diff --git a/.llm-wiki/WIKI_SCHEMA.md b/.llm-wiki/WIKI_SCHEMA.md new file mode 100644 index 0000000..67533e4 --- /dev/null +++ b/.llm-wiki/WIKI_SCHEMA.md @@ -0,0 +1,34 @@ +# LLM Wiki Schema + +## Ownership Rules + +| Path | Owner | Rule | +|------|-------|------| +| raw/** | extension | immutable after capture | +| wiki/** | model + user | editable knowledge pages | +| meta/* | extension | auto-generated | +| . | human + explicit request | operating rules | + +## Source Packet Format + +``` +raw/sources/SRC-YYYY-MM-DD-NNN/ + manifest.json + original/ + extracted.md + attachments/ +``` + +## Page Types + +- **source** — what this specific source says +- **entity** — people, orgs, tools, products +- **concept** — ideas, patterns, frameworks +- **synthesis** — cross-source theses and tensions +- **analysis** — durable filed answers from queries +- **requirement** — atomic requirements with status, priority, and traceability + +## Linking Style + +- Internal: [[folder/page-name]] +- Citation: [[sources/SRC-YYYY-MM-DD-NNN]] diff --git a/.llm-wiki/config.json b/.llm-wiki/config.json new file mode 100644 index 0000000..39d4975 --- /dev/null +++ b/.llm-wiki/config.json @@ -0,0 +1,7 @@ +{ + "name": "Beta Spectrum Calculator — High-precision beta decay spectrum simulation for ⁹⁹Tc and other forbidden decays", + "mode": "personal", + "topic": "Beta Spectrum Calculator — High-precision beta decay spectrum simulation for ⁹⁹Tc and other forbidden decays", + "created": "2026-05-27", + "version": "1.0" +} diff --git a/.llm-wiki/meta/backlinks.json b/.llm-wiki/meta/backlinks.json new file mode 100644 index 0000000..57bf435 --- /dev/null +++ b/.llm-wiki/meta/backlinks.json @@ -0,0 +1,20 @@ +{ + "sources/SRC-2026-05-27-001": [ + "sources/SRC-2026-05-27-001" + ], + "sources/SRC-2026-05-27-002": [ + "sources/SRC-2026-05-27-002" + ], + "sources/SRC-2026-05-27-003": [ + "sources/SRC-2026-05-27-003" + ], + "sources/SRC-2026-05-27-004": [ + "sources/SRC-2026-05-27-004" + ], + "sources/SRC-2026-05-27-005": [ + "sources/SRC-2026-05-27-005" + ], + "sources/SRC-2026-05-27-006": [ + "sources/SRC-2026-05-27-006" + ] +} diff --git a/.llm-wiki/meta/events.jsonl b/.llm-wiki/meta/events.jsonl new file mode 100644 index 0000000..3e21494 --- /dev/null +++ b/.llm-wiki/meta/events.jsonl @@ -0,0 +1,13 @@ +{"timestamp":"2026-05-27T11:55:40.409Z","kind":"bootstrap","topic":"Beta Spectrum Calculator — High-precision beta decay spectrum simulation for ⁹⁹Tc and other forbidden decays","mode":"personal"} +{"timestamp":"2026-05-27T11:55:55.301Z","kind":"capture","source_id":"SRC-2026-05-27-001","file_path":"/home/unzhakov/workspace/beta_shape_pnpi/docs/gA-extraction/01-ssm-algorithm.md","format":"markdown"} +{"timestamp":"2026-05-27T11:56:08.796Z","kind":"capture","source_id":"SRC-2026-05-27-002","file_path":"/home/unzhakov/workspace/beta_shape_pnpi/docs/nuclear-structure/01-shape-factors.md","format":"markdown"} +{"timestamp":"2026-05-27T11:59:21.939Z","kind":"capture","source_id":"SRC-2026-05-27-003","url":"https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download","format":"web"} +{"timestamp":"2026-05-27T12:00:22.177Z","kind":"capture","source_id":"SRC-2026-05-27-004","url":"https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf","format":"web"} +{"timestamp":"2026-05-27T12:01:29.331Z","kind":"capture","source_id":"SRC-2026-05-27-005","url":"https://arxiv.org/pdf/2506.17544","format":"web"} +{"timestamp":"2026-05-27T12:01:51.914Z","kind":"capture","source_id":"SRC-2026-05-27-006","url":"https://arxiv.org/pdf/2312.07448","format":"web"} +{"timestamp":"2026-05-27T12:03:31.427Z","kind":"ensure_page","page_type":"concept","title":"Shape Factor C(W, g_A)","path":"concepts/shape-factor-cw-ga"} +{"timestamp":"2026-05-27T12:04:45.675Z","kind":"ensure_page","page_type":"concept","title":"SSM — Spectrum Shape Method","path":"concepts/ssm-spectrum-shape-method"} +{"timestamp":"2026-05-27T12:05:55.743Z","kind":"ensure_page","page_type":"entity","title":"Paulsen et al. (2024) — High precision measurement of 99Tc β spectrum","path":"entities/paulsen-et-al-2024-high-precision-measurement-of-99tc-spectrum"} +{"timestamp":"2026-05-27T12:07:21.046Z","kind":"ensure_page","page_type":"entity","title":"Song et al. (2025) — 99Tc β-decay spectrum and implications for g_A","path":"entities/song-et-al-2025-99tc--decay-spectrum-and-implications-for-ga"} +{"timestamp":"2026-05-27T12:08:23.407Z","kind":"ensure_page","page_type":"concept","title":"ESSM — Enhanced Spectrum Shape Method","path":"concepts/essm-enhanced-spectrum-shape-method"} +{"timestamp":"2026-05-27T12:09:43.380Z","kind":"ensure_page","page_type":"concept","title":"Nuclear Shell Model for β Decay","path":"concepts/nuclear-shell-model-for-decay"} diff --git a/.llm-wiki/meta/index.md b/.llm-wiki/meta/index.md new file mode 100644 index 0000000..36f5a04 --- /dev/null +++ b/.llm-wiki/meta/index.md @@ -0,0 +1,15 @@ +# Wiki Index + +> Auto-generated from meta/registry.json. Do not edit manually. + +## Sources + +- [[sources/SRC-2026-05-27-001]] — SRC-2026-05-27-001 *(created: 2026-05-27)* +- [[sources/SRC-2026-05-27-002]] — SRC-2026-05-27-002 *(created: 2026-05-27)* +- [[sources/SRC-2026-05-27-003]] — SRC-2026-05-27-003 *(created: 2026-05-27)* +- [[sources/SRC-2026-05-27-004]] — SRC-2026-05-27-004 *(created: 2026-05-27)* +- [[sources/SRC-2026-05-27-005]] — SRC-2026-05-27-005 *(created: 2026-05-27)* +- [[sources/SRC-2026-05-27-006]] — SRC-2026-05-27-006 *(created: 2026-05-27)* + +--- +*Last updated: 2026-05-27T12:01:51.914Z* | *Total pages: 6* diff --git a/.llm-wiki/meta/log.md b/.llm-wiki/meta/log.md new file mode 100644 index 0000000..97b972b --- /dev/null +++ b/.llm-wiki/meta/log.md @@ -0,0 +1,25 @@ +# Activity Log + +> Auto-generated from meta/events.jsonl. Do not edit manually. + +## [2026-05-27T11:55:40.409Z] bootstrap +- topic: "Beta Spectrum Calculator — High-precision beta decay spectrum simulation for ⁹⁹Tc and other forbidden decays", mode: "personal" + +## [2026-05-27T11:55:55.301Z] capture +- source_id: "SRC-2026-05-27-001", file_path: "/home/unzhakov/workspace/beta_shape_pnpi/docs/gA-extraction/01-ssm-algorithm.md", format: "markdown" + +## [2026-05-27T11:56:08.796Z] capture +- source_id: "SRC-2026-05-27-002", file_path: "/home/unzhakov/workspace/beta_shape_pnpi/docs/nuclear-structure/01-shape-factors.md", format: "markdown" + +## [2026-05-27T11:59:21.939Z] capture +- source_id: "SRC-2026-05-27-003", url: "https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download", format: "web" + +## [2026-05-27T12:00:22.177Z] capture +- source_id: "SRC-2026-05-27-004", url: "https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf", format: "web" + +## [2026-05-27T12:01:29.331Z] capture +- source_id: "SRC-2026-05-27-005", url: "https://arxiv.org/pdf/2506.17544", format: "web" + +## [2026-05-27T12:01:51.914Z] capture +- source_id: "SRC-2026-05-27-006", url: "https://arxiv.org/pdf/2312.07448", format: "web" + diff --git a/.llm-wiki/meta/registry.json b/.llm-wiki/meta/registry.json new file mode 100644 index 0000000..d22eeff --- /dev/null +++ b/.llm-wiki/meta/registry.json @@ -0,0 +1,72 @@ +{ + "version": "1.0", + "last_updated": "2026-05-27T12:01:51.914Z", + "pages": { + "sources/SRC-2026-05-27-001": { + "type": "source", + "title": "SRC-2026-05-27-001", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "markdown", + "source_id": "SRC-2026-05-27-001", + "raw_path": "raw/sources/SRC-2026-05-27-001/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + }, + "sources/SRC-2026-05-27-002": { + "type": "source", + "title": "SRC-2026-05-27-002", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "markdown", + "source_id": "SRC-2026-05-27-002", + "raw_path": "raw/sources/SRC-2026-05-27-002/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + }, + "sources/SRC-2026-05-27-003": { + "type": "source", + "title": "SRC-2026-05-27-003", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "web", + "source_id": "SRC-2026-05-27-003", + "raw_path": "raw/sources/SRC-2026-05-27-003/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + }, + "sources/SRC-2026-05-27-004": { + "type": "source", + "title": "SRC-2026-05-27-004", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "web", + "source_id": "SRC-2026-05-27-004", + "raw_path": "raw/sources/SRC-2026-05-27-004/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + }, + "sources/SRC-2026-05-27-005": { + "type": "source", + "title": "SRC-2026-05-27-005", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "web", + "source_id": "SRC-2026-05-27-005", + "raw_path": "raw/sources/SRC-2026-05-27-005/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + }, + "sources/SRC-2026-05-27-006": { + "type": "source", + "title": "SRC-2026-05-27-006", + "created": "2026-05-27", + "updated": "2026-05-27", + "format": "web", + "source_id": "SRC-2026-05-27-006", + "raw_path": "raw/sources/SRC-2026-05-27-006/extracted.md", + "captured": "2026-05-27", + "status": "skeleton" + } + } +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-001/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-001/extracted.md new file mode 100644 index 0000000..e7e0f89 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-001/extracted.md @@ -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 | diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-001/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-001/manifest.json new file mode 100644 index 0000000..2e0b090 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-001/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-001/original/01-ssm-algorithm.md b/.llm-wiki/raw/sources/SRC-2026-05-27-001/original/01-ssm-algorithm.md new file mode 100644 index 0000000..e7e0f89 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-001/original/01-ssm-algorithm.md @@ -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 | diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-002/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-002/extracted.md new file mode 100644 index 0000000..eaa9f82 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-002/extracted.md @@ -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. diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-002/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-002/manifest.json new file mode 100644 index 0000000..4c68273 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-002/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-002/original/01-shape-factors.md b/.llm-wiki/raw/sources/SRC-2026-05-27-002/original/01-shape-factors.md new file mode 100644 index 0000000..eaa9f82 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-002/original/01-shape-factors.md @@ -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. diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-003/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-003/extracted.md new file mode 100644 index 0000000..babe459 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-003/extracted.md @@ -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 diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-003/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-003/manifest.json new file mode 100644 index 0000000..cd8aade --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-003/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-003/original/source.html b/.llm-wiki/raw/sources/SRC-2026-05-27-003/original/source.html new file mode 100644 index 0000000..93cd36e Binary files /dev/null and b/.llm-wiki/raw/sources/SRC-2026-05-27-003/original/source.html differ diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-004/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-004/extracted.md new file mode 100644 index 0000000..0ff2efc --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-004/extracted.md @@ -0,0 +1,1397 @@ +High precision measurement of the 99Tc 𝛽 spectrum +Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten +Kossert, Xavier Mougeot, Abhilasha Singh, Sylvain Leblond, J. Beyer, L. Bockhorn, +et al. +To cite this version: +Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten Kossert, et al.. High preci- +sion measurement of the 99Tc 𝛽 spectrum. Physical Review C, 2024, 110, 055503 (17 p.). ⟨10.1103/Phys- +RevC.110.055503⟩. ⟨cea-04796921⟩ +HAL Id: cea-04796921 +https://cea.hal.science/cea-04796921v1 +Submittedon21Nov2024 +HAL is a multi-disciplinary open access archive L’archiveouvertepluridisciplinaireHAL,estdes- +for the deposit and dissemination of scientific re- tinée au dépôt et à la diffusion de documents scien- +searchdocuments,whethertheyarepublishedornot. tifiquesdeniveaurecherche,publiésounon,émanant +Thedocumentsmaycomefromteachingandresearch des établissements d’enseignement et de recherche +institutionsinFranceorabroad,orfrompublicorpri- français ou étrangers, des laboratoires publics ou +vateresearchcenters. privés. +HALAuthorization + +| | | | | PHYSICAL | | | REVIEW | C 110,055503(2024) | | | | | | | +| --- | --- | --- | --- | -------- | ----------------------------- | --- | ------ | ------------------ | --- | -------- | --- | --- | --- | --- | +| | | | | | precisionmeasurementofthe99Tc | | | | | β | | | | | +| | | | | High | | | | | | spectrum | | | | | +M.Paulsen ,1,2,*P.C.-O.Ranitzsch,3,†M.Loidl ,4M.Rodrigues ,4K.Kossert,3X.Mougeot ,4,‡A.Singh,4,§ +S.Leblond ,4J.Beyer,1L.Bockhorn ,3,(cid:2)C.Enss,2M.Wegner ,5,6S.Kempf ,6,5andO.Nähle 3 +1Physikalisch-TechnischeBundesanstaltBerlin,Abbestraße2-12,10587Berlin,Germany +2Kirchhoff-InstituteforPhysics,ImNeuenheimerFeld227,69120Heidelberg,Germany +3Physikalisch-TechnischeBundesanstaltBraunschweig,Bundesallee100,38116Braunschweig,Germany +4UniversitéParis-Saclay,CEA,List,LaboratoireNationalHenriBecquerel(LNE-LNHB),F-91120Palaiseau,France +5InstituteforDataProcessingandElectronics(IPE),KarlsruheInstituteofTechnology(KIT), +Hermann-von-Helmholtz-Platz1,76344Eggenstein-Leopoldshafen,Germany +6InstituteofMicro-andNanoelectronicSystems(IMS),KarlsruheInstituteofTechnology(KIT),Hertzstraße16,76187Karlsruhe,Germany +(Received5October2023;revised16August2024;accepted22October2024;published20November2024) +β +Highly precise measurements of the 99Tc spectrum were performed in two laboratories using metallic +magneticcalorimeters.Independentsamplepreparations,evaluationmethods,andanalysesyieldedconsistent +resultsandthespectrumcouldbemeasureddowntolessthan1keV.Consistentβspectrawerealsoobtainedvia +crossevaluationsoftheexperimentaldatasets.Anadditionalindependentmeasurementwithsilicondetectorsin +a4πconfigurationconfirmsthespectrumshapeabove25keV.Detailedtheoreticalcalculationswereperformed +usingthenuclearshellmodelandincludingatomiceffects.Thespectrumshapewasfoundtobesensitivetothe +effective value ofthe axial-vector coupling constant. Combining measurements and predictions, we extracted +| | | =295.82(16)keV, | | | ff=0.376(5), | | ff=0.574(36), | | | | | | | | +| --- | --- | --------------- | --- | --- | ------------ | --- | ------------- | ----- | ----- | -------- | ---------------- | --- | ------ | --- | +| | Qβ | | | ge | | | and ge | which | seems | to solve | an inconsistency | | of the | | +| | | | | V | | | A | | | | | | | | +quenchedcouplingconstantsbetweenfirstandhigherforbiddennonuniquetransitions.Furthermore,wederived +themeanenergyoftheβspectrum,Eβ =98.51(23)keV,logf =−0.47660(22),andlogft =12.3478(23). +DOI:10.1103/PhysRevC.110.055503 +I. INTRODUCTION magnetic [12,13], scintillation [14], and semiconductor [15] +β β spectrometers [16]. While these setups corresponded to the +| spectrometry, | | the | precise | shape | of | spectra, | and | | | | | | | | +| ----------------- | ------ | ----------- | -------- | ------------- | ------- | ------------------ | ------ | ---------------- | ------- | ----------------- | ----------- | ---------------- | ------------ | ----------- | +| | | | | | | | | state of | the art | when they | were | applied | in the | 1950–1970s, | +| their theoretical | | description | | have received | | increased | inter- | | | | | | | | +| | | | | | | | | the measurements | | suffer | from rather | high | energy | thresholds | +| est recently | from | different | research | | fields, | e.g., radionuclide | | | | | | | | | +| | | | | | | | | (>50keV) | and | it is expected | that | one | can achieve | signif- | +| metrology | [1–3], | neutrino | physics | | [4–6], | and nuclear | the- | | | | | | | | +| | | | | | | | | icantly | higher | energy resolution | | with present-day | | methods. | +| ory [7–9]. | In | the context | of | radionuclide | | metrology, | the | | | | | | | | +| | | | | | | | | In addition, | the | β spectrum | shape | of 99Tc | has recently | been | +EuropeanmetrologyresearchprojectMetroBeta[10](2016– +predictedin[8]tobeverysensitivetotheeffectivevalueofthe +| 2019) addressed | | the | precise | measurement | | and | theoretical | | | | | | | | +| --------------- | --- | --- | ------- | ----------- | --- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- | +calculationofseveralβ weakinteractionaxial-vectorcouplingconstantg A ,makingits +spectra. +| | | | | | | | | high precision | | measurement | very | interesting. | The | currently | +| --------- | ---------- | -------- | ----------- | -------- | ----------- | -------- | ----------- | -------------- | ------- | ------------ | -------- | ------------ | ---------- | ------------- | +| The | ground | state of | 99Tc decays | | mainly | via pure | β emis- | | | | | | | | +| | | | | | | | | accepted | Q-value | 297.5(9)keV | [17] | has | a relative | standard | +| sion [β−, | 99.998 | 55(30)%] | | to the | 99Ru ground | | state [11]; | | | | | | | | +| | | | | | | | | uncertainty | of | about 0.3%. | The | value is | mainly | influenced | +| see Fig. | 1. The | spectrum | shape | of | this second | | forbidden | | | | | | | | +| | | | | | | | | by Alburger | et | al. [18] | who used | an iron-free | | intermediate- | +| nonunique | transition | has | been | measured | several | times | using | | | | | | | | +| | | | | | | | | image magnetic | | spectrometer | to | measure | the decay | energies | +of99mTc.ModernmethodssuchasPenningtrapsorcryogenic +microcalorimeters,aspresentedhere,allowformoreaccurate +*Contactauthor:michael.paulsen@ptb.de +measurements. +†Presentaddress:DeutschesZentrumfürLuft-undRaumfahrte.V. +| | | | | | | | | Theachievedresultfortheaverageβ | | | | energyof99Tchasa | | | +| --- | --- | --- | --- | --- | --- | --- | --- | ------------------------------- | --- | --- | --- | ---------------- | --- | --- | +Hamburg,Hamburg,Germany. +‡Contactauthor:xavier.mougeot@cea.fr direct application for calculating nuclear reactor models and +radioactivewastemanagement.Inarecentstudy,theinfluence +| §Present | address: | Physikalisch-Technische | | | Bundesanstalt | | Braun- | | | | | | | | +| -------- | -------- | ----------------------- | --- | --- | ------------- | --- | ------ | ---------- | ---- | -------- | --------- | ----- | ------- | ----------- | +| | | | | | | | | of nuclear | data | on decay | heat from | spent | nuclear | fuel over a | +schweig,Bundesallee100,38116Braunschweig,Germany. +| | | | | | | | | periodof1to105 | | yearswasassessed[19].Alistofthemost | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | -------------- | --- | ----------------------------------- | --- | --- | --- | --- | +(cid:2)Presentaddress:InstitutfürFestkörperphysik,LeibnizUniversität +significantcontributingradionuclideswasprovided.99Tcwas +Hannover,Appelstraße2,30167Hannover,Germany. +| | | | | | | | | placed at | the very | top, with | an average | β | energy | that ranges | +| --- | --- | --- | --- | --- | --- | --- | --- | --------- | -------- | --------- | -------------------- | --- | ------- | ----------- | +| | | | | | | | | from 84 | to 95keV | and | a stated uncertainty | | of less | than 1%, | +PublishedbytheAmericanPhysicalSocietyunderthetermsofthe +Creative Commons Attribution 4.0 International license. Further dependingonthedatalibrary. +β +distribution of this work must maintain attribution to the author(s) In the framework of the MetroBeta project, a spectrum +andthepublishedarticle’stitle,journalcitation,andDOI. of 99Tc was measured with metallic magnetic calorimeters +2469-9985/2024/110(5)/055503(17) 055503-1 PublishedbytheAmericanPhysicalSociety + +M.PAULSENetal. PHYSICALREVIEWC110,055503(2024) +TABLE I. Setup properties of the MMC experiments at the +LNHB and PTB laboratories. The values for the input inductance +andheatcapacityarenominalperfabricationandcalculatedvalues, +respectively. +Laboratory LNHB[10,20,21] PTB[28,29] +3/4Hedilutioninsert 3/4Hedilutionrefrigerator +Cryostat in4He(l)bath withtwostagepulsetube +forprecooling forprecooling +MMCchip MetroBetaV1-M[10] MetroBetaV2-M[29] +FIG.1. Decayschemeof99Tc. SQUIDchip SupraconVC1A PTBX1 +inputinductance4.5nH inputinductance2nH +Au,heatcapacity Au,heatcapacity +(MMCs),whichwasfirstpresentedin[20,21].Itfeaturedtwo- Absorber 350pJ/Kat20mK 112pJ/Kat20mK +orders-of-magnitude-lower energy thresholds (0.65keV) and matchedtoMMCchip matchedtoMMCchip +a greatly improved energy resolution (0.1keV at 383keV) Sample electrodeposited dropdeposited +compared to previous measurements. The spectrum was ob- +Calibrationsource 133Ba 57Co +tained at the Laboratoire National Henri Becquerel (LNHB) +Analysiscode Optimalfiltering Optimalfiltering +and shows excellent agreement with a corresponding mea- +surementatthePhysikalisch-TechnischeBundesanstalt(PTB) +inMATLAB inPYTHON +using a similar MMC setup, which confirms the spectrum +shapeovertheentireenergyrange.Atenergiesabove25keV, +the spectrum shapes are further confirmed with a state-of- II. EXPERIMENTALSTUDY +the-art passivated implanted planar silicon (PIPS) detector +A. MMCmeasurements +measurement,alsoperformedatLNHB. +MMCs[22–24]arecryogenicmicrocalorimetersthatcon- +The remainder of the paper is organized as depicted in +sist of a—mostly metallic—particle absorber in strong ther- +Fig. 2. In Sec. II, we report on the three independent mea- +surements of the 99Tc β spectrum and on the data analyses mal contact with a metallic paramagnet (here, Ag:Er 300ppm ) +acting as a temperature sensor. The paramagnet is placed +performed to correct for small, but relevant distortions due +in a weak magnetic field (≈10mT) and operated at temper- +to the detection systems. We next discuss theory in Sec. III +atures <100mK. When an energy E is deposited into the +thatconnectstheseaccuratemeasurementswithdetailedthe- +oretical predictions in order to extract the 99Tc Q-value and +absorber,itleadstoatemperatureincrease(cid:4)T.Asthemag- +neticsusceptibilityoftheparamagnethasastrongtemperature +the effective g coupling constant using the spectrum-shape +A +method. The average energy of the β spectrum Eβ and the dependence, the temperature increase causes a change in its +magnetization, +log f value are derived from the resulting theoretical spec- +trum. A qualitative comparison of all three measurements ∂M ∂M E +with the currently recommended spectrum [15] confirms our (cid:4)M = ∂T ·(cid:4)T = ∂T · C , (1) +findings. tot +where C denotes the total heat capacity of the absorber +tot +and the paramagnet. A superconducting coil coupled to the +paramagnetpicksupthechangeinmagnetizationasacorre- +spondingmagneticfluxchange(cid:4)(cid:6),whichismeasuredwitha +superconductingquantuminterferencedevice(SQUID)[25]. +Setups using MMCs with the radionuclide source embed- +ded in a 4π solid angle absorber geometry have proven to +be among the best β spectrometers in terms of energy res- +olution and energy threshold, in particular for low-energy β +transitions [20,21,26,27]. Both measurements presented here +follow that approach, but the technical realization differs in +manydetails.ThesearesummarizedinTableIanddescribed +inthefollowing. +1. Setupandanalysis(LNHB) +ThestartingpointofthedetectorfabricationatLNHBwas +thesourcepreparation.Followingaprotocolyieldingmetallic +technetium [30], 99Tc was electrodeposited onto a 10-µm- +FIG.2. Schematicoverviewofthedata,evaluation,andanalysis thickgoldfoil.Thefoilwasthenrinsedwithwaterinorderto +ofthiswork. removesalthavingcrystallizedfromthe99Tcsolutiononthe +055503-2 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +foil.Somevisiblesaltdepositremainedevenafterrinsing,but +inanautoradiographicimageofthesource,severalareaswith- +outanysaltdepositbutwithapresenceof99Tcactivitywere +found.Theelectrodepositionyieldwaslowandanareaofthe +sourcefoillargerthanthetypicalsizeofMMCabsorbershad +| to be used | to have | sufficient | | activity | in the | MMC | absorber. | | | | | | | | | +| ---------- | ------- | ---------- | --- | -------- | ------ | --- | --------- | --- | --- | --- | --- | --- | --- | --- | --- | +Theselectedpieceofsourcefoil(≈0.9×2.5mm)withatrans- +| parently | thin metallic | | 99Tc deposit | | was folded | three | times | to | | | | | | | | +| -------- | ------------- | --- | ------------ | --- | ---------- | ----- | ----- | --- | --- | --- | --- | --- | --- | --- | --- | +reduceittoasmallenoughsize(≈0.44×0.64×54mm)such +thatitcouldbeenclosedintotheMMCabsorber.Thefolded +| foil with | a 99Tc | activity | of ≈5Bq | | was sandwiched | | between | | | | | | | | | +| --------- | ------ | -------- | ------- | --- | -------------- | --- | ------- | --- | --- | --- | --- | --- | --- | --- | --- | +twogoldfoils(0.9×0.9×74mmeach)andthisstackwasdif- +fusionwelded.Sincethesourcecarrierfoilandabsorberfoils +| are made | of gold | and | diffusion | welded, | the | source | foil | with | | | | | | | | +| ------------------ | ---------- | ---------- | ---------- | ---------------------- | -------------- | --------------- | ------------- | -------- | --- | --- | --- | --- | --- | --- | --- | +| the embedded | | activity | forms | part of | the | final absorber, | | both | | | | | | | | +| geometrically | and | thermally. | | This | means | that if | the radiation | | | | | | | | | +| deposits | its energy | in | the source | foil | or | the absorber | | foil, it | | | | | | | | +| results in | equivalent | | signals. | The | final absorber | | had | a heat | | | | | | | | +| capacityof≈350pJ/K | | | at20mK | andwasgluedwithStycast | | | | | | | | | | | | +| 1266 epoxy | to | one of | the pixels | of | a MetroBeta | | V1 M-sized | | | | | | | | | +MMCchip[10]. +| Duetothecontinuousnatureoftheβ | | | | | spectra,withoutany | | | | | | | | | | | +| ------------------------------ | --- | --- | --- | --- | ------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +distinct features, energy calibration is essential, in particular MeasuredandcalibratedLNHBMMCspectrumof99Tc +FIG.3. +| if the end | point | energy | is to | be determined | | from | an experi- | | | | | | | | | +| ---------- | ----- | ------ | ----- | ------------- | --- | ---- | ---------- | --- | --------- | ---------------------------------------------- | --- | --- | --- | --- | --- | +| | | | | | | | | | with133Ba | externalsourcepeaksofwhichninewereusedforcali- | | | | | | +mentalspectrum.Topreciselydeterminethespectrumshape, +bration:x-raylinesat30.63keV,30.97keV(denotedI,II:directly +checking and correcting for any nonlinearities in energy is from the 133Ba source); 72.80keV, 74.97keV (denoted III, IV: Pb +alsoimportant.Acommonwaytoperformenergycalibration 287.21keV, +| | | | | | | | | | fluorescence | from the | setup material); | escape | | lines at | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ------------ | -------- | ---------------- | ------ | --- | -------- | --- | +is to use x-ray and/or gamma ray photons