beta_shape_pnpi/docs/corrections/03-screening.md

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title: "04 – Screening Correction" date: 2026-04-26 tags:

• beta-spectrum/correction/screening status: active aliases: [Atomic Screening, S(Z,W)] cssclasses: [] related_components: corrections/08-atomic-overlap, corrections/09-chemical-effects

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Atomic Electron Screening Correction S(Z,W)

Physical Origin

The Fermi function (corrections/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.

The Screening Correction

Screening is implemented as a ratio method:

S(Z,W) = \frac{F_{\text{screened}}(Z, W)}{F_0(Z, W - V_0/m_ec^2)} \tag{144}

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.

Screening Potential

The screening potential V_0 depends on:

• Atomic number Z (parent atom)
• Chemical/physical state (neutral atom, ionized molecule)
• Electronic configuration

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.

Molecular Screening Correction

In molecular environments, the screening potential changes:

\Delta S_{\text{Mol}} \quad (\text{Eq. 176})

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.

Energy Dependence

• Dominant at low energies (T < ~50 keV): where the β particle's wave function is most affected by atomic electron cloud
• Negligible at high energies: S(Z,W) \to 1 as W \gg V_0
• The effect reverses sign between β⁻ and β⁺ (electrons attracted to/repelled from electron cloud differently)

Shake-Up Effects on Screening

Atomic excitations during β decay modify screening:

[!warning] Tritium case 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}.

Shake-off (continuum ejection) has smaller effect:

• 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
• Negligible at target precision for most nuclei

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/screening.py → ScreeningCorrection(FermiFunction) class

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.

API Usage

from beta_spectrum import ScreeningCorrection, FermiFunction

screen = ScreeningCorrection(
    FermiFunction(Z_parent=90),  # bare nucleus reference
    V0=0.015  # screening potential in m_e units (~7.6 keV)
)
values = screen(W_grid)

Size Comparison with Other Corrections

From Hayen et al. Table (Section VIII overview):

Correction	Magnitude	Energy Range Most Relevant
Screening S	10^{-3} – 10^{-4}	T < 50 keV
Exchange X	up to ~20% at lowest E	T < 10 keV
Molecular screening \Delta S_{\text{Mol}}	few 10^{-4}	T < 5 keV (molecular)

Related Notes

• corrections/08-atomic-overlap — Atomic mismatch / overlap correction r(Z,W), Bahcall correction
• corrections/09-chemical-effects — Molecular environment influence on screening and exchange
• corrections/04-exchange — Electron exchange with atomic orbitals (often combined with screening)