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)
2026-05-14 19:46:39 +03:00
parent efcd6a524a
commit 030ae3e4b9
29 changed files with 696 additions and 771 deletions
@@ -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 \;
 | Symbol | Name | Module / Note | Status |
 |---|---|---|---|
-| $F_0(Z,W)$ | Point charge Fermi function | [[01-fermi-function]] | ✓ |
+| $F_0(Z,W)$ | Point charge Fermi function | [[corrections/01-fermi-function]] | ✓ |
-| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[03-finite-size]] | ✓ |
+| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[corrections/02-finite-size]] | ✓ |
-| $U(Z,W)$ | Charge distribution correction | [[03-finite-size]] | ✓ |
+| $U(Z,W)$ | Charge distribution correction | [[corrections/02-finite-size]] | ✓ |
-| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[03-finite-size]] | ✓ |
+| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[corrections/02-finite-size]] | ✓ |
-| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[06-radiative-corrections]] | ✓ |
+| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[corrections/05-radiative]] | ✓ |
-| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[07-recoil-effects]] | ✗ |
+| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[corrections/06-recoil]] | ✗ |
-| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[07-recoil-effects]] | — |
+| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[corrections/06-recoil]] | — |
-| $S(Z,W)$ | Atomic electron screening | [[04-screening-correction]] | ✓ |
+| $S(Z,W)$ | Atomic electron screening | [[corrections/03-screening]] | ✓ |
-| $X(Z,W)$ | Atomic exchange effect | [[05-exchange-correction]] | ✓ |
+| $X(Z,W)$ | Atomic exchange effect | [[corrections/04-exchange]] | ✓ |
-| $r(Z,W)$ | Atomic mismatch / overlap | [[12-atomic-overlap]] | — |
+| $r(Z,W)$ | Atomic mismatch / overlap | [[corrections/08-atomic-overlap]] | — |
-| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[12-atomic-overlap]] | — |
+| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[corrections/08-atomic-overlap]] | — |
-| $C(Z,W)$ | Nuclear structure shape factor | [[10-nuclear-structure]] | — |
+| $C(Z,W)$ | Nuclear structure shape factor | [[nuclear-structure/01-shape-factors]] | — |
-| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[10-nuclear-structure]] | — |
+| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[nuclear-structure/01-shape-factors]] | — |
 ## Baseline: Phase Space
@@ -49,7 +49,7 @@ with $p = \sqrt{W^2-1}$, $W$ total energy in units of $m_e$, and $W_0$ the endpo
 ## Precision Goal
-The paper achieves **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude [[Hayen2017_summary]].
+The Hayen paper achieves **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]].
 > [!key] Why this matters
 > A precision of ~0.5% is required when determining the Fierz term $b_{\text{Fierz}}$ to improve limits on BSM scalar/tensor couplings. Weak magnetism $b_{wm}$ must be included as experimental precision increases, otherwise it limits BSM sensitivity.
@@ -6,7 +6,7 @@ tags:
 status: active
 aliases: [Phase Space, Baseline Spectrum]
 cssclasses: []
-related_components: [[01-fermi-function]], [[07-recoil-effects]]
+related_components: [[corrections/01-fermi-function]], [[corrections/06-recoil]]
 ---
 # Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$
@@ -41,7 +41,7 @@ $$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for
 Deviations from linearity near the endpoint indicate:
 - Non-zero neutrino mass
- Weak magnetism / recoil effects ([[07-recoil-effects]])
+- Weak magnetism / recoil effects ([[corrections/06-recoil]])
 - New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]])
 ## Calculator Implementation
@@ -62,7 +62,7 @@ The `PhaseSpace` calculator accepts a `transition_type` parameter that determine
 | `F2U`, `F3` | 2, 3 | $p_\nu^4 + \frac{3}{10}p_\nu^2 p_e^2 + p_e^4$ |
 | `F3U`, `F4` | 3, 4 | $p_\nu^6 + 7p_\nu^4 p_e^2 + 7p_\nu^2 p_e^4 + p_e^6$ |
-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.
+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.