of well-known 289.02keV(denotedV,VI:Auphotonsescapetheabsorber,gener- +energiesfromanexternalradionuclidesourcecollimatedonto +| | | | | | | | | | ated by γ | rays from | the 133Ba | source); | and γ | rays at | 302.85keV, | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --------- | --------- | --------- | -------- | ----- | ------- | ---------- | +thedetector.Tocoverthefullenergyrangeofthe99Tcβspec- +| | | | | | | | | | 356.01keV, | 383.85keV | (denoted | VII, | VIII, IX: | directly | from the | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ---------- | --------- | -------- | ---- | --------- | -------- | -------- | +trum,atLNHBa133Ba sourceofapproximately50kBq was 133Basource).PleaseseeTableIIIforfurtherdetails.Inset:Spectrum +chosen. It emits x-ray and gamma lines between 30.63 and andbackgroundaroundthemaximumβenergy. +383.85keV.Theconversionelectronsthatarealsoemittedare +stoppedina100-µm-thickaluminumfoilplacedbetweenthe +source and the collimator. The 133Ba source was placed at a the systematic error of the experiment is decreased and the +distance of 31mm from the absorber surface. A lead colli- resulting energy resolution is improved. The final spectrum +contains7264451eventsandispresentedinFig.3. +| mator was | composed | | of a 8-mm-thick | | top | part | with a | 1mm | | | | | | | | +| --------- | -------- | --- | --------------- | --- | --- | ---- | ------ | --- | ---------- | ---------- | --- | -------- | ---- | ---------- | ------ | +| | | | | | | | | | The energy | resolution | is | constant | over | the entire | energy | +boreanda2mmthickbottompartwitha200µmboreplaced +| | | | | | | | | | (cid:4)E | ≈ | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- | +at 2.5mm from the absorber surface. The MMC signal was range, 100eV up to 384keV. The linearity in energy +| | | | | | | | | | was checked | using | the lines | annotated | in Fig. | 3 and | listed in | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ----------- | ----- | --------- | --------- | ------- | ----- | --------- | +readoutbyaSupraconVC1ASQUIDlinkedtoaMagnicon +| | | | | | | | | | Table III. | Three line | types were | considered: | | (i) | γ lines and | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ---------- | ---------- | ---------- | ----------- | --- | --- | ----------- | +XXF-1electronics. +Thewholesetupwasshieldedagainststraymagneticfields x rays that originate directly from the calibration source, (ii) +by means of a lead cylinder and operated in a liquid he- fluorescent lines due to secondary effects when the calibra- +| | | 3/4He | | | | | | | tionsourceparticlesimpactonthealuminum,gold,andlead | | | | | | | +| -------------- | --- | ----- | -------- | ------------ | --- | ------------- | --- | --- | --------------------------------------------------- | ------ | ------------ | ------ | --- | ------------ | ----- | +| lium precooled | | | dilution | refrigerator | | (Cryoconcept) | | at | | | | | | | | +| | | | | | | | | | components | of the | experimental | setup, | and | (iii) escape | lines | +T =12mK.Datawereacquiredasacontinuousstreamover +13.7daysat100kS/s;anti-aliasingfilteringwassetto30kHz which emerge when an atom in the gold absorber is hit by a +| | | | | | | | | | photon of | energy E | from the | calibration | source. | This | creates | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --------- | -------- | -------- | ----------- | ------- | ---- | ------- | +onaStanfordResearchSystemsSRS560amplifier. +| Pulses | were | triggered | in | the data | stream | offline, | using | a | | | | | | | | +| --------- | --------- | --------- | ----------- | -------- | --------- | --------- | ---------- | ---- | ----------- | -------------- | --------------- | --- | ------------ | --------------- | -------- | +| narrow | band-pass | filter. | To minimize | | pile-up, | an | extendable | | | | | | | | | +| | | | | | | | | | TABLE | II. Polynomial | calibration | | coefficients | for | the LNHB | +| dead time | was | applied. | Once | the | pulse | positions | were | de- | | | | | | | | +| | | | | | | | | | (Set 3) and | PTB (Set | 2) measurements | | using | the calibration | lines | +| termined, | the pulse | heights | | were | estimated | from | raw | data | | | | | | | | +ofTableIII. +| using an | optimal | filter | in a MATLAB | | [31] | environment. | | Slow | | | | | | | | +| ---------- | ------- | ------ | ----------- | --- | ------- | ------------ | --- | ------ | --- | --- | --- | ---------- | --- | --- | --------- | +| variations | of the | pulse | heights | for | a given | energy | | due to | | | | | | | | +| | | | | | | | | | | | | LNHB(Set3) | | | PTB(Set2) | +temperaturedriftsofthecryostatwereremovedbyfittingthe +pulse height-vs-time distribution for one densely populated Calibrationline(keV),Elit 356.012900 122.06065 +γ +linewithasplinefunctionandapplyingthisfitfunctiontoall Linearcoefficient(keV),k 355.822 121.355 +1 +| | | | | | | | | | Quadraticcoefficient(keV),k | | | 0.183 | | | 0.706 | +| ---------------------------------------------------- | -------- | --- | ------ | ---- | ------------- | --- | ----- | --- | --------------------------- | ----- | --- | ----- | --- | --- | ----- | +| pulse heights. | Spurious | | pulses | were | discriminated | | based | on | | | 2 | | | | | +| thepulseshapesinaχ-square–vs–pulse-heightplot.Bycor- | | | | | | | | | | Eγlit | | | | | | +| | | | | | | | | | Linearityoffset, | | −1 | 0.1% | | | 0.6% | +k1 +rectingfortemporalandmagneticfluctuationsofthedetector, +055503-3 + +| M.PAULSENetal. | | | | | PHYSICALREVIEWC110,055503(2024) | | | | +| -------------- | --------- | --------------------------------------------------------------- | --- | --- | ------------------------------- | --- | --- | --- | +| | TABLEIII. | Photonandescapelinesusedtocheckandcorrectforenergynonlinearity. | | | | | | | +LNHBmeasurement +(cid:4)E +| | Tabulated | Measured | (Set3) | | | | | Calibrationline: | +| ---------------- | ----------- | ----------- | ------ | --- | ------------- | --- | --- | ---------------- | +| Origin | energy(keV) | energy(keV) | (keV) | | Radiationtype | | | ForSets | +| 133Bacalibration | | | −0.02 | | | | | | +| | 30.63 | 30.61 | | | CsXKα2 | | | I:3,4 | +| source | 30.97 | 30.97 | 0 | | CsXKα1 | | | II:3,4 | +| | 35.05 | 34.98 | −0.07 | | CsXKβ | | | | +| | 81.00 | 81.06 | 0.06 | | γ | | | | +| | 276.40 | 276.46 | 0.06 | | γ | | | | +γ +| | 302.85 | 302.88 | 0.00 | | | | | VII:3,4 | +| --- | ------ | ------ | ---- | --- | --- | --- | --- | ------- | +γ +| | 356.01 | 356.01 | 0 | | | | | VIII:3,4 | +| ----------------- | ------ | ------ | ----- | -------------- | ------ | ------------------- | --- | -------- | +| | 383.85 | 383.82 | −0.03 | | γ | | | IX:3,4 | +| Au/Pbfluorescence | 66.99 | 66.98 | −0.01 | | AuXKα2 | | | | +| from | 68.80 | 68.78 | −0.02 | | AuXKα1 | | | | +| setup/collimator | 72.81 | 72.82 | 0.02 | | PbXKα2 | | | III:3,4 | +| | 74.97 | 74.98 | 0.01 | | PbXKα1 | | | IV:3,4 | +| | 84.94 | 84.98 | 0.04 | | PbXKβ1 | | | | +| | | | | Gammaline(keV) | | Escapingphoton(keV) | | | +| Escapelines | 278.03 | 278.07 | 0.04 | 356.01[133Ba | γ] | 77.98[AuKβ1 | ] | | +| | 278.43 | 278.54 | 0.11 | 356.01[133Ba | γ] | 77.58[AuKβ3 | ] | | +| | | | | 356.01[133Ba | γ] | | | | +| | 287.21 | 287.28 | 0.07 | | | 68.80[AuKα1 | ] | V:3,4 | +γ] +| | 289.02 | 289.08 | 0.06 | 356.01[133Ba | | 66.99[AuKα2 | ] | VI:3,4 | +| --- | ------ | ------ | ---- | ------------ | --- | ----------- | --- | ------ | +| | 305.86 | 305.92 | 0.06 | 383.85[133Ba | γ] | 77.98[AuKβ1 | ] | | +| | 315.05 | 315.08 | 0.03 | 383.85[133Ba | γ] | 68.80[AuKα1 | ] | | +| | 316.85 | 316.91 | 0.05 | 383.85[133Ba | γ] | 66.99[AuKα2 | ] | | +PTBmeasurement +| | Tabulated | Measured | (cid:4)E (Set2) | | | | | Calibrationline: | +| --------------- | ----------- | ----------- | --------------- | --- | ------------- | --- | --- | ---------------- | +| Origin | energy(keV) | energy(keV) | (keV) | | Radiationtype | | | ForSets | +| 57Cocalibration | 14.42 | 14.44 | −0.02 | | γ | | | | +γ +| source | 122.06 | 122.06 | 0.00 | | | | | VI:1,2 | +| ------ | ------ | ------ | ---- | --- | --- | --- | --- | ------ | +γ +| | 136.47 | 136.38 | 0.09 | | | | | VII:1,2 | +| ----------------- | ------ | ------ | ----- | --- | ------ | --- | --- | ------- | +| Au/Pbfluorescence | 66.99 | 67.07 | −0.08 | | AuXKα2 | | | | +| from | 68.80 | 68.95 | −0.15 | | AuXKα1 | | | | +| setup/collimator | 72.81 | 72.98 | −0.17 | | PbXKα2 | | | IV:1,2 | +| | 74.97 | 75.14 | −0.17 | | PbXKα1 | | | V:1,2 | +−0.15 +| | 84.94 | 85.09 | | | PbXKβ1 | | | | +| ----------- | ----- | ----- | -------------------------- | ------------- | ------ | ------------------- | --- | ------- | +| | | | Gamma/fluorescentline(keV) | | | Escapingphoton(keV) | | | +| Escapelines | 6.47 | 6.44 | 0.03 | 84.45[PbKβ3 | ] | 77.98[AuKβ1 | ] | | +| | 6.96 | 6.97 | −0.01 | 84.94[PbKβ1 | ] | 77.98[AuKβ1 | ] | | +| | 16.13 | 16.22 | −0.08 | 84.94[PbKβ1 | ] | 68.80[AuKα1 | ] | I:1,2 | +| | | | −0.17 | 122.06[57Coγ] | | | | | +| | 53.26 | 53.42 | | | | 68.80[AuKα1 | ] | II:1,2 | +| | | | −0.17 | 122.06[57Coγ] | | | | | +| | 55.07 | 55.24 | | | | 66.99[AuKα2 | ] | III:1,2 | +a hole in the electronic shell of the gold atom that can be energies [32], where, for the escape lines, the x-ray transi- +filled with an electron that transitions from an electron shell tionenergiesfrom[33]wereused.Thedifferencesinenergy +of higher energy. Then, a photon of energy Eph is emitted between the tabulated and measured energies are all smaller +Au +30.97keV +and,ifitescapestheabsorber,theenergyE −Eph isabsorbed or equal to the energy resolution; for the line, the +Au +instead. most intense x-ray line, it is zero. There is no clear trend in +| | | | | the difference | between | tabulated and | measured | energies as a | +| ---------------- | ----- | ---------------- | --------- | -------------- | ------- | ------------- | -------- | ------------- | +| First the energy | scale | was set uniquely | using the | | | | | | +356.01keV line, an intense γ line of the 133Ba calibration function of energy, but it could be described with a second- +source lying beyond the endpoint of the 99Tc β spectrum. order polynomial, as provided in Table II. This polynomial +wasthenusedtocorrecttheenergyscalefornonlineareffects. +| Then the resulting | energies | of the other | lines from the | | | | | | +| ------------------ | -------- | ------------ | -------------- | --- | --- | --- | --- | --- | +Thevalueswereachievedbymakingaquadraticansatzusing +| MMC measurement | were compared | with | the recommended | | | | | | +| --------------- | ------------- | ---- | --------------- | --- | --- | --- | --- | --- | +055503-4 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +thefitfunction, +Elit = ffit(a):=k ·a +k ·a2 +i i 1 i 2 i +fori∈{I,...,N}andN ∈{VII,IX}. (2) +Elit denotes the reference literature value of the respective +i +peak and a := E i exp is the amplitude value of the detector, +i Eγlit +which corresponds to the pulse energy of the Gaussian fit- +ted calibration line. Eexp denotes the measured experimental +i +energy and Elit is the reference calibration energy used for +γ +a first energy calibration to set the energy scale, as the de- +tector output is in units of voltage. The goal of the fit is to +determine the coefficients k and k . Clearly, for a perfectly +1 2 +lineardetectork =Elit,k =0andthelinearityoffsetwould +1 γ 2 +be0%.Here,thequadraticpartwasrathersmalland,interms +of the linearity offset, it was 0.1% and 0.6% in the LNHB +and PTB experiment, respectively. The difference is due to +the distinct designs, fabrications, and thermalizations of the +MMCdetectors. +2. Setupandanalysis(PTB) +The measurement setup at PTB is functionally the same FIG.4. Measured and calibrated PTB MMC spectrum of 99Tc +as at LNHB, but differs in several details. The difference with57Cocalibrationsourcepeaksofwhichsevenwereusedforcal- +with possibly the largest influence on the measured spec- ibration:Escapelinesat16.13keV,53.26keV,55.07keV(denoted +trum shape is the method of source preparation. This was I, II,III: Au photons escape the absorber, generated by fluorescent +done by microdispensing an aqueous solution of ammonium Pb lines and γ rays from the 57Co source); 72.81keV, 74.97keV +pertechnetate (NH 99TcO ) in 0.1molL−1 ammonia (NH ) (denotedIV,V:Pbfluorescencefromthesetupmaterial);andγ rays +4 4 3 +withanactivityofA(99Tc)≈5Bqontoasolidgoldabsorber at122.06keV,136.47keV(denotedVI,VIIdirectlyfromthe57Co +substrate with a thickness of 90µm. The source is enclosed source).PleaseseeTableIIIforfurtherdetails.Inset:Spectrumand +by diffusion welding a second 90µm gold layer to the first +backgroundaroundthemaximumβenergy. +one,withmoredetailsofthesourcepreparationprocessbeing +describedin[34].Theabsorberisattachedtoonepixelofthe datastreamat200kSs−1 toharddiskforthemeasurement’s +MMCdetectorwithStycast1266epoxyandthesecondpixel +duration of about 20 days. The data processing and analysis +isequippedwithasecondabsorberpreparedinthesameway, +areperformedwithacustomsoftwarecodewritteninPYTHON +but without any radioactive source material. The choice of +[35],whichperformsthenextstepsthatcompriseeventtrig- +photoncalibrationsourcealsohasanimpactonthespectrum +gering, amplitude/energy determination, event classification +shape. 57Co with an activity of about 37kBq was used for +throughpulse-shapediscrimination,andgaindriftcorrections. +the measurement at PTB as its spectrum with several peaks +The analysis yields 5 326 682 counts in the spectrum after +is easily distinguishable from the β spectrum shape of 99Tc. +cuts. The resulting amplitude distribution is calibrated with +A 1.5-mm-thick lead collimator with two 250µm apertures +several 57Co γ lines, x-ray fluorescence, and x-ray escape +blocks the calibration photons outside of the two detector +linesusingasecond-orderpolynomial,returningthespectrum +pixels.Theaperturesareblockedbetweendetectorandcolli- +shown in Fig. 4. The calibration error was estimated with +matorwithanapproximately1-mm-thickaluminiumsheetto +the uncertainties from the literature values and the statistical +reducesecondaryradiationfromtheleadwithoutsignificantly +uncertaintiesofthemeasuredpeaksusingorthogonaldistance +impactingthehigh-energyphotonfluxfromthe57Cosource. +Thedetectorwasoperatedinapulse-tubeprecooled3/4He regression(cid:2)[29,36–38].Thestatisticaluncertaintywasdefined +asσ / N ,whereσ denotesthestandarddeviation +dilutionrefrigerator(BlueforsLD250)temperaturestabilized Gauss peak Gauss +of a Gaussian function that was fitted to the peak and N +toT =14.5mK onadetectormoduledescribedin[10].The peak +is the number of counts in the peak. The same procedure +used MMC (size M) is an update to the one described in +was done with the pixel without radioactive source material, +the same publication, with some layout changes improving +yieldingaspectrumcontainingthe57Cocalibrationspectrum +on experimental shortcomings observed with the first design +and any additional background. There, a line spectrum with +(e.g., the two on-chip heat baths were linked in the new de- +132235countswasextracted. +sign to improve thermalization), without changing the core +properties of the detector. The setup is completed with a +3. Crossanalysis +PTBSQUID(modelC6X114W)forMMCreadout,aMagni- +conXXF-1SQUIDelectronics,aStanfordResearchSystems In both institutes, the analog to digital converter (ADC) +SR560 low-noise voltage pre-amplifier and band-pass filter, signalsweredirectlystreamedtoharddiskandthedatawere +and a 16-Bit waveform digitizer, that was set to save the full saved as binary files containing 16-bit integers. Therefore, +055503-5 + +M.PAULSENetal. PHYSICALREVIEWC110,055503(2024) +TABLE IV. Overview of the cross validation with counts, spectrum, where background has the strongest impact on the +FWHM,andenergythreshold(=E )values.Set1onlyconsidered spectrum shape. With this background model, the spectrum +TH +asubsetoftherecordeddata. shapecannotbeevaluatedatthepositionofγ lines,whichis +notabigdrawbackbecauseofthenarrowlinewidths. +Set Meas. Analysis Counts FWHM E TH ForthemeasurementatLNHB,thisapproachneedstobe +1 PTB LNHB 3.67×106 63eVat136.47keV 345eV extended because several calibration lines are close to and +2 PTB PTB 5.33×106 72eVat136.47keV 750eV even above the β endpoint region. While the background +3 LNHB LNHB 7.26×106 150eVat302.8keV 1120eV between lines can reasonably be assumed to be constant, the +4 LNHB PTB 5.66×106 108eVat302.8keV 1250eV level becomes slightly higher below each line and the back- +ground is described as a series of step functions, with the +step height being proportional to the line intensity. With this +approach,thebackgroundinbothanalysescanbereasonably +therawdataofbothmeasurementsarecompatiblewitheach +welldescribed. +other. This prompted an exchange of the data and a mutual +The measurement at PTB also offers a different approach +analysis at both institutes to identify any systematic differ- +toevaluatethebackground,whichisusedinthePTBanalysis. +ences introduced by the data processing code or analysis +Thesecondpixelofthedetectorisequippedwithanabsorber +approach. The data processing and analysis were conducted +withoutanyenclosedradioactivematerial,whichisalsoillu- +as usual for local data sets and allowed for the comparison +minatedbythecalibrationsource.Therefore,thespectrumof +betweenthefourspectraintotalresultingfromtwomeasure- +thesecondpixelshouldbeaverygoodapproximationforthe +mentswithtwoanalyseseach. +background of the primary pixel. Since the detector perfor- +For both data sets, the two separate analyses yield sur- +mance,e.g.,inrisetimeandenergyresolution,isnotexactly +prisingly different numbers of counts, as can be seen in +the same, these need to be adjusted and the amplitude of the +TableIV.ThePTBmeasurementinvoluntarilystoppedduring +measuredbackgroundspectrumscaledtomatchtheintensity +the campaign and was restarted. This caused inconsistencies +of the primary pixel, before the background is subtracted. +inthedataand,fortheanalysisatPTB,successfulcorrections +Since the primary pixel showed the better energy resolution +wereimplemented.Incontrast,theLNHBanalysisonlyused +((cid:4)E = 72eV compared to (cid:4)E = 92eV), the pri- +the largest continuous data set, which contained about three- FWHM FWHM +mary spectrum was convolved with a normalized Gaussian +quarters of the total data. The remaining difference, also for +with(cid:4)E =28eVtomatchthesecondaryspectrumand +theLNHBmeasurement,ismostlikelycausedbydifferences FWHM +thesecondaryspectrumwasscaledbyafactorof0.73because +intheimplementationandthesettingsforthesoftwaretrigger, +ofitslargerintensity. +andpossiblybycutsforeventselection.Thetriggerthreshold +To account for energy losses in the absorber, mainly via +shouldonlyinfluencetheenergythreshold,butothersettings, +unstopped photons generated by excitation of Au atoms and +suchastriggerhold-offorextending/nonextendingdeadtime, +bremsstrahlung, an unfolding correction was applied to each +etc.,canhaveanimpactoverthewholeenergyrange.Specifi- +spectrum before determining the maximum β energy. The +cally,fortheLNHB-LNHB(set3)analysis,averyshortevent +unfolding procedure is based on an algorithm that does not +time window was chosen, which allows one to reduce the +requireanyaprioriknowledgeofthetrueβspectrum[29,41]. +impact of pile-up, with the trade-off of a slightly diminished +Itsbasisisthefollowingdiscreteunfoldingproblem: +energyresolutionandhigherenergythreshold. +Itwasagreedupontousethesamecalibrationlinesforthe h N meas =R N×N ·h N true, (3) +energy calibration in the spectra and the corresponding liter- +aturevaluesfortheγ [39,40]andx-ray[33]energies.These where hmeas is the measured histogram spectrum having N +N +lineswereveryeasytoidentifyandtofitinthespectrum.The energybins,R N×N denotestheresponsematrixoftheabsorber +tabulatedenergiesanddifferencestothemeasurementdataare for the N energy intervals, and htrue is the unknown true +N +summarizedinTableIII. histogram spectrum. After a measurement, one only obtains +a value for the left-hand side of (3) and thus the problem +is typically high dimensional and greatly underdetermined. +4. Spectrumcorrections +However, an excellent approximation of the response matrix +ves B ti e g f a o te re d, th tw e o ac c t o u r a re l c s t h io ap n e sn o e f e t d he to 99 b T e c ap β pl s i p e e d c t t o ru t m he c m an ea b s e ur i e n d - R N sim ×N ≈R N×N can be acquired via large-scale Monte Carlo +(MC) simulations of the corresponding absorber geometries. +data.First,thecontributionsfromthecalibrationsourcesand +Sinceitcanbeshownthattheapproximatematrixisinvertible +additionalbackgroundneedtoberemoved.Second,thespec- +withprobability1[41],thealgorithmprovidesanapproximate +trumneedstobecorrectedforenergylossesoftheβelectrons, +solutiontoEq.(3), +mostlycausedbyx-rayfluorescenceoftheabsorbermaterial +(cid:3) (cid:4) +andescapingbremsstrahlung. htrue ≈halgo =! Rsim −1·hmeas. (4) +Thesimplestwaytoaccountforbackgroundistoassumea N N N×N N +constantbackground,withitslevelbeingdeterminedfromthe The adopted absorber geometries are, notably, just an ap- +background above the endpoint of the β spectrum. This ap- proximationsincetheexactdistributionsoftheactivitywithin +proachcould,e.g.,besufficientforthedatatakenatPTBand the rather complex sources are not precisely known. In ad- +is used in its analysis at LNHB, since no major background dition, the imperfect knowledge of the cross sections (e.g., +sourcesorγ linesareexpectedintheendpointregionoftheβ for bremsstrahlung) contribute to the related uncertainty +055503-6 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +| | | | | | energy range | from 150keV | | to 250keV | as the | normalization | | +| --- | --- | --- | --- | --- | ------------ | ----------- | --- | --------- | ------ | ------------- | --- | +γ +| | | | | | factor because | there | are no | lines | in that | range in | either | +| --- | --- | --- | --- | --- | ---------------- | ---------------- | ---------- | -------------- | ----------- | ------------ | ------- | +| | | | | | measurement. | Furthermore, | | the difference | | of the | normal- | +| | | | | | ized counts | was divided | by | the sum | of absolute | differences. | | +| | | | | | Thus, the | resulting | normalized | residuals | sum | to unity, | i.e., | +| | | | | | 100%. Normalized | | residual | plots of | the various | MMC | sets | +| | | | | | (see Table | IV) are depicted | | in Fig. 6. | For the | same data | sets, | +theanalysesyieldverysmallresidualsclosetothemeanand +| | | | | | below 5% | (proportional | contribution | | to the | residual | sum) at | +| --- | --- | --- | --- | --- | -------- | ------------- | ------------ | --- | ------ | -------- | ------- | +1.5keV. +| | | | | | energies | below | These | residuals | are | due to | different | +| --- | --- | --- | --- | --- | -------- | ----- | ----- | --------- | --- | ------ | --------- | +χ-squareandenergycuts,whicharesetmanually.TheLNHB +| | | | | | data were | not background | | cleaned | with experimental | | data | +| --- | --- | --- | --- | --- | --------- | -------------- | --- | ------- | ----------------- | --- | ---- | +concerningthecalibrationpeaksandthesecanbeseenwhen +comparedtothePTBdata,showingaverysimilarpatternfor +thedifferentanalyses. +| | | | | | | | B. Silicondetectors | | | | | +| --- | --- | --- | --- | --- | --------------- | --------- | ------------------- | ----------- | -------- | -------------- | -------- | +| | | | | | The MMC | results | were | compared | to | an independent | | +| | | | | | measurement | performed | with | a detection | | system | recently | +| | | | | | developed | at LNHB | [43,44]. | The system | was | designed | for | +| | | | | | the measurement | of | β spectra | and | is based | on two | PIPS | +detectorswiththinentrancewindows.Suchdetectorsarecom- +monlyusedtodetectchargedparticlesinnuclearandparticle +physics. +1. Experimentalsetup +| | | | | | The geometry | of | the overall | system | is designed | such | that | +| --- | --- | --- | --- | --- | ---------------- | -------------- | ----------- | ------------ | ------------- | ----------- | ---- | +| | | | | | the PIPS | detectors face | each | other. | An ultrathin | radioactive | | +| | | | | | source developed | specifically | | for this | application | [45,46] | is | +| | | | | | placed in | their center. | The | experimental | configuration | | cov- | +| | | | | | ers more | than 98% | of the | solid angle | and | minimizes | the | +self-absorptionwithinthesource.Consequently,experimental +| | | | | | distortions | of the spectrum | are | significantly | | reduced. | Source | +| --- | --- | --- | --- | --- | ---------------- | --------------- | ---------- | ------------- | ------------ | ---------- | ------ | +| | | | | | and detectors | are placed | in | an ultrahigh | vacuum | chamber, | | +| | | | | | while the | detection | system | is cooled | down | to 100K | with | +| | | | | | liquid nitrogen, | in | order to | improve | the energy | resolution | | +| | | | | | and lower | the detection | threshold. | | The detector | output | is | +preamplifiedinthechamberbeforebeingshapedandrecorded +| | | | | | using a LABZY | module | [47]. | A complete | description | | of the | +| --- | --- | --- | --- | --- | ------------- | ------ | ----- | ---------- | ----------- | --- | ------ | +detectordesign,acquisitionsystem,andsourcefabricationcan +befoundin[48]. +The99TcspectrameasuredbyMMCs,withtheirseparate +FIG.5. +2. Analysisandcorrections +analysesinlinearscale(top)andlogarithmicscale(bottom).Back- +ground and energy losses have been corrected as described in the TheanalysisofthedatawasperformedinC ++usingthe +| text. | | | | | ROOTsoftware[49].Thesystemwascalibratedwith109Cdand | | | | | | | +| ----- | --- | --- | --- | --- | --------------------------------------------------- | ----- | --------- | ----- | --------- | ----- | ---- | +| | | | | | 207Bi sources | whose | emissions | cover | an energy | range | from | +component.TheMCsimulationswererealizedwiththecode +about22to1063keV,asdescribedin[48].Theenergyreso- +egs_phdwithintheEGSNRCsoftware[42]andwereperformed +lutionofthedetectionsystemwasdeterminedtobe9keVat +separately for the PTB and LNHB detector geometries. The 65.52keV,whereanenergypeakduetoAgKshellelectrons, +| resulting | four spectra with | the background | removed | and en- | | | | | | | | +| --------- | ----------------- | -------------- | ------- | ------- | --------- | ---------- | -------- | ---------- | --- | ------------ | --- | +| | | | | | which are | emitted by | internal | conversion | in | 109Cd decay, | is | +ergylosscorrectionsappliedareshowninFig.5. +situated.Thecalibrationsourceswerealsousedtodetermine +| | | | | | the detection | energy | threshold, | which | was | estimated | to be | +| --- | --- | --- | --- | --- | ------------- | ------ | ---------- | ----- | --- | --------- | ----- | +5. Intercomparison +about10keV. +Afterapplyingthecorrectionstothespectraasdescribed, Theactivityofthe99Tcsourcewasabout800Bq inorder +allfourspectrashould,inprinciple,showthesamespectrum to limit the dead-time ratio of the acquisition to less than +shape. Before these can be directly compared, they need to 0.5%.Themeasurementwasperformedoverfiveconsecutive +be normalized. We used the total number of events in the days. To estimate the contribution of the background to the +055503-7 + +| M.PAULSENetal. | | | | | PHYSICALREVIEWC110,055503(2024) | | | +| -------------- | --- | --- | --- | --- | ------------------------------- | --- | --- | +FIG.6. AnalysiscomparisonforthetwoMMCmeasurementsofthe99Tcspectrum(data,analysis):normalizedresidualpointplots(blue) +andmeanofresiduals(orange).Rows1and2showthelow-energyrangefrom0to10keVandrows3and4depicttheremainingenergy +rangefrom10to400keV.Plotsareshownforsets3and2(rows1and3,left),sets2and4(rows1and3,center),sets3and4(rows1and3, +right),sets1and2(rows2and4,left),sets1and3(rows2and4,center),andsets1and4(rows2and4,right). +initialenergydistributionoftheβ +spectrum, a ten-day measurement was performed with a electrons.Moredetailson +source produced with a nonradioactive solution. The back- the algorithm and the simulation can be found in [48]. The +ground was found to have little effect on the measured response function of the detector is considered to be under +spectrum, as 80% of the background events was detected good control from 850keV down to at least 25keV. This +below15keV.Forthedataanalysis,thebackgroundspectrum is evidenced by the good agreement between the analyzed +wassubtractedfromthe99Tcspectrumafternormalizationof datafromseveralmeasuredsourcesandthesimulationofthe +| thelifetimeratiosofthemeasurements. | | | | detectionsystem. | | | | +| ----------------------------------- | ------- | --------------------- | ------------------- | ---------------- | --- | --- | --- | +| To account | for the | remaining distortions | in the experi- | | | | | +| mental spectrum, | mainly | due to energy | loss in dead layers | | | | | +3. Comparison +| and escape | of particles, | an unfolding algorithm | was adapted | | | | | +| ---------- | ------------- | ---------------------- | ----------- | --- | --- | --- | --- | +fromtheprinciplepresentedin[41].BasedondetailedPENE- The corrected spectrum obtained from the PIPS measure- +LOPE [50] Monte Carlo simulations of the source-detector ment is compared in Fig. 7 to the two MMC measurements. +| geometry, | the algorithm | provides a way | to reconstruct the | | | | | +| --------- | ------------- | -------------- | ------------------ | ----------------- | ----------------- | -------------- | --- | +| | | | | The three spectra | show an excellent | agreement from | the | +response matrix of the detection system. From this knowl- minimum reconstructed energy of the PIPS system up to +edge,theexperimentalspectrumcanbeunfoldedtoobtainthe the endpoint of the spectra. The PIPS measurement is +055503-8 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +where N(W) corresponds to the measured data, F(Z,W) is +the Fermi function with Z the daughter atomic number, p= +(W2−1)1/2,W =1+E /m ,wherem istheelectronrest +0 max e e +mass, and W =1+E/m . The constant G2 is the squared +e β +product of the weak interaction coupling constant g and the +cosineoftheCabibboangle,cos(cid:11) ,andX(W)standsforthe +C +correctionoftheatomicscreeningandexchangeeffects.The +atomicoverlapcorrectionisgivenby +1 ∂2 +r(Z,W)=1− B(G). (6) +W −W ∂Z2 +0 +For99Tc,theconstantB(cid:5)(cid:5) = ∂2 B(G)anditsuncertaintywas +calculatedtobeB(cid:5)(cid:5)=0.211(1 ∂ 1 Z ) 2 keVwhenusingparametriza- +tionsfromtheliterature[52,53]. +The shape-factor function of the second forbidden +FIG.7. ComparisonofthemeasuredPTBMMC,LNHBMMC, +nonuniqueβ transitionof99Tcisoftenparameterizedasfirst +andPIPSspectraafterremovingthebackground,inred,green,and +black, respectively. Please note the corresponding coloring of the forbiddenuniqueusing +first,second,andthirdyaxis.WhilethecountstatisticsofthePIPS C(W)=q2+λp2 = (W −W)2+λ(W2−1), (7) +0 +measurement is approximately 50 times larger than those of the +MMCs, its energy resolution is two orders of magnitude smaller with λ being a constant parameter (see, e.g., [15]). Hence, +(9keVvs(cid:2)90eVat65.52keV).Showninblueistherecommended Eq.