Add doc summary: 2050_Song 99TcMeasurement
Measurement of the 99Tc beta-spectrum and implications for g_v
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Here is the updated summary with the inline citations integrated as requested.
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# Summary: The measurement of the $^{99}$Tc $\beta$-decay spectrum and its implications for the effective value of weak axial coupling
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## 1. Introduction and Motivation
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The study of $\beta$-spectral shapes is a critical method for determining the effective value of the weak axial coupling constant, $g_A$. In forbidden non-unique $\beta^-$ transitions [5], the spectral shape is directly sensitive to the ratio $g_A/g_V$.
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* **Standard Hypothesis:** The Conserved Vector Current (CVC) hypothesis sets the vector coupling constant $g_V = 1.0$ [8].
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* **Quenching:** The Partially Conserved Axial-Vector Current (PCAC) hypothesis suggests that in finite nuclei, $g_A$ is "quenched" (reduced) to a value of approximately $1.0$, compared to the bare-nucleon value of $1.27$ resulting from the experimental analysis of the decay of an isolated neutron [8]. The quenching of the effective value of $g_A$ has been studied extensively in [2, 8, 9], and in particular its effects on the neutrinoless double beta decay have been addressed in [1, 3].
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* **Contradiction:** While previous studies on the fourth-forbidden non-unique $\beta^-$-decays of $^{113}$Cd [10–13] and $^{115}$In [14–16] supported quenching ($g_A/g_V \approx 1.0$), a recent measurement by Paulsen et al. [17] on the second-forbidden non-unique transition in $^{99}$Tc suggested an *enhanced* ratio of $g_A/g_V = 1.526(92)$.
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This paper aims to resolve this contradiction by performing a high-precision measurement of the $^{99}$Tc $\beta$-decay spectrum using Metallic Magnetic Calorimeters (MMCs) and comparing it with state-of-the-art nuclear theory.
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## 2. Theoretical Framework
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The half-life ($t_{1/2}$) and shape of the $\beta$-spectrum are governed by the integrated shape function $\tilde{C}$.
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**Equation 1:**
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$$ t_{1/2} = \frac{\kappa}{\tilde{C}} $$
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*(where $\kappa$ is a constant [18, 19] and $\tilde{C}$ incorporates phase-space factors and NMEs arising in the next-to-leading-order expansion, as discussed in detail in Refs. [6, 7].)*
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The shape function $C(w_e)$ depends on the weak axial and vector couplings:
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**Equation 2:**
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$$ C(w_e) = g_V^2 C_V(w_e) + g_A^2 C_A(w_e) + g_V g_A C_{VA}(w_e) $$
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*(where $w_e$ is the total energy of the emitted electron. The transition matrix element [6].)*
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**Methodology (ESSM):**
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The authors employ the **Enhanced Spectrum-Shape Method (ESSM)** [19]. This approach simultaneously fits the measured $\beta$-spectrum and the experimental partial half-life to extract $g_A^{eff}$. This allows for the adjustment of the small relativistic vector nuclear matrix element (sNME) [5] to reproduce the half-life, rather than fixing it to the "ideal" CVC value.
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The sNME gathers contributions outside the valence major shell, making its calculation particularly hard for nuclear-theory frameworks used, e.g., in the present work and also in the work of Paulsen et al. [17]. Despite its smallness, the sNME can influence the $\beta$-spectral shapes and half-lives quite strongly [11, 12, 19, 21].
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A realistic value of the sNME can be estimated from its CVC value [5], first used for spectral-shape calculations in Ref. [18] and later by Paulsen et al. [17]. To circumvent the use of the CVC value, the experimental works [11, 13, 15] and theoretical works [12, 16] used sNME as a fitting parameter, together with $g_A^{eff}$.
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In this Letter, we present new theoretical calculations of the $\beta$ spectrum of $^{99}$Tc... In references [21, 23] the related two sets of theoretical $\beta$-spectra of $^{99}$Tc were introduced for two NSM Hamiltonians, namely $jj45pnb$ [24] and $glekpn$ [25].
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## 3. Experimental Setup
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The experiment was conducted at Lawrence Livermore National Laboratory (LLNL).
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* **Detector:** A Metallic Magnetic Calorimeter (MMC) fabricated at KRISS [26, 27].
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* **Sensor:** Paramagnetic ions (420 ppm of Ag:$^{168}$Er) embedded in a metallic host.
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* **Absorber:** A 4$\pi$ solid-angle gold absorber containing the $^{99}$Tc source.
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* **Temperature:** Operated below 100 mK (specifically 15 mK during data taking).
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* **Source Preparation:** $^{99}$Tc solution was dropped onto a gold foil, dried, folded, and rolled (kneading method [28, 29]) to ensure uniform distribution. The final sample mass was 1.75 mg.
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* **Readout:** A Magnicon SQUID (Model X1) and an additional array SQUID (Model X16).
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* **Data Acquisition:** Continuous mode, 200 kS/s sampling rate using a Stanford Research Systems SRS 560 amplifier [30].
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* **Threshold:** A low-energy threshold of 10 keV was applied.
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**Figure 1: Detector Setup**
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*(Description of visual content)*
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The image shows the experimental apparatus. Key components labeled include:
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* **Magnicon SQUID X1:** The signal readout device.
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* **KRISS MMC:** The calorimeter sensor.
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* **$^{99}$Tc source:** The radioactive sample placed on the sensor.
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* **Support washer & Kapton tape:** Used for thermal insulation and alignment.
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## 4. Results and Analysis
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### 4.1 Data Selection
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* **Background Dataset:** 130,011 events recorded.
