Sample preparation

Naturally pure Xe (99.999%, 2.3–5.2 g) pressurized to 120–250 atm was placed in a 1-mm thick spherical stainless steel container 20 mm in diameter. The samples and monitors (niobium foil: Φ = 20 mm, 0.12-mm thick, 99.99% pure; aluminum foil: Φ = 20 mm, 0.3-mm thick,99.999% pure), and zirconium foil (Φ = 20 mm, 0.2-mm thick, 99.95% pure) were combined as ZrNbAl-Xe-AlNbZr for irradiation. The samples were mounted at different angles (0º, 45º, 90º, 110º, 135º) to the incident beam. The neutron-producing target edge was 5 cm from the sample center. Due to its high threshold energy, the ⁹³Nb(n,2n)^92mNb (threshold energy 8.972 MeV) reaction was used to monitor the corresponding threshold energies of ¹³⁴Xe(n,2n)^133m,gXe reactions (E_th = 8.853 MeV and 8.618 MeV). Low-threshold reactions such as (n,α) and (n,p) were monitored via a ²⁷Al(n,α)²⁴Na reaction (threshold energy = 3.249 MeV). The CS of the ⁹³Nb(n,2n)^92mNb reaction was obtained from the IRDFF-II data library [23]. To minimize the influence of the ¹³²Xe(n,γ)^133m,gXe reaction on the ¹³⁴Xe(n,2n)^133m,gXe reaction, the samples were covered with Cd foil (1-mm thick, 99.95% pure). The details of this approach are presented in our previous studies [17,18].

The neutron generator facility was provided by the China Academy of Engineering Physics (CAEP) for sample irradiation at (4-5)×10¹⁰ n/s yield (for 2 h). The d+³H→n+⁴He+17.6 MeV reaction provided neutrons at 14 MeV with deuteron beam current and energy of 200 μA and 134 keV, respectively. The tritium–titanium target thickness was 2.65 mg/cm². A beam of α-particles at 135° was used to correct the neutron flux.

Neutron energy and detection efficiency

In the experiment, Zr and Nb were used to measure the CS ratios of ⁹³Nb(n,2n)^92mNb and ⁹⁰Zr(n,2n)^89m+gZr, respectively, to determine the neutron energy. At irradiation angles of 135°, 110°, 90°, 45°, and 0°, the energies were 13.5 MeV, 13.8 MeV, 14.1 MeV, 14.4 MeV, and 14.8 MeV, respectively. For verification purposes, the neutron energies were determined using different methods [24]. Considering the sample distance from the target, size, and beam radius, the uncertainty in the neutron energy was 0.2 MeV [25]. The results obtained using the CS ratio method were consistent in terms of the experimental errors. The detector efficiency was calibrated using ¹⁵²Eu, ¹³³Ba, ¹³⁷Cs, and ²²⁶Ra standard sources. The Monte Carlo method was used to adjust the data for the geometrical differences between the calibration sources applied to the HPGe detector and the Xe sphere efficiency determination. The details are provided in our previous studies [17,18].

Radioactivity measurement

Five samples (No.1–5) were allowed to decay after irradiation for 2.9 d, 3.9 d, 5.0 d, 4.0 d, and 5.0 d, respectively. The γ-ray activities of ^133mXe, ^133gXe, and ^92mNb nuclei were measured with a coaxial HPGe detector, with crystal diameter, length, relative efficiencies, and energy resolution (1.332 MeV) of 70.1 mm, 72.3 mm, 68%, and 1.69 keV, respectively. Typical γ-ray spectra obtained after 2.9 d of irradiation are shown in Fig. 2. A γ-spectrum analysis was performed using ORTEC® GammaVision® software to estimate the peak area [26].

Fig. 2Full-screen View

Xe γ-ray spectrum after 2.9 d of decay after irradiation. Data were collected for 23.4 h.

Table 2 displays the natural isotope abundance and product decay characteristics.

