beta_shape_pnpi/docs/nuclear-structure/01-shape-factors.md

---

title: "10 – Nuclear Structure Effects" date: 2026-04-26 tags:

• beta-spectrum/nuclear-structure status: active aliases: [Shape Factor, C(Z,W), Form Factors] cssclasses: [] related_components: corrections/06-recoil,

---

Nuclear Structure Effects — The Shape Factor C(Z,W)

Overview

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 (corrections/02-finite-size) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction.

N(W)\,dW \propto p W (W_0 - W)^2 F(Z,W) C(Z,W) \tag{79}

Two Formalisms Compared

Approach	Style	Strengths
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
Holstein (HS)	Decouples leptonic + nuclear parts; manifestly covariant form factors	Clean symmetry properties, intuitive classification

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.

Multipole Expansion (Eq. 78)

Form factors F_{KLs}(q^2) describe nuclear structure model-independently:

\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) \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)!!}

Since (qR)^2 is very small for beta decay, expansion typically stops after first order.

Impulse Approximation (Eq. 80–81)

Nucleons treated as non-interacting particles coupling to weak vertex as free particles:

O_{KLs} = \sum_{\alpha,\beta} \langle \alpha | O_{KLs} | \beta \rangle a^\dagger_\alpha a_\beta \tag{80} \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}

The one-body density matrix elements \langle f | a^\dagger_\alpha a_\beta | i \rangle are calculable via shell model methods.

[!warning] Limitations The impulse approximation neglects many-body effects:

• Meson exchange: long history (Blin-Stoyle, Chemtob & Rho)
• Core polarization: nuclear medium modifications

For allowed decays, results largely hold due to cancellations between core-polarization and meson exchange (Morita 1985), but individual effects can reach ~40%.

Induced Currents — Form Factor Classification

See corrections/06-recoil for detailed equations. Key points:

First-Class Currents (CVC/PCAC)

• Vector g_V: conserved, from EM data via CVC
• Weak magnetism (g_M - g_V)/2M: determined by CVC from M1 transitions
• Axial-vector g_A: measured in neutron decay
• Induced pseudoscalar g_P/2M: PCAC predicts ~−229, quenched ~80% in nuclei

Second-Class Currents (G-parity forbidden)

• Induced scalar g_S/2M: excluded by CVC for Fermi transitions
• Induced tensor g_{II}/2M: experimental null results; reduces to first-class component only

Nuclear Matrix Element Operators (Table V of Hayen)

Operator	Symbol	Type	\Delta J
M_F	Fermi matrix element	Vector	0
M_{GT}	Gamow-Teller matrix element	Axial-vector	0,1 (not 0→0)
M_L	Longitudinal vector	—	depends on K
M_{T}, M_{TL}	Transverse multipoles	—	depends on K

Validity of Impulse Approximation

• Vector current: conserved via CVC → impulse approximation reliable
• Axial current: broken by PCAC → quenching effects significant (up to 80% for g_P)
• Mesonic degrees of freedom absorbed into nuclear potential (Wilkinson 1974)

Nuclear Deformation Correction D_C(Z,W,\beta^2)

The nuclear structure analogue of the electrostatic deformation correction (corrections/02-finite-size):

• Accounts for non-spherical nuclear shape in convolution terms
• Partially cancels with DFS correction (electrostatic)
• Relevant for deformed nuclei (\beta_2 \neq 0)

Calculator Status

[!info] Implementation Planned The shape factor C(Z,W; g_A) is being implemented as ShapeFactorGA component. See .rpiv/artifacts/research/2026-05-27_gA-spectrum-shape-factor-implementation.md for the implementation plan.

Latest Research (May 2026)

Key Findings from Paulsen et al. (2024) Fig. 9

The spectral shape of ⁹⁹Tc β decay is highly sensitive to g_A in the low-energy region (0–100 keV):

• Small g_A (0.5–1.0): Bell-shaped spectrum with low-energy enhancement
• Large g_A (1.4–2.0): Flattened spectrum with low-energy suppression
• Crossover point: All spectra intersect around 100 keV

g_A Values — Resolved

Source	g_A^eff	Method	Status
Paulsen et al. (2024)	1.526(92)	CVC-fixed sNME	Enhanced (controversial)
Song et al. (2025)	1.0–1.2	ESSM (farCVC)	Quenched (recommended)

Resolution: The discrepancy is resolved by proper sNME treatment. ESSM (sNME as fit parameter) gives quenched g_A = 1.0–1.2, consistent with other forbidden decays.

Table Data Status

The energy-dependent tables for C_V(W), C_A(W), C_VA(W) are not publicly available. Priority actions:

1. Obtain from supplementary materials of Kostensalo & Suhonen (2017) or Ramalho & Suhonen (2024)
2. Contact Suhonen group (jouni.t.suhonen@jyu.fi) for table data
3. Use integrated values as proxy for initial implementation

References

• Shape Factor C(W, g_A) — wiki concept page
• SSM — Spectrum Shape Method — wiki concept page
• ESSM — Enhanced Spectrum Shape Method — wiki concept page
• Paulsen et al. (2024) — wiki entity page
• Song et al. (2025) — wiki entity page