||||
|
||||
The four contributions \(\Delta_i\) arise from distinct classes of Feynman diagrams (see Sirlin 1987, Sec. 2). The dominant energy dependence comes from \(\Delta_1^0(W) + \Delta_4(W)\), for which the relativistic limit gives:
|
||||
|
||||
\[\Delta_1^0(W) + \Delta_4(W) = \ln\left(\frac{M}{m_e}\right) - \frac{5}{12}\ln\left(\frac{2W}{m_e}\right) + \frac{43}{18}\]
|
||||
|
||||
where \(M\) is the nucleon mass (use proton mass \(m_p\)).
|
||||
|
||||
### 4.2 Nuclear-Model-Dependent Part
|
||||
|
||||
The finite-nuclear-size contribution \(\Delta^F = \Delta_1^F + \Delta_2 + \Delta_3\) depends on the nuclear charge distribution. For the **modified Gaussian model** (Sirlin 1987, Eq. 10):
|
||||
|
||||
\[\Delta^F(Z) = \ln\left(\frac{\Lambda}{M}\right) - \kappa_1(Z) + O(\Lambda/M)\]
|
||||
|
||||
where:
|
||||
- \(\Lambda = \sqrt{6}/a\), with \(a\) the rms charge radius
|
||||
- \(M/\Lambda = r_0 A^{1/3} / 0.665\) (using Wilkinson's \(r_0\))
|
||||
- \(\kappa_1(Z) = \frac{1}{2}\left[\gamma + \ln(3/2k^2) + 2\alpha/(2+3\alpha)\right]\), with \(\gamma = 0.5772\), \(\alpha = (Z-2)/3\), and \(k^2 = \frac{3}{2}\frac{2+5\alpha}{2+3\alpha}\)
|
||||
|
||||
### 4.3 Practical Simplification
|
||||
|
||||
For light and medium nuclei (\(Z \lesssim 30\)), the total \(O(Z\alpha^2)\) correction can be tabulated once per isotope as an energy-independent shift (the energy dependence from \(\Delta_1^0 + \Delta_4\) is small compared to the order \(\alpha\) term). **For spectrum shape calculations:**
|
||||
|
||||
1. Calculate the energy-dependent part \(\Delta_1^0(W) + \Delta_4(W)\) using the relativistic formula above.
|
||||
2. Add the energy-independent nuclear-structure part \(\Delta^F(Z)\) from precomputed values (see Sirlin 1987, Table II).
|
||||
3. If high precision is required for low-energy electrons \((p \ll m_e)\), use the full non-relativistic expressions from Sirlin 1987.
|
||||
|
||||
For a **uniformly charged sphere**, \(\kappa_2 = \gamma - 4/3 + \ln(\sqrt{10}) = 0.395\), giving results within ~2% of the modified Gaussian model.
|
||||
|
||||
---
|
||||
|
||||
## 5. Order \(Z^2\alpha^3\): Higher-Order Corrections
|
||||
|
||||
The heuristic estimate from Sirlin (1986, 1987) is:
|
||||
|
||||
\[\delta^{(3)}(Z, W) \approx Z^2\alpha^3 \left[a\ln\left(\frac{\Lambda}{W}\right) + b f(W) + \frac{4\pi}{3}g(W) - 0.649\ln(2W_0)\right]\]
|
||||
|
||||
with:
|
||||
\[a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}, \quad b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}\]
|
||||
|
||||
And:
|
||||
\[f(W) = \ln(2W_0/m_e) - 5/6\]
|
||||
\[g(W) = \frac{1}{2}\left[\ln^2(R m_e) - \ln^2(2W/m_e)\right] + \frac{5}{4}\ln(2RW)\]
|
||||
|
||||
where \(R = \sqrt{5/4}\,a\) and \(a\) is the rms charge radius.
|
||||
|
||||
**Practical note:** For \(Z \lesssim 30\), \(\delta^{(3)} < 0.01\%\) and can typically be neglected for spectrum shape. For superallowed Fermi transitions with heavy nuclei, this correction contributes ~0.05–0.12% to the total correction (Sirlin 1987, Table III).
