unzhakov
030ae3e4b9
- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
173 lines
8.9 KiB
Markdown
173 lines
8.9 KiB
Markdown
---
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title: "SSM — Spectrum Shape Method for gA Extraction"
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date: 2026-05-14
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tags:
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- gA/extraction/algorithm
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- gA/ssm
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- beta-spectrum/forbidden
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status: active
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aliases: [Spectrum Shape Method, gA fitting, shape factor fit]
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related_notes:
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- gA-extraction/02-ssnme
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- gA-extraction/03-tc99-case-study
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- gA-extraction/04-implementation-todo
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- nuclear-structure/01-shape-factors
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- corrections/05-radiative
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---
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# Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction
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## 1. Physical Basis
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For **forbidden nonunique β decays**, the shape factor $C(w_e)$ decomposes into three energy-dependent components with distinct coupling-constant dependence:
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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 $w_e$ 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.
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> [!note] Why nonunique decays?
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> 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.
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## 2. The Full Spectrum Model
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The differential spectrum (Paulsen et al., 2024, Eq. 5):
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$$N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)$$
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where:
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- $\mathcal{N}$ — overall normalization (absorbs $G_F^2 \cos^2\theta_C / 2\pi^3$)
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- $F(Z,W)$ — Fermi function (Coulomb correction) — [[corrections/01-fermi-function]]
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- $p W (W_0 - W)^2$ — phase space — [[02-phase-space]]
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- $X(W)$ — atomic exchange correction — [[corrections/04-exchange]]
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- $S(Z,W)$ — atomic screening — [[corrections/03-screening]]
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- $R(W)$ — outer radiative correction — [[corrections/05-radiative]]
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- $C(W; g_A, g_V)$ — **shape factor** — [[nuclear-structure/01-shape-factors]]
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- $\rho(Z,W)$ — atomic overlap correction — [[corrections/08-atomic-overlap]]
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## 3. The Algorithm
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### Step 1: Compute shape factor components from nuclear model
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**Input:** Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.
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**Output:** Three energy-dependent lookup tables: $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $W \in [1, W_0]$.
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**Method:**
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1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
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2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
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3. Sum over all contributing multipole orders $K$ and lepton quantum numbers $k_e, k_\nu$
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**CVC constraint for vector current:**
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The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:
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$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}$$
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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).
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**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.
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### Step 2: Build theoretical spectrum for trial $g_A$
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For each trial value of $g_A$ (with $g_V = 1.0$ fixed by CVC):
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1. **Compute shape factor:**
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$$C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)$$
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(using $g_V = 1.0$)
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2. **Compute full spectrum:**
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$$N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)$$
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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)
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### Step 3: Compare with experimental spectrum
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**Input:** Experimental spectrum data $(W_i, N_i, \sigma_i)$ for $i = 1, \dots, N_{\text{bins}}$.
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**χ² function:**
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$$\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}$$
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**Fit range selection (Paulsen et al., 2024):**
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- **Lower bound:** ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
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- **Upper bound:** ~275 keV for $^{99}\text{Tc}$ — above this, endpoint region with poor statistics and resolution distortion
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- Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range
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### Step 4: Minimize χ² and extract $g_A$
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**Grid search:** Evaluate $\chi^2(g_A)$ on a grid (e.g., $g_A \in [0.4, 1.8]$ with step 0.01).
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**Refinement:** Use `scipy.optimize.minimize_scalar` (Brent's method) around the grid minimum.
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**Uncertainty:** Find $g_A$ values where $\chi^2 = \chi^2_{\text{min}} + 1$. These define the 68% confidence interval.
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**Output:** $(g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})$
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### Step 5 (optional): Simultaneous $g_A$ and $g_V$ fit
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If $g_V$ is not fixed to 1.0 (e.g., testing CVC violation):
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$$\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}$$
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Fit both parameters simultaneously. The correlation between $g_A$ and $g_V$ is typically strong.
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### Step 6 (optional): Half-life normalization — quenched $g_A$
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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:
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1. Compute the theoretical partial half-life for the best-fit $(g_A, g_V)$:
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$$t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW$$
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2. Compare with experimental partial half-life:
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$$t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}$$
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3. Extract quenched values:
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$$g_i^{\text{quenched}} = g_i^{\text{shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
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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 $\approx 0.38$.
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## 4. The sNME Problem and ESSM
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### The Problem
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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.
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### The Solution: Enhanced SSM (ESSM)
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Kumar et al. (2021) proposed treating the sNME as a **fitting parameter** alongside $g_A$. This is the **Enhanced Spectrum Shape Method (ESSM)**:
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1. Fix $g_A$ at a trial value
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2. Vary the sNME to simultaneously fit the **experimental half-life** (not just the shape)
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3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each $g_A$ value:
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- **sNME(c):** closer to the CVC value
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- **sNME(f):** further from the CVC value
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4. Compare both solutions' spectral shapes with the data
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The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See [[gA-extraction/02-ssnme]] for details.
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## 5. Uncertainty Budget
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| Component | Uncertainty on $g_A$ | Relative | Dominant for |
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| Fit method (statistics) | 0.0745 | 4.9% | All SSM analyses |
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| Coulomb displacement energy | 0.0425 | 2.8% | Small-$Q$ decays |
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| Nuclear model | 0.0294 | 1.9% | Model comparison |
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| Lepton current treatment | 0.0127 | 0.8% | High-$Z$ nuclei |
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| Fit range | 0.0061 | 0.4% | All |
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| Endpoint energy | 0.0038 | 0.2% | Precision $Q$-value needed |
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| Radiative corrections | 0.0033 | 0.2% | All |
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| Atomic exchange | 0.0029 | 0.2% | Low-energy region |
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| **Combined** | **0.0919** | **6.0%** | — |
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Source: Paulsen et al. (2024), Table VIII.
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## 6. Key References
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| Ref | Citation | Use |
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| [SSM-orig] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | SSM introduction; 26 decays |
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| [SSM-Tc94-98] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc |
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| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}$Tc MMC measurement; $g_A$ extraction (enhanced value) |
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| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc MMC measurement; quenched $g_A$ via ESSM |
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| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | Enhanced SSM formalism |
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| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME analysis; five second-forbidden decays |
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| [Hayen2017] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | BB formalism; shape factor theory |
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