beta_shape_pnpi/docs/corrections/04-exchange.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: corrections/03-screening, corrections/08-atomic-overlap

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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 Hayen et al., Rev. Mod. Phys. 90, 015008 (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

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.