94 lines
4.5 KiB
Markdown
94 lines
4.5 KiB
Markdown
---
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title: "10 – Nuclear Structure Effects"
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date: 2026-04-26
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tags:
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- beta-spectrum/nuclear-structure
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status: active
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aliases: [Shape Factor, C(Z,W), Form Factors]
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cssclasses: []
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related_components: [[07-recoil-effects]], [[11-isospin-breakdown]]
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---
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# Nuclear Structure Effects — The Shape Factor $C(Z,W)$
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## Overview
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Nuclear structure effects are encapsulated in the shape factor $C(Z,W)$ which multiplies the phase space × Fermi function. This is distinct from electrostatic finite size corrections ([[03-finite-size]]) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction.
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$$N(W)\,dW \propto p W (W_0 - W)^2 F(Z,W) C(Z,W) \tag{79}$$
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## Two Formalisms Compared
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| Approach | Style | Strengths |
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|---|---|---|
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| **Behrens-Bühring** (BB) | Couples leptonic + nuclear parts; rigorous expansion in $r^2$, $(m_eR)^a$, $(WR)^b$, $(\alpha Z)^c$ with tabulated functions $I(k_e,m,n,\rho)$ | Most precise, established computational machinery |
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| **Holstein** (HS) | Decouples leptonic + nuclear parts; manifestly covariant form factors | Clean symmetry properties, intuitive classification |
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Hayen et al. adopt BB's computational approach but present results in Holstein notation for clarity. See Appendix E of Hayen paper for detailed comparison.
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## Multipole Expansion (Eq. 78)
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Form factors $F_{KLs}(q^2)$ describe nuclear structure model-independently:
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$$\langle f | V_\mu - A_\mu | i \rangle \propto \sum_{KMs} \sum_{L=K-1}^{K+1} (-1)^{J_f-M_f+M} (-i)^L F_{KLs}(q^2) $$
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$$\times \sqrt{4\pi(2J_i+1)} \begin{pmatrix} J_f & K & J_i \\ -M_f & M & M_i \end{pmatrix} \frac{(qR)^L}{(2L+1)!!}$$
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Since $(qR)^2$ is very small for beta decay, expansion typically stops after first order.
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## Impulse Approximation (Eq. 80–81)
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Nucleons treated as non-interacting particles coupling to weak vertex as free particles:
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$$O_{KLs} = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle a^\dagger_\alpha a_\beta \tag{80}$$
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$$\langle f | O_{KLs} | i \rangle = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle \langle f | a^\dagger_\alpha a_\beta | i \rangle \tag{81}$$
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The one-body density matrix elements $\langle f | a^\dagger_\alpha a_\beta | i \rangle$ are calculable via shell model methods.
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> [!warning] Limitations
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> The impulse approximation neglects many-body effects:
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> - **Meson exchange**: long history (Blin-Stoyle, Chemtob & Rho)
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> - **Core polarization**: nuclear medium modifications
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>
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> For allowed decays, results largely hold due to cancellations between core-polarization and meson exchange (Morita 1985), but individual effects can reach ~40%.
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## Induced Currents — Form Factor Classification
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See [[07-recoil-effects]] for detailed equations. Key points:
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### First-Class Currents (CVC/PCAC)
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- **Vector** $g_V$: conserved, from EM data via CVC
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- **Weak magnetism** $(g_M - g_V)/2M$: determined by CVC from M1 transitions
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- **Axial-vector** $g_A$: measured in neutron decay
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- **Induced pseudoscalar** $g_P/2M$: PCAC predicts ~−229, quenched ~80% in nuclei
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### Second-Class Currents (G-parity forbidden)
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- **Induced scalar** $g_S/2M$: excluded by CVC for Fermi transitions
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- **Induced tensor** $g_{II}/2M$: experimental null results; reduces to first-class component only
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## Nuclear Matrix Element Operators (Table V of Hayen)
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| Operator | Symbol | Type | $\Delta J$ |
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|---|---|---|---|
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| $M_F$ | Fermi matrix element | Vector | 0 |
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| $M_{GT}$ | Gamow-Teller matrix element | Axial-vector | 0,1 (not 0→0) |
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| $M_L$ | Longitudinal vector | — | depends on K |
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| $M_{T}, M_{TL}$ | Transverse multipoles | — | depends on K |
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## Validity of Impulse Approximation
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- **Vector current**: conserved via CVC → impulse approximation reliable
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- **Axial current**: broken by PCAC → quenching effects significant (up to 80% for $g_P$)
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- Mesonic degrees of freedom absorbed into nuclear potential (Wilkinson 1974)
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## Nuclear Deformation Correction $D_C(Z,W,\beta^2)$
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The nuclear structure analogue of the electrostatic deformation correction ([[03-finite-size]]):
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- Accounts for non-spherical nuclear shape in convolution terms
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- Partially cancels with $DFS$ correction (electrostatic)
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- Relevant for deformed nuclei ($\beta_2 \neq 0$)
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## Calculator Status
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> [!info] Not yet implemented
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> The shape factor $C(Z,W)$ is NOT part of the current calculator. This note documents the theory for future implementation. For superallowed Fermi transitions, $C(Z,W) = |M_F|^2 \approx 1$ (isospin symmetry), so it can be approximated as unity in many cases.
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