unzhakov
876e1e3ab8
- Add neutrino mass parameter to SpectrumConfig and CLI interface - Update PhaseSpace component to handle neutrino mass effects properly - Update CSV headers to show basic endpoint and all transition types - Add --csv-verbosity option to CLI with 'full' and 'concise' options - Fix comparison plots to properly show linear scale variations - Add comprehensive tests for neutrino mass functionality
54 lines
2.1 KiB
Markdown
54 lines
2.1 KiB
Markdown
# Shape Factor C(W, g_A)
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## Overview
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The g_A-dependent nuclear shape factor C(W, g_A) is the key nuclear structure quantity that encodes the energy-dependent modification of the β decay spectrum due to forbidden nonunique transitions. For ⁹⁹Tc (second-forbidden nonunique, 9/2₁⁺ → 5/2₁⁺, Q ≈ 296 keV), the shape factor decomposes as:
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```
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C(W; g_A) = g_V² · C_V(W) + g_A² · C_A(W) + g_V · g_A · C_VA(W)
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```
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with g_V = 1.0 fixed by CVC.
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## Components
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| Component | Description | Sign for ⁹⁹Tc | Magnitude |
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|---|---|---|---|
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| C_V(W) | Vector contribution | + | 2.240 × 10⁻⁹ (integrated) |
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| C_A(W) | Axial-vector contribution | + | 2.130 × 10⁻⁹ (integrated) |
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| C_VA(W) | Vector-axial interference | − | −4.361 × 10⁻⁹ (integrated) |
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**Key insight**: C_VA is negative and twice as large as C_V or C_A. The total integrated shape factor is two orders of magnitude smaller due to destructive interference.
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## g_A Dependence
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The spectral shape is most sensitive to g_A in the **0–100 keV range** (low-energy region). Figure 9 from Paulsen et al. (2024) shows:
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- **Small g_A (0.5–1.0)**: Bell-shaped spectrum with low-energy enhancement
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- **Large g_A (1.4–2.0)**: Flattened spectrum with low-energy suppression
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- **Crossover point**: All spectra intersect around 100 keV
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## Table Data Sources
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| Source | Method | Coverage | Availability |
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|---|---|---|---|
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| Kostensalo & Suhonen (2017) PRC 96, 024317 | MQPM | 26 forbidden decays | Supplementary material (not public) |
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| Ramalho & Suhonen (2024) | NSM (NUSHELLX) | ⁹⁹Tc-specific | Supplementary material (not public) |
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| Song et al. (2025) arXiv:2506.17544 | ESSM + NSM/MQPM | ⁹⁹Tc | Integrated values only |
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## Implementation
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The ShapeFactorGA component should:
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1. Load C_V(W), C_A(W), C_VA(W) tables from CSV files
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2. Interpolate using cubic spline (scipy.interpolate.CubicSpline)
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3. Compute C(W; g_A) = C_V + g_A²·C_A + g_A·C_VA
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4. Return the shape factor for the full spectrum calculation
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## References
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- [[SSM — Spectrum Shape Method]]
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- [[Nuclear Shell Model]]
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- [[CVC Hypothesis]]
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- [[Paulsen et al. (2024)]]
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- [[Song et al. (2025)]]
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