CS calculation

The reaction CSs can be expressed as [17, 18] ${\sigma }_x=\frac{{\left[S\epsilon I_{\gamma }\eta \text{KMD}\right]}_{\text{Nb}}}{{\left[S\epsilon I_{\gamma }\eta \text{KMD}\right]}_x}\frac{{\left[\lambda \text{AFC}\right]}_x}{{\left[\lambda \text{AFC}\right]}_{\text{Nb}}}{\sigma }_{\text{Nb}}$ (1) where Nb and x are the monitored and measured values, respectively; ε is the full-energy peak efficiency; I_γ represents the gamma-ray intensity; η is the abundance of the target nuclide; M is the mass of the sample; $\text{D}=e^{-\lambda t_1}-e^{-\lambda \left(t_1+t_2\right)}$ represents the counting collection factor; $S=1-e^{-\lambda T}$ is the growth factor of the product nuclide; T denotes the irradiation time; t₁ is the cooling time; t₂ is the measurement time; A is the atomic weight; C represents the measured full-energy peak area; λ is the decay constant of the residual nucleus; K is the correction factor for decaying nuclei during irradiation time (T), which is divided into small time intervals Δt_i. This factor is calculated as $K=\left[{\displaystyle \sum _{i=1}^n{\varphi }_i}\left(1-e^{-\lambda \text{Δ}{t}_i}\right)e^{-\lambda T_i}\right]/(S\varphi )$, where φ_i is the mean neutron fluence rate of the i^th part, and is regarded as a constant (when n is sufficiently large). T_i is the time from the end of the i^th part to the end of total irradiation; $\varphi ={\displaystyle \sum _{i=1}^n{\varphi }_i\text{Δ}{t}_i}/T$ is the mean neutron fluence rate within the irradiation time. F is the total correction factor, expressed as $F=F_{\Omega }\times F_{\text{s}}\times F_{\text{g}}$ (2) where F_Ω, F_s, and F_g are the correction factors for the solid angle of the neutron flux, self-absorption of the specific γ-ray energy, and sample-counting geometry, respectively. F_s, F_g, and F_Ω were estimated using the characteristics of spherical samples, as described in Sect. 3.1 of Ref. [18]. Table 3 presents the self-absorption correction factors.

To obtain the pure ground-state CSs for ¹³⁴Xe(n,2n)^133gXe reactions, C_x in Equation (1) was set to a peak area of 80.998 keV minus the contribution from ^133mXe via ${}^{\text{133m}}\text{Xe}{\stackrel{\text{IT}\left(100\%\right)}{\to }}^{\text{133g}}\text{Xe}$ (counting C₁). C₁ can be expressed using the decay of an artificial nuclide, expressed as [18] $C_1=\frac{P_{\text{mg}}{\epsilon }_{\text{g}}{I}_{\text{g}}{C}_{\text{m}}{F}_{\text{m}}\left({\lambda }_{\text{g}}^2{S}_{\text{m}}{D}_{\text{m}}-{\lambda }_{\text{m}}^2{S}_{\text{g}}{D}_{\text{g}}\right)}{\left({\lambda }_{\text{g}}-{\lambda }_{\text{m}}\right)S_{\text{m}}{D}_{\text{m}}{I}_{\text{m}}{\epsilon }_{\text{m}}{\lambda }_{\text{g}}{K}_{\text{m}}{F}_{\text{g}}},$ (3) where subscripts g and m are the ground and metastable states, respectively; P_mg is the percentage of metastable-state disintegrations that produce ground-state nuclides; C_m is the energy peak area of the metastable state; $S_{\text{m}}=1-e^{-{\lambda }_{\text{m}}T}$ and $S_{\text{g}}=1-e^{-{\lambda }_{\text{g}}T}$; I_m and I_g are the intensities of γ-rays, where ε_m and ε_g are the full-energy peak efficiencies of the metastable and ground states, respectively; K_m is the neutron-injection fluctuation indicator of the metastable state; F_m and F_g denote the correction factors of the metastable and ground states, respectively; D_m and D_g are coefficients defined as $D_{\text{m}}=e^{-{\lambda }_{\text{m}}{t}_1}-e^{-{\lambda }_{\text{m}}(t_1+t_2)}$ and ${\text{D}}_{\text{g}}=e^{-{\lambda }_{\text{g}}{t}_1}-e^{-{\lambda }_{\text{g}}\left(t_1+t_2\right)}$.

