75 lines
2.8 KiB
Markdown
75 lines
2.8 KiB
Markdown
---
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title: "01 – Fermi Function"
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date: 2026-04-26
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tags:
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- beta-spectrum/correction/fermi
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status: active
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aliases: [Coulomb Correction, F_0]
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cssclasses: []
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related_components: [[03-finite-size]], [[06-radiative-corrections]]
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---
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# Fermi Function — Coulomb Interaction $F_0(Z,W)$
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## The Point Charge Fermi Function
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The transition matrix element with all-order Coulomb interactions resummed (Hayen et al., Eq. 5):
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$$\mathcal{M}_{fi} = -2\pi i \delta(E_f - E_i) \langle f | T e^{-i\int_0^\infty dt H_Z(t)} H_\beta(0) T e^{-i\int_{-\infty}^0 dt H_{Z'}(t)} | i \rangle$$
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Keeping only the final-state Coulomb interaction (daughter nucleus charge $Z$):
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$$F_0(Z,W) = 4(2pR)^{2(\gamma-1)} e^{\pi y} \frac{|\Gamma(\gamma + iy)|^2}{(\Gamma(1+2\gamma))^2} \tag{6}$$
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with:
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$$\gamma = \sqrt{1 - (\alpha Z)^2}, \quad y = \pm \alpha Z W / p$$
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Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[03-finite-size]]).
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## Coulomb Amplitudes
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Expanding radial Dirac wave functions near the origin (Eq. 12):
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$$\left\{\frac{f_\kappa(r)}{g_\kappa(r)}\right\} = \alpha_\kappa \{(2|\kappa|-1)!!\}^{-1}(pr)^{|\kappa|-1} \sum_{n=0}^\infty \frac{a_{\kappa n}}{b_{\kappa n}} r^n$$
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The Coulomb information is encoded in **Coulomb amplitudes** $\alpha_\kappa$. The dominant electron component at the origin:
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$$F_0 L_0 = \frac{\alpha_{-1} + \alpha_1^2}{2p^2} \tag{13}$$
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This is valid regardless of nuclear charge distribution shape.
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## Alternative Definitions
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| Convention | Prefactor for $L_0$ | Style |
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|---|---|---|
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| **Wilkinson / Blatt & Weisskopf** | $(1+\gamma)/2$ | Absorbs into $L_0$ |
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| **Behrens-Bühring / Schopper** (used in calculator) | 4 | All corrections separate |
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The error from neglecting the $(1+\gamma)/2$ prefactor is ~0.5% for $Z=20$.
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## Numerical Implementation
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> [!tip] Code mapping
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> Module: `beta_spectrum/components/fermi.py` → `FermiFunction(Z, A)` class
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>
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> Uses `scipy.special.loggamma` for numerical stability (avoids overflow of $\Gamma$ function). The implementation follows Eq. (6) with log-space evaluation:
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> ```python
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> log_F0 = ln(4) + 2*(γ-1)*ln(2*p*R) + π*y
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> + 2*Re[loggamma(γ+iy)] - 2*loggamma(1+2γ)
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> F0 = exp(log_F0)
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> ```
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## Energy Dependence
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$F_0$ is a strong function of $Z$:
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- **β⁻ decay** ($y < 0$): wave functions attracted to nucleus → enhancement at low energy
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- **β⁺ decay** ($y > 0$): repulsion → suppression at low energy, grows with energy
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The effect ranges from ~1% for light nuclei to tens of percent for heavy ones.
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## Related Corrections
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After the point charge Fermi function is computed:
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- [[03-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$)
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- [[06-radiative-corrections]] adds QED radiative effects ($R = 1 + \delta_R$)
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