unzhakov/beta_shape_pnpi
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Add massive neutrino support and improve CSV/CLI output

Browse Source 
- Add neutrino mass parameter to SpectrumConfig and CLI interface
- Update PhaseSpace component to handle neutrino mass effects properly
- Update CSV headers to show basic endpoint and all transition types
- Add --csv-verbosity option to CLI with 'full' and 'concise' options
- Fix comparison plots to properly show linear scale variations
- Add comprehensive tests for neutrino mass functionality
This commit is contained in:
unzhakov
2026-07-02 10:33:28 +03:00
parent 406fd57bcd
commit 876e1e3ab8
48 changed files with 5213 additions and 97 deletions
Download Patch File Download Diff File Expand all files Collapse all files
.llm-wiki/raw/sources/SRC-2026-05-27-001/extracted.md
+172
View File
@@ -0,0 +1,172 @@
---
title: "SSM — Spectrum Shape Method for gA Extraction"
date: 2026-05-14
tags:
- gA/extraction/algorithm
- gA/ssm
- beta-spectrum/forbidden
status: active
aliases: [Spectrum Shape Method, gA fitting, shape factor fit]
related_notes:
- gA-extraction/02-ssnme
- gA-extraction/03-tc99-case-study
- gA-extraction/04-implementation-todo
- nuclear-structure/01-shape-factors
- corrections/05-radiative
---
# Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction
## 1. Physical Basis
For **forbidden nonunique β decays**, the shape factor $C(w_e)$ decomposes into three energy-dependent components with distinct coupling-constant dependence:
$$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
where $w_e$ is the total electron energy in units of $m_e c^2$. The components $C_V$, $C_A$, $C_{VA}$ are computed from nuclear many-body theory and are **independent of the coupling constants**. The spectrum shape is sensitive to $g_A$ through the interference between vector, axial-vector, and mixed terms — a purely nuclear-structure effect unique to nonunique forbidden decays.
> [!note] Why nonunique decays?
> Unique forbidden decays have a single dominant NME → universal spectrum shape independent of nuclear structure. Nonunique decays involve multiple NMEs → the relative weight of vector vs. axial contributions depends on $g_A$, creating observable spectral shape differences.
## 2. The Full Spectrum Model
The differential spectrum (Paulsen et al., 2024, Eq. 5):
$$N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)$$
where:
- $\mathcal{N}$ — overall normalization (absorbs $G_F^2 \cos^2\theta_C / 2\pi^3$)
- $F(Z,W)$ — Fermi function (Coulomb correction) — [[corrections/01-fermi-function]]
- $p W (W_0 - W)^2$ — phase space — [[02-phase-space]]
- $X(W)$ — atomic exchange correction — [[corrections/04-exchange]]
- $S(Z,W)$ — atomic screening — [[corrections/03-screening]]
- $R(W)$ — outer radiative correction — [[corrections/05-radiative]]
- $C(W; g_A, g_V)$ — **shape factor** — [[nuclear-structure/01-shape-factors]]
- $\rho(Z,W)$ — atomic overlap correction — [[corrections/08-atomic-overlap]]
## 3. The Algorithm
### Step 1: Compute shape factor components from nuclear model
**Input:** Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.
**Output:** Three energy-dependent lookup tables: $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $W \in [1, W_0]$.
**Method:**
1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
3. Sum over all contributing multipole orders $K$ and lepton quantum numbers $k_e, k_\nu$
**CVC constraint for vector current:**
The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:
$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}$$
where $E_C$ is the Coulomb displacement energy. The sNME is critical for small-$Q$ decays like $^{99}\text{Tc}$ ($[W_0 - (m_n - m_p)] = -0.487$ MeV).
**Practical note:** For $^{99}\text{Tc}$, the dominant transition is $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$. The sub-dominant $1g_{7/2} \to 1g_{9/2}$ transition contributes OBTD $\approx 0.010$. See Kostensalo & Suhonen (2017), Table VII.
### Step 2: Build theoretical spectrum for trial $g_A$
For each trial value of $g_A$ (with $g_V = 1.0$ fixed by CVC):
1. **Compute shape factor:**
$$C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)$$
(using $g_V = 1.0$)
2. **Compute full spectrum:**
$$N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)$$
3. **Normalize:** Choose $\mathcal{N}$ so that $\int_1^{W_0} N_{\text{theo}}(W; g_A)\, dW = 1$ (or match the total counts in the experimental spectrum)
### Step 3: Compare with experimental spectrum
**Input:** Experimental spectrum data $(W_i, N_i, \sigma_i)$ for $i = 1, \dots, N_{\text{bins}}$.
**χ² function:**
$$\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}$$
**Fit range selection (Paulsen et al., 2024):**
- **Lower bound:** ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
- **Upper bound:** ~275 keV for $^{99}\text{Tc}$ — above this, endpoint region with poor statistics and resolution distortion
- Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range
### Step 4: Minimize χ² and extract $g_A$
**Grid search:** Evaluate $\chi^2(g_A)$ on a grid (e.g., $g_A \in [0.4, 1.8]$ with step 0.01).
**Refinement:** Use `scipy.optimize.minimize_scalar` (Brent's method) around the grid minimum.
**Uncertainty:** Find $g_A$ values where $\chi^2 = \chi^2_{\text{min}} + 1$. These define the 68% confidence interval.
**Output:** $(g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})$
### Step 5 (optional): Simultaneous $g_A$ and $g_V$ fit
If $g_V$ is not fixed to 1.0 (e.g., testing CVC violation):
$$\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}$$
Fit both parameters simultaneously. The correlation between $g_A$ and $g_V$ is typically strong.
### Step 6 (optional): Half-life normalization — quenched $g_A$
The $g_A$ extracted from **shape alone** can be enhanced ($>1.27$) because the normalization constant $\mathcal{N}$ absorbs the overall rate. To obtain the **quenched** values consistent with half-life:
1. Compute the theoretical partial half-life for the best-fit $(g_A, g_V)$:
$$t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW$$
2. Compare with experimental partial half-life:
$$t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}$$
3. Extract quenched values:
$$g_i^{\text{quenched}} = g_i^{\text{shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
For $^{99}\text{Tc}$: $t_{1/2}^{\text{theo}} \approx 30 \times 10^3$ a vs. $t_{1/2}^{\text{exp}} \approx 212 \times 10^3$ a → quenching factor $\approx 0.38$.
## 4. The sNME Problem and ESSM
### The Problem
The sNME (small relativistic vector NME) is extremely difficult to compute in nuclear shell model because it involves contributions outside the valence shell. The CVC prediction gives a "canonical" value, but this assumes a perfect many-body theory — which finite valence spaces are not.
### The Solution: Enhanced SSM (ESSM)
Kumar et al. (2021) proposed treating the sNME as a **fitting parameter** alongside $g_A$. This is the **Enhanced Spectrum Shape Method (ESSM)**:
1. Fix $g_A$ at a trial value
2. Vary the sNME to simultaneously fit the **experimental half-life** (not just the shape)
3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each $g_A$ value:
- **sNME(c):** closer to the CVC value
- **sNME(f):** further from the CVC value
4. Compare both solutions' spectral shapes with the data
The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See [[gA-extraction/02-ssnme]] for details.
## 5. Uncertainty Budget
| Component | Uncertainty on $g_A$ | Relative | Dominant for |
|---|---|---|---|
| Fit method (statistics) | 0.0745 | 4.9% | All SSM analyses |
| Coulomb displacement energy | 0.0425 | 2.8% | Small-$Q$ decays |
| Nuclear model | 0.0294 | 1.9% | Model comparison |
| Lepton current treatment | 0.0127 | 0.8% | High-$Z$ nuclei |
| Fit range | 0.0061 | 0.4% | All |
| Endpoint energy | 0.0038 | 0.2% | Precision $Q$-value needed |
| Radiative corrections | 0.0033 | 0.2% | All |
| Atomic exchange | 0.0029 | 0.2% | Low-energy region |
| **Combined** | **0.0919** | **6.0%** | — |
Source: Paulsen et al. (2024), Table VIII.
## 6. Key References
| Ref | Citation | Use |
|---|---|---|
| [SSM-orig] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | SSM introduction; 26 decays |
| [SSM-Tc94-98] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc |
| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}$Tc MMC measurement; $g_A$ extraction (enhanced value) |
| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc MMC measurement; quenched $g_A$ via ESSM |
| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | Enhanced SSM formalism |
| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME analysis; five second-forbidden decays |
| [Hayen2017] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | BB formalism; shape factor theory |
.llm-wiki/raw/sources/SRC-2026-05-27-001/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-001",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "01-ssm-algorithm.md",
"file_path": "/home/unzhakov/workspace/beta_shape_pnpi/docs/gA-extraction/01-ssm-algorithm.md",
"format": "markdown"
}
.llm-wiki/raw/sources/SRC-2026-05-27-001/original/01-ssm-algorithm.md
+172
View File
@@ -0,0 +1,172 @@
---
title: "SSM — Spectrum Shape Method for gA Extraction"
date: 2026-05-14
tags:
- gA/extraction/algorithm
- gA/ssm
- beta-spectrum/forbidden
status: active
aliases: [Spectrum Shape Method, gA fitting, shape factor fit]
related_notes:
- gA-extraction/02-ssnme
- gA-extraction/03-tc99-case-study
- gA-extraction/04-implementation-todo
- nuclear-structure/01-shape-factors
- corrections/05-radiative
---
# Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction
## 1. Physical Basis
For **forbidden nonunique β decays**, the shape factor $C(w_e)$ decomposes into three energy-dependent components with distinct coupling-constant dependence:
$$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
where $w_e$ is the total electron energy in units of $m_e c^2$. The components $C_V$, $C_A$, $C_{VA}$ are computed from nuclear many-body theory and are **independent of the coupling constants**. The spectrum shape is sensitive to $g_A$ through the interference between vector, axial-vector, and mixed terms — a purely nuclear-structure effect unique to nonunique forbidden decays.
> [!note] Why nonunique decays?
> Unique forbidden decays have a single dominant NME → universal spectrum shape independent of nuclear structure. Nonunique decays involve multiple NMEs → the relative weight of vector vs. axial contributions depends on $g_A$, creating observable spectral shape differences.
## 2. The Full Spectrum Model
The differential spectrum (Paulsen et al., 2024, Eq. 5):
$$N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)$$
where:
- $\mathcal{N}$ — overall normalization (absorbs $G_F^2 \cos^2\theta_C / 2\pi^3$)
- $F(Z,W)$ — Fermi function (Coulomb correction) — [[corrections/01-fermi-function]]
- $p W (W_0 - W)^2$ — phase space — [[02-phase-space]]
- $X(W)$ — atomic exchange correction — [[corrections/04-exchange]]
- $S(Z,W)$ — atomic screening — [[corrections/03-screening]]
- $R(W)$ — outer radiative correction — [[corrections/05-radiative]]
- $C(W; g_A, g_V)$ — **shape factor** — [[nuclear-structure/01-shape-factors]]
- $\rho(Z,W)$ — atomic overlap correction — [[corrections/08-atomic-overlap]]
## 3. The Algorithm
### Step 1: Compute shape factor components from nuclear model
**Input:** Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.
**Output:** Three energy-dependent lookup tables: $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $W \in [1, W_0]$.
**Method:**
1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
3. Sum over all contributing multipole orders $K$ and lepton quantum numbers $k_e, k_\nu$
**CVC constraint for vector current:**
The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:
$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}$$
where $E_C$ is the Coulomb displacement energy. The sNME is critical for small-$Q$ decays like $^{99}\text{Tc}$ ($[W_0 - (m_n - m_p)] = -0.487$ MeV).
**Practical note:** For $^{99}\text{Tc}$, the dominant transition is $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$. The sub-dominant $1g_{7/2} \to 1g_{9/2}$ transition contributes OBTD $\approx 0.010$. See Kostensalo & Suhonen (2017), Table VII.
### Step 2: Build theoretical spectrum for trial $g_A$
For each trial value of $g_A$ (with $g_V = 1.0$ fixed by CVC):
1. **Compute shape factor:**
$$C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)$$
(using $g_V = 1.0$)
2. **Compute full spectrum:**
$$N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)$$
3. **Normalize:** Choose $\mathcal{N}$ so that $\int_1^{W_0} N_{\text{theo}}(W; g_A)\, dW = 1$ (or match the total counts in the experimental spectrum)
### Step 3: Compare with experimental spectrum
**Input:** Experimental spectrum data $(W_i, N_i, \sigma_i)$ for $i = 1, \dots, N_{\text{bins}}$.
**χ² function:**
$$\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}$$
**Fit range selection (Paulsen et al., 2024):**
- **Lower bound:** ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
- **Upper bound:** ~275 keV for $^{99}\text{Tc}$ — above this, endpoint region with poor statistics and resolution distortion
- Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range
### Step 4: Minimize χ² and extract $g_A$
**Grid search:** Evaluate $\chi^2(g_A)$ on a grid (e.g., $g_A \in [0.4, 1.8]$ with step 0.01).
**Refinement:** Use `scipy.optimize.minimize_scalar` (Brent's method) around the grid minimum.
**Uncertainty:** Find $g_A$ values where $\chi^2 = \chi^2_{\text{min}} + 1$. These define the 68% confidence interval.
**Output:** $(g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})$
### Step 5 (optional): Simultaneous $g_A$ and $g_V$ fit
If $g_V$ is not fixed to 1.0 (e.g., testing CVC violation):
$$\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}$$
Fit both parameters simultaneously. The correlation between $g_A$ and $g_V$ is typically strong.
### Step 6 (optional): Half-life normalization — quenched $g_A$
The $g_A$ extracted from **shape alone** can be enhanced ($>1.27$) because the normalization constant $\mathcal{N}$ absorbs the overall rate. To obtain the **quenched** values consistent with half-life:
1. Compute the theoretical partial half-life for the best-fit $(g_A, g_V)$:
$$t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW$$
2. Compare with experimental partial half-life:
$$t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}$$
3. Extract quenched values:
$$g_i^{\text{quenched}} = g_i^{\text{shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
For $^{99}\text{Tc}$: $t_{1/2}^{\text{theo}} \approx 30 \times 10^3$ a vs. $t_{1/2}^{\text{exp}} \approx 212 \times 10^3$ a → quenching factor $\approx 0.38$.
## 4. The sNME Problem and ESSM
### The Problem
The sNME (small relativistic vector NME) is extremely difficult to compute in nuclear shell model because it involves contributions outside the valence shell. The CVC prediction gives a "canonical" value, but this assumes a perfect many-body theory — which finite valence spaces are not.
### The Solution: Enhanced SSM (ESSM)
Kumar et al. (2021) proposed treating the sNME as a **fitting parameter** alongside $g_A$. This is the **Enhanced Spectrum Shape Method (ESSM)**:
1. Fix $g_A$ at a trial value
2. Vary the sNME to simultaneously fit the **experimental half-life** (not just the shape)
3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each $g_A$ value:
- **sNME(c):** closer to the CVC value
- **sNME(f):** further from the CVC value
4. Compare both solutions' spectral shapes with the data
The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See [[gA-extraction/02-ssnme]] for details.