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* **Signal Dataset:** After Pulse Shape Discrimination (PSD) cuts, **77,487 events** corresponding to $^{99}$Tc remained for analysis.
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### 4.2 Energy Calibration
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Calibration was performed using an external $^{133}$Ba source and internal X-rays from a $^{148}$Gd source (used for internal calibration checks).
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**Table I: Measured X-ray and gamma-ray lines**
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| Origin | Radiation type | Tabulated energy [keV] | Measured energy [keV] | FWHM resolution [keV] | $\Delta$E [keV] |
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| :--- | :--- | :--- | :--- | :--- | :--- |
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| **Gd fluorescence** | X-ray K$_{\alpha2}$ | 42.3097(13) | 42.30(10) | 0.9(4) | 0.01(10) |
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| | X-ray K$_{\alpha1}$ | 42.9968(12) | 42.9427(31) | 0.68(10) | 0.05(33) |
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| | X-ray K$_{\beta3}$ | 48.5505(21) | 48.29(5)* | 1.05(22) | 0.41(5) |
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| | X-ray K$_{\beta1}$ | 48.6957(21) | | | |
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| **Pt fluorescence** | X-ray K$_{\alpha2}$ | 65.1226(21) | 65.005(22) | 0.9(6) | 0.118(22) |
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| | X-ray K$_{\alpha1}$ | 66.8311(20) | 66.78(5) | 0.65(15) | 0.06(5) |
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| **$^{133}$Ba calibration** | $\gamma$ | 80.9979(11) | 80.881(24) | 0.77(8) | 0.117(25) |
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| | $\gamma$ | 276.3989(12) | 276.12(5) | 0.57(13) | 0.28(5) |
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| | $\gamma$ | 302.8508(5) | 302.764(21) | 0.60(5) | 0.087(21) |
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| | $\gamma$ | 356.0129(7) | 355.926(9) | 0.653(21) | 0.087(9) |
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| | $\gamma$ | 383.8485(12) | 383.69(6) | 0.82(14) | 0.16(6) |
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*(Note: Entries labeled "Not distinguished" in the original text refer to peaks with low relative intensities that could not be individually resolved.)*
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### 4.3 Spectral Comparison
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The measured spectrum was compared against theoretical predictions from three nuclear models:
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1. **$jj45pnb$** (NSM Hamiltonian)
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2. **$glekpn$** (NSM Hamiltonian)
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3. **MQPM** (Microscopic Quasiparticle-Phonon Model)
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**Figure 2: $^{99}$Tc spectrum with external $^{133}$Ba calibration source**
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*(Description)*
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A plot showing "Counts / 200 keV" vs "Energy (keV)". It displays the raw data (black line) with distinct peaks corresponding to Ba-133 $\gamma$-lines, Au X-rays, Pt X-rays, and Gd X-rays [34].
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**Figure 3: Spectrum of calibrated $^{99}$Tc compared with theoretical predictions**
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*(Description)*
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A plot showing "Counts / 1 keV" vs "Energy (keV)" (0-300 keV).
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* **Black line:** Experimental Data.
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* **Blue dashed line:** $jj45pnb$ model ($g_A=1.0$).
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* **Green dotted line:** $glekpn$ model ($g_A=1.2$).
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* **Red dash-dot line:** MQPM model ($g_A=1.0$).
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* **Bottom panel:** Residuals ($\Delta/c$) showing the fit quality over the 0-280 keV range.
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### 4.4 Quantitative Results (Table II)
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The table below summarizes the fit results for the effective axial coupling $g_A^{eff}$ and the goodness of fit ($\chi^2_\nu$ and p-value).
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**Table II: Results for the three considered nuclear models**
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| Model | $g_A^{eff}$ | sNME [fm$^3$] | $\chi^2_\nu$ | p-value |
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| :--- | :--- | :--- | :--- | :--- |
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| **CloseCVC** | | | | |
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| glekpn | 1.1 | 0.0674 | 1.1057 | 0.12 |
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| jj45pnb | 1.0 | 0.0681 | 1.2436 | 0.004 |
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| MQPM | 1.0 | 0.0651 | 1.6437 | <0.0001 |
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| **FarCVC (Best fit)** | | | | |
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| glekpn | 1.2 | -0.0698 | 1.0664 | 0.22 |
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| jj45pnb | 1.0 | -0.0669 | 1.0966 | 0.14 |
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| MQPM | 1.0 | -0.0683 | 1.0951 | 0.14 |
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* **CloseCVC:** Assumes the sNME is fixed to the CVC value. The $jj45pnb$ and MQPM models show poor agreement (low p-values).
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* **FarCVC (Best fit):** Allows sNME to vary. The **$glekpn$ model with $g_A^{eff} = 1.2$** yields the best agreement ($\chi^2_\nu = 1.0664$, p-value = 0.22). The other models with $g_A^{eff} = 1.0$ also show statistically compatible fits (p-values $\approx$ 0.14).
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## 5. Conclusion
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The high-precision measurement of the $^{99}$Tc $\beta$-decay spectrum supports a **quenched axial coupling** ($g_A^{eff} \approx 1.0 - 1.2$).
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* This contradicts the previously reported enhanced value ($g_A/g_V = 1.526$) by Paulsen et al. [17].
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* The discrepancy is attributed to the different treatment of the sNME (small relativistic vector nuclear matrix element) in the theoretical analysis.
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* The results are consistent with quenching patterns observed in heavier nuclei, such as $^{113}$Cd [10–13] and $^{115}$In [14–16].
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