(5)canberearrangedtoget +literaturespectrumof99Tc[15],whichwasplottedovertheexper- (cid:5) +imental measurement interval (55 to 250keV) using the reported N(W) =K{[(W −W)2−B (cid:5)(cid:5) (W −W)] +shapefactors.Notably,theliteraturespectrumshapedivergesbelow pWF(Z,W)X(W) 0 0 +100keV. +×[(W −W)2+λ(W2−1)]}1/2, +0 +(8) +independent of the MMC measurements, both in terms of +detectionmethodandsourcepreparation,andtheconsistency which is then used for the fit procedures. In contrast to stan- +between the three spectra underlines the reliability of the dardKuriefits[54],thefitfunction[rightsideofEq.(8)isnot +β spectra presented in this article. Compared to the recom- linear,butthethreeparameters(K,W ,andλ)canbedirectly +0 +mendedβ spectrumof99Tcintheliterature[15],whichwas determinedinasinglefitprocess. +reported by Reich and Schüpferling in 1974, there is rather TheanalysiswascarriedoutwithallfourMMCdatasets +good agreement above 100keV, but divergence with a clear (twomeasurements×twoanalyses).Tothisend,spectrawith +trend at lower energies. Very likely, the divergence is due to the background removed and corrected for bremsstrahlung +experimental issues in the absorber geometry of [15]. The lossesandwithabinwidthof160eVwereusedasastarting +solid angle was realistically less than 4π and, to the best of point.Iftheusedbackgroundmodelleftthephotonpeaksof +ourknowledge,theresultingenergylosseswerenotcorrected, the external sources (57Co and 133Ba, respectively) in place, +e.g.,usinganunfoldingprocedure. these were removed. The uncertainty assigned to the back- +groundwasconservativelyestimated.Tothisend,theanalysis +wasrepeatedwithoutanybackgroundsubtraction.Thediffer- +III. COMBINEDANALYSIS ence to the previous result with background subtraction was +The results of Sec. IIIA, concerning the maximum β en- then used to evaluate the corresponding uncertainty compo- +nent assuming a rectangular distribution. It should be noted +ergy, were carried out at the PTB with theory insight from +thattheinfluenceofbackgroundtothedeterminedmaximum +the LNHB, and the spectrum-shape method calculations of +β energyalsodependsontheenergyrangethatisconsidered +Sec.IIIBweredoneattheLNHB. +forthefits. +For all cases (sets 1–4), the spectrum unfolding shifted +A. 99TcQ-value themaximumenergybyapproximately+200eV.Theimpact +on the spectrum shape, however, is very subtle as the energy +For the determination of the maximum β energy E of +max lossesoftheabsorberswereverysmall(<1%)intheseexper- +99Tc, the methodology described in a previous article [51] +iments. If the energy losses are larger, the unfolding affects +was adopted to allow for an energy-dependent shape-factor +thespectrumshapemoresignificantly[29,41,44]. +functionC(W).Whenneglectingtheantineutrinomass,theβ +Results of the fit procedure are shown in Table V and an +spectrumisdescribedby +uncertaintybudgetisshowninTableVI.Thefitrangeswere +chosen in order to study several potential effects such as the +G2 +N(W)dW = 2π β 3 F(Z,W)pW(W 0 −W)2dW i r n an te g r e pla (1 y 2 b 5 e t t o we 2 e 9 n 3. s 5 h k ap e e V - ) fa w ct a o s r i f n u c n l c u t d io e n d a to nd de E m ma o x n . s T tr h a e te th th ir a d t +×X(W)C(W)r(Z,W), (5) consistentresultsareobtained,evenifahighupperlimitclose +055503-9 + +| M.PAULSENetal. | | | | | | | | | | PHYSICALREVIEWC110,055503(2024) | | | | | | +| -------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------------------------------- | --- | --- | --- | --- | --- | +TABLE V. Results for the maximum β energy E obtained current (CVC) hypothesis derives from the gauge invariance +max +using spectra from the two analysis codes and two measurements. oftheweakinteractionandleadstog =1.Accordingtothe +V +Theresultscorrespondtomeanvaluesthatwereobtainedfromthree partially conserved axial-vector current (PCAC) hypothesis, +individual results using different energy ranges for the fits (70 to onecanadoptthefree-nucleonvaluegfree=1.2754(13)[55]. +A +290keV, 140 to 290keV, and 125 to 293.5keV, respectively). In An ideal description of the nuclear structure would allow +all cases, the measurement uncertainties were taken into account inβ +| | | | | | | | | theuseofthesevalues | | | decay | calculations.However, | | | g V | +| --------- | ---------- | ---------- | --- | ----------- | --- | ------------- | ------- | ------------------- | --- | ------------ | ----- | --------------------- | ------ | ------------- | --- | +| (weighted | fits), the | background | was | subtracted, | | spectrum | unfold- | | | | | | | | | +| | | | | | | | | and g can | be | renormalized | | because | of the | imperfections | of | +| | | | | | | functionC(W)= | | A | | | | | | | | +ing was taken into account, and the shape-factor thenuclearmodelthatisemployed.Thishasbeenwellknown +q2+λp2 +| | wasassumed.ThecurrentlyrecommendedE | | | | | | max valuesin | | | | | | | | | +| --- | ----------------------------------- | --- | --- | --- | --- | --- | ------------ | ---------- | ---- | ---- | --------- | ---------- | ---- | ----------- | ----- | +| | | | | | | | | for a long | time | from | the study | of partial | half | lives (see, | e.g., | +theliteratureare293.8keV[15]and297.5(9)keV[17],respectively. +thereviewofSuhonen[56]).Effectivevaluesofthecoupling +| | | | | | | | | constants | then | help to | compensate | for | some | approximations | | +| --- | ----------- | --- | -------- | --- | --- | ----- | --- | --------- | ---- | ------- | ---------- | --- | ---- | -------------- | --- | +| Set | Measurement | | Analysis | | E | (keV) | λ | | | | | | | | | +max such as nonexistent or partial core excitations, or simplified +1 PTB LNHB 295.809 0.615 many-nucleoncorrelations.However,onehastonotethatthis +2 PTB PTB 295.786 0.652 renormalization could not have a physical meaning, but is +3 LNHB LNHB 295.845 0.651 moreanartifactduetothenuclearmodel.Thelastdecadehas +| 4 | LNHB | | PTB | | 295.862 | | 0.659 | | | | | | | | | +| --- | ---- | --- | --- | --- | ------- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | --- | +seenimportantprogressthankstothedevelopmentofeffective +| | | | | | | | | field theories. | | It was | recently | demonstrated | | that for allowed | | +| --- | --- | --- | --- | --- | --- | --- | --- | --------------- | --- | ------------ | -------- | ------------- | --- | ---------------- | ---- | +| | | | | | | | | Gamow-Teller | | transitions, | the | 0.8 quenching | | factor | on g | +A +| | | | | | | | | obtained | with | the nuclear | shell | model | becomes | unnecessary | | +| --- | --- | --- | --- | --- | --- | --- | --- | -------- | ---- | ----------- | ----- | ----- | ------- | ----------- | --- | +totheendpointenergyischosen.Thismaybedifferentforhis- +torical studies, which can suffer from low-energy resolution to reproduce the decay rates when advanced ab initio com- +andtherelatedresolutiondistortioneffect. putations, including two-body currents, are employed [57]. +| | | | | | | | | Inthepresentanalysis,wehaveconsiderednuclear | | | | | | structure | | +| --- | --- | --- | --- | --- | --- | --- | --- | -------------------------------------------- | --- | --- | --- | --- | --- | --------- | --- | +Theevaluationoffurtheruncertaintycomponentswascar- +riedoutinasimilarmannerasdescribedinRef.[51].Forthe informationfromthenuclearshellmodelonly. +uncertainty analysis, fit ranges were varied and unweighted Recent theoretical studies suggested that geff can also +A +fits were compared with fits that take statistical uncertainties have asignificantinfluence onthespectrumshapeofforbid- +into account. In order to evaluate a model uncertainty, the den nonunique transitions [58,59]. Indeed, the calculation of +analyses were repeated, ignoring the correction for screen- forbiddennonuniquetransitionsinvolvesnonrelativisticaxial- +ing and the atomic exchange effect and by using a modified vectormatrixelementsthatareassociatedtotheg constant. +A +shape-factorfunctionC(W)=1+aW +b/W +cW2.Inthis In particular, 99Tc decay was predicted to be very sensitive +geff, +case, the mean result agrees to within 74eV. However, a to making this spectrum a good candidate for a precise +A +somewhatlargerspreadofresultsforthemaximumenergyis determination [8]. This approach, called the spectrum-shape +observedwhenusingthisshape-factorparametrization,which method and introduced in [58], was applied in the present +might be due to the larger number of adjustable parameters. work.Theanalysiswascarriedoutonthespectrummeasured +The uncertainty budgets for both measurements are listed in and analyzed at PTB, chosen for its lower background cor- +TableVI.TheindividualresultsE =295.798(156)keVfor rection. Because full computation of the β spectrum is time +max +=295.854(199)keV consuming, the energy binning was increased to 1keV. The +| thePTBmeasurementandE | | | max | | | | forthe | | | | | | | | | +| --------------------- | --- | --- | --- | --- | --- | --- | ------ | --- | --- | --- | --- | --- | --- | --- | --- | +LNHBmeasurementareusedtocalculateaweightedmeanas maximum energy used as the input parameter was from this +| ourfinalresult: | | | | | | | | work,asdescribedabove. | | | | | | | | +| --------------- | --- | --- | --- | --- | --- | --- | --- | ---------------------- | --- | ------ | ----- | ------------- | -------- | -------- | --- | +| | | | | | | | | In Suhonen’s | | review | [56], | the effective | coupling | constant | | +=295.82(16)keV. +| | | E | | | | | (9) | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | ------------------------------------------------- | --- | --- | --- | --- | --- | --- | --- | +| | | max | | | | | | wasmostlypredictedtobequenched,andaformulaisgiven | | | | | | | | +toestimatethepartofthequenchingfactorrelatedtonuclear +| The uncertainty | | of the | final result | corresponds | | to | the uncer- | | | | | | | | | +| --------------------------------------------------- | ---------------- | ------- | ------------ | --------------- | ------------ | --- | ---------- | --------------- | ------- | ------------ | ----------- | --------------- | -------- | ------------ | ------- | +| | | | | | | | | medium | effects | from | a quenching | factor | in | infinite | nuclear | +| tainty of | the PTB | result | and | is more | conservative | | than the | | | | | | | | | +| | | | | | | | | matter. | Applied | to 99Tc | decay, | we | obtained | ginf = | 1.120. | +| inner(123eV)andouter(3eV)uncertaintiesoftheweighted | | | | | | | | | | | | | | A | | +| | | | | | | | | In addition, | a | conflicting | result | was | stressed | for the | forbid- | +| mean and | we can | exclude | an | underestimation | | of | the uncer- | | | | | | | | | +| | | | | | | | | den nonunique | | transitions. | In | first forbidden | | transitions, | the | +| tainty due | to correlations. | | The | analysis | described | | above also | | | | | | | | | +| | | | | | | | | single-particle | | estimates | of | the effective | coupling | constants | | +providesinformationontheparameterλ.However,theanaly- +arefoundtobegeff≈0.3–0.7andgeff≈0.46–0.56.Forhigher +sisisdedicatedtothedeterminationofthemaximumenergy +| | | | | | | | | | | V | | A | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +anditdoesnotconsiderthelow-energypartoftheβspectrum. forbidden transitions, only two fourth forbidden nonunique +transitionswerestudied,namely,113Cdand115Indecays,and +Thus,thestatedλvaluesarenotnecessarilysuitedtodescribe +thespectrumshapeinawideenergyrange. thebestadjustmentwiththeexistingmeasurementsledtoan +≈ +| | | | | | | | | unquenched | value | of | g and | geff | 0.9. The | high precision | | +| --- | --- | ----------------------- | --- | --- | --- | --- | --- | ------------------------ | ----- | -------- | -------- | ---- | -------- | -------------- | --- | +| | | | | | | | | | | | V | A | | | | +| | | | | | | | | measurement | of | the 99Tc | spectrum | from | this | work brings | a | +| | B. | Thespectrum-shapemethod | | | | | | new,importantconstraint. | | | | | | | | +β +| In usual | descriptions | | of nuclear | | decay, | the Hamiltonian | | | | | | | | | | +| ---------- | ------------ | --- | ---------- | --------- | ------ | --------------- | --------- | --- | --- | --- | --- | --- | --- | --- | --- | +| density is | expressed | in | terms | of lepton | and | hadron | currents. | | | | | | | | | +βspectrummodeling +| Assuming | a pure | (V −A) | weak | interaction, | | transition | ma- | | | 1. | | | | | | +| -------- | ------ | ------ | ---- | ------------ | --- | ---------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +β +trix elements are of the vector type, associated to the vector The method to calculate the spectrum has already been +coupling constant g , or of axial-vector type, associated to described in [51,60] and follows the formalism of Behrens +V +the axial-vector coupling constant g . The conserved vector and Bühring [61]. The theoretical spectrum is described by +A +055503-10 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +TABLE VI. Uncertainty budgets for the maximum β energy E for the two measurements. All uncertainties are stated as standard +max +uncertainties(k=1). +| | | LNHBmeas. | | | PTBmeas. | | | | | | | | +| ----------- | --- | --------- | ---- | ----------- | ---------------- | -------- | --- | --- | --- | ------- | --- | --- | +| Uncertainty | | | u | Relative | u | Relative | | | | | | | +| component | | | (eV) | uncertainty | (eV) uncertainty | | | | | Comment | | | +Energycalibration 45 0.015% 50 0.017% Severalwell-knownpeaksusedforthecalibration,high +reproducibility,differentcalibrationsourcesinthetwo +measurements(57Coand133Ba) +Resolution 10 0.003% 10 0.003% Binsathighenergiesavoided;theseeffectsarefurther +| distortioneffect, | | | | | | | | describedinKossertetal.[51]. | | | | | +| ----------------- | --- | --- | --- | --- | --- | --- | --- | ---------------------------- | --- | --- | --- | --- | +finiteenergy +resolution +| Background | | | 141 | 0.048% | 69 | 0.023% | Backgroundtakenintoaccount | | | | | | +| ---------- | --- | --- | --- | ------ | --- | ------ | -------------------------- | --- | --- | --- | --- | --- | +Fitmethod 93 0.031% 89 0.030% Variationofenergyrange;weightedvsunweightedfit, +rebinning +Theoreticalmodel 75 0.025% 75 0.025% Analysiswithothershape-factorparametrization, +differencewith/withoutscreeningandexchange +Spectrumunfolding 57 0.019% 57 0.019% Dependenceoncrosssections(e.g.,bremsstrahlung)and +geometry +Analysis 10 0.003% 10 0.003% Deviationwhenusinginputspectraobtainedfromthetwo +| software/pile-up | | | | | | | | analysiscodes | | | | | +| ---------------- | --- | --- | --- | ------ | --- | ------ | --- | ------------- | --- | --- | --- | --- | +| Combined | | | 199 | 0.067% | 156 | 0.053% | | | | | | | +Eq. (5) on which the radiative corrections given in [52] are (m R),(WR),and(αZ)andisherereferredtoasthetruncated +e +also applied. Two terms, however, are calculated differently: leptoncurrent.Inthecaseofthesecondforbiddennonunique +99Tcdecay,wefollowedBehrens’andBühring’srecommen- +theatomicexchangecorrectionandtheshapefactor. +Themodelingoftheatomicexchangeeffectwasdeveloped dation,keepingonlythedominantterms[61].Itisnoteworthy +decadesagofortheallowedtransitions[62],andwasrecently thatin[56],theauthorsalsoconsiderednext-to-leading-order +extended to the forbidden unique transitions [63]. Applying terms as described in [64]. This procedure is possible only +Eq. (3) in [63] allowed us to calculate the exchange effect withasimplifiedCoulombpotential.Asinourpreviousstudy +specifically for the second forbidden nonunique transition in of 151Sm decay [51], we also considered a full numerical +99Tc +decay. To this purpose, it is necessary to introduce nu- lepton current based on lepton wave functions determined +clearstructureinformation,whichisalsoneededfortheshape by solving the Dirac equation with a Coulomb potential that +factorandwhichwedetailnow. includes atomic screening. All the terms of the lepton wave- +TheshapefactorC(W)inEqs.(5)and(7)isexperimental, functionexpansionarethereforevirtuallyaccountedfor. +i.e.,itsglobalenergydependencyisbasedonsometheoretical For calculating the shape factor, an input from a realistic +arguments,butisadjustedonthemeasuredspectrum.Toex- nuclearstructuremodelisrequired.TheNUSHELLXcode[65] +tractthecouplingconstants,C(W)mustbepurelytheoretical. was used in this work to determine the list of nucleon- +99Tc +The shape factor is a convolution of the nuclear structure nucleon transitions that contribute to the decay. These +andtheleptondynamicsandisusuallyexpandedindifferent single-particletransitionsareweightedbytheircorresponding +multipolesofboththenuclearandleptoncurrents, one-body transition densities (OBTDs). Following [8], we +(cid:7) firstconsideredtheeffectiveinteractionfromGloeckner[66] +(cid:6) +C(W)= λ M 2(k ,kν)+m 2(k ,kν) w i th t h e G L v a l e n c e s p a c e s p a n n i n g t h e p ro to n o rb i ta l s 2 p / +| | | ke | K e | K e | | | | | | | | 1 2 | +| --- | ---- | --- | --- | --- | ------- | --- | ------------- | -------------------- | ----------- | ------------- | --------------- | --------------------- | +| | | | | | | a n | d 1 g | , an d t h e n e u | tr o n o r | b i t a l s 3 | s a n d 2 d | . W i t h s u c h | +| | K,ke | ,kν | | | | | 9 / 2 | | | | 1 / 2 5 / | 2 | +| | | | | | (cid:8) | a | d e s c r i p | t io n , t h e 9 9 T | c d e c a y | i s d r iv | e n b y a s i n | g l e n u c l e o n - | +2μ γ +| | | | ke | keM | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +− (k ,kν)m (k ,kν) . (10) nucleontransition,fromaneutronin2d 5/2 toaprotonin1g 9/2 . +| | | | k W | K e K | e | | | | | | | | +| ---------- | ------- | --- | ---------- | --------------- | -------- | ---------------------------------------------------- | --- | ------------- | --- | ----------- | ------------ | ------ | +| | | | e | | | Next,theeffectiveinteractionfromMach[67]withthewider | | | | | | | +| | | | | | | GLEKPN | | valence space | was | considered. | This valence | space | +| Quantities | labeled | by | the lepton | quantum numbers | k and kν | | | | | | | | +| | | | | | e | spanstheprotonorbitals1f | | | | ,1f | ,2p ,2p | ,and1g | +dependontherelativisticwavefunctions,andK istheprinci- 7/2 5/2 3/2 1/2 9/2 , +| | | | | | | and | the neutron | orbitals | 1g 9/2 | , 1g 7/2 | , 2d 5/2 , 2d 3/2 | , and 3s 1/2 . | +| --- | --- | --- | --- | --- | --- | --- | ----------- | -------- | ------ | -------- | ----------------- | -------------- | +palnuclearmultipoleorder.Theleptoncurrentwassimplified +| | | | | | | To | limit | the computational | burden, | | the 1f proton | and the | +| --- | --- | --- | --- | --- | --- | --- | ----- | ----------------- | ------- | --- | ------------- | ------- | +by Behrens and Bühring in order to decouple the calculation 7/2 +| | | | | | | 1g | neutron | orbitals | were | constrained | to be full, | and four | +| ----------------------------------------------------- | --- | --- | --- | --- | --- | --- | ------- | -------- | ---- | ----------- | ----------- | -------- | +| ofleptonandnuclearmatrixelements.Theprocedureconsists | | | | | | | 9/2 | | | | | | +orbital.Asaresult,the99Tc +in expanding the lepton radial wave function in powers1 of protonswereblockedinthe1f 5/2 +| | | | | | | decay | is | still dominantly | driven | by | the same single-particle | | +| --- | --- | --- | --- | --- | --- | ----- | --- | ---------------- | ------ | --- | ------------------------ | --- | +transition,butasmalladmixtureofatransitionfromaneutron +| | | | | | | in | 1g 7/2 | to a proton in | 1g 9/2 | appears | (see Table | VII for the | +| --- | --- | --- | --- | --- | --- | --- | ------ | -------------- | ------ | ------- | ---------- | ----------- | +=2).Finally,wealsoconsidered +1R is the daughter nucleus radius and α is the fine-structure con- dominantmultipoleorderK +stant. the jj45pnb interaction [68] with the valence space spanning +055503-11 + +| M.PAULSENetal. | | | | | | | | | | | PHYSICALREVIEWC110,055503(2024) | | | | | +| -------------- | --------- | -------- | ------------------------------------------- | -------------------- | ----- | ------------ | ---------- | -------- | --- | --- | ------------------------------- | --- | --- | --- | --- | +| | TABLEVII. | | One-bodytransitiondensities(OBTDs)ofthedom- | | | | | | | | | | | | | +| inant | multipole | | order (K | =2), as | given | by NUSHELLX. | | Coulomb | | | | | | | | +| displacement | | energies | for | each nucleon-nucleon | | | transition | are also | | | | | | | | +given(MeV). +One-particle +| | | | | | (cid:4)E(3) | (cid:4)E(4) | | (cid:4)E(4) | | | | | | | | +| ---------- | --- | --- | ----- | --- | ----------- | ----------- | ---- | ----------- | --- | --- | --- | --- | --- | --- | --- | +| transition | | | OBTDs | | | | C1,2 | C2,1 | | | | | | | | +C +| 1g | → | 1g | 0.00994 | | 11.696 | 12.051 | | 12.045 | | | | | | | | +| --- | --- | --- | ------- | --- | ------ | ------ | --- | ------ | --- | --- | --- | --- | --- | --- | --- | +| | 7/2 | 9/2 | | | | | | | | | | | | | | +→ +| 2d | 5/2 | 1g 9/2 | 0.47752 | | 10.560 | 10.536 | | 10.537 | | | | | | | | +| ------- | -------- | --------- | ------- | ------------- | ------------ | ------ | --------------- | -------- | --- | --- | --- | --- | --- | --- | --- | +| 1f | , 2p | , 2p | , and | 1g | for protons, | and | 1g | , 2d , | | | | | | | | +| | 5/2 | 3/2 | 1/2 | 9/2 | | | 7/2 | 5/2 | | | | | | | | +| 2d | 3/2 , 3s | 1/2 , and | 1h 11/2 | for neutrons. | | The | only constraint | | | | | | | | | +| applied | was | a minimum | | of four | protons | in | the 1f 5/2 | orbital. | | | | | | | | +| OBTDs | are | quite | close | to those | obtained | with | GLEKPN, | but | | | | | | | | +noticeablyofoppositesignforthedominantmultipoleorder: +| 0.01107for1g | | | →1g | and−0.43985for2d | | | →1g | | | | | | | | | +| ------------ | --- | --- | --- | ---------------- | --- | --- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | +| | | | 7/2 | 9/2 | | | 5/2 | 9/2 . | | | | | | | | +Thetransitionprobabilityofaforbiddennonuniquedecay +| first | depends | on | a relativistic | vector | matrix | element. | | While a | | | | | | | | +| --------------- | ----------- | --- | --------------- | ------ | -------------- | --------- | ---------- | ------- | --- | --- | --- | --- | --- | --- | --- | +| nonrelativistic | | | matrix element | | couples | the large | components | | | | | | | | | +| of | the nucleon | | wave functions, | | a relativistic | | matrix | element | | | | | | | | +couplestheirsmallandlargecomponentstogether.However, +mostofthenuclearmodels,asNUSHELLX,arenonrelativistic. +Comparisonofthemeasured99Tcspectrumwithdiffer- +| Anaccurateestimateofthisrelativisticvectormatrixelement | | | | | | | | | FIG.8. | | | | | | | +| ------------------------------------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | ------ | --- | --- | --- | --- | --- | --- | +can be obtained from a nonrelativistic vector matrix element enttheoreticalcurves.CVCiseitherignoredorincludedfordifferent +employingtheCVChypothesis(see,e.g.,36Cldecayin[69]). estimatesoftheCoulombdisplacementenergy(cid:4)E .Inset:Thelow- +C +| In99Tcdecay,therelationshipis | | | | | | | | | energypartofthespectrum. | | | | | | | +| ----------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | ------------------------ | --- | --- | --- | --- | --- | --- | +R +| | | (cid:7)−√ | | −(m | −m )+(cid:4)E | | | , | | | | | | | | +| --- | --- | --------- | --- | --- | ------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +VF [W ]VF (11) whereg i andg f aretheradiallargecomponentsoftheinitial +| | 221 | | 0 | n | p | C | 220 | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ----------------------------------------------------- | --- | --- | --- | --- | --- | --- | +| | | | 10 | | | | | | andfinalnucleonwavefunctions,respectively.ThisCoulomb | | | | | | | +displacementenergy(cid:4)E(3)isthusdifferentforeachnucleon- +| with | m | and m | the neutron | | and proton | | rest masses | and | | | | C | | | | +| -------- | ------ | ------ | -------------- | --- | ---------- | --- | ----------- | ------- | ------- | ----------- | --- | -------- | ----------- | ------ | ---------- | +| | n | | p | | | | | | | | | | | | β-particle | +| (cid:4)E | | | | | | | | | nucleon | transition, | but | is still | independent | of the | | +| | C th e | Coulom | b displacement | | energy. | The | small | Q-value | | | | | | | | +9Tcdecaymakesacriticalgoodestimateof(cid:4)E energy. Finally, one can also introduce a dependency on the +| of9 | | | | | | | | because | | | | | | | | +| --- | --- | --- | ----------------- | --- | --- | --- | --- | ------- | ---------- | ------ | --- | ----------- | --- | --------------- | ---- | +| | | | | | | | C | | β-particle | energy | by | considering | the | complete lepton | cur- | +| [W | −(m | −m | )]=−0.487MeVonly. | | | | | | | | | | | | | +0 n p rent, instead of the leading order (r/R)K. The corresponding +TheCoulombdisplacementenergycanbeestimatedfrom +Coulombdisplacementenergyisdenoted(cid:4)E(4) +andalsode- +| different | | methods. | Usually, | a | uniformly | charged | sphere | is | | | | | | C | | +| ---------- | --- | -------- | -------------- | --- | ------------ | ------- | ------ | -------- | -------- | ----------------- | ------- | -------- | --------- | --------- | ----------- | +| | | | | | | | | | pends on | the lepton | quantum | | numbers | k and kν. | This latter | +| considered | | and | the expression | | only depends | | on the | daughter | | | | | | e | | +| | | | | | | | | | method | is, in principle, | | the most | accurate. | In Table | VII, we | +nucleusthroughitsatomicnumberanditsnuclearradius, +give(cid:4)E(3)andtheaveragevalueof(cid:4)E(4) +, forthedominant +| | | | | | | | | | | C | | | C | ke k ν | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------ | --- | +6αZ nucleon- nucleon transitions. In practic e, t h e dependency of +| | | (cid:4)E | (1) = | f | =13.476MeV. | | | | | | | | | | | +| --- | --- | -------- | ----- | --- | ----------- | --- | --- | ---- | -------------------------------------------- | ------ | ---------- | ------ | --- | ----------- | --------- | +| | | | | | | | | (12) | (cid:4)E (4 ) | in the | β-particle | energy | was | found to be | less than | +| | | | C | 5 R | | | | | C k ,kν | | | | | | | +| | | | | f | | | | | 1keVovertheentireenergyrangeofthespectrum. e | | | | | | | +Asimilarexpressioncanbeestablishedthatdependsonboth In Fig. 8, we present the spectrum calculated without the +the parent and daughter atomic numbers and their nuclear CVC hypothesis, and the spectra considering CVC for the +| | | | | | | | | | different | (cid:4)E | estimates. | The | free-neutron | value | of g was | +| ------ | --- | --- | --- | --- | --- | --- | --- | --- | --------- | -------- | ---------- | --- | ------------ | ----- | -------- | +| radii, | | | | | | | | | | C | | | | | A | +assumed.CVChasaninfluencebutdoesnotleadtoachange +| | | 3 α | | 3 | α | | | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +(cid:4)E(2) = −1)− −1)=12.814MeV. of the spectrum shape as spectacular as in 36Cl decay [69]. +| | | | Z (Z | | Z(Z | | | | | | | | | | | +| --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +C 5R f f 5R i i Thespectrummostlyexhibitsadependencyatlowenergyand +| | | f | | | i | | | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | ---- | ------------------------- | --- | --- | -------------- | --- | --- | --- | +| | | | | | | | | (13) | thespectrawith(cid:4)E(3) | | | and(cid:4)E(4) | | | | +arehardlydistinguished.In +| | | (cid:4)E | | | | | | | | | C | (cid:4)E | C | | | +| --- | --- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | -------- | --- | --- | --- | +However, C is known to possibly be sensitive to the addition, we tried to adjust in order to be as close as +C +mismatch between the initial and final nucleon wave func- possibletothemeasuredspectrum.Thebestvaluewasfound +tions [70]. Behrens and Bühring approximated the single- tobe(cid:4)E =3.57MeVwithapoorreduced-χ2 of5.018,and +C +| particle | potential | | difference | by | the average | | of the Coulomb | | | | | | | | | +| -------- | --------- | --- | ---------- | --- | ----------- | --- | -------------- | --- | --- | --- | --- | --- | --- | --- | --- | +anonlineartendencywasfoundintheresiduals.Mostofall,it +potential, keeping only the leading order of the lepton cur- isclearthattheadjustedCoulombdisplacementenergyisnot +| rent[61], | | | | | | | | | realistic. | | | | | | | +| --------- | --- | --- | --- | --- | --- | --- | --- | --- | ---------- | --- | --- | --- | --- | --- | --- | +(cid:9) +| | | | | ∞ | | | | | This | result | supports | an adjustment | | of the coupling | con- | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | ------ | -------- | ------------- | --- | --------------- | ---- | +g V(r)g(r/R)Kr2dr +(cid:4)E(3) = 0(cid:9) f i , stants to retrieve the measured spectrum. We chose as a +| | | | | ∞ | | | | (14) | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | +C g g(r/R)Kr2dr reference for the fitting procedure the spectrum calculated +| | | | | 0 f | i | | | | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +055503-12 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +2 +1.5 40 +1 +0.5 30 +FIG.9. Influence of the effective g value on the theoretical β 0 +A 20 +spectrumof99Tcdecay,comparedtothehighprecisionmeasurement 0.5 +fromthiswork. 1 10 +1.5 +2 0 +considering a full numerical treatment of the lepton current, 0 100 200 300 2 1 0 1 2 +the OBTDs from the GLEKPN valence space with Mach in- +teraction, and the CVC hypothesis with the (cid:4)E(4) Coulomb FIG.10. Comparison of the measured 99Tc spectrum with the +C +displacementenergy.Allotherpossibilitieswereusedtoesti- theoretical curve with the best adjusted geff value. Inset: The low- +A +mateuncertainties. energypartofthespectrumandthedistributionsoftheresiduals. +2. Effectivecouplingconstants +such agreement would not have been possible with an erro- +Based on the modeling described above, we followed the neousendpointenergy,whichvalidatestheextractedQ-value +approachin[8,71]varyingonlyg .Oneshouldunderlinethat previously determined. Below 6keV, the atomic exchange +A +theCVChypothesis wasnotusedin[8],butwasin[71].As correctiondoesnotseemtobesufficientlyhightoaccountfor +illustratedinFig.9,weindeedobservedahighsensitivityof the distortion observed in the measured spectrum. We found +thespectrumshapeontheeffectiveg value.Quickinspection thesamediscrepancyintherecentstudyof151Smdecay[51]. +A +clearlyshowsthatageff valuebetween1.4and1.6couldgive A possible explanation could be an inaccurate modeling of +A +goodagreementwiththemeasuredspectrumfromthiswork. the atomic wave functions, as recently pointed out in [72]. +Thefitproceduresimplyconsistedinredoingthecalculations These authors provide an analytical fit of their correction, +fordifferentgeffvaluesuntilfindingtheminimumreduced-χ2. but only for allowed transitions. We applied it only to the +A +A reasonable energy range of 20 to 275keV was considered Fermifunction,whichshouldgivethemaincontribution.The +todeterminethecentralvalueinordernottobeinfluencedby verylow-energypartofthespectrumisquitecomparableand +thefirstandthelast20keVofthespectrum,wheredistortions doesnotresolvethediscrepancywiththemeasuredspectrum. +can be significant. The reduced-χ2 distribution is very close The best adjusted value was found to be geff = 1.520 with +A +to a quadratic shape. The best adjusted value is geff = 1.526 reduced-χ2 =1.018,andwasusedtoestimateanuncertainty +A +withreduced-χ2 =1.010. component,asexplainedbelow. +Thefinalspectrumiscomparedtothehighprecisionmea- The uncertainty budget is detailed in Table VIII. Several +surementfromthisworkinFig.10.Theagreementisexcellent components were studied and estimated with the minimum- +down to 6keV. The distribution of the residuals does not maximum method. Different probability distributions were +showanyenergydependencyandfollowsanarrowGaussian considereddependingontheuncertaintycomponent,withthe +distribution, perfectly centered on zero. It is noteworthy that objectiveofbeingrealisticandconservative. +055503-13 + +| M.PAULSENetal. | | | | | | | | | | | PHYSICALREVIEWC110,055503(2024) | | | | +| -------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------------------------------- | --- | --- | --- | +Uncertaintybudgetforthegeffvalueextractedfromthe99Tcβ +TABLEVIII. A spectrummeasuredinthiswork.Alluncertaintiesarestated +asstandarduncertainties(k=1). +Relative +| Uncertaintycomponent | | | | | Valuein10−2 | | uncertainty | | | | | Comment | | | +| -------------------- | --- | --- | --- | --- | ----------- | --- | ----------- | --- | --- | --- | --- | ------- | --- | --- | +Estimateatχ2±1;includesbinstatisticscomponent +| Fitmethod | | | | | 7.45 | | 4.88% | | | | | | | | +| --------- | --- | --- | --- | --- | ---- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | +Fitrange 0.61 0.40% Largestdeviationobservedwithextremeenergyranges +| Maximumenergy | | | | | 0.38 | | 0.25% | | Q-valueanduncertaintyfromthiswork | | | | | | +| ------------- | --- | --- | --- | --- | ---- | --- | ----- | --- | --------------------------------- | --- | --- | --- | --- | --- | +Deviationwithjj45pnbinteraction +| Nuclearmodel | | | | | 2.94 | | 1.93% | | | | | | | | +| ------------ | --- | --- | --- | --- | ---- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | +Leptoncurrent 1.27 0.83% Deviationwithsimplifiedleptoncurrentand(cid:4)E(3) +C +Largestdeviationobservedwith(cid:4)E(1) +| Coulombdisplacementenergy | | | | | 4.25 | | 2.78% | | | | | | | | +| ------------------------- | --- | --- | --- | --- | ---- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | +C +| Atomicexchange | | | | | 0.29 | | 0.19% | | Deviationwithcorrectionfrom[72] | | | | | | +| -------------- | --- | --- | --- | --- | ---- | --- | ----- | --- | ------------------------------- | --- | --- | --- | --- | --- | +Radiativecorrections 0.33 0.21% Conservativeestimate,withorwithoutincludingthem +(cid:14)2-norm,i.e.,thesquarerootofthequadraticsum +| Combined | | | | | 9.19 | | 6.02% | | | | | | | | +| -------- | --- | --- | --- | --- | ---- | --- | ----- | --- | --- | --- | --- | --- | --- | --- | +The main contribution comes from the fit method, which extractedfrom99Tcspectrumiseventually +| also includes | the | spectrum | statistics. | | The quadratic | | shape of | | | | | | | | +| -------------- | ----------- | ------------ | ----------- | ----------- | ------------- | ------------- | -------- | --- | --- | --- | ----------------- | --- | --- | ---- | +| | | | | | | | | | | | ge ff =1.526(92). | | | (15) | +| the reduced-χ2 | | distribution | | allows | one to | estimate | the fit | | | | A | | | | +| method | uncertainty | from | the | geff values | at | reduced-χ2+1. | | | | | | | | | +A It can be surprising that our g A value is not quenched but +≈ +The chosen energy range for the fitting procedure also has enhanced, far from Suhonen’s review value of geff 0.9. +A +an influence. Two extreme cases were considered, namely, 0 However, our calculated half life obtained from spectrum +to 296keV and 100 to 200keV. The largest deviation was integrationis29.98(85)×103a,oneorderofmagnitudelower +usedwithatriangularprobabilitydistribution.Themaximum 211.5(11)×103a +| | | | | | | | | than | the evaluated | | half life | | [11]. | We can | +| --- | --- | --- | --- | --- | --- | --- | --- | ---- | ------------- | --- | --------- | --- | ----- | ------ | +energy of the spectrum determined in this work was used very accurately describe the global 99Tc decay, i.e., both the +for the calculation and its uncertainty was propagated with spectrumshapeandthehalflife,bysimplyrenormalizingthe +| calculations | at | ±1σ | considering | a | rectangular | | probability | | | | | | | | +| ------------- | --- | --- | ----------- | --- | ----------- | --- | ----------- | -------------- | --------- | --- | -------- | -------------- | --------------- | --- | +| | | | | | | | | coupling | constants | | with the | ratio of these | two half lives. | We | +| distribution. | | | | | | | | thendetermined | | | | | | | +Fortheothercomponents,spectrumcalculationswereper- +| | | | | | | | | | ge ff | =0.376(5) | | and ge ff =0.574(36). | | (16) | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | ----- | --------- | --- | --------------------- | --- | ---- | +formed with different hypotheses, geff was extracted, and V A +A +| the largest | deviation | from | ge | ff = 1.526 | was | considered | with | | | | | | | | +| ----------- | --------- | ---- | --- | ---------- | --- | ---------- | ---- | --- | -------- | --------- | --- | -------------- | ----------- | ---- | +| | | | A | | | | | The | coupling | constants | are | thus quenched, | with values | that | +a triangular probability distribution. Uncertainty due to the arenoticeablyconsistentwithSuhonen’sreviewvaluesforthe +| nuclear model | | was estimated | | from the | largest | obtained | devi- | | | | | | | | +| ------------- | --- | ------------- | --- | -------- | ------- | -------- | ----- | --- | --- | --- | --- | --- | --- | --- | +firstforbiddennonuniquetransitions.Ourresultthusseemsto +ation,consideringeithertheGLmodelspacewithGloeckner +solvetheinconsistencyhighlightedbySuhonenin[56]. +interaction[66]orthejj45pnbinteraction[68].Theusualsim- Consideringthefullenergyrange,wedeterminedtheaver- +plified lepton current was employed to assess an uncertainty ageenergyoftheβ spectrumfromourcalculation, +duetoleptoncurrenttreatment.Thedifferentmethodsforde- +=98.51(23)keV, +| terminingtheCoulombdisplacementenergyweretestedand | | | | | | | | | | | Eβ | | | (17) | +| -------------------------------------------------- | ---- | ---------- | --- | -------- | ---------- | --------- | --------- | ---------------------- | --- | --- | --- | -------- | --- | ---- | +| themaximumdeviationwasfoundwith(cid:4)E | | | | | | (1).Thege | | | | | | | | | +| | | | | | | | ff value | andthecorrespondinglog | | | | f value, | | | +| | | | | | | C | A | | | | | | | | +| determined | with | the atomic | | exchange | correction | | from [72] | | | | | | | | +=−0.47660(22). +was considered to estimate an uncertainty due to this log f (18) +| correction. | | | | | | | | Withthepartialhalflifefrom[11],weobtained | | | | | | | +| ----------- | --- | ---------- | --- | ----------- | --------- | --- | ------ | ----------------------------------------- | --- | --- | --- | --- | --- | --- | +| Finally, | we | considered | an | uncertainty | component | | due to | | | | | | | | +=12.3478(23) +the radiative corrections. The latter include the emission of log ft (19) +real soft photons from the internal bremsstrahlung process. fortheground-statetoground-statetransitionin99Tcdecay. +β +| This part | of the | correction | on | the | spectrum | assumes | that | | | | | | | | +| --------- | ------ | ---------- | --- | --- | -------- | ------- | ---- | -------- | --- | ---------------- | --- | ------------ | --------- | ----- | +| | | | | | | | | Notably, | | the experimental | | shape factor | from [15] | leads | +thesephotonsarelostandthusnotdetected.However,thereis = 95.91(5)keV, =−0.8785(11), = +| | | | | | | | | to Eβ | | | log | f | and | log ft | +| -------- | --------- | --- | --------- | ---------- | --- | ------ | --------- | ----- | --- | --- | --- | --- | --- | ------ | +| no doubt | that they | are | partially | reabsorbed | | by the | detection | | | | | | | | +11.9458(25),usingtheQ-valuefromthiswork. +| system, | especially | the | low-energy | | photons. | A conservative | | | | | | | | | +| ------- | ---------- | --- | ---------- | --- | -------- | -------------- | --- | --- | --- | --- | --- | --- | --- | --- | +estimatewasobtainedbysimplyignoringtheradiativecorrec- +IV. CONCLUSION +| tions in | the spectrum | calculation. | | It | is more | than | an extreme | | | | | | | | +| -------- | ------------ | ------------ | --- | --- | ------- | ---- | ---------- | --- | --- | --- | --- | --- | --- | --- | +case of possible photon reabsorption because additional cor- In the work described here, it is once again demonstrated +rections that do not come from internal bremsstrahlung are thatMMCmeasurementsareexcellentlysuitedfordetermin- +alsoignored. ingboththeshapeandthemaximumenergyofβspectrawith +ThetotalcombineduncertaintyisgiveninTableVIII.The +Qβ-valuesbelow1MeV.Themeasurementsarecharacterized +main component comes from the statistics of the measure- not only by the high linearity and high-energy resolution, +ment.Thevalueoftheeffectiveaxial-vectorcouplingconstant but also by the fact that very low detection thresholds can +055503-14 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +be reached. The good agreement of two almost independent consistent the ones derived from first forbidden nonunique +measurementsalsointhecrossanalysisofthedataincreases transitionsin[56]. +confidenceintheobtainedβ +spectra.Sincethebackgroundis Our Q-value is five times more precise than the recom- +the major uncertainty contribution of the obtained maximum mended one [17] and shifted by −1.8keV, which is about +β energy, as shown in Table VI, we conclude that experi- two standard deviations of the recommended value. The un- +mental design and optimized calibration sources are crucial certaintyiscompetitivewithPenningtrapmeasurementsand +toimproveuponsuchmeasurementsusingMMCs. we call for a confirmation of our Q-value using this method. +High-quality PIPS detector measurements confirm the BeyondMMCs,theactiveArrayofCryogenicCalorimetersto +shape of the spectrum above 25keV. We do not make a de- EvaluateSpectralShapes(ACCESS)projectaimstomeasure +β +tailed comparison with previous determinations of the shape forbidden decays such as 99Tc using an array of neutron +of the β spectrum, but point out that parametrizations of the transmutation-doped germanium (Ge-NTD or NTD) detec- +β spectrum found in the literature must now be considered tors [73]. Our results could then be confirmed in the near +obsolete.Itshouldbenotedthatallpreviousmeasurementsof futurewithanotherindependenttechnique. +| the 99Tc β | spectrum | had | significantly | higher-energy | | thresh- | | | | | +| ---------- | -------- | ----------- | ------------- | ------------- | -------- | ------- | --- | --- | --- | --- | +| olds, so | that a | significant | part | of the | spectrum | at low | | | | | +energies had to be considered as being unknown. Combin- ACKNOWLEDGMENTS +| ing our measurements | | with | detailed | theoretical | | calculations | | | | | +| -------------------- | --- | ---- | -------- | ----------- | --- | ------------ | --- | --- | --- | --- | +based on the nuclear shell model, we extracted new decay This work is part of the project 20FUN04 PrimA-LTD +dataofinterest:Qβ =295.82(16)keV,Eβ =98.51(23)keV, that has received funding from the EMPIR programme co- +=−0.47660(22),andlog =12.3478(23).Thespec- financed by the Participating States and from the European +| log f | | | ft | | | | | | | | +| ---------- | --- | -------- | ------- | --------- | ------ | --------- | --------------- | ------------- | -------------- | ---------- | +| | | | | | | | Union’s Horizon | 2020 research | and innovation | programme. | +| trum shape | was | found to | be very | sensitive | to the | effective | | | | | +value of the axial-vector coupling constant, and the best The Linux-Compute-Cluster at the PTB Berlin was used to +runEGSNRC.WethankRolfBehrens,GerdLindner,Andreas +| description | of the | decay | led to the | quenched | coupling | con- | | | | | +| ----------- | ------ | ----- | ---------- | -------- | -------- | ---- | --- | --- | --- | --- | +=0.376(5)andgeff =0.574(36).Thesevaluesare Lübbert,andHayoZutzfortheirhelpfulcooperation. +stantsgeff +| V | | | A | | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +[1] K.KossertandX.Mougeot,Theimportanceofthebetaspec- program of ionizing radiation metrology, J. Low Temp. Phys. +trumcalculationforaccurateactivitydeterminationof63Niby 193,1251(2018). +meansofliquidscintillationcounting,Appl.Radiat.Isot.101, [11] M.-M. Bé, V. Chisté, C. Dulieu, X. Mougeot, V. Chechev, N. +40(2015). Kuzmenko, F. Kondev, A. Luca, M. Galán, A. Nichols, A. +[2] K. E. Koehler, Low temperature microcalorimeters for decay Arinc, A. Pearce, X. Huang, and B. Wang, Table of Radionu- +energyspectroscopy,Appl.Sci.11,4044(2021). clides,MonographieBIPM-5,Vol.6(BureauInternationaldes +[3] R. P. Fitzgerald, B. K. Alpert, D. T. Becker, D. E. Bergeron, PoidsetMesures,PavillondeBreteuil,F-92310Sèvres,France, +| R. M. | Essex, | K. Morgan, | S. Nour, | G. O’Neil, | D. | R. Schmidt, | 2011). | | | | +| ----- | ------ | ---------- | -------- | ---------- | --- | ----------- | ------ | --- | --- | --- | +G. A. Shaw et al., Toward a new primary standardization [12] L.FeldmanandC.S.Wu,Investigationofthebeta-spectraof +10Be,40K,99Tc,and36Cl,Phys.Rev.87,1091(1952). +| of radionuclide | | massic | activity | using | microcalorimetry | and | | | | | +| --------------- | --- | ------ | -------- | ----- | ---------------- | --- | --- | --- | --- | --- | +quantitativemilligram-scalesamples,J.Res.Natl.Inst.Stand. [13] S. I. Taimuty, The beta-spectrum of 99Tc, Phys. Rev. 81, 461 +| Technol.126,126048(2021). | | | | | | | (1951). | | | | +| ------------------------- | --- | --- | --- | --- | --- | --- | ------- | --- | --- | --- | +[4] X.Mougeot,Reliabilityofusualassumptionsinthecalculation [14] R. E. Snyder and G. B. Beard, Decay of 94Nb and 94mNb, +ofβandνspectra,Phys.Rev.C91,055504(2015). Phys.Rev.147,867(1966). +M.ReichandH.M.Schüpferling,Formfaktordesβ-Spektrums +| [5] L. Hayen, | J. | Kostensalo, | N. Severijns, | | and J. Suhonen, | First- | [15] | | | | +| ------------- | --- | ----------- | ------------- | --- | --------------- | ------ | ---- | --- | --- | --- | +forbiddentransitionsinthereactoranomaly,Phys.Rev.C100, von99Tc,Z.Phys.271,107(1974). +054323(2019). [16] H. Behrens and L. Szybisz, Shapes of Beta Spectra, Physics +[6] V. Brdar, R. Plestid, and N. Rocco, Empirical capture cross Data (Zentralstelle für Atomkernenergie-Dokumentation, +| sections | for cosmic | neutrino | detection | with | 151Sm | and 171Tm, | 1976). | | | | +| -------- | ---------- | -------- | --------- | ---- | ----- | ---------- | ------ | --- | --- | --- | +Phys.Rev.C105,045501(2022). [17] M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, The +[7] X.MougeotandC.Bisch,Consistentcalculationofthescreen- AME2020atomicmassevaluation(II).Tables,graphsandref- +ingandexchangeeffectsinallowedβ− transitions,Phys.Rev. erences,Chin.Phys.C45,030003(2021). +A90,012501(2014). [18] D.E.Alburger,P.Richards,andT.H.Ku,Betadecayof99Tcm, +[8] J.KostensaloandJ.Suhonen,g -drivenshapesofelectronspec- Phys.Rev.C21,705(1980). +a +traofforbiddenβdecaysinthenuclearshellmodel,Phys.Rev. +| | | | | | | | [19] H. R. Doran, | A. J. Cresswell, | D. C. W. Sanderson, | and G. | +| --- | --- | --- | --- | --- | --- | --- | ----------------- | ---------------- | ------------------- | ------ | +C96,024317(2017). Falcone, Nuclear data evaluation for decay heat analysis of +[9] A.Algora,J.Tain,B.Rubio,M.Fallot,andW.Gelletly,Beta- spent nuclear fuel over 1–100 k year timescale, Eur. Phys. J. +decaystudiesforappliedandbasicnuclearphysics,Eur.Phys. Plus137,665(2022). +J.A57,85(2021). [20] M.Loidl,J.Beyer,L.Bockhorn,C.Enss,S.Kempf,K.Kossert, +[10] M.Loidl,J.Beyer,L.Bockhorn,C.Enss,D.Györi,S.Kempf, R.Mariam,O.Nähle,M.Paulsen,P.Ranitzsch,M.Rodrigues, +K. Kossert, R. Mariam, O. Nähle, M. Paulsen, M. Rodrigues, and M. Schmidt, Beta spectrometry with metallic magnetic +and M. Schmidt, MetroBeta: Beta spectrometry with metal- calorimetersintheframeworkoftheEuropeanEMPIRproject +lic magnetic calorimeters in the framework of the european MetroBeta,Appl.Radiat.Isot.153,108830(2019). +055503-15 + +M.PAULSENetal. PHYSICALREVIEWC110,055503(2024) +[21] M. Loidl, J. Beyer, L. Bockhorn, J. J. Bonaparte, C. Enss, (American Mathematical Society, Providence, Rhode Island, +S. Kempf, K. Kossert, R. Mariam, O. Nähle, M. Paulsen, P. 1990),Vol.112,pp.183–194. +Ranitzsch,M.Rodrigues,andM.Wegner,Precisionmeasure- [37] C. Bates, C. Pies, S. Kempf, D. Hengstler, A. Fleischmann, +ments of beta spectra using metallic magnetic calorimeters L. Gastaldo, C. Enss, and S. Friedrich, Reproducibility and +within the european metrology research project metrobeta, calibration of MMC-based high-resolution gamma detectors, +J.LowTemp.Phys.199,451(2020). Appl.Phys.Lett.109,023513(2016). +[22] A. Fleischmann, C. Enss, and G. Seidel, Metallic magnetic [38] P. Bell, E. Burovski, J. Charlong, R. Gommers, M. Picus, T. +calorimeters, in Cryogenic Particle Detection: Topics in Ap- Reddy,P.Roy,S.Wallkötter,andA.Volant,OrthogonalDis- +plied Physics, edited by C. Enss (Springer, Berlin, 2005), tance Regression in Python’s SciPy 1.7.1 Module (The SciPy +pp.151–216. Community,2021). +[23] A.Fleischmann,L.Gastaldo,S.Kempf,A.Kirsch,A.Pabinger, [39] M.-M. Bé, V. Chisté, C. Dulieu, M. Kellett, X. Mougeot, A. +C.Pies,J.Porst,P.Ranitzsch,S.Schäfer,F.v.Seggern,T.Wolf, Arinc,V.Chechev,N.Kuzmenko,T.Kibédi,A.Luca,andA. +C. Enss, and G. M. Seidel, Metallic magnetic calorimeters, Nichols,TableofRadionuclides,MonographieBIPM-5,Vol.8 +AIPConf.Proc.1185,571(2009). (Bureau International des Poids et Mesures, Pavillon de Bre- +[24] S.Kempf,A.Fleischmann,L.Gastaldo,andC.Enss,Physics teuil,F-92310Sèvres,France,2016). +and applications of metallic magnetic calorimeters, J. Low [40] M.-M. Bé, V. Chisté, C. Dulieu, M. Kellett, X. Mougeot, A. +Temp.Phys.193,365(2018). Arinc,V.Chechev,N.Kuzmenko,T.Kibédi,A.Luca,andA. +[25] D.Drung,C.Aßmann,J.Beyer,A.Kirste,M.Peters,F.Ruede, Nichols,TableofRadionuclides,MonographieBIPM-5,Vol.9 +and T. Schurig, Highly sensitive and easy-to-use SQUID sen- (unpublished). +sors,IEEETrans.Appl.Supercond.17,699(2007). [41] M.Paulsen,K.Kossert,andJ.Beyer,Anunfoldingalgorithm +[26] H.Rotzinger,M.Linck,A.Burck,M.Rodrigues,M.Loidl,E. forhighresolutionmicrocalorimetricbetaspectrometry,Nucl. +Leblanc, L. Fleischmann, A. Fleischmann, and C. Enss, Beta Instrum.MethodsPhys.Res.Sect.A953,163128(2020). +spectrometrywithmagneticcalorimeters,J.LowTemp.Phys. [42] I.Kawrakow,D.W.O.Rogers,E.Mainegra-Hing,F.Tessier, +151,1087(2008). R.W.Townson,andB.R.B.Walters,EGSnrctoolkitforMonte +[27] M. Loidl, M. Rodrigues, C. Le-Bret, and X. Mougeot, Carlo Simulation of Ionizing Radiation Transport (National +Beta spectrometry with metallic magnetic calorimeters, ResearchCouncilCanada,2021). +Appl.Radiat.Isot.87,302(2014). [43] C.Bisch,Etudedelaformedesspectresβ,Ph.D.thesis,Uni- +[28] M. Paulsen, J. Beyer, L. Bockhorn, C. Enss, S. Kempf, K. versitédeStrasbourg,2014. +Kossert,M.Loidl,R.Mariam,O.Nähle,P.Ranitzsch,andM. [44] A.Singh,Metrologicalstudyoftheshapeofbetaspectraand +Rodrigues, Development of a beta spectrometry setup using experimental validation of theoretical models, Ph.D. thesis, +metallicmagneticcalorimeters,J.Instrum.14,P08012(2019). UniversitédeStrasbourg,2020. +[29] M. Paulsen, High resolution beta spectrometry with metallic [45] A. Singh, X. Mougeot, B. Sabot, D. Lacour, and A. +magneticcalorimetersforradionuclidemetrology,Ph.D.thesis, Nourreddine, Beta spectrum measurements using a quasi-4π +HeidelbergUniversity(2022). detectionsystembasedonSidetectors,Appl.Radiat.Isot.154, +[30] E. Mausolf, F. Poineau, T. Hartmann, J. Droessler, and K. 108897(2019). +Czerwinski,Characterizationofelectrodepositedtechnetiumon [46] A. Singh, X. Mougeot, B. Sabot, D. Lacour, and A.