Associated experimental uncertainties

When calculating the main uncertainties of CS determination, we assumed that only the half-life uncertainty contributed to the uncertainty in the timing factor. The following parameters were used for this purpose: C_x,Nb as the γ-ray counting statistics, M_x,Nb as the target masses, I_x,Nb as the relative γ-ray intensities, η_x, as the target isotopic abundance, σ_Nb as the standard CS, ε_x,Nb as the efficiencies, and S_x,Nb, and D_x,Nb as timing factors. The subscripts Nb and x correspond to the monitored and measured reaction-related terms, respectively.

For generative nuclei ^133mXe and ^133gXe, the uncertainty of correction factor F is estimated as 2.0% and 3.0%, respectively. These uncertainties were applied to assess the total uncertainty of the experimental CSs [27-30]. Table 4 and Table 5 highlight the uncertainties in the parameters contributing to the reaction CS values, which were used to extend the covariance matrix between different energy levels. All samples were investigated using the same setup and standard CS reaction. Thus, the detector efficiency and standard CS accuracy were consistent for all neutron energy correlations. After calculation of fractional uncertainties, covariance analysis calculated the correlation coefficients between each energy-related property. The coefficients and corresponding energies are presented in Table 6. The CS covariance matrix and total uncertainty were generated for energy-level pairs (e.g., (σ_xi,σ_xj)) using the data from Tables 4-6 by adding 15 subset matrices in Equation 4 [29]: $\text{Cor}({\sigma }_{xi},{\sigma }_{xj})={\displaystyle \sum _i{\displaystyle \sum _j\text{Δ}{x}_i\times \text{Cor}(\text{Δ}{x}_i,\text{Δ}{x}_j)\times \text{Δ}{x}_j}}.$ (4)

The generated covariance matrix was [5×5]. The uncertainty in the experimental CS value was determined as (Equation 5) [29] $\text{Cor}({\sigma }_{xi},{\sigma }_{xi})={(\text{Δ}{\sigma }_{xi})}^2.$ (5)

From the covariance matrix and total uncertainty between neutron energies, the correlation matrix [5×5] was derived using Eq. 6 [29]: $\text{Cor}({\sigma }_{xi},{\sigma }_{xj})=\frac{\text{Cor}({\sigma }_{xi},{\sigma }_{xj})}{(\text{Δ}{\sigma }_{xi})\cdot (\text{Δ}{\sigma }_{xj})}$ (6)

The experimentally obtained reaction CS values and their correlation matrices and uncertainties are presented in Table 7 and Table 8.

Table 9 shows experimental values for ¹³⁴Xe(n,2n)^133m+gXe reaction CSs and their IRs, and the monitor reaction CSs at different neutron energies.

¹³⁴Xe(n,2n)^133mXe reaction

Two studies [12,13] have reported ¹³⁴Xe(n,2n) experiments for the ^133mXe reaction. However, they provide the CS at only one energy value (14.4 MeV or 14.6 MeV). For a γ-ray with 233.22 keV energy, Sigg and Kuroda [12] used I_γ□ = 14%, and Kondaiah et al. [13] used I_γ□ = 13.6%. However, other studies indicate that data measurement accuracy was higher with I_γ□ = 10.12%. In this study, γ-ray energy of 233.22 keV with an intensity of I_γ□ = 10.12% emitted during the decay of ^133mXe was used to obtain the ¹³⁴Xe(n,2n)^133mXe reaction CS. Thus, the CS values were modified according to the following equation: ${\sigma }_{\text{corrected}}=\frac{I_{\gamma (\text{old)}}}{I_{\gamma (\text{new})}}{\sigma }_{\text{literature}},$ (8) where σ_literature is the CS from the literature; I_γ(old) is the γ-ray intensity at 233.22 keV, and I_γ(new) is 10.12%. The experimental data are strongly correlated with the corrected experimental results [12, 13], as shown in Fig. 3. However, the data are below the TALYS-1.95 excitation curves (ldmodels 1-6) for neutrons with energies of 13.8-14.8 MeV. These findings were first published at 13.5-MeV, 13.8-MeV, and 14.1-MeV neutron energies.