 ## Energy Range
@@ -1,325 +0,0 @@
 ---
 title: "14 – gA-Driven Shape Factor Parameterization"
 date: 2026-05-13
 tags:
  - beta-spectrum/shape-factor
  - beta-spectrum/gA
  - beta-spectrum/forbidden
 status: active
 aliases: [Spectrum Shape Method, SSM, gA quenching, shape factor parameterization]
 related_components: [[10-nuclear-structure]], [[07-recoil-effects]]
 ---
 # gA-Driven Shape Factor Parameterization — The Spectrum Shape Method (SSM)
 ## Overview
 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).
 > [!tip] Why this matters for 99Tc
 > 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 $g_V^{\text{eff}} = 0.376(5)$, $g_A^{\text{eff}} = 0.574(36)$.
 ## 1. Theoretical Foundation
 ### 1.1 Shape Factor Decomposition
 The shape factor $C(w_e)$ decomposes into vector, axial-vector, and mixed parts:
 $$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
 where $w_e = W$ 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**.
 After integration over the spectrum:
 $$\tilde{C} = g_V^2\, \tilde{C}_V + g_A^2\, \tilde{C}_A + g_V g_A\, \tilde{C}_{VA} \tag{2}$$
 The integrated components $\tilde{C}_i$ are energy-independent constants.
 ### 1.2 The β Spectrum Equation
 The full differential spectrum (Paulsen et al., 2024, Eq. 5):
 $$N(W)\,dW = \frac{G_F^2 \cos^2\theta_C}{2\pi^3}\, F(Z,W)\, p W (W_0 - W)^2\, X(W)\, C(W)\, \rho(Z,W)\, dW$$
 where:
 - $F(Z,W)$ — Fermi function (Coulomb correction)
 - $X(W)$ — screening and exchange corrections
 - $C(W)$ — **shape factor** (nuclear structure)
 - $\rho(Z,W)$ — atomic overlap correction
 - $G_F \cos\theta_C$ — effective Fermi constant
 ### 1.3 Traditional Parametrization (Ad-Hoc)
 Historically, the shape factor for $^{99}\text{Tc}$ has been approximated as **first-forbidden unique** (Reich & Schüpferling, 1974; standard reference [15]):
 $$C_{\text{adhoc}}(W) = q^2 + \lambda p^2 = (W_0 - W)^2 + \lambda (W^2 - 1) \tag{3}$$
 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:
 - Cannot distinguish $g_V$ from $g_A$ effects
 - Provides no insight into nuclear structure
 - Fails below $\sim$100 keV when compared to high-precision data
 ### 1.4 SSM Parametrization (Theory-Driven)
 The SSM replaces Eq. (3) with the **microscopically computed** shape factor:
 $$C_{\text{SSM}}(W; g_A) = g_V^2\, C_V(W) + g_A^2\, C_A(W) + g_V g_A\, C_{VA}(W) \tag{4}$$
 where $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ are computed from shell-model (or MQPM) one-body transition densities (OBTDs).
 ## 2. Algorithm for Parameterization
 ### 2.1 Input Requirements
 | Quantity | Source | Notes |
 |---|---|---|
 | $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ | Nuclear shell model (NuShellX@MSU) or MQPM | Computed for the specific transition |
 | $W_0$ (endpoint) | Independent measurement or fit | Paulsen et al.: $W_0 = 295.82(16)$ keV |
 | Experimental spectrum $N_{\text{exp}}(W)$ | Calorimetry / semiconductor / MMC | Corrected for detector response, background, energy losses |
 | $F(Z,W)$, $X(W)$, $\rho(Z,W)$ | Calculator modules | Already implemented in project |
 | $g_V$ | Fixed (CVC: $g_V = 1.0$, or treated as free) | Paulsen et al. used free $g_V$ fit |
 ### 2.2 Computation of Shape Factor Components
 The shape factor in the **Behrens-Bühring** formalism (truncated lepton current approximation, Paulsen et al. Eq. 10):
 $$C(W) = \sum_{K,k_e,k_\nu} \left[ e_k^2 W k_e\, m_K(k_e,k_\nu) - 2\mu e_k W k_e\, \gamma\, m_K(k_e,k_\nu) \right]$$
 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.