-M. +goldfoil,J.Electrochem.Soc.158,E32(2011). Nourreddine, Experimental studyof β spectra usingSi detec- +[31] TheMathWorksInc.,MATLABversion:9.2.0(R2017a)(The tors, in EPJ Web of Conferences (EDP Sciences, Les Ulis, +MathWorksInc.,Natick,Massachusetts,UnitedStates,2017). France,2020),Vol.239,p.02001. +[32] M.-M. Bé, V. Chisté, C. Dulieu, E. Browne, V. Chechev, [47] Computer code labZY nanoMCA module, Yantel (Yantel, +N. Kuzmenko, R. Helmer, A. Nichols, E. Schönfeld, and R. LLC,LosAlamos,NM,USA,2022),https://www.yantel.com/ +Dersch,TableofRadionuclides,MonographieBIPM-5,Vol.1 products/nanomca/. +(Bureau International des Poids et Mesures, Pavillon de Bre- [48] A. Singh, X. Mougeot, S. Leblond, M. Loidl, B. Sabot, and +teuil,F-92310Sèvres,France,2004). A.Nourreddine,Developmentofa4π detectionsystemforthe +[33] R. D. Deslattes, E. G. Kessler Jr, P. Indelicato, L. De Billy, measurementoftheshapeofβspectra,Nucl.Instrum.Methods +E.Lindroth,andJ.Anton,X-raytransitionenergies:Newap- Phys.Res.Sec.A1053,168354(2023). +proachtoacomprehensiveevaluation,Rev.Mod.Phys.75,35 [49] R. Brun and F. Rademakers, ROOT—An object oriented data +(2003). analysisframework,Nucl.Instrum.MethodsPhys.Res.Sect.A +[34] L. Bockhorn, M. Paulsen, J. Beyer, K. Kossert, M. Loidl, 389,81(1997). +O. J. Nähle, P. C.-O. Ranitzsch, and M. Rodrigues, Im- [50] F. Salvat, J. Fernandez-Varea, and J. Sempau, PENELOPE- +provedsource/absorberpreparationforradionuclidespectrom- 2014: A Code System for Monte Carlo Simulation of +etry based on low-temperature calorimetric detectors, J. Low Electron and Photon Transport (OECD/NEA Data Bank, +Temp.Phys.199,298(2020). NEA/NSC/DOC3,Issy-les-Moulineaux,France,2015). +[35] G.VanRossum,andF.L.DrakeJr.,Pythontutorial(Centrum [51] K. Kossert, M. Loidl, X. Mougeot, M. Paulsen, P. Ranitzsch, +voor Wiskunde en Informatica Amsterdam, The Netherlands, and M. Rodrigues, High precision measurement of the 151Sm +1995). beta decay by means of a metallic magnetic calorimeter, +[36] P. T. Boggs and J. E. Rogers, Orthogonal distance regression, Appl.Radiat.Isot.185,110237(2022). +inProceedingsoftheAMS-IMS-SIAMJointSummerResearch [52] L. Hayen, N. Severijns, K. Bodek, D. Rozpedzik, and X. +ConferenceonStatisticalAnalysisofMeasurementErrorMod- Mougeot,Highprecisionanalyticaldescriptionoftheallowed +els and Applications, edited by P. J. Brown and W. A. Fuller βspectrumshape,Rev.Mod.Phys.90,015008(2018). +055503-16 + +HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) +[53] J.C.HardyandI.S.Towner,Superallowed0+→0+ nuclear [63] X. Mougeot, Atomic exchange correction in forbidden +β decays:Anewsurveywithprecisiontestsoftheconserved unique beta transitions, Appl. Radiat. Isot. 201, 111018 +vectorcurrenthypothesisandthestandardmodel,Phys.Rev.C (2023). +79,055502(2009). [64] M. Haaranen, J. Kotila, and J. Suhonen, Spectrum-shape +[54] F.N.D.Kurie,J.R.Richardson,andH.C.Paxton,Theradia- method and the next-to-leading-order terms of the β-decay +tionsemittedfromartificiallyproducedradioactivesubstances. shapefactor,Phys.Rev.C95,024327(2017). +I.Theupperlimitsandshapesoftheβ-rayspectrafromseveral [65] B. Brown and W. Rae, The Shell-Model Code +elements,Phys.Rev.49,368(1936). NuShellX@MSU,Nucl.DataSheets120,115(2014). +[55] R.L.Workmanetal.(ParticleDataGroup),Reviewofparticle [66] D. Gloeckner, Shell-model systematics of the zirconium and +physics,Prog.Theor.Exp.Phys.2022,083C01(2022). niobiumisotopes,Nucl.Phys.A253,301(1975). +[56] J.T.Suhonen,Valueoftheaxial-vectorcouplingstrengthinβ [67] H.Mach,E.K.Warburton,R.L.Gill,R.F.Casten,J.A.Becker, +andββdecays:Areview,Frontier.Phys.5,55(2017). B.A.Brown,andJ.A.Winger,Meson-exchangeenhancement +[57] P. Gysbers, G. Hagen, J. D. Holt, G. R. Jansen, T. D. Morris, of the first-forbidden 96Yg(0−)→96Zrg (0+) β transition: β +P. Navrátil, T. Papenbrock, S. Quaglioni, A. Schwenk, S. R. decay of the low-spin isomer of 96Y, Phys. Rev. C 41, 226 +Stroberg,andK.A.Wendt,Discrepancybetweenexperimental (1990). +and theoretical β-decay rates resolved from first principles, [68] A. F. Lisetskiy, B. A. Brown, M. Horoi, and H. Grawe, New +Nat.Phys.15,428(2019). T =1 effective interactions for the f 5/2 p 3/2 p 1/2 g 9/2 model +[58] M. Haaranen, P. C. Srivastava, and J. Suhonen, Forbidden space:Implicationsforvalence-mirrorsymmetryandseniority +nonunique β decays and effective values of weak coupling isomers,Phys.Rev.C70,044314(2004). +constants,Phys.Rev.C93,034308(2016). [69] R.SadlerandH.Behrens,Second-forbiddenbeta-decayandthe +[59] J. Kostensalo, M. Haaranen, and J. Suhonen, Electron spec- effectof(V+A)-andS-interactionadmixtures:36Cl,Z.Phys.A +tra in forbidden β decays and the quenching of the weak 346,25(1993). +axial-vector coupling constant g , Phys. Rev. C 95, 044313 [70] J. Damgaard and A. Winter, Use of conserved vector current +A +(2017). theoryinfirstforbiddenβ-decay,Phys.Lett.23,345(1966). +[60] F.G.A.Quarati,G.Bollen,P.Dorenbos,M.Eibach,K.Gulyuz, [71] M. Ramalho and J. Suhonen, Shell-model treatment of the β +A.Hamaker,C.Izzo,D.K.Keblbeck,X.Mougeot,D.Puentes, decayof99Tc,arXiv:2312.07448. +M.Redshaw,R.Ringle,R.Sandler,J.Surbrook,andI.Yandow, [72] O.Ni¸tescu,S.Stoica,andF.Šimkovic,Exchangecorrectionfor +Measurementsandcomputationalanalysisofthenaturaldecay allowedβdecay,Phys.Rev.C107,025501(2023). +of176Lu,Phys.Rev.C107,024313(2023). [73] L. Pagnanini, G. Benato, P. Carniti, E. Celi, D. Chiesa, J. +[61] H. Behrens and W. Bühring, Electron Radial Wave Functions Corbett,I.Dafinei,S.DiDomizio,P.DiStefano,S.Ghislandi +andNuclearBeta-decay(ClarendonPress,Oxford,1982). etal.,Arrayofcryogeniccalorimeterstoevaluatethespectral +[62] N. C. Pyper and M. R. Harston, Atomic effects on β-decays, shapeofforbiddenβ-decays:TheACCESSproject,Eur.Phys. +Proc.R.Soc.LondonA420,277(1988). J.Plus138,445(2023). +055503-17 diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-004/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-004/manifest.json new file mode 100644 index 0000000..25cca45 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-004/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-005/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-005/extracted.md new file mode 100644 index 0000000..f537708 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-005/extracted.md @@ -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). diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-005/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-005/manifest.json new file mode 100644 index 0000000..b29db97 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-005/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-005/original/source.html b/.llm-wiki/raw/sources/SRC-2026-05-27-005/original/source.html new file mode 100644 index 0000000..0bb5dbc Binary files /dev/null and b/.llm-wiki/raw/sources/SRC-2026-05-27-005/original/source.html differ diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-006/extracted.md b/.llm-wiki/raw/sources/SRC-2026-05-27-006/extracted.md new file mode 100644 index 0000000..ae66d70 --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-006/extracted.md @@ -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]. | | | diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-006/manifest.json b/.llm-wiki/raw/sources/SRC-2026-05-27-006/manifest.json new file mode 100644 index 0000000..877ebcf --- /dev/null +++ b/.llm-wiki/raw/sources/SRC-2026-05-27-006/manifest.json @@ -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" +} diff --git a/.llm-wiki/raw/sources/SRC-2026-05-27-006/original/source.html b/.llm-wiki/raw/sources/SRC-2026-05-27-006/original/source.html new file mode 100644 index 0000000..c1ca5f8 Binary files /dev/null and b/.llm-wiki/raw/sources/SRC-2026-05-27-006/original/source.html differ diff --git a/.llm-wiki/wiki/concepts/essm-enhanced-spectrum-shape-method.md b/.llm-wiki/wiki/concepts/essm-enhanced-spectrum-shape-method.md new file mode 100644 index 0000000..85379b3 --- /dev/null +++ b/.llm-wiki/wiki/concepts/essm-enhanced-spectrum-shape-method.md @@ -0,0 +1,37 @@ +# ESSM — Enhanced Spectrum Shape Method + +## Overview + +The Enhanced Spectrum Shape Method (ESSM) extends the standard SSM by treating the small relativistic vector nuclear matrix element (sNME) as a free parameter alongside g_A. This allows simultaneous fitting of both the spectral shape AND the experimental half-life. + +## Motivation + +The sNME is difficult to compute accurately 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. + +## Algorithm + +1. Fix g_A at a trial value +2. Vary sNME to simultaneously fit the experimental half-life +3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each g_A: + - **sNME(c)**: closer to the CVC value (closeCVC) + - **sNME(f)**: further from the CVC value (farCVC) +4. Compare both solutions' spectral shapes with the data + +## sNME Values for ⁹⁹Tc + +| Model | g_A^eff | sNME(c) [fm³] | sNME(f) [fm³] | CVC sNME [fm³] | +|---|---|---|---|---| +| jj45pnb | 1.0 | 0.0666 | −0.0687 | −0.1705 | +| glekpn | 1.0 | 0.1140 | −0.0202 | −0.0080 | + +## Results from Song et al. (2025) + +- **Best fit**: glekpn with g_A^eff = 1.2, sNME(f) = −0.0698 fm³, χ²_ν = 1.066 +- **Conclusion**: Quenched g_A = 1.0–1.2 with farCVC sNME provides the best agreement with data + +## References + +- [[SSM — Spectrum Shape Method]] +- [[Shape Factor C(W, g_A)]] +- [[Song et al. (2025)]] +- [[Kumar et al. (2021)]] diff --git a/.llm-wiki/wiki/concepts/nuclear-shell-model-for-decay.md b/.llm-wiki/wiki/concepts/nuclear-shell-model-for-decay.md new file mode 100644 index 0000000..7866eed --- /dev/null +++ b/.llm-wiki/wiki/concepts/nuclear-shell-model-for-decay.md @@ -0,0 +1,48 @@ +# Nuclear Shell Model for β Decay + +## Overview + +The Nuclear Shell Model (NSM) is the primary theoretical framework for computing nuclear matrix elements (NMEs) in β decay. For forbidden nonunique decays like ⁹⁹Tc, the NSM computes one-body transition densities (OBTDs) that weight the single-particle transitions contributing to the decay. + +## Key Codes + +| Code | Description | Valence Spaces | +|---|---|---| +| NUSHELLX | Standard NSM code | Multiple (pf, sdg, etc.) | +| KSHELL | High-performance NSM | Custom valence spaces | + +## Valence Spaces for ⁹⁹Tc + +### glekpn (Mach interaction) +- **Protons**: 0f₇/₂, 0f₅/₂, 2p₃/₂, 2p₁/₂, 1g₉/₂ +- **Neutrons**: 0g₉/₂, 0g₇/₂, 1d₅/₂, 1d₃/₂, 2s₁/₂ + +### jj45pnb (Brown-Lisetskiy interaction) +- **Protons**: 0f₅/₂, 2p₃/₂, 2p₁/₂, 1g₉/₂ +- **Neutrons**: 0g₇/₂, 1d₅/₂, 1d₃/₂, 2s₁/₂, 0h₁₁/₂ + +## OBTDs for ⁹⁹Tc + +| Transition | OBTD (glekpn) | OBTD (jj45pnb) | +|---|---|---| +| 1g₇/₂ → 1g₉/₂ | 0.00994 | 0.01107 | +| 2d₅/₂ → 1g₉/₂ | 0.47752 | −0.43985 | + +**Key insight**: The dominant transition is 2d₅/₂ → 1g₉/₂ with OBTD ≈ 0.478. The jj45pnb model gives opposite sign for this transition. + +## sNME and CVC + +The small relativistic vector NME (sNME) is related to the large vector NME via the CVC hypothesis: + +``` +sNME = (R/√10) · [W₀ − (m_n − m_p) + E_C] · l-NME +``` + +where E_C is the Coulomb displacement energy. For ⁹⁹Tc, [W₀ − (m_n − m_p)] = −0.487 MeV (small Q-value makes this critical). + +## References + +- [[Shape Factor C(W, g_A)]] +- [[SSM — Spectrum Shape Method]] +- [[Ramalho & Suhonen (2024)]] +- [[Song et al. (2025)]] diff --git a/.llm-wiki/wiki/concepts/shape-factor-cw-ga.md b/.llm-wiki/wiki/concepts/shape-factor-cw-ga.md new file mode 100644 index 0000000..1b8112b --- /dev/null +++ b/.llm-wiki/wiki/concepts/shape-factor-cw-ga.md @@ -0,0 +1,53 @@ +# Shape Factor C(W, g_A) + +## Overview + +The g_A-dependent nuclear shape factor C(W, g_A) is the key nuclear structure quantity that encodes the energy-dependent modification of the β decay spectrum due to forbidden nonunique transitions. For ⁹⁹Tc (second-forbidden nonunique, 9/2₁⁺ → 5/2₁⁺, Q ≈ 296 keV), the shape factor decomposes as: + +``` +C(W; g_A) = g_V² · C_V(W) + g_A² · C_A(W) + g_V · g_A · C_VA(W) +``` + +with g_V = 1.0 fixed by CVC. + +## Components + +| Component | Description | Sign for ⁹⁹Tc | Magnitude | +|---|---|---|---| +| C_V(W) | Vector contribution | + | 2.240 × 10⁻⁹ (integrated) | +| C_A(W) | Axial-vector contribution | + | 2.130 × 10⁻⁹ (integrated) | +| C_VA(W) | Vector-axial interference | − | −4.361 × 10⁻⁹ (integrated) | + +**Key insight**: C_VA is negative and twice as large as C_V or C_A. The total integrated shape factor is two orders of magnitude smaller due to destructive interference. + +## g_A Dependence + +The spectral shape is most sensitive to g_A in the **0–100 keV range** (low-energy region). Figure 9 from Paulsen et al. (2024) shows: + +- **Small g_A (0.5–1.0)**: Bell-shaped spectrum with low-energy enhancement +- **Large g_A (1.4–2.0)**: Flattened spectrum with low-energy suppression +- **Crossover point**: All spectra intersect around 100 keV + +## Table Data Sources + +| Source | Method | Coverage | Availability | +|---|---|---|---| +| Kostensalo & Suhonen (2017) PRC 96, 024317 | MQPM | 26 forbidden decays | Supplementary material (not public) | +| Ramalho & Suhonen (2024) | NSM (NUSHELLX) | ⁹⁹Tc-specific | Supplementary material (not public) | +| Song et al. (2025) arXiv:2506.17544 | ESSM + NSM/MQPM | ⁹⁹Tc | Integrated values only | + +## Implementation + +The ShapeFactorGA component should: +1. Load C_V(W), C_A(W), C_VA(W) tables from CSV files +2. Interpolate using cubic spline (scipy.interpolate.CubicSpline) +3. Compute C(W; g_A) = C_V + g_A²·C_A + g_A·C_VA +4. Return the shape factor for the full spectrum calculation + +## References + +- [[SSM — Spectrum Shape Method]] +- [[Nuclear Shell Model]] +- [[CVC Hypothesis]] +- [[Paulsen et al. (2024)]] +- [[Song et al. (2025)]] diff --git a/.llm-wiki/wiki/concepts/ssm-spectrum-shape-method.md b/.llm-wiki/wiki/concepts/ssm-spectrum-shape-method.md new file mode 100644 index 0000000..4b14f58 --- /dev/null +++ b/.llm-wiki/wiki/concepts/ssm-spectrum-shape-method.md @@ -0,0 +1,48 @@ +# SSM — Spectrum Shape Method + +## Overview + +The Spectrum Shape Method (SSM) is a technique for determining the effective axial-vector coupling constant g_A by comparing theoretically computed β spectral shapes with experimental measurements. It is applicable to forbidden nonunique β decays, where the spectral shape depends nontrivially on g_A through the shape factor C(W; g_A). + +## Algorithm + +1. **Compute shape factor components** C_V(W), C_A(W), C_VA(W) from nuclear model +2. **Build theoretical spectrum** for trial g_A values: N(W; g_A) = PhaseSpace × Fermi × ... × C(W; g_A) +3. **Compare with experimental spectrum** using χ² minimization +4. **Extract g_A** from the minimum χ² position +5. **Estimate uncertainty** from χ² = χ²_min + 1 + +## Key Requirements + +- **Forbidden nonunique transition**: The spectral shape must depend on g_A +- **High-precision measurement**: Low energy threshold (< 10 keV), good energy resolution (< 1 keV) +- **Accurate nuclear model**: Nuclear shell model (NSM) or MQPM with appropriate valence space + +## g_A Values from Literature + +| Source | g_A^eff | Method | Notes | +|---|---|---|---| +| Paulsen et al. (2024) | 1.526(92) | CVC-fixed sNME | Enhanced value | +| Song et al. (2025) | 1.0–1.2 | ESSM (farCVC) | Quenched value | +| Suhonen review | ~0.9 | Half-life comparison | Quenched value | + +## Uncertainty Budget + +| Component | Uncertainty on g_A | Relative | +|---|---|---| +| Fit method (statistics) | 0.0745 | 4.9% | +| Coulomb displacement energy | 0.0425 | 2.8% | +| Nuclear model | 0.0294 | 1.9% | +| Lepton current treatment | 0.0127 | 0.8% | +| Fit range | 0.0061 | 0.4% | +| Endpoint energy | 0.0038 | 0.2% | +| Radiative corrections | 0.0033 | 0.2% | +| Atomic exchange | 0.0029 | 0.2% | +| **Combined** | **0.0919** | **6.0%** | + +## References + +- [[Shape Factor C(W, g_A)]] +- [[ESSM — Enhanced Spectrum Shape Method]] +- [[Paulsen et al. (2024)]] +- [[Song et al. (2025)]] diff --git a/.llm-wiki/wiki/entities/paulsen-et-al-2024-high-precision-measurement-of-99tc-spectrum.md b/.llm-wiki/wiki/entities/paulsen-et-al-2024-high-precision-measurement-of-99tc-spectrum.md new file mode 100644 index 0000000..4512d37 --- /dev/null +++ b/.llm-wiki/wiki/entities/paulsen-et-al-2024-high-precision-measurement-of-99tc-spectrum.md @@ -0,0 +1,42 @@ +# Paulsen et al. (2024) — High precision measurement of 99Tc β spectrum + +## Citation + +Paulsen, M., Ranitzsch, P.C.-O., Loidl, M., et al. (2024). High precision measurement of the ⁹⁹Tc β spectrum. *Physical Review C*, 110, 055503. DOI: 10.1103/PhysRevC.110.055503 + +## Key Results + +| Parameter | Value | Method | +|---|---|---| +| Q_β | 295.82(16) keV | MMC measurement | +| g_A^eff | 1.526(92) | Spectrum shape (CVC-fixed sNME) | +| g_V^eff | 0.376(5) | Half-life normalization | +| g_A^eff (quenched) | 0.574(36) | Half-life normalization | +| E_β | 98.51(23) keV | Theoretical spectrum integration | +| log f | −0.47660(22) | Theoretical spectrum | +| log f t | 12.3478(23) | Partial half-life | + +## Methodology + +- **Detector**: Metallic Magnetic Calorimeters (MMCs) at LNHB and PTB +- **Energy threshold**: 0.65 keV (orders of magnitude lower than previous measurements) +- **Energy resolution**: 0.1 keV at 383 keV +- **Nuclear model**: NUSHELLX with GLEKPN valence space, Mach interaction +- **sNME treatment**: CVC-fixed (not ESSM) + +## Key Figures + +- **Figure 9**: g_A-dependent theoretical β spectrum — shows dramatic shape changes for g_A ∈ {0.5, 0.8, 1.0, 1.2, 1.4, 1.6, 2.0} +- **Figure 10**: Best-fit theoretical spectrum vs. measurement + +## Limitations + +- Used CVC-fixed sNME (not ESSM) → enhanced g_A value +- Song et al. (2025) contradicts this result using ESSM + +## References + +- [[Shape Factor C(W, g_A)]] +- [[SSM — Spectrum Shape Method]] +- [[Song et al. (2025)]] +- [[Ramalho & Suhonen (2024)]] diff --git a/.llm-wiki/wiki/entities/song-et-al-2025-99tc--decay-spectrum-and-implications-for-ga.md b/.llm-wiki/wiki/entities/song-et-al-2025-99tc--decay-spectrum-and-implications-for-ga.md new file mode 100644 index 0000000..ee6639f --- /dev/null +++ b/.llm-wiki/wiki/entities/song-et-al-2025-99tc--decay-spectrum-and-implications-for-ga.md @@ -0,0 +1,43 @@ +# Song et al. (2025) — The measurement of 99Tc β-decay spectrum and implications for g_A + +## Citation + +Song, J.W., Ramalho, M., Lee, M.K., et al. (2025). The measurement of the ⁹⁹Tc β-decay spectrum and its implications for the effective value of weak axial coupling. *arXiv:2506.17544*. + +## Key Results + +| Model | g_A^eff | sNME [fm³] | χ²_ν | p-value | +|---|---|---|---|---| +| glekpn (CloseCVC) | 1.1 | 0.0674 | 1.106 | 0.12 | +| jj45pnb (CloseCVC) | 1.0 | 0.0681 | 1.244 | 0.004 | +| MQPM (CloseCVC) | 1.0 | 0.0651 | 1.644 | <0.0001 | +| **glekpn (FarCVC)** | **1.2** | **−0.0698** | **1.066** | **0.22** | +| jj45pnb (FarCVC) | 1.0 | −0.0669 | 1.097 | 0.14 | +| MQPM (FarCVC) | 1.0 | −0.0683 | 1.095 | 0.14 | + +## Methodology + +- **Detector**: MMC with 4π gold absorber, energy threshold 10 keV +- **Nuclear models**: glekpn, jj45pnb (NSM), MQPM +- **sNME treatment**: ESSM (sNME as fit parameter) — two solutions per g_A (closeCVC and farCVC) +- **Fit range**: 10–41 keV and 50–280 keV (excluding Gd X-ray region) + +## Key Findings + +1. **Contradicts Paulsen et al. (2024)**: The enhanced g_A value (1.526) is not supported when using proper ESSM treatment +2. **Quenched g_A**: Best fit is g_A^eff = 1.0–1.2 with farCVC sNME +3. **Best model**: glekpn with g_A^eff = 1.2, sNME = −0.0698 fm³, χ²_ν = 1.066 + +## Implications + +- Resolves the inconsistency between first-forbidden and higher-forbidden nonunique transitions +- Supports quenched g_A consistent with Suhonen's review values +- Demonstrates the importance of proper sNME treatment in SSM analysis + +## References + +- [[Shape Factor C(W, g_A)]] +- [[SSM — Spectrum Shape Method]] +- [[ESSM — Enhanced Spectrum Shape Method]] +- [[Paulsen et al. (2024)]] +- [[Ramalho & Suhonen (2024)]] diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-001.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-001.md new file mode 100644 index 0000000..2aff7f4 --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-001.md @@ -0,0 +1,40 @@ +--- +type: source +format: markdown +source_id: SRC-2026-05-27-001 +raw_path: raw/sources/SRC-2026-05-27-001/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# 01-ssm-algorithm.md + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: --- 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] relatednotes: - 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 forbi..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-001]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-001/extracted.md](../raw/sources/SRC-2026-05-27-001/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-001/manifest.json](../raw/sources/SRC-2026-05-27-001/manifest.json) diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-002.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-002.md new file mode 100644 index 0000000..c60971a --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-002.md @@ -0,0 +1,40 @@ +--- +type: source +format: markdown +source_id: SRC-2026-05-27-002 +raw_path: raw/sources/SRC-2026-05-27-002/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# 01-shape-factors.md + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: --- 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: [] relatedcomponents: [[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-si..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-002]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-002/extracted.md](../raw/sources/SRC-2026-05-27-002/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-002/manifest.json](../raw/sources/SRC-2026-05-27-002/manifest.json) diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-003.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-003.md new file mode 100644 index 0000000..c68577c --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-003.md @@ -0,0 +1,41 @@ +--- +type: source +format: web +source_id: SRC-2026-05-27-003 +raw_path: raw/sources/SRC-2026-05-27-003/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download +> _Original: [https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download](https://jyx.jyu.fi/bitstreams/eb3a10c8-b94c-4009-8a8e-839bfddc7c58/download)_ + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: 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., &..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-003]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-003/extracted.md](../raw/sources/SRC-2026-05-27-003/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-003/manifest.json](../raw/sources/SRC-2026-05-27-003/manifest.json) diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-004.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-004.md new file mode 100644 index 0000000..8a180c8 --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-004.md @@ -0,0 +1,41 @@ +--- +type: source +format: web +source_id: SRC-2026-05-27-004 +raw_path: raw/sources/SRC-2026-05-27-004/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf +> _Original: [https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf](https://cea.hal.science/cea-04796921/file/PhysRevC.110.055503.pdf)_ + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: High precision measurement of the 99Tc 𝛽 spectrum Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten Kossert, Xavier Mougeot, Abhilasha Singh, Sylvain Leblond, J. Beyer, L. Bockhorn, et al. To cite this version: Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten Kossert, et al.. High preci- sion measurement of the 99Tc 𝛽 spectrum. Physical Review C, 2024, 110, 055503 (17 p.). ⟨10.1103/Phys- RevC.110.055503⟩. ⟨cea-04796921⟩ HAL Id: cea-04796921 ..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-004]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-004/extracted.md](../raw/sources/SRC-2026-05-27-004/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-004/manifest.json](../raw/sources/SRC-2026-05-27-004/manifest.json) diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-005.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-005.md new file mode 100644 index 0000000..6f9bb76 --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-005.md @@ -0,0 +1,41 @@ +--- +type: source +format: web +source_id: SRC-2026-05-27-005 +raw_path: raw/sources/SRC-2026-05-27-005/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# https://arxiv.org/pdf/2506.17544 +> _Original: [https://arxiv.org/pdf/2506.17544](https://arxiv.org/pdf/2506.17544)_ + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: 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, Eng..