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:
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.

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)

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Fermi Function — Coulomb Interaction F_0(Z,W)