 **Practical computation via shell model:**
 1. **OBTD calculation** (NuShellX):
   - Compute one-body transition densities for the transition
   - For $^{99}\text{Tc}$: dominant transition $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$
   - Sub-dominant: $1g_{7/2} \to 1g_{9/2}$ with OBTD $\approx 0.010$
 2. **Coulomb displacement energy** $E_C$ (for relativistic vector current):
   - Leading order: $E_C^{(3)} = \frac{\int_0^\infty g_f V(r) g_i (r/R)^K r^2 dr}{\int_0^\infty g_f g_i (r/R)^K r^2 dr}$
   - Energy-dependent: $E_C^{(4)}$ includes full lepton current dependence
   - For $^{99}\text{Tc}$: $E_C^{(4)} \approx 10.5$ MeV (GLEKPN valence space)
 3. **Vector current from CVC**:
   $$V_F^{221} = -\sqrt{\frac{R}{10}}\left[W_0 - (m_n - m_p) + E_C\right] V_F^{220}$$
   For $^{99}\text{Tc}$, $[W_0 - (m_n - m_p)] = -0.487$ MeV — small Q-value makes $E_C$ critical.
 ### 2.3 Fitting Procedure
 ```
 Step 1: Precompute shape factor components C_i(W) from nuclear model
        → Store as lookup tables: C_V(W), C_A(W), C_VA(W) for W ∈ [1, W0]
 Step 2: For trial (gA, gV) pair:
        a. Compute C(W; gA, gV) = gV²·C_V(W) + gA²·C_A(W) + gV·gA·C_VA(W)
        b. Compute full spectrum:
           N_theo(W) ∝ p·W·(W0−W)²·F(Z,W)·X(W)·C(W; gA, gV)·ρ(Z,W)
        c. Normalize: ∫ N_theo(W) dW = 1
 Step 3: Compare with experimental spectrum:
        χ²(gA, gV) = Σ [N_exp(W_i) − N_theo(W_i; gA, gV)]² / σ_i²
 Step 4: Minimize χ² over (gA, gV) grid
        → Extract best-fit values and confidence intervals
 ```
 ### 2.4 Energy Range Selection
 Paulsen et al. (2024) recommend fitting in the range **20–275 keV** to avoid:
 - **Below ~20 keV**: detector threshold effects, atomic exchange modeling uncertainties
 - **Above ~275 keV**: endpoint region with poor statistics, resolution distortion
 The χ² landscape is approximately quadratic, enabling simple uncertainty estimation at χ² = χ²_min + 1.
 ## 3. Results from Literature
 ### 3.1 $^{99}\text{Tc}$ — Paulsen et al. (2024)
 From the MMC measurement (PTB/LNHB), the extracted values are:
 | Quantity | Value | Method |
 |---|---|---|
 | $Q_\beta$ | 295.82(16) keV | Fit to spectrum shape |
 | $g_V^{\text{eff}}$ | 0.376(5) | From half-life normalization |
 | $g_A^{\text{eff}}$ | 0.574(36) | From spectrum shape fit |
 | $\bar{E}_\beta$ | 98.51(23) keV | From theoretical spectrum |
 | $\log f$ | −0.476 60(22) | From phase space integration |
 | $\log ft$ | 12.3478(23) | Combined |
 **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).
 ### 3.2 $^{87}\text{Rb}$ — Kostensalo et al. (2017)
 Third-forbidden nonunique ($3/2^- \to 9/2^+$):
 - Strong $g_A$ dependence in spectrum shape
 - MQPM and NSM agree on shape evolution
 - $g_A \approx 0.9$ gives best match to available data
 - Best candidate for $g_A$ determination in third-forbidden decays
 ### 3.3 $^{113}\text{Cd}$ and $^{115}\text{In}$ — Kostensalo et al. (2017, 2017a)
 Fourth-forbidden nonunique:
 - Most dramatic $g_A$ dependence: bell-shaped spectrum at $g_A \approx g_V$, monotonic otherwise
 - Consistent $g_A \approx 0.9$ extracted across three nuclear models (MQPM, NSM, IBFM-2)
 - Demonstrates **robustness of SSM** against nuclear model details
 ### 3.4 Summary of $g_A$ Sensitivity by Forbiddenness
 | Transition | Forbiddenness | $g_A$ Sensitivity | SSM Viability |
 |---|---|---|---|
 | $^{99}\text{Tc}$ | 2nd nonunique | **Strong** | ✅ Excellent |
 | $^{94}\text{Nb}$, $^{98}\text{Tc}$ | 2nd nonunique | **Strong** | ✅ Good |
 | $^{87}\text{Rb}$ | 3rd nonunique | **Strong** | ✅ Good |
 | $^{113}\text{Cd}$, $^{115}\text{In}$ | 4th nonunique | **Very strong** | ✅ Excellent |
 | $^{137}\text{Cs}$ (to $11/2^-$) | 1st unique | Weak (NLO only) | ⚠️ Limited |
 | $^{137}\text{Cs}$ (to $3/2^+$) | 2nd nonunique | **None** (cancellation) | ❌ Not applicable |
