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
2.1 KiB
2.1 KiB
Shape Factor C(W, g_A)
Overview
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:
C(W; g_A) = g_V² · C_V(W) + g_A² · C_A(W) + g_V · g_A · C_VA(W)
with g_V = 1.0 fixed by CVC.
Components
| Component | Description | Sign for ⁹⁹Tc | Magnitude |
|---|---|---|---|
| C_V(W) | Vector contribution | + | 2.240 × 10⁻⁹ (integrated) |
| C_A(W) | Axial-vector contribution | + | 2.130 × 10⁻⁹ (integrated) |
| C_VA(W) | Vector-axial interference | − | −4.361 × 10⁻⁹ (integrated) |
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.
g_A Dependence
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:
- Small g_A (0.5–1.0): Bell-shaped spectrum with low-energy enhancement
- Large g_A (1.4–2.0): Flattened spectrum with low-energy suppression
- Crossover point: All spectra intersect around 100 keV
Table Data Sources
| Source | Method | Coverage | Availability |
|---|---|---|---|
| Kostensalo & Suhonen (2017) PRC 96, 024317 | MQPM | 26 forbidden decays | Supplementary material (not public) |
| Ramalho & Suhonen (2024) | NSM (NUSHELLX) | ⁹⁹Tc-specific | Supplementary material (not public) |
| Song et al. (2025) arXiv:2506.17544 | ESSM + NSM/MQPM | ⁹⁹Tc | Integrated values only |
Implementation
The ShapeFactorGA component should:
- Load C_V(W), C_A(W), C_VA(W) tables from CSV files
- Interpolate using cubic spline (scipy.interpolate.CubicSpline)
- Compute C(W; g_A) = C_V + g_A²·C_A + g_A·C_VA
- Return the shape factor for the full spectrum calculation