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docs/05-exchange-correction.md

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+---
+title: "05 – Exchange Correction"
+date: 2026-04-26
+tags:
+  - beta-spectrum/correction/exchange
+status: active
+aliases: [Atomic Exchange, X(Z,W), Hayen-2018]
+cssclasses: []
+related_components: [[04-screening-correction]], [[12-atomic-overlap]]
+---
+
+# Atomic Electron Exchange Correction $X(Z,W)$
+
+## Physical Origin
+
+When a β electron is emitted from the nucleus, it can exchange with bound atomic electrons due to indistinguishability (Pauli exclusion principle). This **electron exchange** effect modifies the β spectrum at low energies where the continuum electron's wave function overlaps significantly with atomic orbitals.
+
+> [!key] Magnitude
+> The exchange contribution can exceed **20% in low-energy regions for higher Z** nuclei — it is one of the largest atomic corrections and absolutely critical for precision work below ~10 keV.
+
+## Theory: Full Potential Approach
+
+Exchange requires radial wave functions of both continuum and bound states integrated over **all space** (not just near origin). This is computationally demanding for analytical description because:
+- Exchange needs integration over entire spatial domain
+- Hydrogenic approach with simple screening is insufficient for high precision
+- Must use realistic atomic potentials (Dirac-Hartree-Fock-Slater)
+
+### Three Potential Models Compared (Hayen Fig. 9)
+
+| Model | Description | Accuracy |
+|---|---|---|
+| **Unoptimized** | Basic Hartree-Fock-Slater, no exchange optimization | Lower bound |
+| **Salvat** | Parametrized atomic potential | Intermediate |
+| **Optimized** | Full potential with exchange optimization included | Target ($10^{-4}$) |
+
+Screening strength is adjusted for best agreement with bound state energies (decreasing importance for higher $n$).
+
+## Analytical Parametrisation — Hayen et al. 2018
+
+The exchange correction $X(W)$ as a function of kinetic energy uses empirical fit coefficients from Table X of [[refs/hayen-2018]]. Coefficients are tabulated for all atoms with $Z = 2$ through $z = 120$.
+
+### Shell Structure Effects
+
+> [!tip] Figure 9 observation
+> Clear shell closure effects observed in orbital exchange contributions: at noble gas Z values, the exchange correction shows characteristic kinks. These must be captured by interpolation between tabulated Z values.
+
+## Calculator Implementation
+
+> [!tip] Code mapping
+> Module: `beta_spectrum/components/exchange.py` → `ExchangeCorrection(Z)` class
+> 
+> Loads empirical fit coefficients from `data/exchange_coeff.csv` (119 rows, $Z = 2$–$120$). Uses interpolation between tabulated values.
+
+### Data File
+
+```
+beta_spectrum/data/exchange_coeff.csv
+├── Row count: 119 (Z=2 to Z=120)
+├── Columns: Z, fit coefficients a₀...aₙ for energy parametrisation
+└── Source: Hayen et al., Phys. Rev. D 97, 013006 (2018), Table X
+```
+
+### API Usage
+
+```python
+from beta_spectrum import ExchangeCorrection
+
+exch = ExchangeCorrection(Z=90)  # Z of parent atom
+values = exch(W_grid)           # returns correction factor at each W
+```
+
+## Energy Dependence
+
+- **Dominant at very low energies** (T < ~10 keV): where continuum electron overlaps with atomic orbitals
+- Decreases rapidly above ~50 keV (continuum wave function too short-ranged for significant overlap)
+- For medium/high Z nuclei, shake-off probability in $s_{1/2}$ or $p_{1/2}$ state is < 0.1% — negligible
+
+## Shake-Up Influence on Exchange
+
+In low-Z nuclei (e.g., tritium):
+- Shake-up probabilities can be significant (~25%)
+- Exchange with higher-lying ns orbitals must be included even if unoccupied in initial state
+
+For most practical cases, inclusion of shake-up changes the exchange correction at the few $10^{-4}$ level down to 0.5 keV for tritium only — otherwise negligible.
+
+## Code Status: ✓ Implemented
+
+The `ExchangeCorrection` class is fully implemented and tested. The CSV data file covers the full range from light (He, Z=2) to transuranic (U, Z=92+) elements.