89 lines
3.8 KiB
Markdown
89 lines
3.8 KiB
Markdown
---
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title: "05 – Exchange Correction"
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date: 2026-04-26
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tags:
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- beta-spectrum/correction/exchange
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status: active
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aliases: [Atomic Exchange, X(Z,W), Hayen-2018]
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cssclasses: []
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related_components: [[04-screening-correction]], [[12-atomic-overlap]]
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---
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# Atomic Electron Exchange Correction $X(Z,W)$
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## Physical Origin
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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.
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> [!key] Magnitude
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> 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.
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## Theory: Full Potential Approach
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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:
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- Exchange needs integration over entire spatial domain
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- Hydrogenic approach with simple screening is insufficient for high precision
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- Must use realistic atomic potentials (Dirac-Hartree-Fock-Slater)
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### Three Potential Models Compared (Hayen Fig. 9)
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| Model | Description | Accuracy |
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|---|---|---|
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| **Unoptimized** | Basic Hartree-Fock-Slater, no exchange optimization | Lower bound |
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| **Salvat** | Parametrized atomic potential | Intermediate |
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| **Optimized** | Full potential with exchange optimization included | Target ($10^{-4}$) |
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Screening strength is adjusted for best agreement with bound state energies (decreasing importance for higher $n$).
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## Analytical Parametrisation — Hayen et al. 2018
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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 $z = 120$.
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### Shell Structure Effects
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> [!tip] Figure 9 observation
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> 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.
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## Calculator Implementation
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> [!tip] Code mapping
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> Module: `beta_spectrum/components/exchange.py` → `ExchangeCorrection(Z)` class
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>
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> Loads empirical fit coefficients from `data/exchange_coeff.csv` (119 rows, $Z = 2$–$120$). Uses interpolation between tabulated values.
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### Data File
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```
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beta_spectrum/data/exchange_coeff.csv
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├── Row count: 119 (Z=2 to Z=120)
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├── Columns: Z, fit coefficients a₀...aₙ for energy parametrisation
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└── Source: Hayen et al., Phys. Rev. D 97, 013006 (2018), Table X
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```
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### API Usage
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```python
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from beta_spectrum import ExchangeCorrection
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exch = ExchangeCorrection(Z=90) # Z of parent atom
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values = exch(W_grid) # returns correction factor at each W
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```
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## Energy Dependence
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- **Dominant at very low energies** (T < ~10 keV): where continuum electron overlaps with atomic orbitals
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- Decreases rapidly above ~50 keV (continuum wave function too short-ranged for significant overlap)
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- For medium/high Z nuclei, shake-off probability in $s_{1/2}$ or $p_{1/2}$ state is < 0.1% — negligible
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## Shake-Up Influence on Exchange
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In low-Z nuclei (e.g., tritium):
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- Shake-up probabilities can be significant (~25%)
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- Exchange with higher-lying ns orbitals must be included even if unoccupied in initial state
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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.
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## Code Status: ✓ Implemented
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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.
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