## 5. Uncertainty Budget
| Component | Uncertainty on $g_A$ | Relative | Dominant for |
|---|---|---|---|
| Fit method (statistics) | 0.0745 | 4.9% | All SSM analyses |
| Coulomb displacement energy | 0.0425 | 2.8% | Small-$Q$ decays |
| Nuclear model | 0.0294 | 1.9% | Model comparison |
| Lepton current treatment | 0.0127 | 0.8% | High-$Z$ nuclei |
| Fit range | 0.0061 | 0.4% | All |
| Endpoint energy | 0.0038 | 0.2% | Precision $Q$-value needed |
| Radiative corrections | 0.0033 | 0.2% | All |
| Atomic exchange | 0.0029 | 0.2% | Low-energy region |
| **Combined** | **0.0919** | **6.0%** | — |
Source: Paulsen et al. (2024), Table VIII.
## 6. Key References
| Ref | Citation | Use |
|---|---|---|
| [SSM-orig] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | SSM introduction; 26 decays |
| [SSM-Tc94-98] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc |
| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}$Tc MMC measurement; $g_A$ extraction (enhanced value) |
| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc MMC measurement; quenched $g_A$ via ESSM |
| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | Enhanced SSM formalism |
| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME analysis; five second-forbidden decays |
| [Hayen2017] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | BB formalism; shape factor theory |
.llm-wiki/raw/sources/SRC-2026-05-27-002/extracted.md
+93
View File
@@ -0,0 +1,93 @@
---
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] 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.
.llm-wiki/raw/sources/SRC-2026-05-27-002/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-002",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "01-shape-factors.md",
"file_path": "/home/unzhakov/workspace/beta_shape_pnpi/docs/nuclear-structure/01-shape-factors.md",
"format": "markdown"
}
.llm-wiki/raw/sources/SRC-2026-05-27-002/original/01-shape-factors.md
+93
View File
@@ -0,0 +1,93 @@
---
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] 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.
.llm-wiki/raw/sources/SRC-2026-05-27-003/extracted.md
+561
View File
@@ -0,0 +1,561 @@
This is a self-archived version of an original article. This version
may differ from the original in pagination and typographic details.
Kostensalo, Joel; Suhonen, Jouni
Author(s):
gA-driven shapes of electron spectra of forbidden β decays in the nuclear shell model
Title:
Year: 2017
Version:
Published version
Copyright: © 2017 American Physical Society. Published in this repository with the kind permission of the publisher.
Rights:
Rights url:
Please cite the original version:
Kostensalo, J., & Suhonen, J. (2017). gA-driven shapes of electron spectra of forbidden β decays
in the nuclear shell model. Physical Review C, 96(2), Article 024317.
https://doi.org/10.1103/PhysRevC.96.024317
PHYSICALREVIEWC96,024317(2017)
g -drivenshapesofelectronspectraofforbiddenβ decaysinthenuclearshellmodel
A
JoelKostensalo* andJouniSuhonen†
DepartmentofPhysics,UniversityofJyvaskyla,P.O.Box35,FI-40014,Finland
(Received10May2017;published21August2017)
The evolution of the shape of the electron spectra of 16 forbidden β− decays as a function of g was
A
studiedusingthenuclearshellmodelinappropriatesingle-particlemodelspaceswithestablished,well-tested
nuclearHamiltonians.Theβspectraof94Nb(6+)→ 94Mo(4+)and98Tc(6+)→ 98Ru(4+)werefoundtodepend
stronglyong ,whichmakesthemexcellentcandidatesforthedeterminationoftheeffectivevalueofg withthe
A A
spectrum-shapemethod(SSM).Astrongg dependenceisalsoseeninthespectrumof96Zr(0+)→ 96Nb(6+).
A
Thisdecaycouldbeusedfordeterminingthequenchingofg insixth-forbiddendecaysinthefuture,whenthe
A
measurementofthespectrumbecomesexperimentallyfeasible.Thecalculatedshell-modelelectronspectraofthe
ground-state-to-ground-statedecaysof87Rb,99Tc,and137Csandthedecayof137Cstotheisomeric11/2−state
in137Bawerefoundtobeinexcellentagreementwiththespectrapreviouslycalculatedusingthemicroscopic
quasiparticle-phononmodel.ThisisfurtherevidenceoftherobustnatureoftheSSMobservedintheprevious
studies.
DOI:10.1103/PhysRevC.96.024317
I. INTRODUCTION valueoftheweakcouplingconstants.IntheSSMtheshapes
of computed and experimental (normalized) electron spectra
At the nuclear level β decay can be considered as a
are compared in order to find the ratio g /g for nonunique
mutual interaction of the hadronic and leptonic currents forbiddenbetaβ−decays,forwhichthes A hap V efactorsdepend
mediated by a massive vector boson W± [1]. The leptonic
ong /g inaverynontrivialway.InRef.[21]theSSMwas
and hadronic currents can be expressed as a mixture of both A V
applied to the fourth-forbidden ground-state-to-ground-state
vector and axial-vector components [2–4]. The weak vector
transitionof113Cdusingthemicroscopicquasiparticle-phonon
andaxial-vectorcouplingconstantsg andg enterthetheory
V A model (MQPM) [22,23], the nuclear shell model (NSM)
when the hadronic current is renormalized at the nucleon
[24–26], the microscopic interacting boson-fermion model
level[5].Theconservedvector-currenthypothesis(CVC)and
(IBFM-2) [27,28], and experimental spectrum of Ref. [29].
partially conserved axial-vector-current hypothesis (PCAC)
The closest match between the theoretical and experimental
yield the free-nucleon values g =1.00 and g =1.27 [6]
V A spectra was found when g /g ≈0.92 for all three nuclear
but inside nuclear matter the value of g is affected by A V
A models.Thehalf-lifecomparisonmethod,ontheotherhand,
many-nucleon correlations and a quenched value might be
gave very different results when different nuclear models
neededtoreproduceexperimentaldata[1].Preciseinformation
were used. In Ref. [30] it was noticed that the shape of the
ontheeffectivevalueofg iscrucialwhenpredictinghalf-lives
A spectrum was largely unaffected by modest changes in the
of neutrinoless double beta decays since the half-lives are
MQPM wave function, even though the predicted half-life
proportionaltothefourthpowerofg [7,8].
A wasaffectedsignificantly.Theobservationsoftheseprevious
The effective value of g has earlier been probed using a
A studiessuggestthattheSSMisveryrobust,quiteinsensitiveto
half-lifecomparisonmethod,inwhichthepredictedandexper-
thedetailsoftheadoptednuclearmeanfieldandHamiltonian.
imentalvaluesarecomparedfordifferentvaluesofg .Thishas
A Itisthusareliabletoolforprobingtheratiog /g .
beendoneforGamow-Tellerandfirst-forbiddendecaysusing A V
The g -driven evolution of the shapes of several exper-
nuclear matrix elements calculated with the proton-neutron A
imentally measurable forbidden nonunique decay spectra in
quasiparticlerandom-phaseapproximation(pnQRPA)[9–13].
medium-heavyodd-AnucleiwasstudiedusingtheMQPMin
Thehalf-lifecomparisonmethodpredictsthatthevalueofg
A Ref.[30].Onlyfourgoodcandidatesfortheapplicationofthe
is quenched significantly in these transitions. Also in the old
spectrum-shape method have been found thus far. These are
studies[14–17]offirst-forbiddenβdecays,strongquenchings
theground-state-to-ground-statedecaysof87Rb,99Tc,113Cd,
werepredicted.Inarecentshell-modelstudy[18]ofr-process
and 115In [20,30]. In the present paper we set out to find
waiting-pointnuclei,astrongquenchingofg wasconfirmed.
A potential new candidates for the application of the SSM in
Thehalf-lifecomparisonmethodcouldbeusedtofindoutif
light to medium-heavy nuclei using the nuclear shell model.
similar quenching of g is seen in highly forbidden unique
A We also compare the NSM and MQPM β spectra of 87Rb,
beta decays by using the half-lives calculated in Ref. [19],
99Tc,and137Cstoseeifthemodelindependenceobservedin
onceexperimentaldatabecomeavailable.
Ref.[21]appliesforthesedecaysaswell.Thedecaysof137Cs
In Ref. [20] the spectrum-shape method (SSM) was tothe3/2+groundstateandthe11/2−isomericstateof137Ba
introduced as a complementary way to study the effective
are particularly interesting for comparing the differences in
the MQPM and NSM spectra, since the MQPM calculations
predictag -independentspectrumshapeforthesedecays.
A
*joel.j.kostensalo@student.jyu.fi This article is organized as follows. In Sec. II we give
†jouni.suhonen@phys.jyu.fi the theoretical background behind the β spectrum shape. In
2469-9985/2017/96(2)/024317(7) 024317-1 ©2017AmericanPhysicalSociety
JOELKOSTENSALOANDJOUNISUHONEN PHYSICALREVIEWC96,024317(2017)
Sec.IIIwedescribetheapplicationofthenuclearshellmodel θ beingtheCabibboangle.Inordertosimplifytheformalism
C
tothespectrumshapeofforbiddenβ− decays.InSec.IVwe it is usual to introduce unitless kinematic quantit(cid:2)ies w
e
=
presentourresultsandinSec.Vwedrawtheconclusions. W /m c2,w =W /m c2,andp =p c/(m c2)= w2−1.
e e 0 0 e e e e
Withtheunitlessquantitiestheintegratedshapefactorcanbe
II. THEORETICALFORMALISM
expressedas
We begin the description of the β− decay by making the (cid:3)
so-calledimpulseapproximation,inwhichattheexactmoment C˜ =
w0
C(w )pw (w −w )2F (Z,w )dw . (6)
of decay the decaying nucleon only interacts via the weak e e 0 e 0 e e
1
interaction and the strong interaction with the other A−1
nucleons can be ignored. Since the vector boson W− has a The shape factor C(w e ) of Eq. (6) contains complicated
largemassandthuspropagatesonlyashortdistance,theflow combinationsofboth(universal)kinematicfactorsandNMEs.
lines of the nucleons, i.e., the hadronic current, and the flow In this paper we use the nuclear shell model to calculate the
lines of the emitted leptons, i.e., the leptonic current, can be one-bodytransitiondensitiesrelatedtotheNMEsoftheshape
considered to interact at a pointlike weak-interaction vertex factor. The details of the shape factor and the constitution
with an effective coupling constant G , the Fermi constant. of its NMEs can be found from Refs. [32,33]. As in the
F
The parity nonconserving nature of the weak interaction is previous SSM studies [20,21,30] we go beyond the earlier
reflected in the fact that the hadronic current can be written studies[32,33]andtakeintoaccountthenext-to-leading-order
at the quark level (up u and down d quarks) as a mixture of
vectorandaxial-vectorcomponentsas[2–4]
Jμ =u¯(x)γμ(1−γ )d(x), (1)
H 5
whereγμ aretheusualDiracmatricesandγ =iγ0γ1γ2γ3.
5
Renormalization effects of strong interactions must be taken
intoaccountwhenmovingfromthequarkleveltothehadron
level.Thehadroniccurrentatthenucleonlevel(neutronnand
protonp)canthenbewrittenas[5,6]
Jμ =p¯(x)γμ(g −g γ )n(x), (2)
H V A 5
where g and g are the weak vector and axial-vector
V A
couplingconstants,respectively.Theconservedvector-current
hypothesisofthestandardmodel(CVC)givesthefree-nucleon
valueg =1.0fortheweakvectorcouplingconstantandthe
V
partially conserved axial-vector-current hypothesis (PCAC)
gives the free-nucleon value g =1.27 for the weak axial-
A
vector coupling constant. Inside the nuclear matter the value
of g is affected by many-nucleon correlations and so the
A
free nucleon value might not be the one to use in practical
calculations [6]. The present paper is a step towards solving
theproblemofwhatvaluetouseforg insidefinitenuclei.
A
Intheimpulseapproximationtheprobabilityoftheelectron
beingemittedwithkineticenergybetweenW andW +dW
e e e
is
G 1
P(W )dW = F C(W )
e e (h¯c)62π3h¯ e
×p cW (W −W )2F (Z,W )dW , (3)
e e 0 e 0 e e
where p is the momentum of the electron, Z is the proton
e
number, F (Z,W ) is the Fermi function, and W is the
0 e 0
end-point energy of the β spectrum. The nuclear structure
informationisburiedintheshapefactorC(w ).
e
Thehalf-lifeofaβ decaycanbewrittenas
κ
t = , (4)
1/2 C˜
whereC˜ istheintegratedshapefactorandtheconstantκ has FIG.1. Normalized electron spectra for the second-forbidden
thevalue[31] nonunique β− decays of 36Cl [panel (a)] and 60Fe [panel (c)], and
thefourth-forbiddennonuniquedecayof50V[panel(b)].Thevalue
κ = 2π3h¯7ln2 =6147s, (5) g V =1.00 was adopted and the color coding represents the value
m5c4(G cosθ )2 ofg .
e F C A
024317-2
g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
A
termsoftheshapefunction.Thedetailsofthenext-to-leading tocalculatetheelectronspectraofthesedecays.TheGXPF1A
ordershapefactorsarediscussedin[21]. interactionisverygoodatpredictingthepropertiesofpf-shell
The shape factor C(w ) can be decomposed into vector, nuclei. However, since the computational burden increases
e
axial-vector, and mixed vector-axial-vector parts. In this dramatically when the number of particles approaches half
decompositiontheshapefactoris of the maximum in the valence space, this interaction could
| | | | | | | | | | 60Fe(0+)→ | | 60Co(2+) | | | |
| --- | --- | ------ | --- | --- | --- | --- | ------ | -------- | --------- | --- | -------- | --- | ----- | ----------- |
| | | | | | | | not be | used for | the | | | | decay | transition. |
| | )=g | 2C )+g | 2C | )+g | | | | | | | | | | |
C(w e V (w e A (w e V g A C VA (w e ). (7) Instead, a model space spanning the π0f , ν1p , ν0f ,
| | | V | A | | | | | | | | | 7/2 | 3/2 | 5/2 |
| --- | --- | --- | --- | --- | --- | --- | ------ | ------------------------------------------ | --- | --- | --- | --- | --- | --- |
| | | | | | | | andν1p | orbitalswasadoptedwiththeHorie-Ogawainter- | | | | | | |
1/2
IntegratingEq.(7)overtheelectronkineticenergy,wegetan
| | | | | | | | action [40,41]. | | pfg | shell: | the jj44b | interaction | | developed |
| ---------------------------------------------- | --- | -------- | ----- | ---- | ---- | --- | --------------- | ------- | ------------ | --------- | --------- | ----------- | -------- | --------- |
| analogousexpressionfortheintegratedshapefactor | | | | | | | | | 9 / 2 | | | −1p−0g | | |
| | | | | | | | by Brown | and | Lise t s kiy | for | the 0f | | | model |
| | | | | | | | | | | | | 5/2 | | 9/2 |
| | | | | | | | space (see | endnote | on | Ref. [28] | of | [42]) | was used | for the |
| | | C˜ =g2C˜ | +g2C˜ | +g g | C˜ , | (8) | | | | | | | | |
V V A A V A VA calculation of the 87Rb(3/2−)→ 87Sr(9/2+) transition. For
A=94–99
wherethefactorsC˜ inEq.(8)arejustconstants,independent the decays of the nuclei, the effective interac-
i
| | | | | | | | tion of | Gloeckner | [43] | in the | model | space | spanning | the |
| --- | --- | --- | --- | --- | --- | --- | ------- | --------- | ---- | ------ | ----- | ----- | -------- | --- |
oftheelectronenergy.