|
||||
|
||||
---
|
||||
|
||||
## 6. Neutrino Spectrum Correction
|
||||
|
||||
For the antineutrino/neutrino spectrum (Sirlin 2011, Eq. 11), define:
|
||||
|
||||
\[\hat{W} \equiv W_0 - W_\nu, \quad \hat{p} \equiv \sqrt{\hat{W}^2 - 1}, \quad \hat{\beta} \equiv \hat{p}/\hat{W}\]
|
||||
|
||||
The correction function \(h(\hat{W}, W_0)\) is:
|
||||
|
||||
\[h(\hat{W}, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) + \frac{23}{4} - \frac{8}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 8\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right)\ln\left(\frac{2\hat{W}\hat{\beta}}{m_e}\right) + 4\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{7+3\hat{\beta}^2}{8} - 2\tanh^{-1}\hat{\beta}\right]\]
|
||||
|
||||
The neutrino spectrum correction is:
|
||||
|
||||
\[\delta_\nu(\hat{W}, W_0) = \frac{\alpha}{2\pi} \, h(\hat{W}, W_0)\]
|
||||
|
||||
**Key difference from the electron correction:** The neutrino correction \(h\) is finite in the \(m_e \to 0\) limit, while the electron correction \(g\) diverges logarithmically. The \(m_e \to 0\) limit of \(h\) is particularly simple:
|
||||
|
||||
\[h(\hat{W}, W_0) \xrightarrow{m_e \to 0} 3\ln\left(\frac{m_p}{2W_0}\right) + \frac{23}{4} - \frac{4\pi^2}{3} - 3\ln\left(\frac{W_\nu}{W_0}\right)\]
|
||||
|
||||
This finite limit is explained theoretically by the Kinoshita-Lee-Nauenberg theorem: summing over the degenerate collinear \(e^-\gamma\) final states (as is done when integrating over electron momenta to obtain the neutrino spectrum) cancels the mass singularities (Sirlin 2011, Sec. II.F).
|
||||
|
||||
---
|
||||
|
||||
## 7. Conversion Between Electron and Neutrino Spectra
|
||||
|
||||
Given the electron spectrum \(f_e(W, W_0)\) for a specific decay, the corresponding neutrino spectrum including radiative corrections is (Sirlin 2011, Eq. 19):
|
||||
|
||||
\[f_\nu(W_\nu, W_0) = f_e(\hat{W}, W_0) \left[1 + \frac{\alpha}{2\pi}\left(h(\hat{W}, W_0) - g(\hat{W}, W_0)\right)\right]\]
|
||||
|
||||
where \(\hat{W} = W_0 - W_\nu\).
|
||||
|
||||
The difference function is given explicitly in Sirlin 2011, Eq. 20:
|
||||
|
||||
\[h(\hat{W}, W_0) - g(\hat{W}, W_0) = \frac{13}{2} - \frac{4}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 4\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right) \times \Bigg[\frac{11}{6} + \ln\left(\frac{2W_0}{m_e}\right) + 2\ln\hat{\beta} + 2\ln\left(\frac{W_\nu}{W_0}\right) - \ln\left(\frac{W_\nu}{W_0}\right) - \frac{1}{3W_\nu/W_0}\Bigg] + \frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{3-\hat{\beta}^2}{2} - 4\tanh^{-1}\hat{\beta} - \frac{(W_\nu/W_0)^2}{6(1-W_\nu/W_0)^2}\right]\]
|
||||
|
||||
This conversion is important for reactor neutrino oscillation studies.