Fig. 3Full-screen View

(Color online) ¹³⁴Xe(n,2n)^133mXe reaction CSs

¹³⁴Xe(n,2n)^133gXe reaction

A thorough literature review yielded only one measurement of the ¹³⁴Xe(n,2n)^133gXe reaction using γ-ray energy of 81 keV (I_γ□ = 35%) [12]. However, many investigations have used I_γ□ = 36.9% for gamma-rays with the same energy. Production of the ^133gXe nucleus resulted in two characteristic gamma-rays with similar energies, 79.61 keV (I_γ□ = 0.44%) and 80.998 keV (I_γ□ = 36.9%). Our detectors could not resolve these energies; the intensity should be equal to their sum (37.34%). Accordingly, Equation (8) was used to modify the CS values from Ref. [12]. During the ^133m,gXe decay, the emitted 80.998-keV γ-ray was used to calculate the ¹³⁴Xe(n,2n)^133gXe reaction CSs after subtracting the contribution of the ¹³⁴Xe(n,2n)^133mXe reaction via the isomeric transition (IT). Almost 50% of the 80.998-keV γ-ray counts were from the isomeric transitions. Figure 4 shows the theoretical assessments of the excitation functions and the experimental results (Ref. [13] does not provide ground-state CS values, which are equal to the total CS value minus the excited-state CS value.). The measured values exceeded those calculated using the TALYS-1.95 software with ldmodels 1-6. However, our experimental and theoretical results show the same dependence on neutron energy. At 14.6 MeV, our measurements agreed with the corrected values obtained by Sigg and Kuroda [12]. At 14.4 MeV, the results in Ref. [13] significantly exceeded our experimental and simulated results (using TALYS-1.95, ldmodels 1-6). For this reaction, the CS values span a wide energy range.

Fig. 4Full-screen View

(Color online) ¹³⁴Xe(n,2n)^133gXe reaction CSs

¹³⁴Xe(n,2n)¹³³Xe reaction

There are two published reports on the ¹³⁴Xe(n,2n)¹³³Xe reaction at a neutron energy of 14 MeV that can be used to validate theoretical calculations and experimental results. The shapes of the excitation curves obtained from the ENDF/B-VIII.0 [35], (JEFF-3.3 [36], CENDL-3 [39]), JENDL-4.0 [37], and ROSFOND [38] databases were almost identical to those obtained by TALYS-1.95 [22] at ~15 MeV (Fig. 5). We conclude that the experimental data agree with the ROSFOND [38] results, TALYS-1.95 with ldmodel 1, and the corrected experimental results of Sigg and Kuroda [12]. At 14.4 MeV, the values [13] were 570 mb higher than ours. The evaluation excitation curve taken from JENDL-4.0 [37] is above the corrected results of Sigg and Kuroda [12], our experimental results, and results from the other databases [35,36,38,39]. Data for the 13.5-14.1 MeV neutron energy range are reported for the first time.

Fig. 5Full-screen View

(Color online) ¹³⁴Xe(n,2n)¹³³Xe reaction CSs

Isomeric CS ratio

The ratios of the isomeric CSs were calculated using the measured ¹³⁴Xe(n,2n)^133m.gXe CSs. The CS ratios (σ_m/σ_g) for the isomeric ^133m,gXe pair created from the ¹³⁴Xe(n,2n) reaction exposed to incident neutrons with energies of 13.5 ± 0.2 MeV, 13.8 ± 0.2 MeV, 14.1 ± 0.2 MeV, 14.4 ± 0.2 MeV, and 14.8 ± 0.2 MeV are 1.07 ± 0.08, 1.10 ± 0.08, 1.16 ± 0.09, 1.20 ± 0.09, and 1.24 ± 0.09, respectively (Table 9). Figure 6 shows the corrected isomeric CS ratios from the literature [12,13] and the excitation curves obtained using TALYS-1.95 calculations for different density-level models. The isomeric CS ratios increased with energy. In the 14-MeV region, the isomeric CS ratio is directly correlated with the neutron energy, indicating that higher excitation energies result in high-spin isomers (11/2^-→3/2⁺) [40-43]. The obtained σ_m/σ_g values were consistent with the modified values from Sigg and Kuroda [12], but slightly lower than the values obtained using ldmodels 1-6 and higher than the results of Ref. [13]. Our data for neutrons with 13.5-14.1 MeV energies are the first available in the literature to the best of our knowledge.

Fig. 6Full-screen View

(Color online) IRs of ¹³⁴Xe(n,2n)^133m,gXe reactions as a function of neutron energy