| | | | | | | | π1p , | π0g | , ν2s | , and 2d | was | used. | sdgh | shell: |
| --- | --- | --- | --- | --- | --- | --- | -------- | ------- | ----------------- | -------- | ---------- | ----- | --------- | ------ |
| | | | | | | | 1/2 | 9/2 | 1/2 | | 5/2 | | | 11/2 |
| | | | | | | | | | | | 126Sn(0+)→ | | 126Sb(2+) | |
| | | | | | | | finally, | for the | decay transitions | | | | | and |
III. ADOPTEDMODELSPACESANDNUCLEAR 137Cs(7/2+)→ 137Ba(11/2−,3/2+) the Sn100pn interaction
HAMILTONIANS
| | | | | | | | [44]inthe0g | 7/2 | 2s1d0h | 11/2 modelspacewasadopted. | | | | |
| --- | -------- | ------- | ----- | --------- | -------------- | ---- | ----------- | --- | ------ | -------------------------- | --- | --- | --- | --- |
| The | electron | spectra | of 16 | forbidden | β− transitions | were | | | | | | | | |
g
| calculated | for | different | values | of the coupling | constant | A | | | | | | | | |
| ---------- | --- | --------- | ------ | --------------- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
IV. RESULTSANDDISCUSSION
| using | the nuclear | matrix | elements | produced | by the | nuclear | | | | | | | | |
| ----- | ----------- | ------ | -------- | -------- | ------ | ------- | --- | --- | --- | --- | --- | --- | --- | --- |
shellmodel.Thecalculationsweredoneusingtheshellmodel Below we present our results: the electron spectra of
code NuShellX@MSU [34], with appropriate model spaces 16 forbidden β− decays (Figs. 1–5) and their integrated
and well established Hamiltonians chosen for each studied shape factors (Table I). The electron spectra are discussed
decay individually. No additional truncations, beyond those inSec.IVAandtheintegratedshapefactorsinSec.IVB.
usedintheoriginalworks,wereintroduced.Thecalculations
wererunonadesktopcomputerwitha3.3GHzprocessor,so
| | | | | | | | | A. Electronspectraandtheeffectivevalueofg | | | | | | |
| ---------------------- | --- | ------ | --- | ----------- | ------------ | ---- | ------------------------------------- | ----------------------------------------- | --- | --- | --- | --- | --- | --- |
| the computational | | burden | had | to be taken | into account | when | | | | | | | | A |
| choosingthemodelspace. | | | | | | | Thestudiedspectrafallintothreegroups. | | | | | | | |
TheadoptedmodelspacesandHamiltoniansareasfollows. GROUP 1. The first group, for which the spectra can be
sdshell:forthecalculationofthewavefunctionsandone-body found in Figs. 1 and 2, consists of six transitions for which
| transitiondensitiesrelatedtothedecay36Cl(2+)→ | | | | | 36Ar(0+) | | | | | | | | | |
| --------------------------------------------- | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
theSSManalysishasnotbeendonepriortothisstudy.These
weadoptedthe0d–1smodelspaceandthephenomenological decaysarepotentiallyinterestingforthepracticalapplication
USDBinteraction[35].pfshell:asintheearliershell-model ofthespectrum-shapemethod,sincemostofthemaresecond-
studiesregardingthehalf-livesofthetransitions48Ca(0+)→
| | | | | | | | forbidden | nonunique | transitions | | with | branching | ratios | above |
| --- | --- | --- | --- | --- | --- | --- | --------- | --------- | ----------- | --- | ---- | --------- | ------ | ----- |
48Sc(4+,5+,6+)[36]and50V(6+)→ 50Cr(2+)[37],the0f– 90%. The only exception is the undetected fourth-forbidden
1pmodelspacewiththeinteractionGXPF1A[38,39]wasused β− decay branch of 50V, for which the branching ratio has
TABLEI. UnitlessintegratedshapefunctionsC˜ ofthestudiedtransitionsandtheirvectorC˜ ,axial-vectorC˜ ,andmixedcomponents
| | | | | | | | | | V | | | A | | |
| -------------------------------------- | --- | --- | --- | ----------------------------------------- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- |
| C˜ .ForthetotalintegratedshapefactorC˜ | | | | | | | | | =g | =1.0. | | | | |
| VA | | | | thevaluesofthecouplingconstantsweresettog | | | | | V A | | | | | |
| | | | | | | C˜ | | C˜ | | | C˜ | | | C˜ |
| Transition | | | | Type | | | | | | | | | | |
| | | | | | | V | | A | | | VA | | | |
137Cs(7/2+)→137Ba(11/2−) 1stuniq. 9.681×10−11 8.322×10−6 3.761×10−8 8.359×10−6
36Cl(2+)→36Ar(0+) 6.081×10−9 3.152×10−10 −2.746×10−9 3.650×10−9
2ndnonuniq.
60Fe(0+)→60Co(2+) 2.347×10−13 5.232×10−11 −7.000×10−12 4.556×10−11
2ndnonuniq.
94Nb(6+)→94Mo(4+) 2ndnonuniq. 1.598×10−8 1.469×10−8 −3.058×10−8 1.029×10−10
98Tc(6+)→98Ru(4+) 2ndnonuniq. 2.723×10−8 2.544×10−8 −5.254×10−8 1.207×10−10
99Tc(9/2+)→99Ru(5/2+) 2ndnonuniq. 2.240×10−9 2.130×10−9 −4.361×10−9 8.777×10−12
126Sn(0+)→126Sb(2+) 1.422×10−8 7.125×10−9 2.011×10−8 4.145×10−8
2ndnonuniq.
137Cs(7/2+)→137Ba(3/2+) 4.211×10−6 2.836×10−6 6.879×10−6 1.392×10−5
2ndnonuniq.
87Rb(3/2−)→87Sr(9/2+) 3rdnonuniq. 1.185×10−13 2.082×10−14 −9.734×10−14 4.202×10−14
48Ca(0+)→48Sc(4+) 4thnonuniq. 8.946×10−28 5.934×10−29 −4.606×10−28 4.934×10−28
50V(6+)→50Cr(2+) 4thnonuniq. 1.024×10−23 9.137×10−25 2.131×10−24 1.329×10−23
96Zr(0+)→96Nb(4+) 3.176×10−27 3.170×10−28 −2.006×10−27 1.486×10−27
4thnonuniq.
48Ca(0+)→48Sc(5+) 2.577×10−30 3.931×10−25 1.334×10−27 3.945×10−25
4thuniq.
96Zr(0+)→96Nb(5+) 4thuniq. 1.765×10−29 1.551×10−24 6.844×10−27 1.558×10−24
48Ca(0+)→48Sc(6+) 6thnonuniq. 1.124×10−32 3.498×10−34 3.947×10−33 1.554×10−32
96Zr(0+)→96Nb(6+) 5.811×10−33 7.494×10−33 −1.318×10−32 1.234×10−34
6thnonuniq.
024317-3
| JOELKOSTENSALOANDJOUNISUHONEN | | | | PHYSICALREVIEWC96,024317(2017) | | |
| ----------------------------- | --- | --- | --- | ------------------------------ | --- | --- |
FIG.2. SameasFig.1butforthesecond-forbiddennonunique FIG.3. Same as Fig. 1 but for the decays 48Ca(0+)→
decaysof94Nb[panel(a)],98Tc[panel(b)],and126Sn[panel(c)]. 48Sc(4+,5+,6+)[panels(a),(b),and(c)].
beenpredictedtobe≈2%byarecentshell-modelcalculation
| | | | beindependentofthevalueofg | A ,whilethedependencewas | | |
| --- | --- | --- | -------------------------- | ------------------------ | --- | --- |
[36]. An earlier study [30] of the g -driven evolution of the significant for 87Rb and 99Tc. The shell-model and MQPM
A
electronspectraofodd-Anucleidemonstratedthatthesecond spectra of these decays are compared in order to see if the
andespeciallythefourth-forbiddennonuniqueβ-decayspectra 113Cd 115In
| | | | previously observed | model independence | of and | |
| --- | --- | --- | ------------------- | ------------------ | ------ | --- |
dependstronglyontheeffectivevalueoftheweakaxial-vector spectra[21]holdsalsoforthesedecays.
couplingconstant. The spectra presented in Figs. 1 and 2 (GROUP 1) fall
GROUP 2. The second group consists of the fourth- into two subgroups. The spectra of 36Cl, 50V, and 60Fe
48Ca(0+)→ 48Sc(4+,5+,6+) [Fig.1,panels(a),(b),and(c)],and126Sn[Fig.2,panel(c)]
| to-sixth-forbidden | decays | | and | | | |
| ------------------------- | ------ | --- | --- | --- | --- | --- |
| 96Zr(0+)→ 96Nb(4+,5+,6+), | | | | | | |
for which the beta spectra are do not exhibit noticeable dependence on the value of the
presented in Figs. 3 and 4. The decays of 48Ca and 96Zr axial-vector coupling constant. This is very surprising since
thedecayof50Visfourthforbidden,andallthesevenstudied
appeartobedominatedbythetwo-neutrinodoublebetadecay
mode [36,45,46], so the study of these decays is somewhat fourth-forbidden decays in Ref. [30] depend heavily on g .
A
challenging experimentally. However, they can shed light on The spectra of the second subgroup, 94Nb and 98Tc [Fig. 2,
whether the spectra of highly forbidden beta decays are g panels (a) and (b)], are significantly g dependent. The g
| | | | A | | A | A |
| --- | --- | --- | --- | --- | --- | --- |
dependentornot. driven evolution of the spectra is not only very similar for
GROUP 3. The third group consists of four de- these two decays, but also nearly identical to the one of
cays: 137Cs(7/2+)→ 137Ba(11/2−,3/2+), 99Tc(9/2+)→ 99Tc, shown in Fig. 5, panel (c), even though the Q value
| 99Ru(5/2+), | 87Rb(3/2−)→ | 87Sr(9/2+). | | | | |
| ----------- | ----------- | ----------- | --- | --- | --- | --- |
and The electron of these decays varies between 300 keV and 450 keV. A
spectra of these decays was studied previously in [30] using similargroupingphenomenonwasseenforseveraldecaysin
the MQPM. The spectra of the 137Cs decays were found to [30]:forexampletheshapesofthespectraoffourth-forbidden
024317-4
g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
A
| | | | in | the present | paper | and | the | fourth-forbidden | | ones | studied |
| --- | --- | --- | ----------- | ------------ | ------------- | ------------- | -------- | ---------------- | --------- | -------- | -------- |
| | | | in | [30]. | The branching | | ratio | is, however, | | so small | that the |
| | | | measurement | | of | this spectrum | | is not | currently | within | reach |
| | | | of | experiments. | | It is | somewhat | striking | that, | from | the 38 |
differentforbiddendecaysstudiedinthepresentpaperandin
| | | | Ref.[30],strongg | | | dependenceisseenonlyineven-forbidden | | | | | |
| --- | --- | --- | ---------------- | --- | --- | ------------------------------------ | --- | --- | --- | --- | --- |
A
decays.
| | | | | In Fig. | 5 the | shell-model | | and | MQPM | spectra | of the |
| --- | --- | --- | --- | ------- | ----- | ----------- | --- | --- | ---- | ------- | ------ |
experimentallyinterestingdecaysfoundinRef.[30]areshown
| | | | for | comparison. | | For 87Rb | [panels | (a) | and (b)], | which | is the |
| --- | --- | --- | ------------------------------------------------- | ----------- | --- | -------- | ------- | --- | --------- | ----- | ------ |
| | | | bestcandidateforthedeterminationofthequenchingofg | | | | | | | | in |
A
| | | | third-forbidden | | | decays, | the MQPM | | and shell-model | | spectra |
| --- | --- | --- | --------------- | --- | --- | ------- | -------- | --- | --------------- | --- | ------- |
99Tc
| | | | are | practically | | identical. | For | | [panels | (c) and | (d)] the |
| --- | --- | --- | --------- | ----------- | --------- | ---------- | ------- | ------ | ------- | ------------- | -------- |
| | | | evolution | | predicted | by | the two | models | is | also similar, | but |
| | | | in | the range | g | =0.9–1.1 | there | is | some | difference | in the |
A
shape,leadingtoapotentialsmalluncertaintyinthevalueof
| | | | g | when | comparing | to | the experimental | | spectrum, | | hopefully |
| --- | --- | --- | --- | ---- | --------- | --- | ---------------- | --- | --------- | --- | --------- |
A
availableinthe(near)future.Thetwostudieddecaysof137Cs
| | | | [panels | (e) | and | (f)] were | found | to be | g A independent | | when |
| --- | --- | --- | ------- | --- | --- | --------- | ----- | ----- | --------------- | --- | ---- |
usingtheMQPMmodelinRef.[30].Theshell-modelspectra
| | | | calculatedforthesedecayswerenotonlyalsog | | | | | | | independent, | |
| --- | --- | --- | ---------------------------------------- | --- | --- | --- | --- | --- | --- | ------------ | --- |
A
butoverlapwiththeMQPMspectraperfectly.Measurements
| | | | of | these | spectra | would | by very | desirable, | | since they | would |
| --- | --- | --- | --- | ----- | ------- | ----- | ------- | ---------- | --- | ---------- | ----- |
shedlightonthequalityofthecalculatedspectra,independent
ofthechosennuclearmodel.
| | | | | | | B. IntegratedshapefactorC˜ | | | | | |
| --- | --- | --- | --- | --------------------------- | --- | -------------------------- | --- | ----------------------- | --- | --- | --- |
| | | | | TheintegratedshapefactorsC˜ | | | | andtheirdecompositionto | | | |
thecomponentsofEq.(8)arepresentedinTableI.Forallthe
studieddecaysthesignofthevectorandaxial-vectorcompo-
nentsispositiveandthesignofthemixedvector-axial-vector
componentvaries.Forthesecond-forbiddennonuniquedecays
| | | | of | 94Nb and | 98,99Tc | which | have | very | similar | spectrum-shape | |
| --- | --- | --- | --- | -------- | ------- | ----- | ---- | ---- | ------- | -------------- | --- |
evolution,thevector-andaxial-vectorcomponentsareroughly
equalandthevector-axial-vectorcomponentisroughlytwice
96Zr(0+)→
| FIG.4. | Same as Fig. 1 | but for the decays | | | | | | | | | |
| ------ | -------------- | ------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- |
96Nb(4+,5+,6+)[panels(a),(b),and(c)]. aslarge,butwithanegativesign.Theresultingtotalintegrated
| | | | shape | factor | is two | magnitudes | | smaller | than | its components. | |
| --- | --- | --- | ----------------------------------- | ------ | ------ | ---------- | --- | ------- | ----------------- | --------------- | --- |
| | | | Forthesixth-forbiddendecay96Zr(0+)→ | | | | | | 96Nb(6+)theaxial- | | |
ground-state-to-ground-stateβ−decaysof97Zr,101Mo,117Cd,
vectorcomponentis30%largerthanthevectorcomponentand
and119Inhaveanalmostidenticalg evolution. thevector-axial-vectorcomponentisagainroughlythesumof
A
Decays of GROUP 2, 48Ca(0+)→ 48Sc(4+,5+,6+) and thetwoothercomponentsandnegativeinsign.
