94 lines
3.9 KiB
Markdown
94 lines
3.9 KiB
Markdown
---
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title: "04 – Screening Correction"
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date: 2026-04-26
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tags:
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- beta-spectrum/correction/screening
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status: active
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aliases: [Atomic Screening, S(Z,W)]
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cssclasses: []
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related_components: [[12-atomic-overlap]], [[13-chemical-effects]]
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---
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# Atomic Electron Screening Correction $S(Z,W)$
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## Physical Origin
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The Fermi function ([[01-fermi-function]]) assumes the β particle moves in the Coulomb field of a bare nucleus. In reality, atomic electrons partially screen this charge at large distances from the nucleus, modifying the effective potential experienced by the emitted electron/positron. This is particularly important at low kinetic energies where the β particle spends more time far from the nucleus.
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## The Screening Correction
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Screening is implemented as a ratio method:
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$$S(Z,W) = \frac{F_{\text{screened}}(Z, W)}{F_0(Z, W - V_0/m_ec^2)} \tag{144}$$
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where $V_0$ is the screening potential (binding energy of the outermost atomic electrons). The denominator evaluates the Fermi function at a shifted energy representing the unscreened case.
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### Screening Potential
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The screening potential $V_0$ depends on:
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- Atomic number $Z$ (parent atom)
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- Chemical/physical state (neutral atom, ionized molecule)
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- Electronic configuration
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For neutral atoms, $V_0$ is typically a few eV to tens of eV — small compared to nuclear potentials but significant at β energies below ~10 keV.
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### Molecular Screening Correction
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In molecular environments, the screening potential changes:
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$$\Delta S_{\text{Mol}} \quad (\text{Eq. 176})$$
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Since screening is typically at sub-percent level maximum, molecular deviations are expected at the **few $10^{-4}$** level — relevant for precision work down to 1 keV.
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## Energy Dependence
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- **Dominant at low energies** (T < ~50 keV): where the β particle's wave function is most affected by atomic electron cloud
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- **Negligible at high energies**: $S(Z,W) \to 1$ as $W \gg V_0$
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- The effect reverses sign between β⁻ and β⁺ (electrons attracted to/repelled from electron cloud differently)
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## Shake-Up Effects on Screening
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Atomic excitations during β decay modify screening:
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> [!warning] Tritium case
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> In tritium β⁻ decay, the screening potential can change by as much as **20%** because only a single electron contributes to atomic Coulomb interaction. For heavier systems, relative changes decrease but scale with $Z$. Neglecting shake-up introduces errors of order $\sim 1 \times 10^{-4}$.
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Shake-off (continuum ejection) has smaller effect:
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- With $\Delta Z_{\text{eff}} \approx 0.3$–$0.4$, yields ~**0.1%** relative change in screening potential → few $10^{-5}$ level in spectral shape
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- **Negligible at target precision** for most nuclei
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## Calculator Implementation
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> [!tip] Code mapping
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> Module: `beta_spectrum/components/screening.py` → `ScreeningCorrection(FermiFunction)` class
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>
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> Uses ratio method with smooth logistic switching function. The screening potential $V_0$ is passed as a parameter and depends on the parent atom's atomic number.
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### API Usage
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```python
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from beta_spectrum import ScreeningCorrection, FermiFunction
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screen = ScreeningCorrection(
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FermiFunction(Z_parent=90), # bare nucleus reference
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V0=0.015 # screening potential in m_e units (~7.6 keV)
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)
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values = screen(W_grid)
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```
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## Size Comparison with Other Corrections
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From Hayen et al. Table (Section VIII overview):
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| Correction | Magnitude | Energy Range Most Relevant |
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|---|---|---|
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| Screening $S$ | $10^{-3}$ – $10^{-4}$ | T < 50 keV |
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| Exchange $X$ | up to ~20% at lowest E | T < 10 keV |
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| Molecular screening $\Delta S_{\text{Mol}}$ | few $10^{-4}$ | T < 5 keV (molecular) |
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## Related Notes
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- [[12-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction
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- [[13-chemical-effects]] — Molecular environment influence on screening and exchange
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- [[05-exchange-correction]] — Electron exchange with atomic orbitals (often combined with screening)
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