# Nuclear Shell Model for β Decay

## Overview

The Nuclear Shell Model (NSM) is the primary theoretical framework for computing nuclear matrix elements (NMEs) in β decay. For forbidden nonunique decays like ⁹⁹Tc, the NSM computes one-body transition densities (OBTDs) that weight the single-particle transitions contributing to the decay.

## Key Codes

| Code | Description | Valence Spaces |
|---|---|---|
| NUSHELLX | Standard NSM code | Multiple (pf, sdg, etc.) |
| KSHELL | High-performance NSM | Custom valence spaces |

## Valence Spaces for ⁹⁹Tc

### glekpn (Mach interaction)
- **Protons**: 0f₇/₂, 0f₅/₂, 2p₃/₂, 2p₁/₂, 1g₉/₂
- **Neutrons**: 0g₉/₂, 0g₇/₂, 1d₅/₂, 1d₃/₂, 2s₁/₂

### jj45pnb (Brown-Lisetskiy interaction)
- **Protons**: 0f₅/₂, 2p₃/₂, 2p₁/₂, 1g₉/₂
- **Neutrons**: 0g₇/₂, 1d₅/₂, 1d₃/₂, 2s₁/₂, 0h₁₁/₂

## OBTDs for ⁹⁹Tc

| Transition | OBTD (glekpn) | OBTD (jj45pnb) |
|---|---|---|
| 1g₇/₂ → 1g₉/₂ | 0.00994 | 0.01107 |
| 2d₅/₂ → 1g₉/₂ | 0.47752 | −0.43985 |

**Key insight**: The dominant transition is 2d₅/₂ → 1g₉/₂ with OBTD ≈ 0.478. The jj45pnb model gives opposite sign for this transition.

## sNME and CVC

The small relativistic vector NME (sNME) is related to the large vector NME via the CVC hypothesis:

```
sNME = (R/√10) · [W₀ − (m_n − m_p) + E_C] · l-NME
```

where E_C is the Coulomb displacement energy. For ⁹⁹Tc, [W₀ − (m_n − m_p)] = −0.487 MeV (small Q-value makes this critical).

## References

- [[Shape Factor C(W, g_A)]]
- [[SSM — Spectrum Shape Method]]
- [[Ramalho & Suhonen (2024)]]
- [[Song et al. (2025)]]