| 96Zr(0+)→ | 96Nb(4+,5+,6+),werestudiedinordertoseeif | | | 87Rb | | | | | | | |
| --------- | ---------------------------------------- | --- | --- | ---- | --- | ------------- | --- | ---------- | --- | -------- | ------ |
| | | | | For | the | decomposition | | is similar | to | the case | of the |
C˜
thereappearssomeinterestingsystematicbehavior.Sincethe MQPM: the vector component is the largest, the axial-
V
shape of the spectra of fourth-forbidden nonunique decays vectorcomponentC˜ isabout20%ofC˜ ,andthevector-axial-
| | | | | | | A | | | V | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
of odd-A nuclei depends on g in a significant way, it vectorcomponentC˜ isthesmallestandhasanegativesign.
| | | A | | | | VA | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
is interesting to see if this holds for the even-A nuclei. However,thecomponentsandthetotalintegratedshapefactor
Since the spectrum on 50V is largely independent of the are approximately 30% larger for the MQPM. In the case of
valuesoftheweakcouplingconstants,thedependenceisnot 99TcthedifferenceismuchlargerandtheMQPMresultsare
strong for at least some fourth-forbidden decays. The shape roughlytwicetheshellmodelones.Forthetwostudieddecays
48Ca(0+)→ 48Sc(4+,5+,6+)
of the electron spectra of and of137Csthereisnog dependence.Thedecaytotheisomeric
A
96Zr(0+)→ 96Nb(4+,5+)transitionsisindependentofg ,as 11/2−stateisuniqueandthus(practically)g independent.On
| | | A | | | | | | | A | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
showninFig.3,panels(a),(b),and(c)andinFig.4,panels(a) theotherhand,thedecaytothe3/2+groundstateisnonunique
and(b),revealingasecondexception,48Ca(0+)→ 48Sc(4+), independentsincethethreecomponentsofC˜
| | | | butg | A | | | | | | | havethe |
| --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | ------- |
totheg -dependentfourth-forbiddennonuniquedecays.The samesignandnointerferencebetweenthecomponentsoccurs.
A
sixth-forbidden decay 96Zr(0+)→ 96Nb(6+) [Fig. 4, panel ForthesetwodecaysthenormalizedshellmodelandMQPM
C˜
(c)],ontheotherhand,exhibitsastrongg A dependence.The electron spectra overlap perfectly but the difference in is
evolutionisverydifferentfromtheoneofthe94Nband98Tc considerable. Again, the relative fractional decomposition is
[Fig.2,panels(a)and(b)],and99Tc[Fig.5,panel(c)]studied almost identical for the two nuclear models, but the absolute
024317-5
| JOELKOSTENSALOANDJOUNISUHONEN | | | | | | PHYSICALREVIEWC96,024317(2017) | | | |
| ----------------------------- | --- | --- | --- | --- | --- | ------------------------------ | --- | --- | --- |
FIG.5. Presentlycalculatedshell-modelspectraandtheMQPMspectra,firstpublishedinRef.[30],fortheground-state-to-ground-state
decaysof87Rb[panels(a)and(b)],99Tc[panels(c)and(d)],and137Cs[panel(f)],andthedecayof137Cstotheisomeric11/2−statein137Ba
[panel(e)].Thecolorcodinginthespectraof87Rband99Tcreferstothevalueofg .Theshapeoftheelectronspectraofthetwo137Csdecays
A
isg independent,andthecolorcodinginthesedecaysreferstotheadoptednuclearmodel.Thefree-nucleonvalueg =1.00wasusedinall
| A | | | | | | | V | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
calculations.
valueoftheMQPMC˜ is40-foldtheshellmodeloneforthe objective of the study was to find experimentally detectable
decaytothe11/2−state,and8-foldforthedecaytothe3/2+ transitionsforwhichtheshapeoftheelectronspectradepends
state.DuetotheC˜−1dependenceofthehalf-life,thehalf-lives sensitively on the value of the weak coupling constants.
predicted by the two models do not agree with each other at Comparingthecalculatedandmeasuredspectratheeffective
all.Thisisastrongindicationthatthespectrum-shapemethod valueofg canbeextracted.
A
| | | | | | The electron | spectra | of 94Nb(6+)→ | 94Mo(4+) | and |
| --------------------------------------------------- | --- | --- | --- | --- | ------------ | -------- | ------------ | -------- | --- |
| isamorerobusttoolfordeterminingtheeffectivevalueofg | | | | A | | | | | |
| | | | | | 98Tc(6+)→ | 98Ru(4+) | | | |
thanjustasimplehalf-lifecomparison. depend significantly on the effective
| | | | | | value of | g and these transitions | are | thus excellent | new |
| --- | --- | --- | --- | --- | -------- | ----------------------- | --- | -------------- | --- |
A
| | | | | | candidates | for the spectrum-shape | method | (SSM). | Both |
| --- | --- | --- | --- | --- | ---------- | ---------------------- | ------ | ------ | ---- |
V. CONCLUSIONS
| | | | | | of these | decays have almost | a 100% | branching | ratio, |
| --- | --- | --- | --- | --- | -------- | ------------------ | ------ | --------- | ------ |
The evolution of the electron-spectrum shapes of 16 and have been experimentally observed. The shell-model
| | β− | | | | spectra of | the previously | found candidates, | the | ground- |
| --------- | -------------- | ------ | ------------------------- | --- | ---------- | -------------- | ----------------- | ----- | ------- |
| forbidden | decays, driven | by the | value of the axial-vector | | | | | | |
| | | | | | | | 87Rb | 99Tc, | |
coupling constant g , were studied using nuclear matrix state-to-ground-state decays of and agree well
A
elements derived from the nuclear shell model. Established with the microscopic quasiparticle-phonon model predicted
| | | | | | spectra of | a previous study, | which makes | the experimen- | |
| --------------- | ------- | ------------ | --------- | -------- | ---------- | ----------------- | ----------- | -------------- | --- |
| and well-tested | nuclear | Hamiltonians | were used | in these | | | | | |
investigations.Intheβ-decayshapefactorstheusuallyomitted tal measurement of these spectra very desirable. The g A -
next-to-leading-ordertermsweretakenintoaccount.Themain independent shell model and MQPM spectrum shapes of the
024317-6
g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
A
137Cs(7/2+)→ 137Ba(11/2−,3/2+) A nuclei with A(cid:2)98 have now been studied. The potential
| | | | transitions overlap | per- | | | | |
| --- | --- | --- | ------------------- | ---- | --- | --- | --- | --- |
fectly.Themeasurementofthesespectrawouldshedlighton new candidates for the spectrum-shape method are heavier,
thequalityofthecalculatedspectra. and the study of these decays requires the adoption of some
othernuclearmodelduetothelargecomputationalburdenand
| Thetransitionsforwhichaheavyg | | | A dependencehasbeen | | | | | |
| ----------------------------- | ------------ | ------------------- | ------------------- | ----------- | ------------------ | ---------------- | ------------ | --------- |
| | | | | | the other problems | with shell-model | calculations | for heavy |
| recorded | thus far are | all even forbidden. | Unlike | in the case | | | | |
ofthefourth-forbiddentransitionsinodd-Anucleistudiedin deformednuclei.
| Ref. [30], | which all | showed a strong | g dependence, | such a | | | | |
| ---------- | --------- | --------------- | ------------- | ------ | --- | --- | --- | --- |
A
dependenceisnotseeninthecorrespondingtransitionsinthe
ACKNOWLEDGMENTS
studiedeven-Anuclei.Thusfarallthefoundtransitionswith
averystronglyg -dependentspectrumshapeareinthemass ThisworkhasbeenpartiallysupportedbytheAcademyof
A
A=94–119.
region Most of the experimentally accessible Finland under the Finnish Centre of Excellence Programme
forbiddennonuniquetransitionsinodd-AnucleiwithA(cid:2)169
| | | | | | 2012-2017 (Nuclear | and Accelerator | Based | Programme at |
| ------------------------------------------------------ | --- | --- | --- | --- | ------------------ | --------------- | ----- | ------------ |
| andsecondorhigher-forbiddennonuniquetransitionsineven- | | | | | JYFL). | | | |
[1] J.Suhonen,FromNucleonstoNucleus:ConceptsofMicroscopic [29] P.Belli,R.Bernabei,N.Bukilic,F.Cappella,R.Cerulli,C.J.
NuclearTheory(Springer,Berlin,2007). Dai, F. A. Danevich, J. R. de Laeter, A. Incicchitti, V. V.
[2] R.P.FeynmannandM.Gell-Mann,Phys.Rev.109,193(1958). Kobychev, S. S. Nagorny, S. Nisi, F. Nozzoli, D. V. Poda, D.
[3] R.P.FeynmannandM.Gell-Mann,Phys.Rev.111,362(1958). Prosperi,V.I.Tretyak,andS.S.Yurchenko,Phys.Rev.C76,
| [4] W.Theis,Z.Phys.150,590(1958). | | | | | 064603(2007). | | | |
| --------------------------------- | --- | --- | --- | --- | ------------- | --- | --- | --- |
[5] K.Zuber,NeutrinoPhysics(InstituteofPhysicsPublishingLtd., [30] J.Kostensalo,M.Haaranen,andJ.Suhonen,Phys.Rev.C95,
| London,2004). | | | | | 044313(2017). | | | |
| ------------- | --- | --- | --- | --- | ------------- | --- | --- | --- |
[6] E.D.Commins,WeakInteractions(McGraw-Hill,NewYork, [31] J. C. Hardy, I. S. Towner, V. Koslowsky, E. Hagberg, and H.
| 2007). | | | | | Schmeing,Nucl.Phys.A509,429(1990). | | | |
| ------ | --- | --- | --- | --- | ---------------------------------- | --- | --- | --- |
[7] J.SuhonenandO.Civitarese,Phys.Rep.300,123(1998). [32] H. Behrens and W. Bühring, Electron Radial Wave Functions
[8] J. Maalampi and J. Suhonen, Adv. High Energy Phys. 2013, andNuclearBetaDecay(Clarendon,Oxford,1982).
505874(2013). [33] M.T.Mustonen,M.Aunola,andJ.Suhonen,Phys.Rev.C73,
[9] H.EjiriandJ.Suhonen,J.Phys.G:Nucl.Part.Phys.42,055201 054301(2006).
(2015). [34] B. A. Brown and W. D. M. Rae, Nucl. Data Sheets 120, 115
| [10] H. Ejiri, | N. Soukouti, | and J. Suhonen, | Phys. Lett. | B 729, 27 | (2014). | | | |
| -------------- | ------------ | --------------- | ----------- | --------- | ------- | --- | --- | --- |
(2014). [35] B.A.BrownandW.A.Richter,Phys.Rev.C74,034315(2006).
[11] P.PirinenandJ.Suhonen,Phys.Rev.C91,054309(2015). [36] M. Haaranen, M. Horoi, and J. Suhonen, Phys. Rev. C 89,
[12] D.S.DelionandJ.Suhonen,Europhys.Lett.107,52001(2014). 034315(2014).
[13] F.F.DeppischandJ.Suhonen,Phys.Rev.C94,055501(2016). [37] M.Haaranen,P.C.Srivastava,J.Suhonen,andK.Zuber,Phys.
[14] E.K.Warburton,Phys.Rev.C42,2479(1990). Rev.C90,044314(2014).
[15] E.K.Warburton,Phys.Rev.C44,233(1991). [38] M.Honma,T.Otsuka,B.A.Brown,andT.Mizusaki,Phys.Rev.
[16] E.K.Warburton,I.S.Towner,andB.A.Brown,Phys.Rev.C C69,034335(2004).
49,824(1994). [39] M.Honma,T.Otsuka,B.A.Brown,andT.Mizusaki,Eur.Phys.
[17] E.K.WarburtonandI.S.Towner,Phys.Rep.242,103(1994). J.A25,499(2005).
[18] Q. Zhi, E. Caurier, J. J. Cuenca-García, K. Langanke, G. [40] H.HorieandK.Ogawa,Prog.Theor.Phys.46,439(1971).
Martínez-Pinedo,andK.Sieja,Phys.Rev.C87,025803(2013). [41] H.HorieandK.Ogawa,Nucl.Phys.A216,407(1973).
[19] J.KostensaloandJ.Suhonen,Phys.Rev.C95,014322(2017). [42] B.Cheal,E.Mané,J.Billowes,M.L.Bissell,K.Blaum,B.A.
[20] M.Haaranen,P.C.Srivastava,andJ.Suhonen,Phys.Rev.C93, Brown, F. C. Charlwood, K. T. Flanagan, D. H. Forest, C.
034308(2016). Geppert,M.Honma,A.Jokinen,M.Kowalska,A.Krieger,J.
[21] M.Haaranen,J.Kotila,andJ.Suhonen,Phys.Rev.C95,024327 Krämer,I.D.Moore,R.Neugart,G.Neyens,W.Nörtershäuser,
| (2017). | | | | | M.Schug,H.H.Stroke,P.Vingerhoets,D.T.Yordanov,andM. | | | |
| ------- | --- | --- | --- | --- | --------------------------------------------------- | --- | --- | --- |
[22] J.ToivanenandJ.Suhonen,J.Phys.G.21,1491(1995). Žáková,Phys.Rev.Lett.104,252502(2010).
[23] J.ToivanenandJ.Suhonen,Phys.Rev.C57,1237(1998). [43] D.H.Gloeckner,Nucl.Phys.A253,301(1975).
[24] B.H.Wildenthal,M.S.Curtin,andB.A.Brown,Phys.Rev.C [44] B. A. Brown, N. J. Stone, J. R. Stone, I. S. Towner, and M.
| 28,1343(1983). | | | | | Hjorth-Jensen,Phys.Rev.C71,044317(2005). | | | |
| -------------- | --- | --- | --- | --- | ---------------------------------------- | --- | --- | --- |
[25] G.Martínez-Pinedo,A.Poves,E.Caurier,andA.P.Zuker,Phys. [45] H. Heiskanen, M. T. Mustonen, and J. Suhonen, J. Phys. G:
| Rev.C53,R2602(R)(1996). | | | | | Nucl.Part.Phys.34,837(2007). | | | |
| ----------------------- | --- | --- | --- | --- | ---------------------------- | --- | --- | --- |
[26] E. Caurier, F. Nowacki, and A. Poves, Phys. Lett. B 711, 62 [46] M. Alanssari, D. Frekers, T. Eronen, L. Canete, J. Dilling,
(2012). M. Haaranen, J. Hakala, M. Holl, M. Ješkovský, A. Jokinen,
[27] F. Iachello and A. Arima, The Interacting Boson Model A. Kankainen, J. Koponen, A. J. Mayer, I. D. Moore, D.
