# 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:
1. Load C_V(W), C_A(W), C_VA(W) tables from CSV files
2. Interpolate using cubic spline (scipy.interpolate.CubicSpline)
3. Compute C(W; g_A) = C_V + g_A²·C_A + g_A·C_VA
4. Return the shape factor for the full spectrum calculation

## References

- [[SSM — Spectrum Shape Method]]
- [[Nuclear Shell Model]]
- [[CVC Hypothesis]]
- [[Paulsen et al. (2024)]]
- [[Song et al. (2025)]]