|
||||
|
||||
---
|
||||
|
||||
## 8. Complete Calculation Algorithm
|
||||
|
||||
### 8.1 Full Correction Factor for Electron Spectrum
|
||||
|
||||
```
|
||||
Algorithm: RadiativeCorrection(W, W0, Z)
|
||||
Input: W — array of electron total energies [m_e c^2]
|
||||
W0 — endpoint energy [m_e c^2]
|
||||
Z — nuclear charge of daughter nucleus
|
||||
Output: R — multiplicative correction factor at each W
|
||||
|
||||
Constants:
|
||||
alpha = 1/137.035999084
|
||||
m_p = 938.272 / 0.51099895 # proton mass in m_e units
|
||||
gamma_E = 0.5772156649
|
||||
pi = 3.141592653589793
|
||||
|
||||
For each W_i in W:
|
||||
1. Compute derived quantities:
|
||||
p = sqrt(W_i^2 - 1)
|
||||
beta = p / W_i
|
||||
|
||||
2. Order α correction (Section 3):
|
||||
a. Calculate tanh^{-1}(beta) / beta
|
||||
If beta < 1e-6: use Taylor series 1 + beta^2/3 + beta^4/5
|
||||
Else: tanh_inv_beta_beta = 0.5 * ln((1+beta)/(1-beta)) / beta
|
||||
|
||||
b. Calculate dilogarithm L(2*beta/(1+beta)) = -Li_2(2*beta/(1+beta))
|
||||
Use robust numerical implementation
|
||||
|
||||
c. Compute g(W_i, W0) per Eq. (20b) of Sirlin 1967
|
||||
|
||||
d. If use_endpoint_resummation and (W0 - W_i) < delta_cut:
|
||||
t_beta = (2*alpha/pi) * (tanh_inv_beta_beta - 1)
|
||||
C_beta = 4 * (tanh_inv_beta_beta - 1)
|
||||
Replace: C_beta * ln(W0 - W_i)
|
||||
-> C_beta * ((W0 - W_i)^t_beta - 1) / t_beta
|
||||
|
||||
e. delta_1 = (alpha / (2*pi)) * g(W_i, W0)
|
||||
|
||||
3. Order Zα² correction (Section 4):
|
||||
a. Energy-dependent part:
|
||||
Delta_1_4 = ln(m_p) - (5/12)*ln(2*W_i) + 43/18
|
||||
|
||||
b. Energy-independent nuclear part:
|
||||
Look up Delta_F for the given (Z, A) from precomputed table
|
||||
(Use Sirlin 1987 Table II or compute from nuclear model)
|
||||
|
||||
c. delta_2 = Z * alpha^2 * (Delta_1_4 + Delta_F)
|
||||
|
||||
4. Order Z²α³ correction (Section 5, optional):
|
||||
For Z >= 20 or precision better than 0.01% required:
|
||||
Compute delta_3 from Sirlin heuristic formula
|
||||
Else:
|
||||
delta_3 = 0
|
||||
|
||||
5. Combine corrections:
|
||||
R_i = 1 + delta_1 + delta_2 + delta_3
|
||||
|
||||
Return R array
|
||||
```
|
||||
|
||||
### 8.2 Neutrino Spectrum Correction
|
||||
|
||||
```
|
||||
Algorithm: NeutrinoRadiativeCorrection(W_nu, W0, Z)
|
||||
Input: W_nu — array of neutrino energies [m_e c^2]
|
||||
W0 — endpoint energy [m_e c^2]
|
||||
Z — nuclear charge
|
||||
Output: R_nu — multiplicative correction factor
|
||||
|
||||
For each W_nu in array:
|
||||
W_hat = W0 - W_nu
|
||||
p_hat = sqrt(W_hat^2 - 1)
|
||||
beta_hat = p_hat / W_hat
|
||||
|
||||
Compute h(W_hat, W0) per Sirlin 2011 Eq. (11)
|
||||
(Use same numerical stability measures as for g-function)
|
||||
|
||||
R_nu_i = 1 + (alpha/(2*pi)) * h(W_hat, W0)
|
||||
```
|
||||
|
||||
### 8.3 Conversion Algorithm (Electron → Neutrino Spectrum)
|
||||
|
||||
```
|
||||
Algorithm: ElectronToNeutrinoConversion(W_nu, W0, Z, f_e)
|
||||
Input: W_nu — neutrino energies
|
||||
W0 — endpoint
|
||||
Z — nuclear charge
|
||||
f_e — function returning electron spectrum values
|
||||
Output: f_nu — neutrino spectrum values
|
||||
|
||||
For each W_nu in array:
|
||||
W_hat = W0 - W_nu
|
||||
|
||||
# Evaluate electron spectrum at W_hat
|
||||
f_e_val = f_e(W_hat)
|
||||
|
||||
# Compute difference correction
|
||||
Compute delta_diff = (alpha/(2*pi)) * [h(W_hat, W0) - g(W_hat, W0)]
|
||||
per Sirlin 2011 Eq. (20)
|
||||
|
||||
f_nu_i = f_e_val * (1 + delta_diff)
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## 9. Physical Interpretation and Validation
|
||||
|
||||
### 9.1 Dominant Terms
|
||||
|
||||
The leading \(O(\alpha)\) correction is the **universal Sirlin function** \(g(W, W_0)\). Its dominant contribution comes from the logarithm:
|
||||
|
||||
\[3\ln\left(\frac{m_p}{m_e}\right) \approx 15.4\]
|
||||
|
||||
multiplied by \(\alpha/2\pi \approx 1.16 \times 10^{-3}\), giving a typical ~1.8% correction. The energy-dependent terms modulate this by ~±1% across the spectrum.