(CambridgeUniversityPress,Cambridge,UK,1987). A. Nesterenko, I. Pohjalainen, P. Povinec, J. Reinikainen, S.
[28] F. Iachello and P. V. Isacker, The Interacting Boson-Fermion Rinta-Antila,P.C.Srivastava,J.Suhonen,R.I.Thompson,A.
Model(CambridgeUniversityPress,NewYork,1991). Voss,andM.E.Wieser,Phys.Rev.Lett.116,072501(2016).
024317-7
.llm-wiki/raw/sources/SRC-2026-05-27-003/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-003",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download",
"url": "https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download",
"format": "web"
}
.llm-wiki/raw/sources/SRC-2026-05-27-003/original/source.html
BIN
View File
Binary file not shown.
.llm-wiki/raw/sources/SRC-2026-05-27-004/extracted.md
+1397
View File
File diff suppressed because it is too large Load Diff
.llm-wiki/raw/sources/SRC-2026-05-27-004/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-004",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf",
"url": "https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf",
"format": "web"
}
.llm-wiki/raw/sources/SRC-2026-05-27-005/extracted.md
+504
View File
@@ -0,0 +1,504 @@
5202
nuJ
12
]xe-lcun[
1v44571.6052:viXra
The measurement of the 99Tc β-decay spectrum and its implications for the effective
value of weak axial coupling
J. W. Song,1,2 M. Ramalho,3,4 M. K. Lee,1 G. B. Kim,5 I. Kim,5 H. L. Kim,6 Y. C.
Lee,6,7 K. R. Woo,6 J. Kotila,8 J. Kostensalo,9 J. Suhonen,4,10,∗ and H. J. Kim2,†
1Korea Research Institute of Standards and Science, Daejeon 34113, South Korea
2Department of Physics, Kyungpook National University, Daegu 41566, South Korea
3School of Physics, Engineering and Technology,
University of York, Heslington, York YO10 5DD, United Kingdom
4Department of Physics, University of Jyva¨skyla¨,
P. O. Box 35 (YFL), FI-40014, Jyva¨skyl¨a, Finland
5Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
6Institute for Basic Science, Daejeon 34126, South Korea
7Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
8Finnish Institute for Educational Research, University of Jyva¨skyl¨a, Jyv¨askyla¨, Finland
9Natural Resources Institute Finland, Yliopistokatu 6B, FI-80100 Joensuu, Finland
10International Centre for Advanced Training and Research in Physics (CIFRA), 077125 Bucharest-Magurele, Romania
(Dated: June 24, 2025)
Measurements of β-spectral shapes is an important way to examine the effective value of the
weakaxialcouplingg . Thesestudiesfocusspecificallyonforbiddennon-uniqueβ− transitions,as
A
only in these cases is the spectral shape directly sensitive to the ratio g /g . Here, the value of
A V
the weak vector coupling constant, g , is fixed at 1.0 according to the Conserved Vector Current
V
(CVC) hypothesis. In previous studies for the fourth-forbidden non-unique β− decays of 113Cd
[J. Kostensalo et al., Phys. Lett. B 822, 136652 (2021)] and 115In [A. F. Leder et al., Phys.
Rev. Lett. 129, 232502 (2022) and L. Pagnanini et al., Phys. Rev. Lett. 133, 122501 (2024)] a
quenchedvaluewasdeterminedfortheratiog /g usingg =1.0. Anotableexceptionistherecent
A V V
measurementandanalysisofthesecond-forbiddennon-uniqueβ-decaytransitionin99Tc,performed
byM.Paulsenet al.,Phys. Rev. C110,05503(2024). Whereanenhancedratiog /g =1.526(92)
A V
was suggested. To resolve this apparently contradictory situation with the effective value of g , we
A
haveperformedcalculationsbasedonthenuclearshellmodel(NSM)Hamiltoniansglekpn,jj45pnb,
andtheMQPMapproachwithacarefulconsiderationofthesmallrelativisticvectornuclearmatrix
element(sNME).Thetheoreticalspectrawerecomparedtothe99Tcβ-decayspectrumbyusingthe
4πgoldabsorberwithaMetallicMagneticCalorimeter(MMC).Inallcases,wefoundthatthedata
matches well with reduced g /g values of 1.0–1.2. Our result contradicts the previously reported
A V
measurementfor99Tcandinsteadsupportsaquenchedaxialcouplingasreportedforotherisotopes.
Introduction.—Electrons and electron antineutrinos turebeyond-the-standard-modelexperiments;seethere-
are emitted in β− decays as a result of a transmuta- views [1–3].
tion of a parent nucleus (Z,N) to a daughter nucleus
In allowed β decays, such as Fermi and Gamow–Teller
(Z +1,N −1), Z being the number of protons and N
transitions, the emitted lepton pair (electron and an-
the number of neutrons. Studies of β-electrons (elec-
tineutrino)carrieszeroorbitalangularmomentum[4]. In
trons emitted in β− decays) have become an important
contrast,forbiddendecaysoccurwhentheemittedlepton
subject of study, e.g., in connection with nuclear reac-
pair carries nonzero orbital angular momentum [5]. The
tors and the anomalies in their fluxes of antineutrinos,
degreeofforbiddennesscanrangefromfirst-forbiddento
and rare-events experiments searching for beyond-the-
n:th-forbidden, with each additional level typically in-
standard-modelphysics,likethosetryingtomeasurerare
creasing the decay half-life by a factor of approximately
β-decays and double beta decays. The β-electrons can 104. Forbidden β-transitions are generally classified into
also be severe backgrounds in these experiments. In all
two types: unique and non-unique. Unique transitions
the aforementioned cases, measurements of the associ-
are characterized by a single dominant nuclear matrix
atedβ-spectralshapeisessential. Distortionsinβ-decay
element (NME) and thus exhibit a universal β-spectral
spectral shapes also provide important input for under-
shapethatisindependentofnuclearstructure[4]. Incon-
standing the weak axial coupling constant g , which
A trast, non-uniquetransitionsinvolveacomplexinterplay
is relevant for sensitivity estimates in current and fu-
of several NMEs, making the β-decay spectrum highly
sensitivetothenuclearwavefunctionsofboththeinitial
and final states [5–7].
∗ jouni.t.suhonen@jyu.fi The (partial) half-life of a forbidden non-unique β-
† hongjoo@knu.ac.kr transitionsalsodependsonthevaluesoftheweakvector
2
and axial-vector couplings, g and g , respectively [5– plings by writing
V A
7]. The conserved vector current (CVC) hypothesis
sets g V = 1.0 whereas the partially conserved axial- C(w e )=g V 2C V (w e )+g A 2C A (w e )+g V g A C VA (w e ) (2)
vector current (PCAC) hypothesis leads to an effective
where w is the total (rest-mass plus kinetic) energy of
(quenched) value of geff in finite nuclei [8], its bare- e
A the emitted electron. The shape of the β-electron spec-
nucleon value g =1.27 resulting from the experimental
A trum is sensitive to the value of g due to subtle in-
studies of the decay of an isolated neutron. The quench- A
terference effects among the vector C (w ), axial-vector
ing of the effective value of g has recently been studied V e
A C (w), and mixed vector–axial-vector C (w) contribu-
extensively in [2, 8, 9], and in particular its effects on A VA
tionstothetransitionmatrixelement[6]. Forsomenuclei
the neutrinoless double beta decay have been addressed
(veryfew,ascharteduptopresentdays[2,18,19,21,22])
in [1, 3]. To quantify the quenched value of g , com-
A this causes measurable changes in the β-spectra. Thus,
binedexperimentalandtheoreticalanalysesofβ-spectral
the effective value geff can be found by computing the-
shapes of individual β− transitions have recently been A
oretical β-spectra for a set of geff values and comparing
performed for the fourth-forbidden non-unique β-decays A
themwiththemeasuredone. Thismethod,coinedasthe
of 113Cd [10–13] and 115In [14–16].
spectrum-shape method (SSM), was originally proposed
A recent measurement of the β-decay spectrum
in [6].
for the second-forbidden non-unique ground-state-to-
Another factor influencing theoretical analyses of β-
ground-state transition 99Tc(9/2+) → 99Ru(5/2+),
spectral shapes and half-lives is the so-called small rela-
which has a 100% branching ratio, was reported by
tivistic vector nuclear matrix element (sNME) [5]. The
Paulsen et al. [17]. The accompanying theoretical analy-
sNME gathers contributions outside the valence major
sisyieldedacontradictoryresult,suggestinganenhance-
shell that contains the proton and neutron Fermi sur-
ment of the axial coupling with a value of g /g =
A V faces, which makes its calculation particularly hard for
1.526(92) using g =1.0.
V nuclear-theoryframeworksused,e.g.,inthepresentwork
ThegA-EXPERimentandTheory(gA-EXPERT)col-
and also in the work of Paulsen et al. [17]. Despite its
laboration is an international effort dedicated to quan-
smallness, the sNME can influence the β-spectral shapes
tifying the effective axial-vector coupling constant, geff,
A and half-lives quite strongly [11, 12, 19, 21].
by precisely measuring β-decays of various nuclei and
ArealisticvalueofthesNMEcanbeestimatedfromits
comparing the results with state-of-the-art nuclear the-
CVC value [5], first used for spectral-shape calculations
ory calculations. To address the tension raised by the
in Ref. [18] and later by Paulsen et al. [17]. It should be
enhanced value of geff = 1.526(92) reported in Ref. [17],
A noted,though,thattheCVCvaluerepresentsan“ideal”,
we performed a high-precision measurement of the 99Tc
pertaining to a perfect many-body theory, which is not
β-decay spectrum using metallic magnetic calorimeter
the case for the nuclear-structure frameworks typically
quantum sensors with a low-energy threshold of 10 keV.
used for β-decay calculations of heavy open-shell nuclei,
The measured spectrum was compared to predictions
like the presently discussed 99Tc. To circumvent the use
from three nuclear models, each based on distinct nu-
of the CVC value, the experimental works [11, 13, 15]
clear Hamiltonians. In this Letter, we present a quan-
and theoretical works [12, 16] used sNME as a fitting
titative comparison between data and modern nuclear
parameter, together with geff. By this, it was possible to
theory,providingnewconstraintsongeff anddemonstrat- A
A reproduce both the measured β-spectral shape and the
ing the sensitivity of precision β-spectroscopy to nuclear
experimental (partial) half-life of the corresponding β-
structure effects.
transitionwithoutlosingtheCVCvalueg =1.0forthe
Theory.—Tofirstorder,thehalf-lifeofaβ-decaytran- A
vector coupling. This approach, termed the enhanced
sition can be written as:
spectrum-shape method (ESSM) in Ref. [19], is adopted
κ in the present work.
t = , (1)
1/2 C˜ In this Letter, we present new theoretical calculations
of the β spectrum of 99Tc (with Z = 43 and N = 56)
where κ is a constant [18, 19] and C˜ is the integrated and compare them to the measured spectrum in order
shapefunctionC,whichincorporatesvariousphase-space to extract the geff, using ESSM. To gain preliminary in-
A
factors and NMEs arising in the next-to-leading-order sightsintotheeffectiveaxialcouplinggeff intheβ-decay
A
expansion, as discussed in detail in Refs. [6, 7]. Our β- of 99Tc, we performed nuclear shell model (NSM) cal-
spectral shape and half-life analyses employ screening, culations and compared the results with the measured
radiative, and atomic-exchange corrections. For electron spectrum reported in Ref. [17]. As the β-decay rate
energies below some tens of keV, the distortion in the β- has a quadratic dependence on the sNME, two solutions
decay spectral shape is mostly dominated by the atomic werefoundforeachvalueofgeff,onecloserto[sNME(c)]
A
exchange correction, as originally studied for allowed β- and one further away from [sNME(f)] CVC value. In
decays [20]. This correction is responsible for the sharp references [21, 23] the related two sets of theoretical β-
upward trend of the spectral shape at low energies. spectra of 99Tc were introduced for two NSM Hamilto-
The complexity of the shape function C can, however, nians, namely jj45pnb [24] and glekpn [25]; In that pre-
be cast in a very simple dependence on the weak cou- liminary study, for the glekpn interaction we’ve used a
3
set of single particle energies fit to the A=94−98 mass below 1 Bq.
region, however here we use the default single particle
energies since it yields better experimental agreement
for this isotope. Comparison with the measured spec-
trum[17]indicatedthatbothsNMEvaluesusedwiththe
jj45pnb Hamiltonian yielded good agreement, while the
fit is somewhat poorer with sNME(c), and significantly
worsewithsNME(f)whenusingtheglekpn Hamiltonian,
further details of this preliminary study and its calcula-
tions can be found in Refs [21, 23].
In addition to the two NSM Hamiltonians described
above, we also employ the nuclear many-body method
known as the Microscopic Quasiparticle-Phonon Model
(MQPM). A concise presentation of the essentials of
NSM, and MQPM can be found in Ref. [2]. It should
be pointed out here that Ref. [17] employed the NSM
in their analysis, adopting the CVC value for the sNME
withoutfittingittotheexperimentalhalf-lifeofβ-decay.
A systematic comparison of various theoretical models
with the measured spectrum offers a more robust frame-
work for extracting the geff from high-precision spectral
A
data.