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-005]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-005/extracted.md](../raw/sources/SRC-2026-05-27-005/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-005/manifest.json](../raw/sources/SRC-2026-05-27-005/manifest.json) diff --git a/.llm-wiki/wiki/sources/SRC-2026-05-27-006.md b/.llm-wiki/wiki/sources/SRC-2026-05-27-006.md new file mode 100644 index 0000000..00f239c --- /dev/null +++ b/.llm-wiki/wiki/sources/SRC-2026-05-27-006.md @@ -0,0 +1,41 @@ +--- +type: source +format: web +source_id: SRC-2026-05-27-006 +raw_path: raw/sources/SRC-2026-05-27-006/extracted.md +captured: 2026-05-27 +status: skeleton +--- + +# https://arxiv.org/pdf/2312.07448 +> _Original: [https://arxiv.org/pdf/2312.07448](https://arxiv.org/pdf/2312.07448)_ + +## Summary + +[LLM: Replace with 2-3 paragraph summary of key content] + +> _Auto-preview: 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,withintheframewo..._ + +## Key Takeaways + +- [LLM: Most important point] +- [LLM: Second important point] +- [LLM: Third important point] + +## Entities Mentioned + +- [[entity-name]] + +## Concepts Mentioned + +- [[concept-name]] + +## Notable Quotes + +> [LLM: Important quote] — attribution + +## Source Packet + +- **ID:** `[[sources/SRC-2026-05-27-006]]` +- **Extracted:** [raw/sources/SRC-2026-05-27-006/extracted.md](../raw/sources/SRC-2026-05-27-006/extracted.md) +- **Manifest:** [raw/sources/SRC-2026-05-27-006/manifest.json](../raw/sources/SRC-2026-05-27-006/manifest.json) diff --git a/beta_spectrum/cli.py b/beta_spectrum/cli.py index d47e9be..da62731 100644 --- a/beta_spectrum/cli.py +++ b/beta_spectrum/cli.py @@ -85,6 +85,12 @@ def _build_parser() -> argparse.ArgumentParser: default=0.001, help="Energy step size in MeV (default: 0.001). All energies are in MeV throughout the toolkit.", ) + parser.add_argument( + "--neutrino-mass", + type=float, + default=0.0, + help="Neutrino mass in MeV (default: 0.0). Set to non-zero value to include neutrino mass effects in phase space calculation.", + ) # Output parser.add_argument( @@ -139,6 +145,13 @@ def _build_parser() -> argparse.ArgumentParser: "(default: 0.0 = include all branches). " "Branches below this threshold are excluded from the calculation and plots.", ) + log_group.add_argument( + "--csv-verbosity", + type=str, + default="full", + choices=["full", "concise"], + help="CSV output verbosity: 'full' (all components) or 'concise' (total spectrum and normalized branch components only) (default: full)", + ) return parser @@ -237,6 +250,7 @@ def _run(args: argparse.Namespace) -> None: endpoint_MeV=decay_info.endpoint_MeV, transition_type=decay_info.transition_type, e_step_MeV=args.e_step, + neutrino_mass_MeV=args.neutrino_mass, branches=branches, intensity_cutoff=args.intensity_cutoff, ) @@ -246,6 +260,7 @@ def _run(args: argparse.Namespace) -> None: nuclide=source_value, e_step_MeV=args.e_step, intensity_cutoff=args.intensity_cutoff, + neutrino_mass_MeV=args.neutrino_mass, ) else: config = create_config_from_source( @@ -253,6 +268,7 @@ def _run(args: argparse.Namespace) -> None: nuclide=source_value, e_step_MeV=args.e_step, intensity_cutoff=args.intensity_cutoff, + neutrino_mass_MeV=args.neutrino_mass, ) else: logger.info("Source: JSON file=%s", source_value) @@ -288,7 +304,9 @@ def _run(args: argparse.Namespace) -> None: # Export to CSV if args.output: analyzer = BetaSpectrumAnalyzer(spectrum, config, logger=logger) - analyzer.export_to_csv(args.output, source_type=source_type) + analyzer.export_to_csv( + args.output, source_type=source_type, csv_verbosity=args.csv_verbosity + ) logger.info("Spectrum exported to %s", args.output) # Generate plot @@ -333,6 +351,7 @@ def _print_dry_run_output(config, source_type: str) -> None: print(f"Endpoint: {config.endpoint_MeV * 1000:.1f} keV") print(f"Transition: {config.transition_type}") print(f"e_step: {config.e_step_MeV:.4f} MeV") + print(f"neutrino_mass_MeV: {config.neutrino_mass_MeV:.6f}") print(f"Corrections: {', '.join(enabled) if enabled else 'none'}") if config.branches and len(config.branches) > 1: diff --git a/beta_spectrum/components/phase_space.py b/beta_spectrum/components/phase_space.py index 92f334d..4c50362 100644 --- a/beta_spectrum/components/phase_space.py +++ b/beta_spectrum/components/phase_space.py @@ -50,10 +50,26 @@ class PhaseSpace(SpectrumComponent): w_e = W p_e = momentum(W, self.m_e) w_nu = self.W0 - W - p_nu = momentum(w_nu, self.m_nu) + # Calculate p_nu, handling the case where w_nu < self.m_nu + # In this case, the neutrino energy is less than its mass, which is unphysical + # The phase space should be 0 in this region + if self.m_nu > 0: + # Check where w_nu < self.m_nu + below_threshold = w_nu < self.m_nu + # Calculate p_nu only where it's physically valid + p_nu = np.zeros_like(w_nu) + valid_mask = ~below_threshold + if np.any(valid_mask): + p_nu[valid_mask] = momentum(w_nu[valid_mask], self.m_nu) + else: + below_threshold = np.zeros_like(w_nu, dtype=bool) + p_nu = momentum(w_nu, self.m_nu) + + # Calculate phase space factor phase_space = p_e * w_e * p_nu * w_nu + # Calculate forbid factor if self.tr_type in ["A", "F1"]: forbid_factor: float | np.ndarray = 1 elif self.tr_type in ["F1U", "F2"]: @@ -66,6 +82,11 @@ class PhaseSpace(SpectrumComponent): ) result = np.asarray(phase_space * forbid_factor, dtype=np.float64) + + # For regions below the neutrino mass threshold, the phase space should be 0 + if self.m_nu > 0: + result[below_threshold] = 0.0 + if self._logger: self._logger.debug( "PhaseSpace: output range=[%.6e, %.6e]", diff --git a/beta_spectrum/nuclear_data.py b/beta_spectrum/nuclear_data.py index 7e1a19a..171c558 100644 --- a/beta_spectrum/nuclear_data.py +++ b/beta_spectrum/nuclear_data.py @@ -470,6 +470,7 @@ def decay_info_to_config( info: DecayInfo, e_step_MeV: float = 0.001, intensity_cutoff: float = 0.0, + neutrino_mass_MeV: float = 0.0, ) -> SpectrumConfig: """ Convert DecayInfo to a SpectrumConfig. @@ -482,6 +483,8 @@ def decay_info_to_config( Energy step size in MeV. intensity_cutoff : float Minimum branch intensity fraction to include. + neutrino_mass_MeV : float + Neutrino mass in MeV. Returns ------- @@ -510,6 +513,7 @@ def decay_info_to_config( endpoint_MeV=info.endpoint_MeV, transition_type=info.transition_type, e_step_MeV=e_step_MeV, + neutrino_mass_MeV=neutrino_mass_MeV, branches=branches, intensity_cutoff=intensity_cutoff, ) @@ -529,6 +533,7 @@ DEFAULT_JSON_SCHEMA = { "A_number": 99, "transition_type": "F2", "e_step_MeV": 0.001, + "neutrino_mass_MeV": 0.0, "detector_model": "gaussian", "detector_sigma_a_keV": 1.0, "detector_sigma_b": 0.0, @@ -638,6 +643,7 @@ def json_to_config(data: Dict[str, Any]) -> SpectrumConfig: endpoint_MeV=float(data["endpoint_MeV"]), transition_type=str(data.get("transition_type", "A")), e_step_MeV=float(data.get("e_step_MeV", 0.001)), + neutrino_mass_MeV=float(data.get("neutrino_mass_MeV", 0.0)), ) @@ -683,11 +689,13 @@ def create_config_from_source( decay_index = kwargs.get("decay_index", None) e_step = kwargs.get("e_step_MeV", 0.001) intensity_cutoff = kwargs.get("intensity_cutoff", 0.0) + neutrino_mass = kwargs.get("neutrino_mass_MeV", 0.0) info = get_decay_info_from_paceENSDF(nuclide, decay_type, decay_index) return decay_info_to_config( info, e_step_MeV=e_step, intensity_cutoff=intensity_cutoff, + neutrino_mass_MeV=neutrino_mass, ) elif source == "json": if not json_path: diff --git a/beta_spectrum/spectrum.py b/beta_spectrum/spectrum.py index 3a9602e..f244f7c 100644 --- a/beta_spectrum/spectrum.py +++ b/beta_spectrum/spectrum.py @@ -2,7 +2,7 @@ from __future__ import annotations import logging -from typing import TYPE_CHECKING, Dict, List, Optional, Tuple +from typing import Dict, List, Optional, Tuple from dataclasses import dataclass import numpy as np @@ -15,7 +15,6 @@ from beta_spectrum.components.screening import ScreeningCorrection from beta_spectrum.components.exchange import ExchangeCorrection from beta_spectrum.components.radiative import RadiativeCorrection from beta_spectrum.utils import T_to_W -from beta_spectrum.constants import ME_MEV @dataclass @@ -71,6 +70,7 @@ class SpectrumConfig: endpoint_MeV: float transition_type: str = "A" # should be [A, F1, F1U, F2, F2U, F3, F3U, F4] e_step_MeV: float = 0.001 + neutrino_mass_MeV: float = 0.0 # Toggle components and corrections use_phase_space: bool = True @@ -251,7 +251,12 @@ class BetaSpectrum: if config.use_phase_space: components.append( - PhaseSpace(W0=W0, transition_type=config.transition_type, logger=logger) + PhaseSpace( + W0=W0, + transition_type=config.transition_type, + m_nu=config.neutrino_mass_MeV, + logger=logger, + ) ) if config.use_fermi: @@ -331,6 +336,7 @@ class BetaSpectrum: endpoint_MeV=branch.endpoint_MeV, transition_type=branch.transition_type, e_step_MeV=config.e_step_MeV, + neutrino_mass_MeV=config.neutrino_mass_MeV, # Branch-dependent components use_phase_space=config.use_phase_space, use_radiative=config.use_radiative, @@ -451,7 +457,3 @@ class BetaSpectrum: branch_spectrum = branch_spec(W) result.append(self.branches[i].intensity * np.maximum(0.0, branch_spectrum)) return result - - - - diff --git a/beta_spectrum/visualize/__init__.py b/beta_spectrum/visualize/__init__.py index a5b06a0..52ff51a 100644 --- a/beta_spectrum/visualize/__init__.py +++ b/beta_spectrum/visualize/__init__.py @@ -107,7 +107,6 @@ class BetaSpectrumAnalyzer: return total - # --------------------------------------------------------------------------- # Unified branch view # --------------------------------------------------------------------------- @@ -162,17 +161,42 @@ class BetaSpectrumAnalyzer: f"| ΔE={e_step_keV:.1f} keV" ) fig.text( - 0.01, 0.98, header, - transform=fig.transFigure, fontsize=9, - verticalalignment="top", horizontalalignment="left", + 0.01, + 0.98, + header, + transform=fig.transFigure, + fontsize=9, + verticalalignment="top", + horizontalalignment="left", fontfamily="monospace", ) + + # Add neutrino mass info to header if non-zero + if self.config.neutrino_mass_MeV > 0: + nu_mass_text = f"| m_ν={self.config.neutrino_mass_MeV:.6f} MeV" + fig.text( + 0.01, + 0.96, + nu_mass_text, + transform=fig.transFigure, + fontsize=8, + verticalalignment="top", + horizontalalignment="left", + fontfamily="monospace", + color="red", + ) + id_text = f"{get_git_short_hash(6)} | {self._timestamp}" fig.text( - 0.01, 0.01, id_text, - transform=fig.transFigure, fontsize=8, - verticalalignment="bottom", horizontalalignment="left", - fontfamily="monospace", color="gray", + 0.01, + 0.01, + id_text, + transform=fig.transFigure, + fontsize=8, + verticalalignment="bottom", + horizontalalignment="left", + fontfamily="monospace", + color="gray", ) def _build_figure_title(self, fig: "Figure") -> str: @@ -190,7 +214,7 @@ class BetaSpectrumAnalyzer: branch = self._unified_branches[i] e0 = branch.endpoint_MeV * 1000 intensity_pct = branch.intensity * 100 - return f"Br. {i+1}: {e0:.1f} keV ({intensity_pct:.1f}%)" + return f"Br. {i + 1}: {e0:.1f} keV ({intensity_pct:.1f}%)" # --------------------------------------------------------------------------- # Plotting — unified multi-branch approach @@ -231,9 +255,16 @@ class BetaSpectrumAnalyzer: fig, ax = plt.subplots(figsize=(12, 7)) branch_colors = [ - "#1f77b4", "#ff7f0e", "#2ca02c", "#d62728", - "#9467bd", "#8c564b", "#e377c2", "#7f7f7f", - "#bcbd22", "#17becf", + "#1f77b4", + "#ff7f0e", + "#2ca02c", + "#d62728", + "#9467bd", + "#8c564b", + "#e377c2", + "#7f7f7f", + "#bcbd22", + "#17becf", ] e_step_keV = self.config.e_step_MeV * 1000 @@ -242,8 +273,11 @@ class BetaSpectrumAnalyzer: branch_contribs = [] # Universal components are shared (no per-branch key prefix) universal_names = [ - "Fermi", "Screening", "FiniteSizeL0", - "ChargeDistributionU", "Exchange", + "Fermi", + "Screening", + "FiniteSizeL0", + "ChargeDistributionU", + "Exchange", ] for i in range(self._n_branches): universal_product = np.ones_like(self.W) @@ -260,28 +294,33 @@ class BetaSpectrumAnalyzer: total_raw = sum(branch_contribs) ax.plot( - self.energies_MeV, total_raw, "k-", lw=2.5, - label="Total spectrum", zorder=10, + self.energies_MeV, + total_raw, + "k-", + lw=2.5, + label="Total spectrum", + zorder=10, ) for i, contrib in enumerate(branch_contribs): color = branch_colors[i % len(branch_colors)] ax.plot( - self.energies_MeV, contrib, color=color, lw=1.5, - alpha=0.7, label=self._format_branch_label(i), + self.energies_MeV, + contrib, + color=color, + lw=1.5, + alpha=0.7, + label=self._format_branch_label(i), ) ax.set_xlabel(r"Electron kinetic energy $E$ [MeV]", fontsize=12) - ax.set_ylabel( - f"Normalized Counts / {e_step_keV:.1f} keV", fontsize=12 - ) - fig.suptitle( - self._build_figure_title(fig), fontsize=14, fontweight="bold" - ) + ax.set_ylabel(f"Normalized Counts / {e_step_keV:.1f} keV", fontsize=12) + fig.suptitle(self._build_figure_title(fig), fontsize=14, fontweight="bold") ax.set_yscale("log") ax.grid(True, alpha=0.3) ax.legend( - loc="best", fontsize=9, + loc="best", + fontsize=9, ncol=min(2, self._n_branches + 1), ) @@ -304,21 +343,29 @@ class BetaSpectrumAnalyzer: e_step_keV = self.config.e_step_MeV * 1000 fig, axes = plt.subplots(4, 1, figsize=(12, 16), sharex=True) - fig.suptitle( - self._build_figure_title(fig), fontsize=14, fontweight="bold" - ) + fig.suptitle(self._build_figure_title(fig), fontsize=14, fontweight="bold") branch_colors = [ - "#1f77b4", "#ff7f0e", "#2ca02c", "#d62728", - "#9467bd", "#8c564b", "#e377c2", "#7f7f7f", - "#bcbd22", "#17becf", + "#1f77b4", + "#ff7f0e", + "#2ca02c", + "#d62728", + "#9467bd", + "#8c564b", + "#e377c2", + "#7f7f7f", + "#bcbd22", + "#17becf", ] # Panel 1: Total spectrum + per-branch spectra branch_contribs = [] universal_names = [ - "Fermi", "Screening", "FiniteSizeL0", - "ChargeDistributionU", "Exchange", + "Fermi", + "Screening", + "FiniteSizeL0", + "ChargeDistributionU", + "Exchange", ] for i in range(n_branches): universal_product = np.ones_like(self.W) @@ -336,19 +383,25 @@ class BetaSpectrumAnalyzer: ax = axes[0] ax.plot( - self.energies_MeV, total_raw, "k-", lw=2, - label="Total spectrum", zorder=10, + self.energies_MeV, + total_raw, + "k-", + lw=2, + label="Total spectrum", + zorder=10, ) for i, contrib in enumerate(branch_contribs): color = branch_colors[i % len(branch_colors)] ax.plot( - self.energies_MeV, contrib, color=color, lw=1.5, - alpha=0.7, label=self._format_branch_label(i), + self.energies_MeV, + contrib, + color=color, + lw=1.5, + alpha=0.7, + label=self._format_branch_label(i), ) ax.set_yscale("log") - ax.set_ylabel( - f"Normalized Counts / {e_step_keV:.1f} keV", fontsize=10 - ) + ax.set_ylabel(f"Normalized Counts / {e_step_keV:.1f} keV", fontsize=10) ax.set_title("Spectrum", fontsize=11) ax.grid(True, alpha=0.3) ax.legend(fontsize=8, loc="best", ncol=min(2, n_branches + 1)) @@ -359,8 +412,12 @@ class BetaSpectrumAnalyzer: fermi_vals = components["Fermi"] fermi_scaled = fermi_vals / np.max(fermi_vals) ax.plot( - self.energies_MeV, fermi_scaled, "r-", lw=1.5, - label="Fermi", alpha=0.8, + self.energies_MeV, + fermi_scaled, + "r-", + lw=1.5, + label="Fermi", + alpha=0.8, ) for i in range(n_branches): color = branch_colors[i % len(branch_colors)] @@ -377,8 +434,12 @@ class BetaSpectrumAnalyzer: else: ps_scaled = ps_masked ax.plot( - self.energies_MeV, ps_scaled, color=color, lw=1.5, - alpha=0.7, label=f"PS Br. {i+1}", + self.energies_MeV, + ps_scaled, + color=color, + lw=1.5, + alpha=0.7, + label=f"PS Br. {i + 1}", ) ax.set_ylabel("Normalized Factor", fontsize=10) ax.set_title("Fermi & Phase Space", fontsize=11) @@ -388,8 +449,10 @@ class BetaSpectrumAnalyzer: # Panel 3: Universal W0-independent corrections ax = axes[2] universal_components = [ - "Screening", "FiniteSizeL0", - "ChargeDistributionU", "Exchange", + "Screening", + "FiniteSizeL0", + "ChargeDistributionU", + "Exchange", ] colors = ["green", "orange", "purple", "brown"] for name, color in zip(universal_components, colors): @@ -397,8 +460,12 @@ class BetaSpectrumAnalyzer: values = components[name] values_scaled = values / np.max(values) ax.plot( - self.energies_MeV, values_scaled, color=color, lw=1.5, - label=name, alpha=0.8, + self.energies_MeV, + values_scaled, + color=color, + lw=1.5, + label=name, + alpha=0.8, ) ax.set_ylabel("Normalized Factor", fontsize=10) ax.set_title("Universal Corrections", fontsize=11) @@ -413,8 +480,12 @@ class BetaSpectrumAnalyzer: if key in components: rad_vals = components[key] ax.plot( - self.energies_MeV, rad_vals, color=color, lw=1.5, - alpha=0.7, label=f"Br. {i+1}", + self.energies_MeV, + rad_vals, + color=color, + lw=1.5, + alpha=0.7, + label=f"Br. {i + 1}", ) ax.set_ylabel("Radiative Correction", fontsize=10) ax.set_xlabel(r"Electron kinetic energy $E$ [MeV]", fontsize=10) @@ -433,16 +504,106 @@ class BetaSpectrumAnalyzer: def _element_symbol(self, Z: int) -> str: """Return chemical symbol for atomic number Z.""" symbols = [ - "", "H", "He", "Li", "Be", "B", "C", "N", "O", "F", - "Ne", "Na", "Mg", "Al", "Si", "P", "S", "Cl", "Ar", "K", - "Ca", "Sc", "Ti", "V", "Cr", "Mn", "Fe", "Co", "Ni", "Cu", - "Zn", "Ga", "Ge", "As", "Se", "Br", "Kr", "Rb", "Sr", "Y", - "Zr", "Nb", "Mo", "Tc", "Ru", "Rh", "Pd", "Ag", "Cd", "In", - "Sn", "Sb", "Te", "I", "Xe", "Cs", "Ba", "La", "Ce", "Pr", - "Nd", "Pm", "Sm", "Eu", "Gd", "Tb", "Dy", "Ho", "Er", "Tm", - "Yb", "Lu", "Hf", "Ta", "W", "Re", "Os", "Ir", "Pt", "Au", - "Hg", "Tl", "Pb", "Bi", "Po", "At", "Rn", "Fr", "Ra", "Ac", - "Th", "Pa", "U", "Np", "Pu", "Am", "Cm", "Bk", "Cf", "Es", + "", + "H", + "He", + "Li", + "Be", + "B", + "C", + "N", + "O", + "F", + "Ne", + "Na", + "Mg", + "Al", + "Si", + "P", + "S", + "Cl", + "Ar", + "K", + "Ca", + "Sc", + "Ti", + "V", + "Cr", + "Mn", + "Fe", + "Co", + "Ni", + "Cu", + "Zn", + "Ga", + "Ge", + "As", + "Se", + "Br", + "Kr", + "Rb", + "Sr", + "Y", + "Zr", + "Nb", + "Mo", + "Tc", + "Ru", + "Rh", + "Pd", + "Ag", + "Cd", + "In", + "Sn", + "Sb", + "Te", + "I", + "Xe", + "Cs", + "Ba", + "La", + "Ce", + "Pr", + "Nd", + "Pm", + "Sm", + "Eu", + "Gd", + "Tb", + "Dy", + "Ho", + "Er", + "Tm", + "Yb", + "Lu", + "Hf", + "Ta", + "W", + "Re", + "Os", + "Ir", + "Pt", + "Au", + "Hg", + "Tl", + "Pb", + "Bi", + "Po", + "At", + "Rn", + "Fr", + "Ra", + "Ac", + "Th", + "Pa", + "U", + "Np", + "Pu", + "Am", + "Cm", + "Bk", + "Cf", + "Es", ] return symbols[Z] if Z < len(symbols) else f"Z{Z}" @@ -450,8 +611,20 @@ class BetaSpectrumAnalyzer: # Export # --------------------------------------------------------------------------- - def export_to_csv(self, filename: str, source_type: str = "unknown") -> None: - """Export spectrum data to CSV with metadata header.""" + def export_to_csv( + self, filename: str, source_type: str = "unknown", csv_verbosity: str = "full" + ) -> None: + """Export spectrum data to CSV with metadata header. + + Parameters + ---------- + filename : str + Output CSV file path. + source_type : str + Source type (paceENSDF, json, etc.). + csv_verbosity : str + Verbosity level: 'full' (all components) or 'concise' (total spectrum and normalized branch components only). + """ total = self.total_spectrum(normalize=True) components = self.components @@ -479,16 +652,32 @@ class BetaSpectrumAnalyzer: f"# timestamp: {datetime.now(timezone.utc).isoformat()}", f"# source: {source_type}", f"# nuclide: {parent_symbol}{self.config.A_number} -> {daughter_symbol}{self.config.A_number} (Z={self.config.Z_parent}->{self.config.Z_daughter})", - f"# endpoint: {self.config.endpoint_MeV * 1000:.1f} keV", - f"# transition: {self.config.transition_type}", - f"# corrections: {', '.join(enabled)}", - f"# e_step: {self.config.e_step_MeV:.4f} MeV", + f"# basic_endpoint: {self.config.endpoint_MeV * 1000:.1f} keV", ] + # Add neutrino mass info if non-zero + if self.config.neutrino_mass_MeV > 0: + header_lines.append( + f"# neutrino_mass_MeV: {self.config.neutrino_mass_MeV:.6f} (non-zero neutrino mass effects included)" + ) + else: + header_lines.append( + f"# neutrino_mass_MeV: {self.config.neutrino_mass_MeV:.6f}" + ) + + header_lines.append(f"# corrections: {', '.join(enabled)}") + header_lines.append(f"# e_step: {self.config.e_step_MeV:.4f} MeV") + + # Add transition types for all branches + transition_types = [] + for i, branch in enumerate(self._unified_branches): + transition_types.append(f"branch_{i + 1}:{branch.transition_type}") + header_lines.append(f"# transition_types: {'; '.join(transition_types)}") + # Branch info — unified view for i, branch in enumerate(self._unified_branches): header_lines.append( - f"# branch_{i+1}: E₀={branch.endpoint_MeV*1000:.1f} keV, " + f"# branch_{i + 1}: E₀={branch.endpoint_MeV * 1000:.1f} keV, " f"transition={branch.transition_type}, " f"intensity={branch.intensity:.4f}" ) @@ -500,18 +689,27 @@ class BetaSpectrumAnalyzer: "spectrum": total, } - # Per-branch spectra and components - for i in range(self._n_branches): - data[f"branch_{i+1}_spectrum"] = self._get_branch_spectrum(i) - for comp_name in ["PhaseSpace", "Radiative"]: - key = self._get_component_key(i, comp_name) - if key in components: - data[f"branch_{i+1}_{comp_name}"] = components[key] + if csv_verbosity == "full": + # Per-branch spectra and components + for i in range(self._n_branches): + data[f"branch_{i + 1}_spectrum"] = self._get_branch_spectrum(i) + for comp_name in ["PhaseSpace", "Radiative"]: + key = self._get_component_key(i, comp_name) + if key in components: + data[f"branch_{i + 1}_{comp_name}"] = components[key] - # Universal components (no prefix) - for name, values in components.items(): - if not name.startswith("branch_"): - data[name] = values + # Universal components (no prefix) + for name, values in components.items(): + if not name.startswith("branch_"): + data[name] = values + elif csv_verbosity == "concise": + # Only total spectrum and normalized branch components + for i in range(self._n_branches): + data[f"branch_{i + 1}_spectrum_normalized"] = ( + self._get_branch_spectrum(i) / np.sum(self._get_branch_spectrum(i)) + if np.sum(self._get_branch_spectrum(i)) > 0 + else self._get_branch_spectrum(i) + ) df = pd.DataFrame(data) df.to_csv(filename, index=False, float_format="%.4e") @@ -538,6 +736,7 @@ class BetaSpectrumAnalyzer: @dataclass class _UnifiedBranch: """Unified branch view for both single- and multi-branch modes.""" + endpoint_MeV: float transition_type: str intensity: float diff --git a/create_comparison_plot.py b/create_comparison_plot.py new file mode 100644 index 0000000..88a2123 --- /dev/null +++ b/create_comparison_plot.py @@ -0,0 +1,144 @@ +#!/usr/bin/env python3 +""" +Script to create a comparison plot of total beta spectra for edge neutrino masses (1.5 keV and 2.5 keV) +""" + +import pandas as pd +import matplotlib.pyplot as plt +import numpy as np + +# Input files +mass_1_5_csv = "/home/unzhakov/workspace/massive_nu_pr/pr144_nu_mass_1_50keV.csv" +mass_2_45_csv = "/home/unzhakov/workspace/massive_nu_pr/pr144_nu_mass_2_45keV.csv" + +# Output file +output_png = ( + "/home/unzhakov/workspace/massive_nu_pr/pr144_comparison_1_5_vs_2_45_keV.png" +) + +print(f"Reading data from {mass_1_5_csv} and {mass_2_45_csv}...") + + +# Read CSV files, skipping metadata headers +# The metadata headers start with '#' +def read_csv_with_metadata(filepath): + with open(filepath, "r") as f: + lines = f.readlines() + + # Find the line where data starts (not starting with '#') + data_start_line = 0 + for i, line in enumerate(lines): + if not line.startswith("#"): + data_start_line = i + break + + # Read the data + df = pd.read_csv(filepath, skiprows=data_start_line) + return df + + +# Read the data +df_1_5 = read_csv_with_metadata(mass_1_5_csv) +df_2_45 = read_csv_with_metadata(mass_2_45_csv) + +print("Data loaded successfully.") + +# Extract energy and spectrum data +energies_1_5 = df_1_5["energy_MeV"].values +spectrum_1_5 = df_1_5["spectrum"].values + +energies_2_45 = df_2_45["energy_MeV"].values +spectrum_2_45 = df_2_45["spectrum"].values + +print( + f"Energy range: {energies_1_5[0] * 1000:.1f} eV to {energies_1_5[-1] * 1000:.1f} eV" +) + +# Create comparison plot +fig, ax = plt.subplots(figsize=(12, 8)) + +# Plot both spectra +ax.plot(energies_1_5 * 1000, spectrum_1_5, "b-", lw=2, label="m_ν = 1.5 keV") +ax.plot(energies_2_45 * 1000, spectrum_2_45, "r--", lw=2, label="m_ν = 2.45 keV") + +# Set labels and title +ax.set_xlabel(r"Electron kinetic energy $E$ [keV]", fontsize=12) +ax.set_ylabel("Normalized Spectrum", fontsize=12) +ax.set_title( + "Pr144 -> Nd144 Beta Decay: Comparison of Neutrino Mass Effects\n" + "(1.5 keV vs 2.45 keV)", + fontsize=14, +) + +# Set log scale for y-axis to see the endpoint region clearly +ax.set_yscale("log") + +# Add grid +ax.grid(True, alpha=0.3) + +# Add legend +ax.legend(fontsize=12) + +# Focus on the endpoint region (last part of the spectrum) +# Find the energy range near the endpoint +max_energy_keV = max(energies_1_5) * 1000 +# Show last 10 keV of the spectrum +start_energy_keV = max_energy_keV - 10.0 +ax.set_xlim(start_energy_keV, max_energy_keV) + +# Save the plot +plt.tight_layout() +plt.savefig(output_png, dpi=150, bbox_inches="tight") +print(f"Comparison plot saved to {output_png}") + +# Also create a linear scale plot for the endpoint region +fig_linear, ax_linear = plt.subplots(figsize=(12, 6)) + +# Plot both spectra in linear scale +ax_linear.plot(energies_1_5 * 1000, spectrum_1_5, "b-", lw=2, label="m_ν = 1.5 keV") +ax_linear.plot(energies_2_45 * 1000, spectrum_2_45, "r--", lw=2, label="m_ν = 2.45 keV") + +# Set labels and title +ax_linear.set_xlabel(r"Electron kinetic energy $E$ [keV]", fontsize=12) +ax_linear.set_ylabel("Normalized Spectrum", fontsize=12) +ax_linear.set_title( + "Pr144 -> Nd144 Beta Decay: Comparison of Neutrino Mass Effects (Endpoint Region)\n" + "(1.5 keV vs 2.45 keV) - Linear Scale", + fontsize=14, +) + +# Set linear scale for y-axis +ax_linear.set_yscale("linear") + +# Set y-axis limits to show the variation in the spectrum +# Find the min and max values in the endpoint region to set appropriate limits +min_spec_1_5 = np.min(spectrum_1_5[-100:]) # Last 100 points +max_spec_1_5 = np.max(spectrum_1_5[-100:]) +min_spec_2_45 = np.min(spectrum_2_45[-100:]) +max_spec_2_45 = np.max(spectrum_2_45[-100:]) + +overall_min = ( + min(min_spec_1_5, min_spec_2_45) * 0.9 + if min(min_spec_1_5, min_spec_2_45) > 0 + else 0 +) +overall_max = max(max_spec_1_5, max_spec_2_45) * 1.1 + +ax_linear.set_ylim(overall_min, overall_max) + +# Add grid +ax_linear.grid(True, alpha=0.3) + +# Add legend +ax_linear.legend(fontsize=12) + +# Focus on the endpoint region (last part of the spectrum) +ax_linear.set_xlim(start_energy_keV, max_energy_keV) + +# Save the linear plot +plt.tight_layout() +linear_output_png = ( + "/home/unzhakov/workspace/massive_nu_pr/pr144_comparison_1_5_vs_2_45_keV_linear.png" +) +plt.savefig(linear_output_png, dpi=150, bbox_inches="tight") +print(f"Linear comparison plot saved to {linear_output_png}") diff --git a/docs/nuclear-structure/01-shape-factors.md b/docs/nuclear-structure/01-shape-factors.md index eaa9f82..1f114b9 100644 --- a/docs/nuclear-structure/01-shape-factors.md +++ b/docs/nuclear-structure/01-shape-factors.md @@ -89,5 +89,39 @@ The nuclear structure analogue of the electrostatic deformation correction ([[co ## 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. +> [!info] Implementation Planned +> The shape factor $C(Z,W; g_A)$ is being implemented as `ShapeFactorGA` component. See `.rpiv/artifacts/research/2026-05-27_gA-spectrum-shape-factor-implementation.md` for the implementation plan. + +## Latest Research (May 2026) + +### Key Findings from Paulsen et al. (2024) Fig. 9 + +The spectral shape of ⁹⁹Tc β decay is **highly sensitive to g_A** in the low-energy region (0–100 keV): + +- **Small g_A (0.5–1.0)**: Bell-shaped spectrum with low-energy enhancement +- **Large g_A (1.4–2.0)**: Flattened spectrum with low-energy suppression +- **Crossover point**: All spectra intersect around 100 keV + +### g_A Values — Resolved + +| Source | g_A^eff | Method | Status | +|---|---|---|---| +| Paulsen et al. (2024) | 1.526(92) | CVC-fixed sNME | Enhanced (controversial) | +| Song et al. (2025) | 1.0–1.2 | ESSM (farCVC) | Quenched (recommended) | + +**Resolution**: The discrepancy is resolved by proper sNME treatment. ESSM (sNME as fit parameter) gives quenched g_A = 1.0–1.2, consistent with other forbidden decays. + +### Table Data Status + +The energy-dependent tables for C_V(W), C_A(W), C_VA(W) are **not publicly available**. Priority actions: +1. Obtain from supplementary materials of Kostensalo & Suhonen (2017) or Ramalho & Suhonen (2024) +2. Contact Suhonen group (jouni.t.suhonen@jyu.fi) for table data +3. Use integrated values as proxy for initial implementation + +### References + +- [[Shape Factor C(W, g_A)]] — wiki concept page +- [[SSM — Spectrum Shape Method]] — wiki concept page +- [[ESSM — Enhanced Spectrum Shape Method]] — wiki concept page +- [[Paulsen et al. (2024)]] — wiki entity page +- [[Song et al. (2025)]] — wiki entity page diff --git a/run_pr144_calculations.py b/run_pr144_calculations.py new file mode 100644 index 0000000..6abe9f1 --- /dev/null +++ b/run_pr144_calculations.py @@ -0,0 +1,86 @@ +#!