beta_shape_pnpi/docs/03-finite-size.md

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title: "03 – Finite Size Effects" date: 2026-04-26 tags:

• beta-spectrum/correction/finite-size status: active aliases: [Finite Nuclear Size, L_0, U, DFS] cssclasses: [] related_components: 01-fermi-function, 10-nuclear-structure

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Finite Nuclear Size Effects — L_0, U(Z,W), and DFS

Overview

The Fermi function (01-fermi-function) assumes a point charge nucleus. Real nuclei have finite extent and possibly deformation — these corrections account for the difference in electric potential inside the nuclear volume.

[!note] Two types of "finite size"

1. Electrostatic (L_0, U, DFS): Potential difference when moving from point charge to realistic nuclear shape — applied after evaluating wave function at origin
2. Convolution: Integration of leptonic wave functions over the nuclear volume — appears in shape factor C(Z,W) (10-nuclear-structure)

Nuclear Radius

R = \sqrt{5/3} \, \langle r^2 \rangle^{1/2} \tag{14}

where \langle r^2 \rangle^{1/2} is the root-mean-square charge radius from electron scattering experiments.

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L_0(Z,W) — Uniformly Charged Sphere

Low-Z Expansion (Behrens & Bühring)

L_0 \simeq 1 \mp \alpha Z W R + (\alpha Z)^2 - \frac{13}{60} - \frac{\alpha Z R}{2W} \tag{15}

Wilkinson's Analytical Parametrization (Eq. 16)

Accurate to 10^{-4} for p \leq 45, |Z| \leq 60:

L_0(Z,W) = 1 + \frac{13}{60}(\alpha Z)^2 \mp \frac{\alpha Z W R(41 - 26\gamma)}{[15(2\gamma-1)]} \mp \frac{\alpha Z R \gamma(17 - 2\gamma)}{[30W(2\gamma-1)]} + \frac{R}{W}\sum_{n=0}^5 a_n (WR)^n + 0.41(R-0.0164)(\alpha Z)^{4.5}

with a_n = \sum_{x=1}^6 b_{x,n} (\alpha Z)^x (Eq. 17). Coefficients in Tables I (electrons) and II (positrons); signs of odd powers flipped for positrons.

[!warning] Magnitude The effect ranges from a few 0.1% up to several percent — everywhere highly significant. Cannot be neglected for precision work at 10^{-4} level.

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/finite_size.py → FiniteSizeL0(Z, A) class

Uses Wilkinson parametrization (Eq. 16) with coefficients from Hayen Tables I-II. The nuclear radius is computed via utils.nuclear_radius(A).

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U(Z,W) — More Realistic Charge Distributions

The uniform sphere approximation is insufficient for high precision. Two common models:

Modified Gaussian (MG)

\rho_{MG}(r) = N_0\left[1 + A\left(\frac{r}{a}\right)^2\right]e^{-(r/a)^2} \tag{18}

with N_0 and a determined by normalization and RMS radius (Eqs. 19–20).

Fermi Distribution

\rho_F(r) \propto [1 + \exp((r-c)/a)]^{-1}, \quad a \simeq 0.55\text{ fm}

The correction is defined as ratio:

\frac{U(Z,W)}{L_0} = \frac{\alpha_{02}' + \alpha_{-1}'^2}{\alpha_{1}^2 + \alpha_{-1}^2} \tag{23}

Analytical Approximation for Low Z (Eq. 25)

U(Z,W) \approx 1 + \alpha Z W R \Delta_1 + \frac{\gamma}{W}\alpha Z R \Delta_2 + (\alpha Z)^2 \Delta_3 - (WR)^2 \Delta_4

Wilkinson's Fermi Distribution Fit (Eqs. 29–30)

U(Z,W) = 1 + \sum_{n=0}^\infty a_n p^n

with a_0, a_1, a_2 given as quadratic polynomials in Z.

[!tip] Magnitude The effect of U(Z,W) is ~0.1% for medium-high masses — cannot be neglected at the 10^{-4} precision level.

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/finite_size.py → ChargeDistributionU(Z, A) class

Implements Eq. (25) with \Delta_i coefficients computed from nuclear model parameters.

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DFS(Z,W,\beta^2) — Nuclear Deformation

For axially symmetric deformations:

R(\theta,\phi) = R_0\left[1 + \beta_2 Y_{20}(\theta,\phi)\right] \tag{32}

with \beta_2 > 0 prolate, \beta_2 < 0 oblate. The intrinsic quadrupole moment:

Q_0 = \frac{\alpha}{\pi}\sqrt{\frac{5}{3}} R Z \beta_2(1 + 0.16\beta_2) \tag{33}

The correction is a ratio of angle-averaged potentials (Eq. 40):

DFS(Z,W,\beta) = \frac{L_0^*(Z,W)}{L_0(Z,W)} \tag{40}

where L_0^* integrates over a continuous superposition of uniformly charged spheres (Eq. 39).

[!tip] Magnitude Effect can reach several parts in 10^3. Partially canceled by nuclear structure deformation correction D_C (10-nuclear-structure). Figure 1 of Hayen shows energy dependence reversal between β⁻ and β⁺ for \beta_2 = 0.2.

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/finite_size.py → includes DFS in the finite size module (combined with L₀)

The deformation parameter \beta_2 is typically provided by nuclear data tables or shell model calculations.

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Summary of Corrections Flow

Fermi function F₀(Z,W)   [point charge]
        ↓
    × L₀(Z,W)            [uniform sphere correction]
        ↓
    × U(Z,W)             [realistic charge distribution]
        ↓
    × DFS(Z,W,β²)         [nuclear deformation]

References in Hayen Paper

• Section IV.A.1 (Eq. 15–17): L_0 uniform sphere
• Section IV.A.2 (Eq. 18–30): U charge distributions
• Section IV.A.3 (Eq. 32–40): DFS deformation