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 (03-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 07-recoil-effects 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 (03-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] Not yet implemented 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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Nuclear Structure Effects — The Shape Factor C(Z,W)