/usr/bin/env python3 +""" +Script to run Pr144 -> Nd144 beta decay calculations for neutrino mass range 1.5-2.5 keV +""" + +import subprocess +import os + +# Create output directory +output_dir = "/home/unzhakov/workspace/massive_nu_pr" +os.makedirs(output_dir, exist_ok=True) + +# Neutrino mass range: 1.5 to 2.5 keV with step of 50 eV +# Convert to MeV: 1.5 keV = 0.0015 MeV, 2.5 keV = 0.0025 MeV, 50 eV = 0.00005 MeV +start_mass_meV = 1.5 # keV +end_mass_meV = 2.5 # keV +step_mass_meV = 0.05 # keV (50 eV) + +# Convert to MeV +start_mass_MeV = start_mass_meV / 1000.0 +end_mass_MeV = end_mass_meV / 1000.0 +step_mass_MeV = step_mass_meV / 1000.0 + +# Energy step: 100 eV = 0.0001 MeV +e_step_MeV = 0.0001 + +# Intensity cutoff: 0.001 +intensity_cutoff = 0.001 + +# Generate neutrino mass values +mass_values = [] +mass = start_mass_MeV +while mass <= end_mass_MeV: + mass_values.append(mass) + mass += step_mass_MeV + +print(f"Running calculations for {len(mass_values)} neutrino mass values...") +print( + f"Neutrino mass range: {start_mass_meV} - {end_mass_meV} keV with step {step_mass_meV} keV" +) +print(f"Energy step: {e_step_MeV * 1000} eV") +print(f"Intensity cutoff: {intensity_cutoff}") + +# Run calculations for each neutrino mass +for i, mass_MeV in enumerate(mass_values): + mass_keV = mass_MeV * 1000.0 + print( + f"\n[{i + 1}/{len(mass_values)}] Calculating for neutrino mass = {mass_keV:.2f} keV ({mass_MeV:.6f} MeV)" + ) + + # Generate filename + mass_str = f"{mass_keV:.2f}".replace(".", "_") + csv_filename = f"{output_dir}/pr144_nu_mass_{mass_str}keV.csv" + png_filename = f"{output_dir}/pr144_nu_mass_{mass_str}keV.png" + + # Command to run + cmd = [ + "python", + "-m", + "beta_spectrum.cli", + "--nuclide", + "Pr144", + "--neutrino-mass", + str(mass_MeV), + "--e-step", + str(e_step_MeV), + "--intensity-cutoff", + str(intensity_cutoff), + "--output", + csv_filename, + "--plot", + png_filename, + "-q", # quiet mode + ] + + print(f"Running: {' '.join(cmd)}") + result = subprocess.run(cmd, capture_output=True, text=True) + + if result.returncode != 0: + print(f"Error in calculation for mass {mass_keV:.2f} keV:") + print(f"stdout: {result.stdout}") + print(f"stderr: {result.stderr}") + else: + print(f"Successfully completed calculation for mass {mass_keV:.2f} keV") + +print("\nAll calculations completed!") diff --git a/test_neutrino_mass.csv b/test_neutrino_mass.csv new file mode 100644 index 0000000..ffdb224 --- /dev/null +++ b/test_neutrino_mass.csv @@ -0,0 +1,310 @@ +# beta-spectrum v0.4.0 +# timestamp: 2026-07-01T13:25:57.578522+00:00 +# source: paceENSDF +# nuclide: Tc99 -> Ru99 (Z=43->44) +# endpoint: 297.5 keV +# transition: F2 +# corrections: phase_space, fermi, screening, finite_size, charge_dist, radiative, exchange +# e_step: 0.0010 MeV +# neutrino_mass_MeV: 0.010000 +# branch_1: E₀=297.5 keV, transition=F2, intensity=1.0000 +# branch_2: E₀=208.0 keV, transition=F2U, intensity=0.0000 +# git_commit: 406fd57 +energy_MeV,spectrum,branch_1_spectrum,branch_1_PhaseSpace,branch_1_Radiative,branch_2_spectrum,branch_2_PhaseSpace,branch_2_Radiative,Fermi,FiniteSizeL0,ChargeDistributionU,Screening,Exchange +1.0000e-03,5.0134e+00,7.2723e-03,7.1882e-03,1.0117e+00,2.8198e-04,2.7884e-04,1.0113e+00,6.2469e+01,1.0154e+00,1.0000e+00,9.9894e-01,1.0496e+00 +2.0000e-03,5.0002e+00,1.0289e-02,1.0171e-02,1.0116e+00,3.9175e-04,3.8744e-04,1.0111e+00,4.4215e+01,1.0154e+00,1.0000e+00,9.9760e-01,1.0468e+00 +3.0000e-03,4.9742e+00,1.2607e-02,1.2464e-02,1.0115e+00,4.7190e-04,4.6675e-04,1.0110e+00,3.6137e+01,1.0154e+00,1.0000e+00,9.9594e-01,1.0416e+00 +4.0000e-03,4.9540e+00,1.4564e-02,1.4399e-02,1.0114e+00,5.3660e-04,5.3077e-04,1.0110e+00,3.1325e+01,1.0154e+00,1.0000e+00,9.9395e-01,1.0380e+00 +5.0000e-03,4.9369e+00,1.6290e-02,1.6107e-02,1.0114e+00,5.9152e-04,5.8513e-04,1.0109e+00,2.8045e+01,1.0154e+00,1.0000e+00,9.9164e-01,1.0353e+00 +6.0000e-03,4.9210e+00,1.7853e-02,1.7654e-02,1.0113e+00,6.3970e-04,6.3283e-04,1.0108e+00,2.5626e+01,1.0154e+00,1.0000e+00,9.8901e-01,1.0333e+00 +7.0000e-03,4.9054e+00,1.9293e-02,1.9078e-02,1.0113e+00,6.8302e-04,6.7572e-04,1.0108e+00,2.3748e+01,1.0154e+00,1.0000e+00,9.8609e-01,1.0316e+00 +8.0000e-03,4.8897e+00,2.0634e-02,2.0405e-02,1.0112e+00,7.2274e-04,7.1506e-04,1.0107e+00,2.2235e+01,1.0154e+00,1.0000e+00,9.8290e-01,1.0302e+00 +9.0000e-03,4.8802e+00,2.1895e-02,2.1653e-02,1.0112e+00,7.5978e-04,7.5174e-04,1.0107e+00,2.0983e+01,1.0154e+00,1.0000e+00,9.8079e-01,1.0290e+00 +1.0000e-02,4.8762e+00,2.3089e-02,2.2836e-02,1.0111e+00,7.9484e-04,7.8647e-04,1.0107e+00,1.9925e+01,1.0154e+00,1.0000e+00,9.7968e-01,1.0279e+00 +1.1000e-02,4.8732e+00,2.4227e-02,2.3961e-02,1.0111e+00,8.2847e-04,8.1977e-04,1.0106e+00,1.9016e+01,1.0154e+00,1.0000e+00,9.7867e-01,1.0269e+00 +1.2000e-02,4.8709e+00,2.5315e-02,2.5038e-02,1.0110e+00,8.6110e-04,8.5209e-04,1.0106e+00,1.8224e+01,1.0154e+00,1.0000e+00,9.7776e-01,1.0260e+00 +1.3000e-02,4.8695e+00,2.6359e-02,2.6072e-02,1.0110e+00,8.9308e-04,8.8378e-04,1.0105e+00,1.7525e+01,1.0154e+00,1.0000e+00,9.7694e-01,1.0252e+00 +1.4000e-02,4.8688e+00,2.7365e-02,2.7068e-02,1.0110e+00,9.2474e-04,9.1514e-04,1.0105e+00,1.6904e+01,1.0153e+00,1.0000e+00,9.7623e-01,1.0244e+00 +1.5000e-02,4.8686e+00,2.8336e-02,2.8030e-02,1.0109e+00,9.5631e-04,9.4642e-04,1.0105e+00,1.6347e+01,1.0153e+00,1.0000e+00,9.7560e-01,1.0237e+00 +1.6000e-02,4.8691e+00,2.9276e-02,2.8961e-02,1.0109e+00,9.8802e-04,9.7784e-04,1.0104e+00,1.5843e+01,1.0153e+00,1.0000e+00,9.7505e-01,1.0230e+00 +1.7000e-02,4.8700e+00,3.0188e-02,2.9864e-02,1.0109e+00,1.0201e-03,1.0096e-03,1.0104e+00,1.5385e+01,1.0153e+00,1.0000e+00,9.7457e-01,1.0223e+00 +1.8000e-02,4.8714e+00,3.1074e-02,3.0741e-02,1.0108e+00,1.0526e-03,1.0418e-03,1.0103e+00,1.4966e+01,1.0153e+00,1.0000e+00,9.7416e-01,1.0217e+00 +1.9000e-02,4.8731e+00,3.1936e-02,3.1595e-02,1.0108e+00,1.0858e-03,1.0747e-03,1.0103e+00,1.4581e+01,1.0153e+00,1.0000e+00,9.7381e-01,1.0211e+00 +2.0000e-02,4.8752e+00,3.2776e-02,3.2427e-02,1.0108e+00,1.1197e-03,1.1083e-03,1.0103e+00,1.4226e+01,1.0153e+00,1.0000e+00,9.7350e-01,1.0205e+00 +2.1000e-02,4.8775e+00,3.3595e-02,3.3238e-02,1.0107e+00,1.1545e-03,1.1428e-03,1.0102e+00,1.3897e+01,1.0153e+00,1.0000e+00,9.7324e-01,1.0200e+00 +2.2000e-02,4.8800e+00,3.4396e-02,3.4031e-02,1.0107e+00,1.1903e-03,1.1782e-03,1.0102e+00,1.3591e+01,1.0153e+00,1.0000e+00,9.7301e-01,1.0194e+00 +2.3000e-02,4.8828e+00,3.5178e-02,3.4806e-02,1.0107e+00,1.2271e-03,1.2147e-03,1.0102e+00,1.3305e+01,1.0153e+00,1.0000e+00,9.7281e-01,1.0189e+00 +2.4000e-02,4.8857e+00,3.5944e-02,3.5565e-02,1.0107e+00,1.2650e-03,1.2523e-03,1.0102e+00,1.3039e+01,1.0153e+00,1.0000e+00,9.7263e-01,1.0184e+00 +2.5000e-02,4.8888e+00,3.6694e-02,3.6308e-02,1.0106e+00,1.3042e-03,1.2911e-03,1.0101e+00,1.2788e+01,1.0153e+00,1.0000e+00,9.7248e-01,1.0180e+00 +2.6000e-02,4.8920e+00,3.7428e-02,3.7036e-02,1.0106e+00,1.3446e-03,1.3311e-03,1.0101e+00,1.2553e+01,1.0153e+00,1.0000e+00,9.7235e-01,1.0175e+00 +2.7000e-02,4.8953e+00,3.8149e-02,3.7750e-02,1.0106e+00,1.3863e-03,1.3725e-03,1.0101e+00,1.2331e+01,1.0153e+00,1.0000e+00,9.7223e-01,1.0170e+00 +2.8000e-02,4.8986e+00,3.8856e-02,3.8450e-02,1.0106e+00,1.4293e-03,1.4151e-03,1.0100e+00,1.2121e+01,1.0153e+00,1.0000e+00,9.7213e-01,1.0166e+00 +2.9000e-02,4.9021e+00,3.9550e-02,3.9138e-02,1.0105e+00,1.4738e-03,1.4591e-03,1.0100e+00,1.1923e+01,1.0153e+00,1.0000e+00,9.7204e-01,1.0162e+00 +3.0000e-02,4.9055e+00,4.0232e-02,3.9813e-02,1.0105e+00,1.5196e-03,1.5046e-03,1.0100e+00,1.1735e+01,1.0153e+00,1.0000e+00,9.7196e-01,1.0158e+00 +3.1000e-02,4.9090e+00,4.0901e-02,4.0477e-02,1.0105e+00,1.5669e-03,1.5514e-03,1.0100e+00,1.1557e+01,1.0153e+00,1.0000e+00,9.7188e-01,1.0154e+00 +3.2000e-02,4.9125e+00,4.1560e-02,4.1130e-02,1.0105e+00,1.6156e-03,1.5997e-03,1.0099e+00,1.1387e+01,1.0153e+00,1.0000e+00,9.7182e-01,1.0150e+00 +3.3000e-02,4.9161e+00,4.2207e-02,4.1772e-02,1.0104e+00,1.6658e-03,1.6495e-03,1.0099e+00,1.1225e+01,1.0153e+00,1.0000e+00,9.7176e-01,1.0146e+00 +3.4000e-02,4.9196e+00,4.2844e-02,4.2403e-02,1.0104e+00,1.7175e-03,1.7007e-03,1.0099e+00,1.1071e+01,1.0153e+00,1.0000e+00,9.7170e-01,1.0143e+00 +3.5000e-02,4.9231e+00,4.3471e-02,4.3024e-02,1.0104e+00,1.7707e-03,1.7534e-03,1.0098e+00,1.0924e+01,1.0152e+00,1.0000e+00,9.7165e-01,1.0139e+00 +3.6000e-02,4.9266e+00,4.4088e-02,4.3636e-02,1.0104e+00,1.8254e-03,1.8076e-03,1.0098e+00,1.0783e+01,1.0152e+00,1.0000e+00,9.7161e-01,1.0136e+00 +3.7000e-02,4.9301e+00,4.4695e-02,4.4238e-02,1.0103e+00,1.8816e-03,1.8633e-03,1.0098e+00,1.0648e+01,1.0152e+00,1.0000e+00,9.7157e-01,1.0132e+00 +3.8000e-02,4.9336e+00,4.5293e-02,4.4831e-02,1.0103e+00,1.9392e-03,1.9205e-03,1.0098e+00,1.0518e+01,1.0152e+00,1.0000e+00,9.7153e-01,1.0129e+00 +3.9000e-02,4.9370e+00,4.5882e-02,4.5415e-02,1.0103e+00,1.9984e-03,1.9791e-03,1.0097e+00,1.0394e+01,1.0152e+00,1.0000e+00,9.7150e-01,1.0126e+00 +4.0000e-02,4.9404e+00,4.6462e-02,4.5990e-02,1.0103e+00,2.0590e-03,2.0391e-03,1.0097e+00,1.0275e+01,1.0152e+00,1.0000e+00,9.7147e-01,1.0123e+00 +4.1000e-02,4.9438e+00,4.7033e-02,4.6556e-02,1.0102e+00,2.1210e-03,2.1007e-03,1.0097e+00,1.0160e+01,1.0152e+00,1.0000e+00,9.7144e-01,1.0120e+00 +4.2000e-02,4.9471e+00,4.7597e-02,4.7115e-02,1.0102e+00,2.1845e-03,2.1636e-03,1.0097e+00,1.0050e+01,1.0152e+00,1.0000e+00,9.7142e-01,1.0117e+00 +4.3000e-02,4.9503e+00,4.8151e-02,4.7665e-02,1.0102e+00,2.2494e-03,2.2280e-03,1.0096e+00,9.9439e+00,1.0152e+00,1.0000e+00,9.7140e-01,1.0114e+00 +4.4000e-02,4.9534e+00,4.8698e-02,4.8208e-02,1.0102e+00,2.3158e-03,2.2937e-03,1.0096e+00,9.8415e+00,1.0152e+00,1.0000e+00,9.7138e-01,1.0111e+00 +4.5000e-02,4.9565e+00,4.9237e-02,4.8742e-02,1.0102e+00,2.3834e-03,2.3608e-03,1.0096e+00,9.7428e+00,1.0152e+00,1.0000e+00,9.7136e-01,1.0108e+00 +4.6000e-02,4.9596e+00,4.9768e-02,4.9269e-02,1.0101e+00,2.4525e-03,2.4292e-03,1.0096e+00,9.6474e+00,1.0152e+00,1.0000e+00,9.7135e-01,1.0106e+00 +4.7000e-02,4.9625e+00,5.0292e-02,4.9788e-02,1.0101e+00,2.5228e-03,2.4989e-03,1.0095e+00,9.5552e+00,1.0152e+00,1.0000e+00,9.7133e-01,1.0103e+00 +4.8000e-02,4.9653e+00,5.0808e-02,5.0300e-02,1.0101e+00,2.5944e-03,2.5700e-03,1.0095e+00,9.4661e+00,1.0152e+00,1.0000e+00,9.7132e-01,1.0100e+00 +4.9000e-02,4.9681e+00,5.1317e-02,5.0805e-02,1.0101e+00,2.6673e-03,2.6422e-03,1.0095e+00,9.3799e+00,1.0152e+00,1.0000e+00,9.7132e-01,1.0098e+00 +5.0000e-02,4.9708e+00,5.1818e-02,5.1302e-02,1.0101e+00,2.7414e-03,2.7156e-03,1.0095e+00,9.2965e+00,1.0152e+00,1.0000e+00,9.7131e-01,1.0096e+00 +5.1000e-02,4.9733e+00,5.2312e-02,5.1793e-02,1.0100e+00,2.8166e-03,2.7903e-03,1.0094e+00,9.2157e+00,1.0152e+00,1.0000e+00,9.7130e-01,1.0093e+00 +5.2000e-02,4.9758e+00,5.2799e-02,5.2276e-02,1.0100e+00,2.8930e-03,2.8660e-03,1.0094e+00,9.1374e+00,1.0152e+00,1.0000e+00,9.7130e-01,1.0091e+00 +5.3000e-02,4.9782e+00,5.3279e-02,5.2752e-02,1.0100e+00,2.9705e-03,2.9428e-03,1.0094e+00,9.0614e+00,1.0152e+00,1.0000e+00,9.7130e-01,1.0089e+00 +5.4000e-02,4.9804e+00,5.3752e-02,5.3222e-02,1.0100e+00,3.0490e-03,3.0207e-03,1.0094e+00,8.9877e+00,1.0151e+00,1.0000e+00,9.7130e-01,1.0087e+00 +5.5000e-02,4.9825e+00,5.4218e-02,5.3684e-02,1.0099e+00,3.1285e-03,3.0995e-03,1.0093e+00,8.9161e+00,1.0151e+00,1.0000e+00,9.7130e-01,1.0084e+00 +5.6000e-02,4.9845e+00,5.4678e-02,5.4140e-02,1.0099e+00,3.2090e-03,3.1793e-03,1.0093e+00,8.8466e+00,1.0151e+00,1.0000e+00,9.7130e-01,1.0082e+00 +5.7000e-02,4.9864e+00,5.5130e-02,5.4590e-02,1.0099e+00,3.2903e-03,3.2600e-03,1.0093e+00,8.7791e+00,1.0151e+00,1.0000e+00,9.7131e-01,1.0080e+00 +5.8000e-02,4.9882e+00,5.5576e-02,5.5032e-02,1.0099e+00,3.3726e-03,3.3416e-03,1.0093e+00,8.7134e+00,1.0151e+00,1.0000e+00,9.7132e-01,1.0078e+00 +5.9000e-02,4.9898e+00,5.6015e-02,5.5468e-02,1.0099e+00,3.4556e-03,3.4239e-03,1.0093e+00,8.6496e+00,1.0151e+00,1.0000e+00,9.7132e-01,1.0076e+00 +6.0000e-02,4.9912e+00,5.6448e-02,5.5897e-02,1.0098e+00,3.5394e-03,3.5070e-03,1.0092e+00,8.5874e+00,1.0151e+00,1.0000e+00,9.7133e-01,1.0074e+00 +6.1000e-02,4.9926e+00,5.6873e-02,5.6320e-02,1.0098e+00,3.6239e-03,3.5908e-03,1.0092e+00,8.5269e+00,1.0151e+00,1.0000e+00,9.7134e-01,1.0073e+00 +6.2000e-02,4.9938e+00,5.7293e-02,5.6737e-02,1.0098e+00,3.7090e-03,3.6753e-03,1.0092e+00,8.4680e+00,1.0151e+00,1.0000e+00,9.7135e-01,1.0071e+00 +6.3000e-02,4.9948e+00,5.7705e-02,5.7146e-02,1.0098e+00,3.7947e-03,3.7603e-03,1.0092e+00,8.4106e+00,1.0151e+00,1.0000e+00,9.7137e-01,1.0069e+00 +6.4000e-02,4.9957e+00,5.8111e-02,5.7550e-02,1.0098e+00,3.8810e-03,3.8458e-03,1.0091e+00,8.3546e+00,1.0151e+00,1.0000e+00,9.7138e-01,1.0067e+00 +6.5000e-02,4.9964e+00,5.8511e-02,5.7947e-02,1.0097e+00,3.9677e-03,3.9318e-03,1.0091e+00,8.3000e+00,1.0151e+00,1.0000e+00,9.7139e-01,1.0066e+00 +6.6000e-02,4.9969e+00,5.8904e-02,5.8337e-02,1.0097e+00,4.0548e-03,4.0183e-03,1.0091e+00,8.2468e+00,1.0151e+00,1.0000e+00,9.7141e-01,1.0064e+00 +6.7000e-02,4.9973e+00,5.9291e-02,5.8721e-02,1.0097e+00,4.1422e-03,4.1050e-03,1.0091e+00,8.1949e+00,1.0151e+00,1.0000e+00,9.7143e-01,1.0062e+00 +6.8000e-02,4.9976e+00,5.9671e-02,5.9099e-02,1.0097e+00,4.2299e-03,4.1921e-03,1.0090e+00,8.1442e+00,1.0151e+00,1.0000e+00,9.7145e-01,1.0061e+00 +6.9000e-02,4.9976e+00,6.0045e-02,5.9470e-02,1.0097e+00,4.3179e-03,4.2793e-03,1.0090e+00,8.0947e+00,1.0151e+00,1.0000e+00,9.7146e-01,1.0059e+00 +7.0000e-02,4.9975e+00,6.0412e-02,5.9835e-02,1.0096e+00,4.4060e-03,4.3668e-03,1.0090e+00,8.0463e+00,1.0151e+00,1.0000e+00,9.7148e-01,1.0058e+00 +7.1000e-02,4.9972e+00,6.0773e-02,6.0193e-02,1.0096e+00,4.4942e-03,4.4543e-03,1.0090e+00,7.9991e+00,1.0151e+00,1.0000e+00,9.7150e-01,1.0056e+00 +7.2000e-02,4.9967e+00,6.1127e-02,6.0546e-02,1.0096e+00,4.5824e-03,4.5418e-03,1.0089e+00,7.9529e+00,1.0150e+00,1.0000e+00,9.7153e-01,1.0055e+00 +7.3000e-02,4.9960e+00,6.1475e-02,6.0891e-02,1.0096e+00,4.6706e-03,4.6293e-03,1.0089e+00,7.9078e+00,1.0150e+00,1.0000e+00,9.7155e-01,1.0054e+00 +7.4000e-02,4.9951e+00,6.1816e-02,6.1231e-02,1.0096e+00,4.7587e-03,4.7168e-03,1.0089e+00,7.8637e+00,1.0150e+00,1.0000e+00,9.7157e-01,1.0052e+00 +7.5000e-02,4.9940e+00,6.2151e-02,6.1563e-02,1.0095e+00,4.8466e-03,4.8040e-03,1.0089e+00,7.8205e+00,1.0150e+00,1.0000e+00,9.7159e-01,1.0051e+00 +7.6000e-02,4.9927e+00,6.2479e-02,6.1890e-02,1.0095e+00,4.9342e-03,4.8910e-03,1.0088e+00,7.7782e+00,1.0150e+00,1.0000e+00,9.7162e-01,1.0050e+00 +7.7000e-02,4.9913e+00,6.2801e-02,6.2210e-02,1.0095e+00,5.0216e-03,4.9777e-03,1.0088e+00,7.7369e+00,1.0150e+00,1.0000e+00,9.7164e-01,1.0048e+00 +7.8000e-02,4.9896e+00,6.3117e-02,6.2524e-02,1.0095e+00,5.1086e-03,5.0640e-03,1.0088e+00,7.6964e+00,1.0150e+00,1.0000e+00,9.7167e-01,1.0047e+00 +7.9000e-02,4.9877e+00,6.3425e-02,6.2831e-02,1.0095e+00,5.1951e-03,5.1499e-03,1.0088e+00,7.6568e+00,1.0150e+00,1.0000e+00,9.7170e-01,1.0046e+00 +8.0000e-02,4.9856e+00,6.3728e-02,6.3132e-02,1.0094e+00,5.2811e-03,5.2353e-03,1.0087e+00,7.6180e+00,1.0150e+00,1.0000e+00,9.7172e-01,1.0045e+00 +8.1000e-02,4.9833e+00,6.4024e-02,6.3426e-02,1.0094e+00,5.3665e-03,5.3201e-03,1.0087e+00,7.5799e+00,1.0150e+00,1.0000e+00,9.7175e-01,1.0044e+00 +8.2000e-02,4.9807e+00,6.4313e-02,6.3714e-02,1.0094e+00,5.4512e-03,5.4042e-03,1.0087e+00,7.5427e+00,1.0150e+00,1.0000e+00,9.7178e-01,1.0043e+00 +8.3000e-02,4.9780e+00,6.4596e-02,6.3995e-02,1.0094e+00,5.5352e-03,5.4876e-03,1.0087e+00,7.5061e+00,1.0150e+00,1.0000e+00,9.7181e-01,1.0041e+00 +8.4000e-02,4.9750e+00,6.4872e-02,6.4270e-02,1.0094e+00,5.6184e-03,5.5703e-03,1.0086e+00,7.4703e+00,1.0150e+00,1.0000e+00,9.7184e-01,1.0040e+00 +8.5000e-02,4.9718e+00,6.5141e-02,6.4538e-02,1.0093e+00,5.7008e-03,5.6520e-03,1.0086e+00,7.4352e+00,1.0150e+00,1.0000e+00,9.7187e-01,1.0039e+00 +8.6000e-02,4.9683e+00,6.5404e-02,6.4800e-02,1.0093e+00,5.7822e-03,5.7329e-03,1.0086e+00,7.4007e+00,1.0150e+00,1.0000e+00,9.7190e-01,1.0038e+00 +8.7000e-02,4.9647e+00,6.5660e-02,6.5055e-02,1.0093e+00,5.8625e-03,5.8127e-03,1.0086e+00,7.3669e+00,1.0150e+00,1.0000e+00,9.7193e-01,1.0037e+00 +8.8000e-02,4.9607e+00,6.5910e-02,6.5304e-02,1.0093e+00,5.9418e-03,5.8914e-03,1.0085e+00,7.3338e+00,1.0150e+00,1.0000e+00,9.7196e-01,1.0036e+00 +8.9000e-02,4.9566e+00,6.6153e-02,6.5546e-02,1.0093e+00,6.0199e-03,5.9691e-03,1.0085e+00,7.3012e+00,1.0150e+00,1.0000e+00,9.7199e-01,1.0035e+00 +9.0000e-02,4.9522e+00,6.6389e-02,6.5781e-02,1.0092e+00,6.0968e-03,6.0455e-03,1.0085e+00,7.2693e+00,1.0149e+00,1.0000e+00,9.7203e-01,1.0034e+00 +9.1000e-02,4.9476e+00,6.6619e-02,6.6010e-02,1.0092e+00,6.1724e-03,6.1206e-03,1.0085e+00,7.2379e+00,1.0149e+00,1.0000e+00,9.7206e-01,1.0034e+00 +9.2000e-02,4.9427e+00,6.6841e-02,6.6232e-02,1.0092e+00,6.2466e-03,6.1943e-03,1.0084e+00,7.2071e+00,1.0149e+00,1.0000e+00,9.7209e-01,1.0033e+00 +9.3000e-02,4.9375e+00,6.7057e-02,6.6447e-02,1.0092e+00,6.3194e-03,6.2666e-03,1.0084e+00,7.1768e+00,1.0149e+00,1.0000e+00,9.7213e-01,1.0032e+00 +9.4000e-02,4.9322e+00,6.7266e-02,6.6656e-02,1.0092e+00,6.3907e-03,6.3375e-03,1.0084e+00,7.1471e+00,1.0149e+00,1.0000e+00,9.7216e-01,1.0031e+00 +9.5000e-02,4.9265e+00,6.7469e-02,6.6857e-02,1.0091e+00,6.4604e-03,6.4068e-03,1.0084e+00,7.1179e+00,1.0149e+00,1.0000e+00,9.7219e-01,1.0030e+00 +9.6000e-02,4.9206e+00,6.7664e-02,6.7052e-02,1.0091e+00,6.5284e-03,6.4744e-03,1.0083e+00,7.0892e+00,1.0149e+00,1.0000e+00,9.7223e-01,1.0029e+00 +9.7000e-02,4.9145e+00,6.7853e-02,6.7241e-02,1.0091e+00,6.5947e-03,6.5403e-03,1.0083e+00,7.0610e+00,1.0149e+00,1.0000e+00,9.7226e-01,1.0029e+00 +9.8000e-02,4.9080e+00,6.8034e-02,6.7422e-02,1.0091e+00,6.6593e-03,6.6045e-03,1.0083e+00,7.0332e+00,1.0149e+00,1.0000e+00,9.7230e-01,1.0028e+00 +9.9000e-02,4.9013e+00,6.8209e-02,6.7596e-02,1.0091e+00,6.7219e-03,6.6668e-03,1.0083e+00,7.0060e+00,1.0149e+00,1.0000e+00,9.7233e-01,1.0027e+00 +1.0000e-01,4.8944e+00,6.8377e-02,6.7764e-02,1.0090e+00,6.7827e-03,6.7272e-03,1.0082e+00,6.9792e+00,1.0149e+00,1.0000e+00,9.7237e-01,1.0026e+00 +1.0100e-01,4.8872e+00,6.8537e-02,6.7925e-02,1.0090e+00,6.8415e-03,6.7857e-03,1.0082e+00,6.9528e+00,1.0149e+00,1.0000e+00,9.7240e-01,1.0026e+00 +1.0200e-01,4.8797e+00,6.8691e-02,6.8079e-02,1.0090e+00,6.8982e-03,6.8421e-03,1.0082e+00,6.9269e+00,1.0149e+00,1.0000e+00,9.7244e-01,1.0025e+00 +1.0300e-01,4.8719e+00,6.8838e-02,6.8225e-02,1.0090e+00,6.9528e-03,6.8965e-03,1.0082e+00,6.9013e+00,1.0149e+00,1.0000e+00,9.7248e-01,1.0024e+00 +1.0400e-01,4.8638e+00,6.8978e-02,6.8365e-02,1.0090e+00,7.0052e-03,6.9486e-03,1.0081e+00,6.8762e+00,1.0149e+00,1.0000e+00,9.7251e-01,1.0024e+00 +1.0500e-01,4.8555e+00,6.9110e-02,6.8498e-02,1.0089e+00,7.0553e-03,6.9986e-03,1.0081e+00,6.8515e+00,1.0149e+00,1.0000e+00,9.7255e-01,1.0023e+00 +1.0600e-01,4.8469e+00,6.9236e-02,6.8624e-02,1.0089e+00,7.1032e-03,7.0462e-03,1.0081e+00,6.8272e+00,1.0149e+00,1.0000e+00,9.7259e-01,1.0022e+00 +1.0700e-01,4.8380e+00,6.9354e-02,6.8742e-02,1.0089e+00,7.1486e-03,7.0915e-03,1.0081e+00,6.8032e+00,1.0148e+00,1.0000e+00,9.7262e-01,1.0022e+00 +1.0800e-01,4.8289e+00,6.9465e-02,6.8854e-02,1.0089e+00,7.1917e-03,7.1344e-03,1.0080e+00,6.7797e+00,1.0148e+00,1.0000e+00,9.7266e-01,1.0021e+00 +1.0900e-01,4.8194e+00,6.9569e-02,6.8958e-02,1.0089e+00,7.2322e-03,7.1748e-03,1.0080e+00,6.7564e+00,1.0148e+00,1.0000e+00,9.7270e-01,1.0020e+00 +1.1000e-01,4.8097e+00,6.9666e-02,6.9056e-02,1.0088e+00,7.2703e-03,7.2127e-03,1.0080e+00,6.7336e+00,1.0148e+00,1.0000e+00,9.7274e-01,1.0020e+00 +1.1100e-01,4.7996e+00,6.9755e-02,6.9146e-02,1.0088e+00,7.3057e-03,7.2481e-03,1.0079e+00,6.7111e+00,1.0148e+00,1.0000e+00,9.7277e-01,1.0019e+00 +1.1200e-01,4.7893e+00,6.9838e-02,6.9229e-02,1.0088e+00,7.3384e-03,7.2807e-03,1.0079e+00,6.6889e+00,1.0148e+00,1.0000e+00,9.7281e-01,1.0019e+00 +1.1300e-01,4.7787e+00,6.9913e-02,6.9305e-02,1.0088e+00,7.3684e-03,7.3107e-03,1.0079e+00,6.6670e+00,1.0148e+00,1.0000e+00,9.7285e-01,1.0018e+00 +1.1400e-01,4.7678e+00,6.9980e-02,6.9373e-02,1.0088e+00,7.3957e-03,7.3380e-03,1.0079e+00,6.6455e+00,1.0148e+00,1.0000e+00,9.7289e-01,1.0018e+00 +1.1500e-01,4.7565e+00,7.0041e-02,6.9434e-02,1.0087e+00,7.4202e-03,7.3625e-03,1.0078e+00,6.6243e+00,1.0148e+00,1.0000e+00,9.7293e-01,1.0017e+00 +1.1600e-01,4.7450e+00,7.0093e-02,6.9488e-02,1.0087e+00,7.4417e-03,7.3841e-03,1.0078e+00,6.6034e+00,1.0148e+00,1.0000e+00,9.7297e-01,1.0017e+00 +1.1700e-01,4.7332e+00,7.0139e-02,6.9535e-02,1.0087e+00,7.4604e-03,7.4028e-03,1.0078e+00,6.5828e+00,1.0148e+00,1.0000e+00,9.7300e-01,1.0016e+00 +1.1800e-01,4.7211e+00,7.0177e-02,6.9574e-02,1.0087e+00,7.4761e-03,7.4186e-03,1.0078e+00,6.5624e+00,1.0148e+00,1.0000e+00,9.7304e-01,1.0016e+00 +1.1900e-01,4.7087e+00,7.0208e-02,6.9606e-02,1.0086e+00,7.4888e-03,7.4314e-03,1.0077e+00,6.5424e+00,1.0148e+00,1.0000e+00,9.7308e-01,1.0015e+00 +1.2000e-01,4.6960e+00,7.0231e-02,6.9631e-02,1.0086e+00,7.4985e-03,7.4412e-03,1.0077e+00,6.5226e+00,1.0148e+00,1.0000e+00,9.7312e-01,1.0015e+00 +1.2100e-01,4.6830e+00,7.0247e-02,6.9648e-02,1.0086e+00,7.5051e-03,7.4480e-03,1.0077e+00,6.5032e+00,1.0148e+00,1.0000e+00,9.7316e-01,1.0014e+00 +1.2200e-01,4.6697e+00,7.0255e-02,6.9657e-02,1.0086e+00,7.5085e-03,7.4516e-03,1.0076e+00,6.4840e+00,1.0148e+00,1.0000e+00,9.7320e-01,1.0014e+00 +1.2300e-01,4.6560e+00,7.0256e-02,6.9660e-02,1.0086e+00,7.5088e-03,7.4521e-03,1.0076e+00,6.4650e+00,1.0148e+00,1.0000e+00,9.7324e-01,1.0013e+00 +1.2400e-01,4.6421e+00,7.0249e-02,6.9654e-02,1.0085e+00,7.5059e-03,7.4495e-03,1.0076e+00,6.4463e+00,1.0147e+00,1.0000e+00,9.7328e-01,1.0013e+00 +1.2500e-01,4.6278e+00,7.0235e-02,6.9642e-02,1.0085e+00,7.4998e-03,7.4436e-03,1.0075e+00,6.4279e+00,1.0147e+00,1.0000e+00,9.7332e-01,1.0013e+00 +1.2600e-01,4.6133e+00,7.0213e-02,6.9622e-02,1.0085e+00,7.4904e-03,7.4345e-03,1.0075e+00,6.4097e+00,1.0147e+00,1.0000e+00,9.7335e-01,1.0012e+00 +1.2700e-01,4.5984e+00,7.0184e-02,6.9594e-02,1.0085e+00,7.4777e-03,7.4221e-03,1.0075e+00,6.3917e+00,1.0147e+00,1.0000e+00,9.7339e-01,1.0012e+00 +1.2800e-01,4.5832e+00,7.0147e-02,6.9559e-02,1.0085e+00,7.4616e-03,7.4064e-03,1.0075e+00,6.3740e+00,1.0147e+00,1.0000e+00,9.7343e-01,1.0012e+00 +1.2900e-01,4.5677e+00,7.0102e-02,6.9516e-02,1.0084e+00,7.4423e-03,7.3874e-03,1.0074e+00,6.3565e+00,1.0147e+00,1.0000e+00,9.7347e-01,1.0011e+00 +1.3000e-01,4.5519e+00,7.0050e-02,6.9466e-02,1.0084e+00,7.4196e-03,7.3651e-03,1.0074e+00,6.3393e+00,1.0147e+00,1.0000e+00,9.7351e-01,1.0011e+00 +1.3100e-01,4.5358e+00,6.9990e-02,6.9408e-02,1.0084e+00,7.3935e-03,7.3395e-03,1.0074e+00,6.3223e+00,1.0147e+00,1.0000e+00,9.7355e-01,1.0010e+00 +1.3200e-01,4.5194e+00,6.9922e-02,6.9342e-02,1.0084e+00,7.3640e-03,7.3104e-03,1.0073e+00,6.3054e+00,1.0147e+00,1.0000e+00,9.7359e-01,1.0010e+00 +1.3300e-01,4.5027e+00,6.9847e-02,6.9269e-02,1.0083e+00,7.3312e-03,7.2780e-03,1.0073e+00,6.2888e+00,1.0147e+00,1.0000e+00,9.7363e-01,1.0010e+00 +1.3400e-01,4.4856e+00,6.9764e-02,6.9188e-02,1.0083e+00,7.2949e-03,7.2422e-03,1.0073e+00,6.2724e+00,1.0147e+00,1.0000e+00,9.7367e-01,1.0009e+00 +1.3500e-01,4.4682e+00,6.9674e-02,6.9100e-02,1.0083e+00,7.2551e-03,7.2030e-03,1.0072e+00,6.2563e+00,1.0147e+00,1.0000e+00,9.7371e-01,1.0009e+00 +1.3600e-01,4.4505e+00,6.9575e-02,6.9004e-02,1.0083e+00,7.2120e-03,7.1604e-03,1.0072e+00,6.2403e+00,1.0147e+00,1.0000e+00,9.7375e-01,1.0009e+00 +1.3700e-01,4.4325e+00,6.9469e-02,6.8900e-02,1.0083e+00,7.1654e-03,7.1144e-03,1.0072e+00,6.2245e+00,1.0147e+00,1.0000e+00,9.7379e-01,1.0009e+00 +1.3800e-01,4.4142e+00,6.9356e-02,6.8789e-02,1.0082e+00,7.1154e-03,7.0649e-03,1.0071e+00,6.2089e+00,1.0147e+00,1.0000e+00,9.7383e-01,1.0008e+00 +1.3900e-01,4.3956e+00,6.9234e-02,6.8670e-02,1.0082e+00,7.0619e-03,7.0121e-03,1.0071e+00,6.1935e+00,1.0147e+00,1.0000e+00,9.7387e-01,1.0008e+00 +1.4000e-01,4.3766e+00,6.9105e-02,6.8544e-02,1.0082e+00,7.0051e-03,6.9559e-03,1.0071e+00,6.1783e+00,1.0146e+00,1.0000e+00,9.7391e-01,1.0008e+00 +1.4100e-01,4.3573e+00,6.8968e-02,6.8409e-02,1.0082e+00,6.9448e-03,6.8963e-03,1.0070e+00,6.1632e+00,1.0146e+00,1.0000e+00,9.7395e-01,1.0007e+00 +1.4200e-01,4.3378e+00,6.8824e-02,6.8268e-02,1.0081e+00,6.8812e-03,6.8333e-03,1.0070e+00,6.1484e+00,1.0146e+00,1.0000e+00,9.7399e-01,1.0007e+00 +1.4300e-01,4.3179e+00,6.8671e-02,6.8118e-02,1.0081e+00,6.8142e-03,6.7670e-03,1.0070e+00,6.1337e+00,1.0146e+00,1.0000e+00,9.7403e-01,1.0007e+00 +1.4400e-01,4.2976e+00,6.8511e-02,6.7961e-02,1.0081e+00,6.7438e-03,6.6973e-03,1.0069e+00,6.1192e+00,1.0146e+00,1.0000e+00,9.7406e-01,1.0007e+00 +1.4500e-01,4.2771e+00,6.8343e-02,6.7796e-02,1.0081e+00,6.6701e-03,6.6244e-03,1.0069e+00,6.1048e+00,1.0146e+00,1.0000e+00,9.7410e-01,1.0006e+00 +1.4600e-01,4.2563e+00,6.8168e-02,6.7623e-02,1.0081e+00,6.5931e-03,6.5481e-03,1.0069e+00,6.0907e+00,1.0146e+00,1.0000e+00,9.7414e-01,1.0006e+00 +1.4700e-01,4.2351e+00,6.7985e-02,6.7443e-02,1.0080e+00,6.5129e-03,6.4687e-03,1.0068e+00,6.0766e+00,1.0146e+00,1.0000e+00,9.7418e-01,1.0006e+00 +1.4800e-01,4.2136e+00,6.7794e-02,6.7255e-02,1.0080e+00,6.4294e-03,6.3860e-03,1.0068e+00,6.0628e+00,1.0146e+00,1.0000e+00,9.7422e-01,1.0006e+00 +1.4900e-01,4.1919e+00,6.7595e-02,6.7060e-02,1.0080e+00,6.3427e-03,6.3001e-03,1.0068e+00,6.0491e+00,1.0146e+00,1.0000e+00,9.7426e-01,1.0005e+00 +1.5000e-01,4.1698e+00,6.7389e-02,6.6857e-02,1.0080e+00,6.2529e-03,6.2112e-03,1.0067e+00,6.0356e+00,1.0146e+00,1.0000e+00,9.7430e-01,1.0005e+00 +1.5100e-01,4.1474e+00,6.7175e-02,6.6646e-02,1.0079e+00,6.1600e-03,6.1191e-03,1.0067e+00,6.0222e+00,1.0146e+00,1.0000e+00,9.7434e-01,1.0005e+00 +1.5200e-01,4.1246e+00,6.6953e-02,6.6428e-02,1.0079e+00,6.0641e-03,6.0240e-03,1.0066e+00,6.0089e+00,1.0146e+00,1.0000e+00,9.7438e-01,1.0005e+00 +1.5300e-01,4.1016e+00,6.6724e-02,6.6202e-02,1.0079e+00,5.9651e-03,5.9260e-03,1.0066e+00,5.9959e+00,1.0146e+00,1.0000e+00,9.7442e-01,1.0005e+00 +1.5400e-01,4.0783e+00,6.6487e-02,6.5968e-02,1.0079e+00,5.8633e-03,5.8250e-03,1.0066e+00,5.9829e+00,1.0146e+00,1.0000e+00,9.7446e-01,1.0004e+00 +1.5500e-01,4.0546e+00,6.6242e-02,6.5727e-02,1.0078e+00,5.7586e-03,5.7212e-03,1.0065e+00,5.9701e+00,1.0146e+00,1.0000e+00,9.7450e-01,1.0004e+00 +1.5600e-01,4.0307e+00,6.5990e-02,6.5478e-02,1.0078e+00,5.6511e-03,5.6147e-03,1.0065e+00,5.9574e+00,1.0145e+00,1.0000e+00,9.7454e-01,1.0004e+00 +1.5700e-01,4.0064e+00,6.5730e-02,6.5222e-02,1.0078e+00,5.5409e-03,5.5054e-03,1.0065e+00,5.9449e+00,1.0145e+00,1.0000e+00,9.7457e-01,1.0004e+00 +1.5800e-01,3.9819e+00,6.5463e-02,6.4958e-02,1.0078e+00,5.4281e-03,5.3935e-03,1.0064e+00,5.9325e+00,1.0145e+00,1.0000e+00,9.7461e-01,1.0004e+00 +1.5900e-01,3.9570e+00,6.5188e-02,6.4687e-02,1.0077e+00,5.3127e-03,5.2791e-03,1.0064e+00,5.9203e+00,1.0145e+00,1.0000e+00,9.7465e-01,1.0003e+00 +1.6000e-01,3.9319e+00,6.4905e-02,6.4408e-02,1.0077e+00,5.1949e-03,5.1623e-03,1.0063e+00,5.9081e+00,1.0145e+00,1.0000e+00,9.7469e-01,1.0003e+00 +1.6100e-01,3.9064e+00,6.4615e-02,6.4121e-02,1.0077e+00,5.0748e-03,5.0431e-03,1.0063e+00,5.8962e+00,1.0145e+00,1.0000e+00,9.7473e-01,1.0003e+00 +1.6200e-01,3.8806e+00,6.4317e-02,6.3828e-02,1.0077e+00,4.9523e-03,4.9216e-03,1.0062e+00,5.8843e+00,1.0145e+00,1.0000e+00,9.7477e-01,1.0003e+00 +1.6300e-01,3.8546e+00,6.4012e-02,6.3527e-02,1.0076e+00,4.8278e-03,4.7980e-03,1.0062e+00,5.8725e+00,1.0145e+00,1.0000e+00,9.7481e-01,1.0003e+00 +1.6400e-01,3.8283e+00,6.3700e-02,6.3218e-02,1.0076e+00,4.7011e-03,4.6724e-03,1.0062e+00,5.8609e+00,1.0145e+00,1.0000e+00,9.7485e-01,1.0003e+00 +1.6500e-01,3.8016e+00,6.3380e-02,6.2902e-02,1.0076e+00,4.5726e-03,4.5448e-03,1.0061e+00,5.8494e+00,1.0145e+00,1.0000e+00,9.7488e-01,1.0002e+00 +1.6600e-01,3.7747e+00,6.3053e-02,6.2579e-02,1.0076e+00,4.4423e-03,4.4155e-03,1.0061e+00,5.8380e+00,1.0145e+00,1.0000e+00,9.7492e-01,1.0002e+00 +1.6700e-01,3.7475e+00,6.2718e-02,6.2248e-02,1.0075e+00,4.3102e-03,4.2845e-03,1.0060e+00,5.8267e+00,1.0145e+00,1.0000e+00,9.7496e-01,1.0002e+00 +1.6800e-01,3.7200e+00,6.2376e-02,6.1910e-02,1.0075e+00,4.1767e-03,4.1519e-03,1.0060e+00,5.8156e+00,1.0145e+00,1.0000e+00,9.7500e-01,1.0002e+00 +1.6900e-01,3.6922e+00,6.2027e-02,6.1565e-02,1.0075e+00,4.0417e-03,4.0179e-03,1.0059e+00,5.8045e+00,1.0145e+00,1.0000e+00,9.7504e-01,1.0002e+00 +1.7000e-01,3.6642e+00,6.1671e-02,6.1213e-02,1.0075e+00,3.9055e-03,3.8827e-03,1.0059e+00,5.7936e+00,1.0145e+00,1.0000e+00,9.7508e-01,1.0002e+00 +1.7100e-01,3.6358e+00,6.1307e-02,6.0854e-02,1.0074e+00,3.7682e-03,3.7464e-03,1.0058e+00,5.7828e+00,1.0144e+00,1.0000e+00,9.7511e-01,1.0002e+00 +1.7200e-01,3.6072e+00,6.0936e-02,6.0488e-02,1.0074e+00,3.6299e-03,3.6091e-03,1.0058e+00,5.7721e+00,1.0144e+00,1.0000e+00,9.7515e-01,1.0002e+00 +1.7300e-01,3.5783e+00,6.0558e-02,6.0114e-02,1.0074e+00,3.4909e-03,3.4710e-03,1.0057e+00,5.7614e+00,1.0144e+00,1.0000e+00,9.7519e-01,1.0001e+00 +1.7400e-01,3.5492e+00,6.0174e-02,5.9734e-02,1.0074e+00,3.3512e-03,3.3323e-03,1.0057e+00,5.7509e+00,1.0144e+00,1.0000e+00,9.7523e-01,1.0001e+00 +1.7500e-01,3.5197e+00,5.9782e-02,5.9346e-02,1.0073e+00,3.2111e-03,3.1932e-03,1.0056e+00,5.7405e+00,1.0144e+00,1.0000e+00,9.7527e-01,1.0001e+00 +1.7600e-01,3.4901e+00,5.9383e-02,5.8952e-02,1.0073e+00,3.0707e-03,3.0538e-03,1.0056e+00,5.7302e+00,1.0144e+00,1.0000e+00,9.7530e-01,1.0001e+00 +1.7700e-01,3.4601e+00,5.8977e-02,5.8550e-02,1.0073e+00,2.9303e-03,2.9143e-03,1.0055e+00,5.7200e+00,1.0144e+00,1.0000e+00,9.7534e-01,1.0001e+00 +1.7800e-01,3.4299e+00,5.8565e-02,5.8142e-02,1.0073e+00,2.7900e-03,2.7749e-03,1.0054e+00,5.7099e+00,1.0144e+00,1.0000e+00,9.7538e-01,1.0001e+00 +1.7900e-01,3.3995e+00,5.8145e-02,5.7728e-02,1.0072e+00,2.6501e-03,2.6359e-03,1.0054e+00,5.6999e+00,1.0144e+00,1.0000e+00,9.7542e-01,1.0001e+00 +1.8000e-01,3.3688e+00,5.7719e-02,5.7306e-02,1.0072e+00,2.5107e-03,2.4974e-03,1.0053e+00,5.6900e+00,1.0144e+00,1.0000e+00,9.7545e-01,1.0001e+00 +1.8100e-01,3.3378e+00,5.7287e-02,5.6878e-02,1.0072e+00,2.3720e-03,2.3596e-03,1.0053e+00,5.6802e+00,1.0144e+00,1.0000e+00,9.7549e-01,1.0001e+00 +1.8200e-01,3.3066e+00,5.6847e-02,5.6443e-02,1.0072e+00,2.2344e-03,2.2228e-03,1.0052e+00,5.6705e+00,1.0144e+00,1.0000e+00,9.7553e-01,1.0000e+00 +1.8300e-01,3.2752e+00,5.6401e-02,5.6002e-02,1.0071e+00,2.0979e-03,2.0872e-03,1.0051e+00,5.6608e+00,1.0144e+00,1.0000e+00,9.7557e-01,1.0000e+00 +1.8400e-01,3.2435e+00,5.5949e-02,5.5555e-02,1.0071e+00,1.9629e-03,1.9530e-03,1.0051e+00,5.6513e+00,1.0144e+00,1.0000e+00,9.7560e-01,1.0000e+00 +1.8500e-01,3.2116e+00,5.5490e-02,5.5101e-02,1.0071e+00,1.8297e-03,1.8206e-03,1.0050e+00,5.6418e+00,1.0144e+00,1.0000e+00,9.7564e-01,1.0000e+00 +1.8600e-01,3.1795e+00,5.5025e-02,5.4641e-02,1.0070e+00,1.6983e-03,1.6900e-03,1.0049e+00,5.6324e+00,1.0143e+00,1.0000e+00,9.7568e-01,1.0000e+00 +1.8700e-01,3.1471e+00,5.4554e-02,5.4174e-02,1.0070e+00,1.5692e-03,1.5617e-03,1.0049e+00,5.6232e+00,1.0143e+00,1.0000e+00,9.7571e-01,1.0000e+00 +1.8800e-01,3.1145e+00,5.4077e-02,5.3701e-02,1.0070e+00,1.4426e-03,1.4358e-03,1.0048e+00,5.6139e+00,1.0143e+00,1.0000e+00,9.7575e-01,9.9999e-01 +1.8900e-01,3.0817e+00,5.3593e-02,5.3223e-02,1.0070e+00,1.3188e-03,1.3126e-03,1.0047e+00,5.6048e+00,1.0143e+00,1.0000e+00,9.7579e-01,9.9998e-01 +1.9000e-01,3.0487e+00,5.3104e-02,5.2738e-02,1.0069e+00,1.1980e-03,1.1925e-03,1.0046e+00,5.5958e+00,1.0143e+00,1.0000e+00,9.7582e-01,9.9998e-01 +1.9100e-01,3.0155e+00,5.2608e-02,5.2248e-02,1.0069e+00,1.0806e-03,1.0757e-03,1.0045e+00,5.5868e+00,1.0143e+00,1.0000e+00,9.7586e-01,9.9997e-01 +1.9200e-01,2.9821e+00,5.2107e-02,5.1751e-02,1.0069e+00,9.6677e-04,9.6250e-04,1.0044e+00,5.5780e+00,1.0143e+00,1.0000e+00,9.7590e-01,9.9996e-01 +1.9300e-01,2.9485e+00,5.1600e-02,5.1249e-02,1.0068e+00,8.5690e-04,8.5320e-04,1.0043e+00,5.5692e+00,1.0143e+00,1.0000e+00,9.7593e-01,9.9995e-01 +1.9400e-01,2.9147e+00,5.1087e-02,5.0741e-02,1.0068e+00,7.5129e-04,7.4811e-04,1.0042e+00,5.5605e+00,1.0143e+00,1.0000e+00,9.7597e-01,9.9995e-01 +1.9500e-01,2.8807e+00,5.0569e-02,5.0228e-02,1.0068e+00,6.5023e-04,6.4755e-04,1.0041e+00,5.5518e+00,1.0143e+00,1.0000e+00,9.7601e-01,9.9994e-01 +1.9600e-01,2.8466e+00,5.0045e-02,4.9709e-02,1.0068e+00,5.5403e-04,5.5182e-04,1.0040e+00,5.5432e+00,1.0143e+00,1.0000e+00,9.7604e-01,9.9993e-01 +1.9700e-01,2.8122e+00,4.9516e-02,4.9185e-02,1.0067e+00,4.6300e-04,4.6121e-04,1.0039e+00,5.5348e+00,1.0143e+00,1.0000e+00,9.7608e-01,9.9993e-01 +1.9800e-01,2.7777e+00,4.8982e-02,4.8656e-02,1.0067e+00,3.7740e-04,3.7599e-04,1.0038e+00,5.5263e+00,1.0143e+00,1.0000e+00,9.7611e-01,9.9992e-01 +1.9900e-01,2.7430e+00,4.8442e-02,4.8121e-02,1.0067e+00,2.9744e-04,2.9637e-04,1.0036e+00,5.5180e+00,1.0143e+00,1.0000e+00,9.7615e-01,9.9992e-01 +2.0000e-01,2.7082e+00,4.7897e-02,4.7582e-02,1.0066e+00,2.2316e-04,2.2239e-04,1.0035e+00,5.5097e+00,1.0143e+00,1.0000e+00,9.7619e-01,9.9991e-01 +2.0100e-01,2.6732e+00,4.7348e-02,4.7037e-02,1.0066e+00,1.5417e-04,1.5367e-04,1.0033e+00,5.5015e+00,1.0142e+00,1.0000e+00,9.7622e-01,9.9990e-01 +2.0200e-01,2.6381e+00,4.6793e-02,4.6488e-02,1.0066e+00,8.8228e-05,8.7957e-05,1.0031e+00,5.4934e+00,1.0142e+00,1.0000e+00,9.7626e-01,9.9990e-01 +2.0300e-01,2.6028e+00,4.6234e-02,4.5934e-02,1.0065e+00,0.0000e+00,0.0000e+00,1.0028e+00,5.4853e+00,1.0142e+00,1.0000e+00,9.7629e-01,9.9989e-01 +2.0400e-01,2.5674e+00,4.5670e-02,4.5375e-02,1.0065e+00,0.0000e+00,0.0000e+00,1.0026e+00,5.4773e+00,1.0142e+00,1.0000e+00,9.7633e-01,9.9989e-01 +2.0500e-01,2.5318e+00,4.5102e-02,4.4812e-02,1.0065e+00,0.0000e+00,0.0000e+00,1.0022e+00,5.4694e+00,1.0142e+00,1.0000e+00,9.7636e-01,9.9988e-01 +2.0600e-01,2.4961e+00,4.4530e-02,4.4244e-02,1.0064e+00,0.0000e+00,0.0000e+00,1.0017e+00,5.4615e+00,1.0142e+00,1.0000e+00,9.7640e-01,9.9988e-01 +2.0700e-01,2.4603e+00,4.3953e-02,4.3672e-02,1.0064e+00,0.0000e+00,0.0000e+00,1.0009e+00,5.4537e+00,1.0142e+00,1.0000e+00,9.7644e-01,9.9987e-01 +2.0800e-01,2.4244e+00,4.3372e-02,4.3097e-02,1.0064e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4460e+00,1.0142e+00,1.0000e+00,9.7647e-01,9.9987e-01 +2.0900e-01,2.3884e+00,4.2787e-02,4.2517e-02,1.0064e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4383e+00,1.0142e+00,1.0000e+00,9.7651e-01,9.9986e-01 +2.1000e-01,2.3523e+00,4.2198e-02,4.1933e-02,1.0063e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4307e+00,1.0142e+00,1.0000e+00,9.7654e-01,9.9986e-01 +2.1100e-01,2.3161e+00,4.1606e-02,4.1346e-02,1.0063e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4232e+00,1.0142e+00,1.0000e+00,9.7658e-01,9.9985e-01 +2.1200e-01,2.2798e+00,4.1009e-02,4.0754e-02,1.0063e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4157e+00,1.0142e+00,1.0000e+00,9.7661e-01,9.9985e-01 +2.1300e-01,2.2435e+00,4.0410e-02,4.0160e-02,1.0062e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4083e+00,1.0142e+00,1.0000e+00,9.7665e-01,9.9985e-01 +2.1400e-01,2.2071e+00,3.9807e-02,3.9562e-02,1.0062e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.4009e+00,1.0142e+00,1.0000e+00,9.7668e-01,9.9984e-01 +2.1500e-01,2.1706e+00,3.9201e-02,3.8961e-02,1.0062e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3936e+00,1.0142e+00,1.0000e+00,9.7671e-01,9.9984e-01 +2.1600e-01,2.1341e+00,3.8592e-02,3.8357e-02,1.0061e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3864e+00,1.0141e+00,1.0000e+00,9.7675e-01,9.9984e-01 +2.1700e-01,2.0975e+00,3.7980e-02,3.7751e-02,1.0061e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3792e+00,1.0141e+00,1.0000e+00,9.7678e-01,9.9983e-01 +2.1800e-01,2.0608e+00,3.7366e-02,3.7141e-02,1.0061e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3720e+00,1.0141e+00,1.0000e+00,9.7682e-01,9.9983e-01 +2.1900e-01,2.0242e+00,3.6749e-02,3.6529e-02,1.0060e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3650e+00,1.0141e+00,1.0000e+00,9.7685e-01,9.9982e-01 +2.2000e-01,1.9875e+00,3.6129e-02,3.5915e-02,1.0060e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3579e+00,1.0141e+00,1.0000e+00,9.7689e-01,9.9982e-01 +2.2100e-01,1.9508e+00,3.5508e-02,3.5298e-02,1.0059e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3510e+00,1.0141e+00,1.0000e+00,9.7692e-01,9.9982e-01 +2.2200e-01,1.9142e+00,3.4884e-02,3.4679e-02,1.0059e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3441e+00,1.0141e+00,1.0000e+00,9.7695e-01,9.9982e-01 +2.2300e-01,1.8775e+00,3.4259e-02,3.4059e-02,1.0059e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3372e+00,1.0141e+00,1.0000e+00,9.7699e-01,9.9981e-01 +2.2400e-01,1.8408e+00,3.3632e-02,3.3437e-02,1.0058e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3304e+00,1.0141e+00,1.0000e+00,9.7702e-01,9.9981e-01 +2.2500e-01,1.8042e+00,3.3003e-02,3.2813e-02,1.0058e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3236e+00,1.0141e+00,1.0000e+00,9.7706e-01,9.9981e-01 +2.2600e-01,1.7675e+00,3.2373e-02,3.2188e-02,1.0058e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3169e+00,1.0141e+00,1.0000e+00,9.7709e-01,9.9980e-01 +2.2700e-01,1.7310e+00,3.1742e-02,3.1562e-02,1.0057e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3103e+00,1.0141e+00,1.0000e+00,9.7712e-01,9.9980e-01 +2.2800e-01,1.6944e+00,3.1110e-02,3.0934e-02,1.0057e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.3037e+00,1.0141e+00,1.0000e+00,9.7716e-01,9.9980e-01 +2.2900e-01,1.6580e+00,3.0478e-02,3.0306e-02,1.0056e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2971e+00,1.0141e+00,1.0000e+00,9.7719e-01,9.9980e-01 +2.3000e-01,1.6216e+00,2.9844e-02,2.9678e-02,1.0056e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2906e+00,1.0141e+00,1.0000e+00,9.7722e-01,9.9979e-01 +2.3100e-01,1.5852e+00,2.9211e-02,2.9049e-02,1.0056e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2842e+00,1.0140e+00,1.0000e+00,9.7726e-01,9.9979e-01 +2.3200e-01,1.5490e+00,2.8577e-02,2.8420e-02,1.0055e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2778e+00,1.0140e+00,1.0000e+00,9.7729e-01,9.9979e-01 +2.3300e-01,1.5129e+00,2.7943e-02,2.7791e-02,1.0055e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2714e+00,1.0140e+00,1.0000e+00,9.7732e-01,9.9979e-01 +2.3400e-01,1.4768e+00,2.7310e-02,2.7162e-02,1.0055e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2651e+00,1.0140e+00,1.0000e+00,9.7735e-01,9.9979e-01 +2.3500e-01,1.4409e+00,2.6677e-02,2.6533e-02,1.0054e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2588e+00,1.0140e+00,1.0000e+00,9.7739e-01,9.9978e-01 +2.3600e-01,1.4052e+00,2.6045e-02,2.5906e-02,1.0054e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2526e+00,1.0140e+00,1.0000e+00,9.7742e-01,9.9978e-01 +2.3700e-01,1.3695e+00,2.5413e-02,2.5279e-02,1.0053e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2465e+00,1.0140e+00,1.0000e+00,9.7745e-01,9.9978e-01 +2.3800e-01,1.3340e+00,2.4783e-02,2.4653e-02,1.0053e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2403e+00,1.0140e+00,1.0000e+00,9.7749e-01,9.9978e-01 +2.3900e-01,1.2987e+00,2.4154e-02,2.4028e-02,1.0052e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2342e+00,1.0140e+00,1.0000e+00,9.7752e-01,9.9978e-01 +2.4000e-01,1.2635e+00,2.3527e-02,2.3405e-02,1.0052e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2282e+00,1.0140e+00,1.0000e+00,9.7755e-01,9.9978e-01 +2.4100e-01,1.2286e+00,2.2902e-02,2.2784e-02,1.0052e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2222e+00,1.0140e+00,1.0000e+00,9.7758e-01,9.9977e-01 +2.4200e-01,1.1938e+00,2.2278e-02,2.2165e-02,1.0051e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2162e+00,1.0140e+00,1.0000e+00,9.7762e-01,9.9977e-01 +2.4300e-01,1.1592e+00,2.1657e-02,2.1548e-02,1.0051e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2103e+00,1.0140e+00,1.0000e+00,9.7765e-01,9.9977e-01 +2.4400e-01,1.1249e+00,2.1039e-02,2.0934e-02,1.0050e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.2045e+00,1.0140e+00,1.0000e+00,9.7768e-01,9.9977e-01 +2.4500e-01,1.0908e+00,2.0423e-02,2.0322e-02,1.0050e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1986e+00,1.0139e+00,1.0000e+00,9.7771e-01,9.9977e-01 +2.4600e-01,1.0569e+00,1.9811e-02,1.9713e-02,1.0049e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1928e+00,1.0139e+00,1.0000e+00,9.7774e-01,9.9977e-01 +2.4700e-01,1.0233e+00,1.9201e-02,1.9108e-02,1.0049e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1871e+00,1.0139e+00,1.0000e+00,9.7778e-01,9.9977e-01 +2.4800e-01,9.8995e-01,1.8596e-02,1.8506e-02,1.0048e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1814e+00,1.0139e+00,1.0000e+00,9.7781e-01,9.9977e-01 +2.4900e-01,9.5688e-01,1.7994e-02,1.7908e-02,1.0048e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1757e+00,1.0139e+00,1.0000e+00,9.7784e-01,9.9976e-01 +2.5000e-01,9.2411e-01,1.7396e-02,1.7314e-02,1.0047e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1701e+00,1.0139e+00,1.0000e+00,9.7787e-01,9.9976e-01 +2.5100e-01,8.9165e-01,1.6803e-02,1.6724e-02,1.0047e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1645e+00,1.0139e+00,1.0000e+00,9.7790e-01,9.9976e-01 +2.5200e-01,8.5950e-01,1.6214e-02,1.6139e-02,1.0046e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1590e+00,1.0139e+00,1.0000e+00,9.7793e-01,9.9976e-01 +2.5300e-01,8.2769e-01,1.5630e-02,1.5559e-02,1.0046e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1534e+00,1.0139e+00,1.0000e+00,9.7796e-01,9.9976e-01 +2.5400e-01,7.9623e-01,1.5052e-02,1.4984e-02,1.0045e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1480e+00,1.0139e+00,1.0000e+00,9.7800e-01,9.9976e-01 +2.5500e-01,7.6512e-01,1.4479e-02,1.4414e-02,1.0045e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1425e+00,1.0139e+00,1.0000e+00,9.7803e-01,9.9976e-01 +2.5600e-01,7.3439e-01,1.3911e-02,1.3850e-02,1.0044e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1371e+00,1.0139e+00,1.0000e+00,9.7806e-01,9.9976e-01 +2.5700e-01,7.0405e-01,1.3350e-02,1.3292e-02,1.0044e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1318e+00,1.0139e+00,1.0000e+00,9.7809e-01,9.9976e-01 +2.5800e-01,6.7411e-01,1.2795e-02,1.2741e-02,1.0043e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1264e+00,1.0139e+00,1.0000e+00,9.7812e-01,9.9976e-01 +2.5900e-01,6.4459e-01,1.2247e-02,1.2196e-02,1.0042e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1211e+00,1.0139e+00,1.0000e+00,9.7815e-01,9.9976e-01 +2.6000e-01,6.1550e-01,1.1706e-02,1.1658e-02,1.0042e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1159e+00,1.0138e+00,1.0000e+00,9.7818e-01,9.9976e-01 +2.6100e-01,5.8686e-01,1.1173e-02,1.1127e-02,1.0041e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1106e+00,1.0138e+00,1.0000e+00,9.7821e-01,9.9976e-01 +2.6200e-01,5.5867e-01,1.0647e-02,1.0604e-02,1.0041e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1054e+00,1.0138e+00,1.0000e+00,9.7824e-01,9.9976e-01 +2.6300e-01,5.3096e-01,1.0129e-02,1.0088e-02,1.0040e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.1003e+00,1.0138e+00,1.0000e+00,9.7827e-01,9.9976e-01 +2.6400e-01,5.0375e-01,9.6190e-03,9.5813e-03,1.0039e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0952e+00,1.0138e+00,1.0000e+00,9.7830e-01,9.9975e-01 +2.6500e-01,4.7704e-01,9.1179e-03,9.0828e-03,1.0039e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0901e+00,1.0138e+00,1.0000e+00,9.7833e-01,9.9975e-01 +2.6600e-01,4.5085e-01,8.6257e-03,8.5931e-03,1.0038e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0850e+00,1.0138e+00,1.0000e+00,9.7836e-01,9.9975e-01 +2.6700e-01,4.2520e-01,8.1429e-03,8.1126e-03,1.0037e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0800e+00,1.0138e+00,1.0000e+00,9.7839e-01,9.9975e-01 +2.6800e-01,4.0011e-01,7.6697e-03,7.6417e-03,1.0037e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0750e+00,1.0138e+00,1.0000e+00,9.7842e-01,9.9975e-01 +2.6900e-01,3.7559e-01,7.2065e-03,7.1807e-03,1.0036e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0701e+00,1.0138e+00,1.0000e+00,9.7846e-01,9.9975e-01 +2.7000e-01,3.5165e-01,6.7536e-03,6.7300e-03,1.0035e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0651e+00,1.0138e+00,1.0000e+00,9.7848e-01,9.9975e-01 +2.7100e-01,3.2832e-01,6.3114e-03,6.2898e-03,1.0034e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0602e+00,1.0138e+00,1.0000e+00,9.7851e-01,9.9975e-01 +2.7200e-01,3.0560e-01,5.8803e-03,5.8606e-03,1.0034e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0554e+00,1.0138e+00,1.0000e+00,9.7854e-01,9.9975e-01 +2.7300e-01,2.8353e-01,5.4606e-03,5.4428e-03,1.0033e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0505e+00,1.0138e+00,1.0000e+00,9.7857e-01,9.9975e-01 +2.7400e-01,2.6211e-01,5.0527e-03,5.0367e-03,1.0032e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0457e+00,1.0137e+00,1.0000e+00,9.7860e-01,9.9975e-01 +2.7500e-01,2.4135e-01,4.6570e-03,4.6426e-03,1.0031e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0410e+00,1.0137e+00,1.0000e+00,9.7863e-01,9.9975e-01 +2.7600e-01,2.2129e-01,4.2738e-03,4.2610e-03,1.0030e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0362e+00,1.0137e+00,1.0000e+00,9.7866e-01,9.9975e-01 +2.7700e-01,2.0193e-01,3.9035e-03,3.8922e-03,1.0029e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0315e+00,1.0137e+00,1.0000e+00,9.7869e-01,9.9975e-01 +2.7800e-01,1.8330e-01,3.5465e-03,3.5366e-03,1.0028e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0268e+00,1.0137e+00,1.0000e+00,9.7872e-01,9.9975e-01 +2.7900e-01,1.6541e-01,3.2033e-03,3.1946e-03,1.0027e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0222e+00,1.0137e+00,1.0000e+00,9.7875e-01,9.9975e-01 +2.8000e-01,1.4828e-01,2.8741e-03,2.8666e-03,1.0026e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0175e+00,1.0137e+00,1.0000e+00,9.7878e-01,9.9975e-01 +2.8100e-01,1.3192e-01,2.5593e-03,2.5530e-03,1.0025e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0129e+00,1.0137e+00,1.0000e+00,9.7881e-01,9.9975e-01 +2.8200e-01,1.1636e-01,2.2595e-03,2.2541e-03,1.0024e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0084e+00,1.0137e+00,1.0000e+00,9.7884e-01,9.9975e-01 +2.8300e-01,1.0161e-01,1.9748e-03,1.9704e-03,1.0022e+00,0.0000e+00,0.0000e+00,1.0000e+00,5.0038e+00,1.0137e+00,1.0000e+00,9.7887e-01,9.9975e-01 +2.8400e-01,8.7693e-02,1.7058e-03,1.7022e-03,1.0021e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9993e+00,1.0137e+00,1.0000e+00,9.7890e-01,9.9975e-01 +2.8500e-01,7.4619e-02,1.4528e-03,1.4499e-03,1.0020e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9948e+00,1.0137e+00,1.0000e+00,9.7892e-01,9.9975e-01 +2.8600e-01,6.2405e-02,1.2160e-03,1.2138e-03,1.0018e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9904e+00,1.0137e+00,1.0000e+00,9.7895e-01,9.9975e-01 +2.8700e-01,5.1064e-02,9.9590e-04,9.9425e-04,1.0017e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9859e+00,1.0137e+00,1.0000e+00,9.7898e-01,9.9975e-01 +2.8800e-01,4.0602e-02,7.9253e-04,7.9136e-04,1.0015e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9815e+00,1.0136e+00,1.0000e+00,9.7901e-01,9.9975e-01 +2.8900e-01,3.1012e-02,6.0587e-04,6.0509e-04,1.0013e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9771e+00,1.0136e+00,1.0000e+00,9.7904e-01,9.9975e-01 +2.9000e-01,2.2259e-02,4.3523e-04,4.3477e-04,1.0011e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9728e+00,1.0136e+00,1.0000e+00,9.7907e-01,9.9975e-01 +2.9100e-01,1.4207e-02,2.7804e-04,2.7781e-04,1.0008e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9685e+00,1.0136e+00,1.0000e+00,9.7910e-01,9.9975e-01 +2.9200e-01,6.1264e-03,1.1999e-04,1.1993e-04,1.0005e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9642e+00,1.0136e+00,1.0000e+00,9.7912e-01,9.9975e-01 +2.9300e-01,0.0000e+00,0.0000e+00,0.0000e+00,1.0002e+00,0.0000e+00,0.0000e+00,1.0000e+00,4.9599e+00,1.0136e+00,1.0000e+00,9.7915e-01,9.9976e-01 +2.9400e-01,0.0000e+00,0.0000e+00,0.0000e+00,9.9978e-01,0.0000e+00,0.0000e+00,1.0000e+00,4.9556e+00,1.0136e+00,1.0000e+00,9.7918e-01,9.9976e-01 +2.9500e-01,0.0000e+00,0.0000e+00,0.0000e+00,9.9923e-01,0.0000e+00,0.0000e+00,1.0000e+00,4.9514e+00,1.0136e+00,1.0000e+00,9.7921e-01,9.9976e-01 +2.9600e-01,0.0000e+00,0.0000e+00,0.0000e+00,9.9841e-01,0.0000e+00,0.0000e+00,1.0000e+00,4.9472e+00,1.0136e+00,1.0000e+00,9.7924e-01,9.9976e-01 +2.9700e-01,0.0000e+00,0.0000e+00,0.0000e+00,9.9673e-01,0.0000e+00,0.0000e+00,1.0000e+00,4.9430e+00,1.0136e+00,1.0000e+00,9.7926e-01,9.9976e-01 diff --git a/tests/physics/test_neutrino_mass_effects.py b/tests/physics/test_neutrino_mass_effects.py new file mode 100644 index 0000000..14a1f3b --- /dev/null +++ b/tests/physics/test_neutrino_mass_effects.py @@ -0,0 +1,127 @@ +""" +Physics tests for neutrino mass effects on beta spectrum. + +Following TDD principles: write failing tests first, then ensure implementation fixes them. +""" + +import numpy as np + +from beta_spectrum.components.phase_space import PhaseSpace +from beta_spectrum.spectrum import SpectrumConfig, BetaSpectrum + + +class TestNeutrinoMassPhaseSpaceEffects: + """Test neutrino mass effects on phase space calculation.""" + + def test_phase_space_with_zero_neutrino_mass(self): + """Test that phase space calculation works with zero neutrino mass.""" + ps = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.0) + W_test = np.array([2.0, 3.0, 4.0]) + result = ps(W_test) + + # Should not produce NaN values + assert not np.any(np.isnan(result)), ( + "Phase space with m_nu=0 should not produce NaN values" + ) + assert np.all(result > 0), ( + "Phase space with m_nu=0 should be positive in physical range" + ) + + def test_phase_space_with_nonzero_neutrino_mass(self): + """Test that phase space calculation works with non-zero neutrino mass.""" + ps = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.1) + W_test = np.array([2.0, 3.0, 4.0]) # W0-W = 3.0, 2.0, 1.0 > m_nu=0.1 + result = ps(W_test) + + # Should not produce NaN values for energies above threshold + assert not np.any(np.isnan(result)), ( + "Phase space with m_nu>0 should not produce NaN values above threshold" + ) + assert np.all(result > 0), ( + "Phase space with m_nu>0 should be positive above threshold" + ) + + def test_phase_space_with_neutrino_mass_near_endpoint(self): + """Test that phase space calculation handles energies near endpoint with neutrino mass.""" + ps = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.1) + + # Test energies where W0-W > m_nu (should be finite) + W_above_threshold = np.array([4.8]) # W0-W = 0.2 > m_nu=0.1 + result_above = ps(W_above_threshold) + assert not np.any(np.isnan(result_above)), ( + "Should produce finite value above threshold" + ) + + # Test energies where W0-W < m_nu (should be handled properly) + # With our fix, this should return 0 instead of NaN + W_below_threshold = np.array([4.95]) # W0-W = 0.05 < m_nu=0.1 + result_below = ps(W_below_threshold) + # After our fix, NaN values should be replaced with 0 + assert np.all(result_below == 0.0) or np.all(np.isnan(result_below)), ( + "Should handle below-threshold energies properly (either 0 or NaN)" + ) + + def test_spectrum_config_with_neutrino_mass(self): + """Test that SpectrumConfig accepts and stores neutrino mass parameter.""" + config = SpectrumConfig( + Z_parent=43, + Z_daughter=44, + A_number=99, + endpoint_MeV=0.2111, + transition_type="F2", + e_step_MeV=0.001, + neutrino_mass_MeV=0.01, + ) + assert config.neutrino_mass_MeV == 0.01, ( + "SpectrumConfig should store neutrino mass parameter" + ) + + def test_beta_spectrum_with_neutrino_mass(self): + """Test that BetaSpectrum works with neutrino mass in configuration.""" + config = SpectrumConfig( + Z_parent=43, + Z_daughter=44, + A_number=99, + endpoint_MeV=0.2111, + transition_type="F2", + e_step_MeV=0.001, + neutrino_mass_MeV=0.001, # Small neutrino mass + ) + + # This should not raise an error + spectrum = BetaSpectrum.from_config(config) + W, kinetic_MeV = spectrum.get_energy_grid(config) + + # Calculate spectrum + result = spectrum(W) + + # Should not produce NaN values + assert not np.any(np.isnan(result)), ( + "BetaSpectrum with neutrino mass should not produce NaN values" + ) + assert np.all(result >= 0), ( + "BetaSpectrum with neutrino mass should be non-negative" + ) + + +class TestNeutrinoMassCLIIntegration: + """Test CLI integration with neutrino mass parameter.""" + + def test_cli_neutrino_mass_parameter_parsing(self): + """Test that CLI properly parses neutrino mass parameter.""" + from beta_spectrum.cli import _build_parser + + parser = _build_parser() + args = parser.parse_args(["--nuclide", "Tc99", "--neutrino-mass", "0.01"]) + + assert hasattr(args, "neutrino_mass"), "CLI should have neutrino_mass argument" + assert args.neutrino_mass == 0.01, "CLI should parse neutrino_mass correctly" + + def test_cli_neutrino_mass_default_value(self): + """Test that CLI neutrino mass parameter defaults to 0.0.""" + from beta_spectrum.cli import _build_parser + + parser = _build_parser() + args = parser.parse_args(["--nuclide", "Tc99"]) + + assert args.neutrino_mass == 0.0, "CLI neutrino_mass default should be 0.0" diff --git a/tests/physics/test_phase_space.py b/tests/physics/test_phase_space.py index 10cf5e9..1adf3a2 100644 --- a/tests/physics/test_phase_space.py +++ b/tests/physics/test_phase_space.py @@ -74,15 +74,15 @@ class TestPhaseSpaceNeutrinoMass: val_massive = ps_massive(W_near_endpoint)[0] assert not np.isnan(val_massive), "Should produce finite value" - assert ( - val_massive < val_zero - ), "Massive neutrino must suppress spectrum near endpoint" + assert val_massive < val_zero, ( + "Massive neutrino must suppress spectrum near endpoint" + ) @pytest.mark.physics def test_massive_neutrino_below_threshold(self): - """With m_nu > 0, below the effective threshold (W₀−W < m_nu), sqrt gives NaN. + """With m_nu > 0, below the effective threshold (W₀−W < m_nu), the spectrum should be 0. - This is expected physical behavior — the spectrum simply doesn't exist there. + This is the correct physical behavior — the spectrum doesn't exist below the kinematic threshold. """ ps = PhaseSpace(W0=5.0, m_nu=0.1) @@ -94,6 +94,4 @@ class TestPhaseSpaceNeutrinoMass: W_deep_below = np.array([4.95]) # W₀ − W = 0.05 < m_nu=0.1 — below threshold result_below = ps(W_deep_below) - assert np.isnan( - result_below[0] - ), "Below kinematic threshold, spectrum should be NaN" + assert result_below[0] == 0.0, "Below kinematic threshold, spectrum should be 0" diff --git a/tests/quality/test_neutrino_mass_cli.py b/tests/quality/test_neutrino_mass_cli.py new file mode 100644 index 0000000..6f102d3 --- /dev/null +++ b/tests/quality/test_neutrino_mass_cli.py @@ -0,0 +1,85 @@ +""" +Tests for neutrino mass functionality in CLI and configuration. +""" + +import numpy as np +from beta_spectrum.spectrum import SpectrumConfig +from beta_spectrum.components.phase_space import PhaseSpace + + +class TestNeutrinoMassConfig: + """Test neutrino mass parameter in SpectrumConfig.""" + + def test_neutrino_mass_default_value(self): + """Test that neutrino_mass_MeV defaults to 0.0 in SpectrumConfig.""" + config = SpectrumConfig( + Z_parent=43, + Z_daughter=44, + A_number=99, + endpoint_MeV=0.2111, + transition_type="F2", + e_step_MeV=0.001, + ) + assert config.neutrino_mass_MeV == 0.0, "Default neutrino mass should be 0.0" + + def test_neutrino_mass_custom_value(self): + """Test that neutrino_mass_MeV can be set to a custom value.""" + config = SpectrumConfig( + Z_parent=43, + Z_daughter=44, + A_number=99, + endpoint_MeV=0.2111, + transition_type="F2", + e_step_MeV=0.001, + neutrino_mass_MeV=0.01, + ) + assert config.neutrino_mass_MeV == 0.01, "Neutrino mass should be 0.01 MeV" + + +class TestNeutrinoMassPhaseSpace: + """Test neutrino mass parameter in PhaseSpace component.""" + + def test_phase_space_neutrino_mass_parameter(self): + """Test that PhaseSpace accepts and uses neutrino mass parameter.""" + ps = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.01) + assert ps.m_nu == 0.01, "PhaseSpace should store neutrino mass parameter" + + def test_phase_space_neutrino_mass_effect(self): + """Test that neutrino mass affects the phase space calculation.""" + ps_zero = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.0) + ps_massive = PhaseSpace(W0=5.0, transition_type="A", m_nu=0.1) + + # Test near endpoint where neutrino mass effect should be visible + W_near_endpoint = np.array([4.8]) # W₀−W = 0.2 > m_nu=0.1 + + val_zero = ps_zero(W_near_endpoint)[0] + val_massive = ps_massive(W_near_endpoint)[0] + + assert not np.isnan(val_massive), "Should produce finite value" + assert val_massive < val_zero, ( + "Massive neutrino must suppress spectrum near endpoint" + ) + + +class TestCLIArguments: + """Test CLI argument parsing for neutrino mass.""" + + def test_neutrino_mass_cli_argument_exists(self): + """Test that --neutrino-mass argument is available in CLI parser.""" + from beta_spectrum.cli import _build_parser + + parser = _build_parser() + + # Check that the argument is in the parser + args = parser.parse_args(["--nuclide", "Tc99", "--neutrino-mass", "0.01"]) + assert hasattr(args, "neutrino_mass"), "CLI should have neutrino_mass argument" + assert args.neutrino_mass == 0.01, "CLI neutrino_mass should be 0.01" + + def test_neutrino_mass_cli_default_value(self): + """Test that --neutrino-mass defaults to 0.0 in CLI.""" + from beta_spectrum.cli import _build_parser + + parser = _build_parser() + + args = parser.parse_args(["--nuclide", "Tc99"]) + assert args.neutrino_mass == 0.0, "CLI neutrino_mass default should be 0.0"