Experiment.— Metallic magnetic calorimeters
(MMCs) are some of the most advanced cryogenic FIG. 1. The detector setup used for the β-decay measure-
ment. The gold sample containing the 99Tc source is ther-
detectors for precision energy measurements. These
mally coupled to the MMC sensor via gold wire bonds. A
devices utilize paramagnetic ions embedded in metal-
support washer with a Kapton tape on top for thermal insu-
lic hosts—typically Au:Er or Ag:Er—as temperature
lation was placed below the sample to align its height with
sensors thermally coupled to metallic absorbers. The
that of the MMC sensor.
paramagnetic material is placed in a weak magnetic
field (<10 mT) and operated at temperatures below The detector setup consists of an MMC and a Mag-
100 mK. When an incident particle deposits energy nicon SQUID (Model X1) for signal readout (see Fig-
E in the absorber, it induces a temperature rise ∆T ure 1). The detector was installed and operated in a
in the paramagnetic bath. This temperature increase pulsed tube pre-cooled 3He/4He dilution refrigerator at
leads to a decrease in the total magnetization of the Lawrence Livermore National Laboratory (LLNL). The
paramagneticions,whichissensedbyasuperconducting signal was amplified using an additional array SQUID
pickupcoil. Themagneticsignalisthentransmittedtoa (Model X16) installed at the 4 K stage.
superconducting quantum interference device (SQUID) The output was low-pass filtered using a Stanford Re-
for precise readout. In this work, we employed an search Systems SRS 560 amplifier [30] with a 30 kHz
MMC fabricated at the Korea Research Institute of cutoff frequency. Data acquisition was performed over
Standards and Science (KRISS), incorporating 420 ppm 34 hours at 15 mK in un-triggered continuous mode to
of Ag:168Er as the sensor material Refs [26, 27]. capture the full information, with a sampling rate of
Radionuclide sources were prepared in the form of 4π 200 kS/s. A software trigger was applied using a trape-
solid-angle absorbers. A solution of 99Tc dissolved in ni- zoidal filter with a 1-ms shaping time.
tric acid was dropped onto a gold foil (2 mm × 2 mm For energy calibration, an external 133Ba source was
× 33 µm) and subsequently dried. The foil, containing used as a reference, while a 148Gd source was introduced
approximately 10 Bq of 99Tc activity, was then folded inside the setup for internal calibration. Although the α
androlledthreetimesinaprocessknownasthekneading signals from 148Gd were not observed due to slight mis-
method[28,29]. Thistechniqueenhancestheuniformity alignment, characteristic X-rays from 148Gd interactions
of 99Tc distribution within the absorber and improves weresuccessfullydetected. Thesimultaneousobservation
energyresolution. Aportionofthekneadedfoilwascut, ofγraysfrom133BaandX-raysresultingfrom148Gdpro-
encapsulated between two 25 µm thick gold foils, and videdmultiplereferencepointsforaccuratecalibrationof
compressed using a rolling machine. The final encapsu- the energy scale.
lated sample, measuring 1.2 mm × 1.3 mm × 50 µm, Two trapezoidal filters with different shaping parame-
was thermally coupled to the MMC sensor. The total ters were applied to the raw pulses. The long trapezoid
absorber mass was 1.75 mg, and the total heat capacity, was used to obtain information about the entire DC sig-
includingthecontributionfromtheparamagneticionsin nal, while the short trapezoid was designed to capture
the MMC sensor, was approximately 400 pJ in the 15- information corresponding only to the rise time of the
50 mK temperature range. The final 99Tc activity was DC signal [31, 32]. All data were analyzed using custom
4
MATLAB○R
scripts [33]. five most prominent γ peaks from 133Ba: 383.8 keV,
Results.—Inthebackgrounddataset,atotalof130,001 356.0 keV, 302.8 keV, 276.4 keV, and 80.9 keV. The re-
sultingcalibrationfunctionwasthenappliedtotheanal-
| events | were | recorded. | Among | them, | 80,903 | events ex- | | | | | | | | |
| ------ | ---- | --------- | ----- | ----- | ------ | ---------- | --- | --- | --- | --- | --- | --- | --- | --- |
ceededtheenergythresholddefinedfortheanalysisas10 ysis dataset. The positions of the Gd X-ray peaks were
keV. A pulse shape discrimination (PSD) cut was sub- compared between the calibration and analysis datasets,
sequently applied to select events consistent with 99Tc andnosignificantdiscrepancieswereobserved. Thiscon-
signal-like pulses. After that, 77,487 events remained for sistency validated the use of the same calibration func-
further analysis. tion for both datasets. Specifically, the 42.3097 keV,
Similarly, the calibration dataset obtained with an ex- 42.9968 keV, and 48.6957 keV peaks in the two datasets
| | 133Ba | | | | | | differedby0.14(11)keV,0.093(18)keV,and0.38(5)keV, | | | | | | | |
| ------ | ----- | ------ | --------- | ------ | ------- | ------ | ------------------------------------------------- | --- | --- | --- | --- | --- | --- | --- |
| ternal | | source | contained | 47,491 | events. | Out of | | | | | | | | |
these, 35,821 events passed the energy threshold. Af- respectively—well within the corresponding uncertain-
| ter applying | | the same | PSD | criteria, | 23,596 | events were | ties. | | | | | | | |
| ------------ | --- | -------- | --- | --------- | ------ | ----------- | ----- | --- | --- | --- | --- | --- | --- | --- |
retained.
FIG.2. 99Tcspectrumwithexternal133Bacalibrationsource.
| Details | of the | X-ray peaks | are | provided | in Table | I. | | | | | | | | |
| ------- | ---------- | ----------- | -------- | -------- | -------- | ---- | -------- | -------- | ---- | ----------- | ----- | ----------- | ----- | ------- |
| | | | | | | | FIG. 3. | Spectrum | of | calibrated | 99Tc, | plotted | with | a 1 keV |
| Figure | 2 presents | | the 99Tc | spectrum | measured | with | | | | | | | | |
| | | | | | | | binning, | compared | with | theoretical | | predictions | based | on the |
the external 133Ba source in place. Characteristic X-ray farCVC variants of the glekpn, jj45pnb, and MQPM models.
peaksfromGdandPtfluorescence,aswellasγlinesfrom Themodelswerefittedusingaχ2 method. Theresidualgraph
ν
133Ba, are clearly visible [34]. These peaks, along with below was calculated over the energy range from 0–280 keV,
| | | 133Ba | | | | | corresponding | | to the | χ2 fitting | range. | | | |
| -------- | ----------- | ----- | --------------- | ------------ | ----------- | ---------- | ------------- | ---- | ------ | ---------- | ------ | --------- | --- | --------- |
| the γ | lines from | | and | escape peaks | due | to partial | | | | ν | | | | |
| energy | deposition, | were | used | for energy | calibration | [35]. | | | | | | | | |
| The peak | positions | | were determined | | by Gaussian | fitting | | | | | | | | |
| | | | | | | | In our | ESSM | theory | framework, | | templates | | of the β- |
withalinearbackgroundmodel,andtheresultsaresum- spectra were generated for a set of g values (with
A
marized in Table I. g = 1.0), adjusting simultaneously the sNME value(s)
V
Mostpeaksexhibitexcellentagreementwithliterature to correctly reproduce the decay half-life. Then, reduced
values,with∆E typicallybelow0.3keVandenergyreso- chi-square (χ2) fitting was performed using the provided
ν
lutionsbetterthan1keVinthe66–383keVenergyrange. models,applyingsimpleshapecomparisonsandcalculat-
TheentriesinTableIlabeledas“Notdistinguished,”in- ingresidualstoidentifytheenergyregionswiththemost
| cluding | K and | K | transitions | from | Gd X-ray | and Au | | | | | | | | |
| ------- | ----- | --- | ----------- | ---- | -------- | ------ | ----------- | ----------- | --- | --- | --- | -------- | ---- | -------- |
| | β3 | β1 | | | | | significant | deviations. | | The | bin | size for | each | spectrum |
escape lines, correspond to peaks with low relative in- was set to 1 keV.
tensities [36]. Given the limited event statistics, these Areducedchi-squared(χ2)fittingprocedurewasthen
ν
| peaks | could | not be | individually | resolved | in | the spectral | | | | | | | | |
| ----- | ----- | ------ | ------------ | -------- | --- | ------------ | --- | --- | --- | --- | --- | --- | --- | --- |
appliedtocomparethetheoreticalmodelswiththemea-
fit. Theuncertaintiesreportedforeachvaluereflectonly suredspectrum. Thebinsizeforeachspectrumwassetto
thestatisticalerrorsderivedfromtheGaussianfits,with- 1keV.Thefitswereperformedbyminimizingχ2 overthe
ν
out accounting for potential systematic contributions. energy ranges 10–41 keV and 50–280 keV, excluding the
148Gd
The Gd X-ray peaks from the internal source GdX-rayregion. Simpleshapecomparisonsandresidual
and the 133Ba γ lines were used as primary reference analyseswereusedtoidentifyenergyregionswithsignifi-
points for the calibration. Additional fluorescence lines cantdeviations. Figure3visuallydemonstrateshowwell
| fromPtareattributedtomaterialsinthedetectorsetup | | | | | | | | | | | 99Tc | | | |
| ------------------------------------------------ | --- | --- | --- | --- | --- | --- | ---------- | ------ | ---- | --- | ---- | ----- | --- | --- |
| | | | | | | | each model | aligns | with | the | | data. | | |
and collimator. Several low-energy escape peaks (below We performed the fitting using geff with both the
A
20keV)showbroaderFWHManddegradedrelativereso- closeCVC[sNME(c)]andfarCVC[sNME(f)]models,and
lutionduetoincreasednoiseandlimitedeventstatistics. the results are summarized in Table II. For the farCVC
modelatgeff
A linear energy calibration was performed using the cases,theglekpn =1.2yieldsthebestagree-
A
5
TABLE I. Measured X-ray and gamma-ray lines, including their tabulated and measured energies.
Tabulated Measured FWHM
Origin Radiation type ∆E [keV]
energy [keV] [34, 35] energy [keV] resolution [keV]
X-rayK 42.3097(13) 42.30(10) 0.9(4) 0.01(10)
α2
Gdfluorescence X-rayK 42.9968(12) 42.9427(31) 0.686(10) 0.0541(33)
α1
frominternalsource X-rayK 48.5505(21)
β3 48.29(5)a 1.05(22) 0.41(5)
X-rayK 48.6957(21)
β1
Ptfluorescence X-rayK 65.1226(21) 65.005(22) 0.96(6) 0.118(22)
α2
fromcollimator X-rayK 66.8311(20) 66.78(5) 0.65(15) 0.06(5)
α1
γ 80.9979(11) 80.881(24) 0.77(8) 0.117(25)
γ 276.3989(12) 276.12(5) 0.57(13) 0.28(5)
133Bacalibrationlines γ 302.8508(5) 302.764(21) 0.60(5) 0.087(21)
γ 356.0129(7) 355.926(9) 0.653(21) 0.087(9)
γ 383.8485(12) 383.69(6) 0.82(14) 0.16(6)
Absorbed Escaping
133Ba γ line [keV] Au X-ray [keV]
66.993(23)[K ] 316.8555(26) 316.817(18) 0.37(5) 0.038(18)
383.8485 α2
68.8069(22)[K ] 315.0416(25) 315.09(4) 0.72(9) 0.05(4)
α1
66.993(23)[K ] 289.0199(24) 288.943(21) 0.51(5) 0.077(21)
α2
68.8069(22)[K ] 287.206(23) 287.051(26) 0.59(6) 0.155(26)
356.0129 α1
77.5773(35)[K ] 278.4356(36)
β3 277.911(31)a 0.88(8) 0.122(31)
77.983(37)[K ] 278.0329(38)
Escapelines β1
66.993(23)[K ] 235.8578(24) 235.659(23) 0.47(6) 0.199(23)
302.8508 α2
68.8069(22)[K ] 234.0439(23) 233.99(4) 0.63(11) 0.05(4)
α1
66.993(23)[K ] 209.4059(26) 209.137(12) 0.72(4) 0.269(12)
276.3989 α2
68.8069(22)[K ] 207.592(25) 207.36(5) 0.62(13) 0.24(5)
α1
66.993(23)[K ] 14.0049(25) 13.87(5) 1.49(21) 0.13(5)
80.9979 α2
68.8069(22)[K ] 12.191(25) 12.144(17) 0.97(6) 0.046(17)
α1
a Notdistinguished
considered, glekpn slightly outperforms the others, but
TABLE II. Results for the three considered nuclear models
allthreesuggestthataquenchedaxial-vectorcouplingof
ontheparametersofinterestg . Thereducedchi-squareand
A geff =1.0–1.2providesthebestagreementwiththedata.
p-value are also reported as an estimator of the goodness of A
Our result shows a clear deviation from that reported in
fit.
Ref. [17], and instead exhibits better consistency with a
Model geff sNME[fm3] χ2 p-value
A ν quenched g value comparable to those extracted from
CloseCVC A
glekpn 1.1 0.0674 1.1057 0.12 other nuclei.
jj45pnb 1.0 0.0681 1.2436 0.004 Conclusions.—In this work, we performed a high-
MQPM 1.0 0.0651 1.6437 <0.0001 resolutionmeasurementofthe β-decay spectrum of99Tc
usingmetallicmagneticcalorimeterquantumsensorsand
FarCVC(Bestfit)
glekpn 1.2 -0.0698 1.0664 0.22 compared the results with predictions from multiple nu-
jj45pnb 1.0 -0.0669 1.0966 0.14 clear models. The extracted value of the effective axial-
MQPM 1.0 -0.0683 1.0951 0.14 vector coupling, geff, deviates from the enhanced value
A
reported in Ref. [17] and is more consistent with values
observed in other nuclei. We attribute this discrepancy
ment with the experimental spectrum, with a reduced primarilytothedifferenttreatmentofthesNMEadopted
chi-squared value of χ2 = 1.0664 and a corresponding in Ref. [17]. By applying the ESSM, which simultane-
ν
p-value of 0.22. The jj45pnb model at geff = 1.0 also ously fits the measured β-spectrum and reproduces the
A
provides a good match with χ2 = 1.0966 and p-value of partialhalf-life,weplacedmorerobustconstraintsongeff
ν A
0.14. Similarly, the MQPM model at geff = 1.0 gives χ2 based on three nuclear theory approaches.
A ν
= 1.0951 and p-value of 0.14. All fits were performed Our results for the second-forbidden non-unique β−
with ν = 261 degrees of freedom. decay of 99Tc provide valuable insight into the effective
The reduced chi-squared values close to unity and p- axial-vector coupling strength geff in the mass region.
A
values well above 0.05 indicate that all three farCVC Theextractedquenchingpatternisconsistentwiththose
models are statistically compatible with the measured observed in heavier nuclei, such as 113Cd [10–13] and
spectrum. This consistency supports the reliability of 115In [14–16], suggesting that the quenching of g may
A
the farCVC scenario, where the sNME is adjusted to re- follow a systematic trend across mass regions. This im-
produce the experimental half-life. Among the models pliesthatthesamelevelofquenchingcouldbeappliedto
6
bothallowedandforbiddenvirtualtransitionscontribut- istryofResearch,InnovationandDigitization(theNEP-
ing to 0νββ NMEs, thereby simplifying the theoretical TUN project). M. Ramalho acknowledges support by
evaluation of NMEs. theOskarHuttunenFoundation. Weacknowledgegrants
In particular, our findings can inform the modeling of of computer capacity from the Finnish Grid and Cloud
100Mo, Infrastructure (persistent identifier urn:nbn: fi:research-
| 0νββ decay | in | | which | lies | in the | same | mass re- | | | | | | | |
| ---------- | ----- | ------ | -------- | --------- | ------ | -------- | -------- | ------------------ | --- | --- | ----------- | --- | ------ | ---- |
| | | | | | | | | infras-2016072533) | | and | the support | by | CSC–IT | Cen- |
| gion as | 99Tc. | If the | observed | quenching | | behavior | per- | | | | | | | |
sists across neighboring nuclei, it could facilitate a more ter for Science, Finland, for generous computational re-
| | | | | | | | | sources. | G. B. | Kim | and I. Kim | acknowledge | | support |
| ----------- | ------------ | --------- | ------- | ------------ | ---- | ------- | ------ | -------- | --------------- | --- | ---------- | ----------- | --- | ------- |
| unified and | robust | framework | | for | 0νββ | nuclear | matrix | | | | | | | |
| | | | | | | | | from the | U.S. Department | | of Energy, | performed | | under |
| element | calculations | | in this | mass region. | | | | | | | | | | |
theauspicesofLawrenceLivermoreNationalLaboratory
| These | results | demonstrate | | the sensitivity | | of | precision | | | | | | | |
| -------------- | ------- | ----------- | --- | --------------- | ------- | --- | --------- | -------------- | -------------- | ------------------ | -------- | -------- | --------- | ------ |
| | | | | | | | | under Contract | | DE-AC52-07NA27344. | | | This work | was |
| β-spectroscopy | | to nuclear | | structure | effects | and | provide | | | | | | | |
| | | | | | | | | funded by | the Laboratory | | Directed | Research | and | Devel- |
valuable benchmarks for the development and validation opment program of Lawrence Livermore National Labo-
| of nuclear | theory—an | | effort | central | to the | mission | of the | | | | | | | |
| ---------- | --------- | --- | ------ | ------- | ------ | ------- | ------ | --- | --- | --- | --- | --- | --- | --- |
ratory (23-LW-043).
| gA-EXPERT | | collaboration. | | | | | | | | | | | | |
| --------- | --- | -------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
ThisworkwascarriedoutwithinthegA-EXPERiment
Acknowledgments.— H. J. Kim acknowledges support and Theory (gA-EXPERT) collaboration—an interna-
by the National Research Foundation of Korea (NRF) tional effort involving Kyungpook National University
grant, funded by the Korean government (MSIT), Con- (KNU), the Institute for Basic Science (IBS), the Korea
tract No. RS-2024-00348317. J. W. Song acknowledges Research Institute of Standards and Science (KRISS),
support by the National Research Foundation of Korea the University of Jyv¨askyl¨a, and the Finnish Institute
(NRF)grant,fundedbytheMinistryofEducation,Con- of Natural Resources (Luke)—dedicated to determining
tract No. RS-2023-00274589. J. Kotila and J. Suhonen theeffectiveaxial-vectorcouplingconstant, geff, through
A
acknowledge support from project PNRR-I8/C9-CF264, precision β-decay measurements and state-of-the-art nu-
Contract No. 760100/23.05.2023 of the Romanian Min- clear theory comparisons.