 | First-forbidden nonunique | 1st nonunique | **Weak** | ❌ Not useful |
 **Systematic pattern**: Even-forbidden decays show strong $g_A$ dependence; odd-forbidden decays are mostly insensitive.
 ## 4. Implementation Plan
 ### 4.1 New Module: `components/shape_factor_gA.py`
 ```
 ShapeFactorGA
 ├── __init__: load C_V(W), C_A(W), C_VA(W) from precomputed tables or nuclear model
 ├── evaluate(W, gA, gV=1.0): compute C(W; gA, gV)
 ├── evaluate_components(W): return (C_V, C_A, C_VA) separately
 ├── chi2_spectrum(gA, gV, W_grid, N_exp, sigma): χ² for spectrum fit
 ├── fit_gA(W_grid, N_exp, sigma, W_min, W_max): minimize χ², return (gA_best, gA_err)
 ├── fit_gA_gV(W_grid, N_exp, sigma, W_min, W_max): fit both, return (gA, gV, cov)
 └── mean_energy(gA, gV, W0): compute ⟨E⟩ from theoretical spectrum
 ```
 ### 4.2 Integration with BetaSpectrum
 The shape factor becomes a new `SpectrumComponent`:
 ```python
 class ShapeFactorGA(SpectrumComponent):
    """gA-dependent shape factor for forbidden nonunique decays."""
    def __init__(self, C_V_table, C_A_table, C_VA_table, gA=1.27, gV=1.0):
        # Load precomputed shape factor components
        # C_i tables: W array → C_i(W) values
    def evaluate(self, W):
        return gV**2 * self.C_V(W) + gA**2 * self.C_A(W) + gV*gA * self.C_VA(W)
    def integrate(self, W0):
        """Compute integrated shape factor for half-life prediction."""
        return integrate(C(W) * p * W * (W0-W)**2 * F(Z,W) dW, 1, W0)
 ```
 ### 4.3 Precomputed Tables from Nuclear Model
 For $^{99}\text{Tc}$, the shape factor components should be precomputed from NuShellX@MSU (GLEKPN valence space, Mach interaction):
 | W (keV) | $C_V(W)$ | $C_A(W)$ | $C_{VA}(W)$ |
 |---|---|---|---|
 | 1.001 | ... | ... | ... |
 | ... | ... | ... | ... |
 | 1.581 | ... | ... | ... |
 (Endpoint: $W_0 = 1 + 295.82/510.998950 \approx 1.579$)
 ### 4.4 Fitting Pipeline
 ```python
 def fit_gA_for_Tc95(data_file, W_min=20*keV, W_max=275*keV):
    """Fit gA to 99Tc beta spectrum data."""
    W, N_exp, sigma = load_spectrum(data_file)
    # Load precomputed shape factor components
    sf = ShapeFactorGA("Tc95_CV_CA_CVA_tables.npy")
    # Grid search + refinement
    best_chi2 = inf
    for gA in np.linspace(0.4, 1.8, 141):
        chi2 = sf.chi2_spectrum(gA=gA, gV=1.0, W=W, N_exp=N_exp, sigma=sigma,
                                W_min=W_min, W_max=W_max)
        if chi2 < best_chi2:
            best_chi2 = chi2
            best_gA = gA
    # Refine around minimum
    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
@@ -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.
