Add Obsidian knowledge base (docs/) and update README with docs structure

docs/01-fermi-function.md

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