[1] J. Engel and J. Menendez, Rep. Prog. Phys. 60, 046301 [19] A. Kumar, P. C. Srivastava, and J. Suhonen, Eur. Phys.
| (2017). | | | | | | | | J. A | 57, 225 | (2021). | | | | |
| ------- | --- | --- | --- | --- | --- | --- | --- | ---- | ------- | ------- | --- | --- | --- | --- |
O.Nitescu,S.Stoica,andF.Sˇimkovic,Phys.Rev.C107
| [2] H. Ejiri, | J. | Suhonen, | and | K. Zuber, | Phys. | Rep. | 797, 1 | [20] | | | | | | |
| ------------- | --- | -------- | --- | --------- | ----- | ---- | ------ | ------ | ------- | --- | --- | --- | --- | --- |
| (2019). | | | | | | | | (2023) | 025501. | | | | | |
[3] M. Agostini, G. Benato, J. A. Detwiler, J. Men´endez, [21] M. Ramalho and J. Suhonen, Phys. Rev. C 109, 034321
| and | F. Vissani, | Rev. | Mod. | Phys. | 95, 025002 | | (2023). | (2024). | | | | | | |
| --- | ----------- | ---- | ---- | ----- | ---------- | --- | ------- | ------- | --- | --- | --- | --- | --- | --- |
[4] J.Suhonen,From Nucleons to Nucleus: Concepts of Mi- [22] M. Ramalho, J. Suhonen, A. Neacsu, and S. Stoica,
croscopic Nuclear Theory (Springer, Berlin, 2007). Front. Phys. 12:1455778 (2024).
[5] H. Behrens and W. Bu¨hring, Electron Radial Wave [23] M. Ramalho and J. Suhonen, Il Nuovo Cimento 47 C,
| Functions | | and Nuclear | Beta-decay | | (International | | Series | 377 (2024). | | | | | | |
| --------- | --- | ----------- | ---------- | --- | -------------- | --- | ------ | ----------- | --- | --- | --- | --- | --- | --- |
of Monographs on Physics) (Clarendon Press, Oxford, [24] A. F. Lisetskiy, B. A. Brown, M. Horoi, and H. Grawe,
| 1982). | | | | | | | | Phys. | Rev. C | 70, 0444314 | (2004). | | | |
| ------ | --- | --- | --- | --- | --- | --- | --- | ----- | ------ | ----------- | ------- | --- | --- | --- |
[6] M. Haaranen, P. C. Srivastava, and J. Suhonen, Phys. [25] H. Mach et al., Phys. Rev. C 41, 226 (1990).
Rev. C 93, 034308 (2016). [26] J. W. Song, S. G. Kim, H. S. Kim, H. J. Kim, and
[7] M. Haaranen, J. Kotila, and J. Suhonen, Phys. Rev. C M. K. Lee, J. Low Temp. Phys. 216, 436 (2024).
95, 024327 (2017). [27] J. W. Song, Y. C. Cho, H. J. Kim, and M. K. Lee, J.
[8] J. Suhonen, Front. Phys. 5, 55 (2017). Low Temp. Phys. 218, 110 (2025).
[9] J.SuhonenandJ.Kostensalo,Front.Phys.7,29(2019). [28] M. P. Croce, A. S. Hoover, M. W. Rabin, E. M. Bond,
[10] L. Bodenstein-Dresler et al. (The COBRA Collabora- L.E.Wolfsberg,D.R.Schmidt,andJ.N.Ullom,J.Low
tion), Phys. Lett. B 800, 135092 (2020). Temp. Phys. 184, 938 (2016).
[11] J.Kostensalo, J.Suhonen,J.Volkmer,S.Zatschler,and [29] A. S. Hoover et al., Anal. Chem. 87, 3996 (2015).
K. Zuber, Phys. Lett. B 822, 136652 (2021). [30] Stanford Research Systems, SR560 Low-
[12] J.Kostensalo,E.Lisi,A.Marrone,andJ.Suhonen,Phys. Noise Preamplifier, Rev. 2.8 (Stanford Re-
Rev. C 107, 055502 (2023). search Systems, Sunnyvale, CA, 2006),
[13] I. Bandac et al., Eur. Phys. J. C (2024) 84:1158. https://www.thinksrs.com/products/sr560.html
[14] A. F. Leder et al., Phys. Rev. Lett. 129, 232502 (2022). [31] V. T. Jordanov, G. F. Knoll, A. C. Huber, and J.
[15] L. Pagnanini et al. (The ACCESS Collaboration), Phys. A. Pantazis, Nucl. Instrum. Methods Phys. Res., Sect.
| Rev. | Lett. | 133, 122501 | (2024). | | | | | A 353, | 261 (1994). | | | | | |
| ---- | ----- | ----------- | ------- | --- | --- | --- | --- | ------ | ----------- | --- | --- | --- | --- | --- |
[16] J.Kostensalo,E.Lisi,A.Marrone,andJ.Suhonen,Phys. [32] A. R. L. Kavner,D. Lee,S. T. P. Boyd,S. Friedrich,
Rev. C 110, 045503 (2024). I. Jovanovic, and G. B. Kim, J. Low Temp. Phys. 209,
| [17] M. Paulsen | | et al., | Phys. | Rev. C | 110, 055503(2024). | | | 1070 | (2022). | | | | | |
| --------------- | --- | ------- | ----- | ------ | ------------------ | --- | --- | ---- | ------- | --- | --- | --- | --- | --- |
[18] A. Kumar, P. C. Srivastava, J. Kostensalo, and J. Suho- [33] The MathWorks, Inc., MATLAB, version R2016b
nen, Phys. Rev. C 101, 064304 (2020). (The MathWorks, Inc., Natick, MA, 2016),
7
https://www.mathworks.com/products/matlab.html [35] Y. Khazov, A. Rodionov, and F. G. Kondev, Nucl.
[34] R. D. Deslattes, E. G. Kessler Jr., P. Indelicato, L. Data Sheets 112, 855 (2011).
De Billy, E. Lindroth, and J. Anton, Rev. Mod. Phys. [36] A. C. Thompson et al., X-Ray Data Booklet (Center
75, 35 (2003). forX-RayOpticsandAdvancedLightSource, Lawrence
Berkeley National Laboratory, Berkeley, CA, 2009).
.llm-wiki/raw/sources/SRC-2026-05-27-005/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-005",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "https://arxiv.org/pdf/2506.17544",
"url": "https://arxiv.org/pdf/2506.17544",
"format": "web"
}
.llm-wiki/raw/sources/SRC-2026-05-27-005/original/source.html
BIN
View File
Binary file not shown.
.llm-wiki/raw/sources/SRC-2026-05-27-006/extracted.md
+342
View File
@@ -0,0 +1,342 @@
IL NUOVO CIMENTO Vol. ?, N. ? ?
Shell-model treatment of the β decay of 99Tc
M. Ramalho(1) and J. Suhonen(1)(2)
(1) DepartmentofPhysics,UniversityofJyva¨skyla¨,P.O.Box35,FI-40014,Jyva¨skyl¨a,Finland
(2) International Centre for Advanced Training and Research in Physics (CIFRA), P.O. Box
MG12, 077125 Bucharest-Magurele, Romania
Summary.— Inthepresentpaperwetreatthesecond-forbiddennon-unique(2nd-
nu)ground-state-to-ground-stateβ−decay99Tc(9/2+)→ 99Ru(5/2+),witha100%
branchingratio,withintheframeworkofthenuclearshellmodel(NSM).Theenergy
spectrum of the electrons emitted in this β-decay transition (β-electron spectrum)
is sensitive to the wave functions of the involved initial (9/2+) and final (5/2+)
nuclear ground states through the many involved nuclear matrix elements (NME).
The β-electron spectrum of this transition is potentially indicative of the effective
value, geff, of the weak axial coupling, g , of crucial importance for extraction of
A A
information on beyond-the-standard-model physics from the results of the present
and future rare-events experiments. We describe the β spectral shape of this de-
cay by using a state-of-the-art β-decay formalism and compute the many involved
NME using the well established NSM Hamiltonians jj45pnb and glekpn. We have
found a strong dependence of the β spectral shape on the value of g making it a
A
good candidate for determination of the value of geff through comparison with the
A
corresponding experimental β spectral shape. We have also found an interesting
dependence of the β spectral shape on the value of the so-called small relativistic
vector NME, sNME, used to match the computed half-life with the measured one.
1. – Introduction
Rare-eventsexperimentstypicallylookforbeyond-the-standard-model(BSM)physics
by,e.g.,measuringrarenuclearβ decays,doubleβ decays,and(anti)neutrinoandWIMP
(weakly interacting massive particle) scatterings off nuclei in searches for astrophysical
neutrinos and cold dark matter of the Universe. In recent years, a booming interest in
these experiments has concentrated on studies of β electrons (electrons emitted in β
−
decays) and their energy distributions, the so-called β-electron spectra. Experimental
and/ortheoreticalinformationonthesespectraiscrucialfor,e.g.,resolvingtheanomalies
related to the antineutrino flux from nuclear reactors and for resolving the common
backgroundsintherare-eventsexperimentsthemselves. Also,pinningdowntheeffective
'Societ`aItalianadiFisica 1
2 M.RAMALHOand J.SUHONEN
values of weak couplings is a considerable incentive for those present and future β-decay
experiments able to tackle the spectral shapes of β electrons, see the review [1].
Nuclearβdecaysvaryincomplexity: fromallowedtohighlyforbiddenones[2,3]. Like
in the case of allowed β decays, also in the case of forbidden unique β decays the lepton
phase space can be separated from the nuclear part, resulting in a universal β spectral
shape,independentofnuclear-structuredetails[2]. Ofspecialinterestfortherare-events
experiments are the forbidden non-unique β decays for which the β spectral shapes can
bestronglynuclear-structuredependentthroughseveralnuclearmatrixelements(NME).
ThevaluesoftheseNMEarenon-triviallydeterminedbythewavefunctionsoftheinitial
and final states of a β-decay transition.
In addition to the many NME, the (partial) half-life of a forbidden non-unique β
transition depends on the values of the weak vector and axial-vector couplings, g and
V
g [3]. In atomic nuclei, the CVC (conserved vector current) hypothesis sets the value
A
g =1.0, whereas the PCAC (partially conserved axial-vector current) hypothesis leads
V
only to the so-called effective value of g , recently discussed in the reviews [1, 4, 5].
A
On the other hand, the so-called bare-nucleon value g = 1.27 stems from data on the
A
decay of an isolated neutron. Typically, the effective value of g is quenched relative
A
to the bare-nucleon value which can have drastic effects on the sensitivity estimates of
rare-events experiments trying to detect the neutrinoless double beta decay [6, 7], of
crucial importance in the search for the BSM physics. Only in rare cases there is an
enhancement of g present [8].
A
As discussed in [9, 10], information on the value of g can be gained by using the
A
so-called spectrum-shape method (SSM). Use of SSM requires a β-electron spectrum
with a notable g dependence in its shape. In this case information on the effective
A
value of g can be gained through the comparison of computed template β spectra,
A
for different g values, with the measured one in the aim to find a match. Such SSM
A
analyses of β-spectral shapes of individual β transitions have been done recently for
−
the fourth-forbidden non-unique β decays of 113Cd and 115In in [11, 12, 13].
Anadditionalingredientinthetheoreticalanalysesofβspectralshapesistheso-called
small relativistic vector NME, the sNME, used to fix the measured (partial) half-life in
the so-called enhanced SSM [14, 15]. In spite of its smallness, sNME can influence the
(partial) half-lives and shapes of β-electron spectra quite strongly, see [12, 16, 17]. The
sNME gathers its major contributions outside the proton (neutron) valence major shell
thatcontainstheproton(neutron)Fermisurfacemakingithardtobecalculatedinmany
nuclear models.
In the present paper we investigate the g and sNME dependence of the second-
A
forbidden non-unique (2nd-nu) β decay of the 9/2+ ground state of 99Tc to the 5/2+
−
ground state of 99Ru, with a branching ratio of 100%.
2. – Theoretical Background
.
21. NSM Hamiltonians. – The NSM calculations were performed using the software
KSHELL [18] with the interactions jj45pnb [19] and glekpn [20] with the set of single-
particle energies fit for the mass region of A = 94 98. Their model spaces consist of
−
the proton 0f (2)-1p -1p -1g orbitals and the neutron 0g -1d -1d -2s -
5/2 3/2 1/2 9/2 7/2 5/2 3/2 1/2
0h orbitals for jj45pnb, as well as 0f (1)-0f (2)-1p -1p -1g proton orbitals
11/2 7/2 5/2 3/2 1/2 9/2
and 0g (1)-0g -1d -1d -2s neutron orbitals for glekpn. The restrictions in the
9/2 7/2 5/2 3/2 1/2
orbitaloccupationsareimposedinordertoreducethecomputationalburden. Webelieve,
SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 3
99Tc 0 0 .7 . 5 5 ( 3 2 5 2 3 2 7 2 − − − )+ ( ( 3 2 9 2 7 2 5 2 ) ) + + + + 1 2 1+ 1 2 5 2 3 2 1 2 3 − − + + 1 2 1+ 3 2 7 2 − + 1 9 2 2 1 + − 9 2−
0.25
7 2 + 1 2 5 2 − + 9
5
2 +
+
1 2−
7 2−
2
5+ 2 0 9 2 + 7 2 + 5 2−
Exp. jj45pnb glekpn
)VeM(ygrenEnoitaticxE 99Ru 1 2 3 2+ + 5 2 + 9 2 + 5 3 2+ + 9+ (5 2)+ 9 2 + 2 2 7+ (1 2)+ (5 2 7 2 ) + + 2 1 2 1 2 1+ + 5 2 + (3 2)+ 1 2 +
7+
3 2 + 2 7 2 + 3 2 + 3+ 2
3+
2 5+ 7+ 5+ 2 3+ 5+ 2 2 2 2
Exp. jj45pnb glekpn
Fig.1.: Energylevelschemesfor99Tcand99RuwiththeircorrespondingNSM-computed
values using the interactions jj45pnb and glekpn. Experimental data are taken from the
ENDSF [21] evaluation. Levels in blue are the initial and final states relevant for this
work.
however,thattheseposedlimitationsdonotaffectmuchthewavefunctionsoftheinitial
and final ground states, relevant for the present work.