|
||||
|
||||
### 9.2 Sign Conventions
|
||||
|
||||
- The correction factor \(R(W, W_0)\) **multiplies** the uncorrected spectrum.
|
||||
- \(R > 1\) means the correction increases the transition probability at that energy.
|
||||
- For the electron spectrum: the correction typically **decreases** the probability at high \(W\) (\(W/W_0 \gtrsim 0.44\)) and **increases** it at low \(W\) (Sirlin 2011, Sec. III).
|
||||
- For the neutrino spectrum: the correction is **positive and monotonically increasing** with \(W_\nu\) (Sirlin 2011, Sec. II.G).
|
||||
|
||||
### 9.3 Consistency Checks
|
||||
|
||||
1. **Lifetime check:** Integrate the corrected and uncorrected spectra:
|
||||
|
||||
\[\int_1^{W_0} \frac{d\Gamma_0}{dW} R(W, W_0) dW\]
|
||||
|
||||
The relative correction to the lifetime should be finite (the logarithmic divergence at the endpoint integrable over the spectrum).
|
||||
|
||||
2. **Electron-neutrino consistency:** The lifetime calculated from the corrected electron spectrum must equal that from the corrected neutrino spectrum.
|
||||
|
||||
3. **Zero-charge limit:** For \(Z = 0\), \(R(W, W_0) \to 1 + \frac{\alpha}{2\pi}g(W, W_0)\).
|
||||
|
||||
4. **Low-energy limit:** As \(W \to 1\) (non-relativistic), the correction should approach the values tabulated in Sirlin 1967.
|
||||
|
||||
---
|
||||
|
||||
## 10. Summary of Required Special Functions
|
||||
|
||||
| Function | Definition | Numerical Source |
|
||||
|----------|------------|------------------|
|
||||
| \(\text{Li}_2(x)\) | \(\sum_{k=1}^\infty x^k/k^2\) | Standard library (e.g., `scipy.special.spence` in Python, `gsl_sf_dilog` in C) |
|
||||
| \(\tanh^{-1}(x)\) | \(\frac{1}{2}\ln\frac{1+x}{1-x}\) | Built-in or \(\text{arctanh}(x)\) |
|
||||
| \(\gamma_E\) | Euler-Mascheroni constant | 0.57721566490153286 |
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. A. Sirlin, Phys. Rev. 164, 1767 (1967) — Universal correction function \(g(E, E_m, m)\) and proof of model independence.
|
||||
2. A. Sirlin and R. Zucchini, Phys. Rev. Lett. 57, 1994 (1986) — Analytic \(O(Z\alpha^2)\) calculation resolving CVC discrepancy.
|
||||
3. A. Sirlin, Phys. Rev. D 35, 3420 (1987) — Refined finite-nuclear-size effects with realistic charge distributions.
|
||||
4. A. Sirlin, Phys. Rev. D 84, 014021 (2011) — Neutrino spectrum radiative correction and conversion formula.
|
||||
5. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959) — Mass singularity cancellation theorems.
|
||||
Reference in New Issue
Block a user