+```
-
+docs/
-### Core Theory
+├── README.md                          ← You are here
-
+├── 00-beta-spectrum-overview.md       ← Master equation & full correction list
- [[00-beta-spectrum-overview]] — Master equation and overview of all corrections
+├── 02-phase-space.md                  ← Baseline spectral shape
- [[02-phase-space]] — Baseline spectral shape $pW(W_0-W)^2$
+├── corrections/                       ← All correction components
-
+│   ├── 01-fermi-function.md           ← Fermi function F₀(Z,W)
-### Correction Components (mapped to calculator modules)
+│   ├── 02-finite-size.md              ← Nuclear size: L₀, U, DFS
-
+│   ├── 03-screening.md                ← Atomic screening S(Z,W)
-| # | Note | Calculator Module | Status |
+│   ├── 04-exchange.md                 ← Atomic exchange X(Z,W)
-|---|---|---|---|
+│   ├── 05-radiative.md                ← Outer radiative R(W,W₀)
-| 1 | [[01-fermi-function]] | `components/fermi.py` | ✓ implemented |
+│   ├── 06-recoil.md                   ← Recoil: R_N, Q, weak magnetism
-| 2 | [[03-finite-size]] | `components/finite_size.py` | ✓ implemented |
+│   ├── 07-detector-response.md        ← Detector convolution
-| 3 | [[04-screening-correction]] | `components/screening.py` | ✓ implemented |
+│   ├── 08-atomic-overlap.md           ← Atomic mismatch r(Z,W)
-| 4 | [[05-exchange-correction]] | `components/exchange.py` | ✓ implemented |
+│   └── 09-chemical-effects.md         ← Molecular environment
-| 5 | [[06-radiative-corrections]] | `components/radiative.py` | ✓ implemented |
+├── nuclear-structure/                 ← Nuclear theory
-| 6 | [[07-recoil-effects]] | `components/recoil.py` | ✗ not yet implemented |
+│   ├── index.md                       ← Overview
-
+│   └── 01-shape-factors.md            ← C(Z,W), NMEs, impulse approximation
-### Nuclear Structure & Atomic Effects
+├── gA-extraction/                     ← gA extraction (new, in development)
-
+│   ├── 01-ssm-algorithm.md            ← SSM step-by-step algorithm
- [[10-nuclear-structure]] — Shape factors, form factors, impulse approximation
+│   ├── 02-ssnme.md                    ← sNME and ESSM formalism
- [[11-isospin-breakdown]] — Shake-off/shake-up processes, endpoint shift
+│   ├── 03-tc99-case-study.md          ← 99Tc literature results
- [[12-atomic-overlap]] — Bahcall correction, bound-state β decay
+│   └── 04-implementation-todo.md      ← Implementation TODO list
- [[13-chemical-effects]] — Molecular environment influence on spectrum
+└── refs/                              ← Paper summaries
-
+    ├── electromagnetic/               ← CVC, radiative corrections
-### References
+    ├── forbidden-decay/               ← Forbidden decay studies
-
+    └── (organized by topic)
- [[refs/hayen2017]] — Metadata and key equations from Hayen et al. (2017)
+```
 ## Calculator Architecture
 ```
 SpectrumComponent (ABC, base.py)
- ├── PhaseSpace              → phase_space.py
+ ├── PhaseSpace              → 02-phase-space.md
- ├── FermiFunction           → fermi.py
+ ├── FermiFunction           → corrections/01-fermi-function.md
- ├── FiniteSizeL0            → finite_size.py
+ ├── FiniteSizeL0            → corrections/02-finite-size.md
- ├── ChargeDistributionU     → finite_size.py
+ ├── ChargeDistributionU     → corrections/02-finite-size.md
- ├── ScreeningCorrection     → screening.py
+ ├── ScreeningCorrection     → corrections/03-screening.md
- ├── ExchangeCorrection      → exchange.py
+ ├── ExchangeCorrection      → corrections/04-exchange.md
- └── RadiativeCorrection     → radiative.py
+ ├── 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$)
+- [[corrections/02-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$)
- [[06-radiative-corrections]] adds QED radiative effects ($R = 1 + \delta_R$)
+- [[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
+- [[corrections/08-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction
- [[13-chemical-effects]] — Molecular environment influence on screening and exchange
+- [[corrections/09-chemical-effects]] — Molecular environment influence on screening and exchange
- [[05-exchange-correction]] — Electron exchange with atomic orbitals (often combined with screening)
+- [[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
+- [[corrections/03-screening]] — Screening modification from shake-up processes
- [[11-isospin-breakdown]] — Shake-off process correction
+-  — Shake-off process correction
- [[13-chemical-effects]] — Molecular environment influence on atomic overlap
+- [[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
+- [[corrections/08-atomic-overlap]] — Atomic mismatch / Bahcall correction
- [[04-screening-correction]] — Baseline atomic screening $S(Z,W)$
+- [[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.