With the considered truncations we have calculated the excitation spectra for both
interactions and compare them with the experimental spectrum in Fig 1. It can be
seen that the jj45pnb Hamiltonian works, overall, much better than the glekpn one, in
particular for the ground state of 99Tc.
To further assess the reliability of the Hamiltonians employed, we have analyzed the
magnetic dipole and electric quadrupole moments of the 99Tc and 99Ru ground states.
The analysis can be seen in Table I, where once again the jj45pnb Hamiltonian does
better overall than the glekpn Hamiltonian.
.
22. β spectral shapes and half-lives. – The half-life of a β transition can be obtained
fromt =κ/C˜,whereκisaconstant[14,15]andC˜ istheintegratedshapefunctionC,
1/2
(1) Orbital fixed with maximum amount of nucleons, e.g, 8 for 0f .
7/2
(2) Orbital with at least 4 particles.
TableI.: Comparisonofexperimentalandtheoretical(jj45pnbandglekpnHamiltonians)
nuclear magnetic dipole (in units of nuclear magneton) and electric quadrupole (in units
of barns) moments for 99Tc and 99Ru, for the experimental ground states and their
correspondents in theory.
99Tc(9/2+) 99Ru(5/2+)
Exp. jj45pnb glekpn Exp. jj45pnb glekpn
µ (µ ) +5.678(2) +5.662 +5.4160 -0.641(5) -0.2875 +1.0826
N
Q (b) -0.129(6) +0.0218 +0.2428 +0.079(4) -0.0018 -0.3222
| 4 | | | | | | | | M.RAMALHOand | | |
| --- | --- | --- | --- | --- | --- | --- | --- | ------------ | --- | --- |
J.SUHONEN
Table
II.: Values of the small relativistic NME emerging from the fitting procedure for
the 99Tc(9/2+) Ru(5/2+) decay. The sNME values (1) and (2) shown reproduce the
→
| experimental | | half-life | for | their | given geff. | | | | | |
| ------------ | --- | --------- | --- | ----- | ----------- | --- | --- | --- | --- | --- |
A
| | jj45pnb | | - CVC =−0.1705 | | | | glekpn | - CVC | =−0.0080 | |
| ---- | ------- | ------- | -------------- | ------- | --- | ---- | ------ | ------- | -------- | ------- |
| geff | | sNME(1) | | sNME(2) | | geff | | sNME(1) | | sNME(2) |
| A | | | | | | A | | | | |
| 0.8 | | 0.0496 | | -0.0856 | | 0.8 | | 0.1042 | | -0.0311 |
| 0.9 | | 0.0582 | | -0.0772 | | 0.9 | | 0.1091 | | -0.0257 |
| 1.0 | | 0.0666 | | -0.0687 | | 1.0 | | 0.1140 | | -0.0202 |
| 1.1 | | 0.0750 | | -0.0601 | | 1.1 | | 0.1189 | | -0.0148 |
| 1.2 | | 0.0833 | | -0.0515 | | 1.2 | | 0.1237 | | -0.0093 |
fullofphase-spacefactorsandNMEsinthenext-to-leading-orderexpansion,asdiscussed
in detail in [9, 10]. Our calculations employ screening, radiative, and atomic exchange
corrections. The most essential to the current study, due to its small β-decay Q value,
is the atomic exchange correction which was originally derived for allowed β decays [22];
the effects of this correction are the most important for electron energies below 50 keV.
The complexity of the shape function C can, however, be cast in a very simple de-
| pendence | | on the | weak couplings | | by writing | | | | | |
| -------- | --- | ------ | -------------- | ----- | ---------- | ---------- | --- | --------- | ---- | --- |
| | | | | )=g2C | )+g2C | | | | | |
| (1) | | | C(w e | | V (w e | A (w e )+g | V g | A C VA (w | e ), | |
| | | | | V | | A | | | | |
where w is the total (mass plus kinetic) energy of the emitted electron. The variation
e
of the shape of the β-electron spectrum with the value of g comes from the subtle
A
interference of the combined vector C (w ) and axial C (w ) parts with the mixed
| | | | | | V | e | | A e | | |
| ------------ | --- | ---- | ---- | ------ | --- | --- | --- | --- | --- | --- |
| vector-axial | | part | C (w | ) [9]. | | | | | | |
| | | | VA e | | | | | | | |
Forthecurrentlystudied2nd-nudecaytransition99Tc(9/2+) 99Ru(5/2+)wetake
→
theCVC-compatiblevalueg =1.0fortheweakvectorcoupling,andexploretheimpact
V
ofthequenchedgeff
| | | | values,rangingfrom0.8 | | | 1.2(seeTableII),ontheβ | | | spectralshape. | |
| --- | --- | --- | --------------------- | --- | --- | ---------------------- | --- | --- | -------------- | --- |
| | | | A | | | − | | | | |
.
23.SmallrelativisticNuclearElements.–Animportantcontributortotheβ spectral
shape and the half-life is the small relativistic NME (sNME) [12, 15, 16, 17]. In these
works, by fitting the value of the sNME, alongside with geff, one is able to reproduce
A
both the β spectral shape and the half-life of a β transition. The computation of the
value of the sNME is a challenge for the ISM. This is why its CVC value (also presented
in Table II) can be used as a reference [3]. It should be noted, though, that the CVC
value represents an ”ideal”, pertaining to a perfect many-body theory, which is not the
case for NSM, mainly due to its restricted proton and neutron valence spaces.
Following the philosophy adopted in [12, 15, 16, 17], we search for the values of the
sNME that reproduce the experimental half-life of the 99Tc(9/2+) Ru(5/2+) transi-
→
tion. Sincetheβ-decayratehasaquadraticdependencyonthesNME,twosolutionsare
geff,
| usually | found | for | each value | of | as presented | in | Table | II. | | |
| ------- | ----- | --- | ---------- | --- | ------------ | --- | ----- | --- | --- | --- |
A
| 3. – | Results | for | the β | spectral | shapes | | | | | |
| ---- | ------- | --- | ----- | -------- | ------ | --- | --- | --- | --- | --- |
Theβ-decaytransitionofinterest,99Tc(9/2+) Ru(5/2+),is2nd-nuandissensitive
→
to the Q value of the decay. Thus, to ensure reliability of our β spectral-shape calcula-
SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 5
10 3
−
×
8
6
4
2
0
0 50 100 150 200 250 300
Electronenergy(keV)
).mron(Ed/Nd
jj45pnb glekpn
10 3
−
×
12
8
4
0
0 50 100 150 200 250 300
Electronenergy(keV)
Fig. 2.: Computed β spectral shapes for the decay transition 99Tc(9/2+) Ru(5/2+)
→
using the Hamiltonians jj45pnb (left panel) and glekpn (right panel). The crossed-blue
curvescorrespondtothefittedsNMEclosesttotheCVC-valueandthecircled-redtothe
farthest(seeTableII),withgeff =1.0. Thegray-hatched(gray-crossed)areacorresponds
A
to the range geff = 0.8 1.2 with all the fitted sNME values chosen closest to (farthest
A −
from)thecorrespondingCVCvalues. Allcurvesarenormalizedtounitareaunderthem.
tions, we have fixed the energy difference of the initial 9/2+ and final 5/2+ state by the
experimental Q value. We plot our results for the β spectral shapes of this transition in
Fig. 2. There the crossed-blue (circled-red) curves correspond to a typical moderately
quenched value geff =1.0 [4] and the fitted sNME value closer to (farther from) its CVC
A
value, see Table II. The gray-hatched (gray-crossed) area, in turn, corresponds to the
range geff =0.8 1.2 with all the sNME values chosen closest to (farthest from) the cor-
A −
respondingCVCvalues,thesNMEvaluesbeingshowninTableII. Interestinglyenough,
thetwodifferentoptionsforthesNMEvalue(thetwodifferenthatchedregionsinFig.2)
sometimes produce drastically different shapes, as can be seen for glekpn, a pattern also
recorded in [17]. As for the interaction jj45pnb, both hatched regions overlap nicely and
there seems not to be any strong dependency of the β spectral shape on the choice of
the value of sNME.
Since the β spectral shapes produced by the two interactions clearly disagree, it is
hard to judge which one of the shapes is the more realistic one. Based on both the level
schemesinFig. 1andthemagneticandelectricmomentsofTableIitistemptingtosay
that the jj45pnb Hamiltonian is better suited for these particular nuclei. Furthermore,
looking at a recently measured experimental β spectral shape for this transition in [23],
especially in their Fig. 5, one can see a striking resemblance of it to the spectral shapes
forbothfittedsNME(1,2) ofthejj45pnb Hamiltonian(seeTableII)andabitlessstriking
forthesolutionsNME(2) fortheglekpn Hamiltonian. However,onlyapropercomparison
with data will allow us to draw any definite conclusions concerning the effective value of
the axial coupling and the proper value of sNME. This we leave as a future exercise.
| 6 | | | | | M.RAMALHOand | |
| --- | --- | --- | --- | --- | ------------ | --- |
J.SUHONEN
| 4. – Summary | and | conclusions | | | | |
| ------------ | --- | ----------- | --- | --- | --- | --- |
We have performed shell-model calculations of the β spectral shape related to the
second-forbidden non-unique β -decay transition 99Tc(9/2+) 99Ru(5/2+), with a
−
→
100% branching ratio. We use a state-of-the-art β-decay formalism and compute the
many involved NME using the well established NSM Hamiltonians jj45pnb and glekpn.
When compared with future available data, the β-electron spectrum of this transition
is potentially indicative of the effective value of the weak axial coupling, g , of crucial
A
importance for extraction of information on beyond-the-standard-model physics from
the results of the present and future rare-events experiments. We also point out the
important role played by the small relativistic vector matrix element, used to reproduce
the measured half-life of the transition, in shaping the β-electron spectrum.
5. – Acknowledgments
WeacknowledgethesupportbyCSC–ITCenterforScience,Finland,forthegenerous
| computational | resources, | which | were | essential to | this project. | |
| ------------- | ---------- | ----- | ---- | ------------ | ------------- | --- |
REFERENCES
| [1] Ejiri, | H., Suhonen, | J., and | Zuber, | K., Phys. Rep., | 797 (2019) 1. | |
| ---------- | ------------ | ------- | ------ | --------------- | ------------- | --- |
[2] Suhonen,J.,FromNucleonstoNucleus: ConceptsofMicroscopicNuclearTheory,Springer,
| Berlin, | 2007. | | | | | |
| ------- | ----- | --- | --- | --- | --- | --- |
[3] Behrens, H. and Bu¨hring, W., Electron Radial Wave Functions and Nuclear Beta-decay,
| Clarendon | Press, | Oxford, | 1982. | | | |
| --------------- | ------------------ | --------- | ---------- | ----------------- | -------------- | --- |
| [4] Suhonen, | J., Front. | Phys., | 5 (2017) | 55. | | |
| [5] Suhonen, | J. and Kostensalo, | | J., Front. | Phys., 7 (2019) | 29. | |
| [6] Engel, | J. and Men´endez, | J., | Rep. Prog. | Phys., 60 | (2017) 046301. | |
| [7] Agostini, | M. et al., | Rev. Mod. | Phys., | 95 (2023) 025002. | | |
| [8] Kostensalo, | J. and | Suhonen, | J., Phys. | Lett. B, 781 | (2018) 480. | |
[9] Haaranen, M., Srivastava, P. C., and Suhonen, J., Phys. Rev. C, 93 (2016) 034308.
| [10] Haaranen, | M., Kotila, | J., | and Suhonen, | J., Phys. | Rev. C, 95 (2017) | 024327. |
| -------------- | ----------- | --- | ------------ | --------- | ----------------- | ------- |
[11] Bodenstein-Dresler, L. et al. (The COBRA Collaboration), Phys. Lett. B, 800 (2020)
135092.
[12] Kostensalo, J., Suhonen, J., Volkmer, J., Zatschler, S., and Zuber, K., Phys. Lett. B, 822
| (2021) | 136652. | | | | | |
| ----------- | ------------- | ----- | ----------- | ---------- | ------- | --- |
| [13] Leder, | A. F. et al., | Phys. | Rev. Lett., | 129 (2022) | 232502. | |
[14] Kumar, A., Srivastava, P. C., Kostensalo, J., and Suhonen, J., Phys. Rev. C, 101 (2020)
064304.
[15] Kumar, A., Srivastava, P. C., and Suhonen, J., Eur. Phys. J. A, 57 (2021) 225.
[16] Kostensalo, J., Lisi, E., Marrone, A., and Suhonen, J., Phys. Rev. C, 107 (2023) 055502.
| [17] Ramalho, | M. and | Suhonen, | J., Submitted | to Phys. | Rev. C (2023). | |
| ------------- | ---------- | -------- | ------------- | ---------- | -------------- | --- |
| [18] Shimizu, | N. et al., | Comp. | Phys. Comm., | 244 (2019) | 372. | |
[19] Lisetskiy,A.F.,Brown,B.A.,Horoi,M.,andGrawe,H.,Phys. Rev. C,70(2004)044314.
| [20] Mach, | H. and Warburton, | | E. K. et | al., Phys. Rev. | C, 41 (1990) | 226. |
| ---------- | ----------------- | --- | -------- | --------------- | ------------ | ---- |
[21] From ENSDF database as of Nov, 2023. Version available at
http://www.nndc.bnl.gov/ensarchivals/.
| [22] Nitescu, | O., Stoica, | S., and | Simkovic, | F., Phys. Rev. | C, 107 (2023) | 025501. |
| ------------- | ----------- | ---------------- | --------- | -------------- | ------------- | ------- |
| [23] Paulsen, | M. et al., | arXiv:2309.14014 | | [nucl-ex]. | | |
.llm-wiki/raw/sources/SRC-2026-05-27-006/manifest.json
+8
View File
@@ -0,0 +1,8 @@
{
"id": "SRC-2026-05-27-006",
"captured": "2026-05-27",
"packet_version": "1.0",
"title": "https://arxiv.org/pdf/2312.07448",
"url": "https://arxiv.org/pdf/2312.07448",
"format": "web"
}
.llm-wiki/raw/sources/SRC-2026-05-27-006/original/source.html
BIN
View File
Binary file not shown.