Measurement of ¹³⁴Xe(n,2n)^133m,gXe reaction cross-sections in 14-MeV region with detailed uncertainty quantification

Measurement of 134 Xe(n,2n) 133m,g Xe reaction cross-sections in 14-MeV region with detailed uncertainty quantification 1 2 corresp first-author * luojh71@163.com * luojh71@163.com luojh71@163.com https://orcid.org/0000-0002-6892-9380 3 4 5 https://orcid.org/0000-0003-4236-0189 2 1 Institute of New Energy, Hexi University, Zhangye 734000, China Institute of New Energy, Hexi University, Zhangye 734000, China School of Physics and Electromechanical Engineering, Hexi University, Zhangye 734000, China School of Physics and Electromechanical Engineering, Hexi University, Zhangye 734000, China National Institute of Metrology, Beijing 100029, China National Institute of Metrology, Beijing 100029, China Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621900, China Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621900, China National Institute for Radiological Protection, China CDC, Beijing 100088, China National Institute for Radiological Protection, China CDC, Beijing 100088, China A lead-shielded HPGe detector and offline γ–ray spectra of the residual product were used to measure the cross-section (CS) and ratios of isomeric CS (σ m /σ g ) in 134 Xe(n,2n) 133m,g Xe reactions at different energies (13.5 MeV, 13.8 MeV, 14.1 MeV, 14.4 MeV, 14.8 MeV) relative to the 93 Nb(n,2n) 92m Nb reaction CS. The target was high-purity natural Xe gas under high pressure. The T(d,n) 4 He reaction produces neutrons. TALYS code (version 1.95) for nuclear reactions was used for calculations, with default parameters and nuclear level density models. The uncertainties in the measured CS data were thoroughly analyzed using the covariance analysis method. The results were compared with theoretical values, evaluation data, and previous experimental findings. CS data of the 134 Xe(n,2n) 133m Xe and 134 Xe(n,2n) 133g Xe reactions and the corresponding isomeric CS ratios at 13.5 MeV, 13.8 MeV, and 14.1 MeV neutron energies are reported for the first time. This research advances our knowledge of pre-equilibrium emission in the (n,2n) reaction channel by resolving inconsistencies in the Xe data. 134 Xe (n,2n) reactions 14-MeV neutrons Cross-sections (CSs) Isomeric ratios (IRs) Covariance analysis Introduction 1 Reliable knowledge of neutron-induced cross-sections (CSs) is required in nuclear power plants for energy and environmental protection, and in nuclear astrophysics reaction network calculations for vital input parameters [ 1 1 - 5 5 1-5 ]. Accurate measurements of CSs induced by neutrons at 14 MeV are desired for fusion reactors, nuclear transmutation, and nuclear model analyses [ 6 6 - 8 8 6-8 ]. Xe has nine stable isotopes [ 9 9 ]; 14-MeV neutrons can trigger nuclear reactions of Xe isotopes, including (n,α), (n,d), (n,p), (n,t), and (n,2n) [ 10 10 , 11 11 ]. The 134 Xe(n,2n) 133m,g Xe CSs for such reactions were experimentally obtained by two laboratories [ 12 12 , 13 13 ] using sodium perxenate (Na 4 XeO 6 ·2H 2 O) and quinol clathrate targets, respectively. In general, preparation of this compound is complicated; the composition is complex, and it is difficult to ensure purity (Ref. (12) reported an Na 4 XeO 6 ·2H 2 O purity of 95.8%. Only one laboratory reported pure ground-state CSs in the 134 Xe(n,2n) 133g Xe reaction [ 12 12 ]. The excited-state, ground-state, total CS, and CS ratio for 134 Xe(n,2n) 133m,g Xe reactions correspond to a single energy value (14.4 MeV or 14.6 MeV). Extensive research on the excitation functions for 124,126,128,130,132,136 Xe(n, 2n) and 130,131,132 Xe(n, p) reactions in the range of 13-15 MeV was recently reported by Bhatia et al. [ 14 14 ], Bhike et al. [ 15 15 , 16 16 ], and Luo et al. [ 17 17 , 18 18 ]. Our goal was to analyze four aspects of the 134 Xe(n,2n) reaction CS. First, the target 134 Xe isotope belongs to a transitional region ( N = 82, below the closed neutron shell); an even 134 Xe nucleus for such shell closure is a good indicator of single-particle excitations [ 17 17 ]. Second, the half-life of the metastable state of the 133m,g Xe pair is shorter than that of the ground state. This pair exhibits isomeric transition (IT) decay ( Fig. 1 Fig. 1 ). Third, published data on the 134 Xe(n,2n) reaction CS are scarce owing to difficulties in preparing gaseous targets for irradiation. Fourth, significant discrepancies have been observed in published results [ 12 12 , 13 13 ], mainly due to the following three factors: i. Systematic inconsistencies are caused by differences in experimental techniques and conditions (neutron field characteristics, target sample, neutron monitoring method). Previous studies have used sodium perxenate (Na 4 XeO 6 ·2H 2 O) and the solid quinol clathrate of Xe as the target materials [ 12 12 , 13 13 ]. For neutron flux monitoring, Sigg and Kuroda (1976) [ 12 12 ] used a low-threshold reaction: 27 Al(n,α) 24 Na ( E th =3.25 MeV). Kondaiah et al . (1968) [ 13 13 ] used 136 Xe(n,2n) 135(m+g) Xe and 128 Xe(n,2n) 127(m+g) Xe reactions to monitor 134 Xe(n,2n) 133m Xe and 134 Xe(n,2n) 133(m+g) Xe nuclear reactions ( Table 1 Table 1 ). a: corrected results using characteristic γ-ray intensity b: Reference 13 does not provide ground-state CS values. This value was obtained by subtracting the excited-state CS value from the total CS value. 1 1 Reaction 1 1 Decay data 1 1 Sample 1 1 Detector 1 1 Monitor reaction 1 1 E n (MeV) 1 1 σ (mb) 1 1 Ref. 1 1 134 Xe(n,2n) 133m Xe 1 1 T 1/2 =2.26 d, E γ=232.8 keV, I γ=14 % 1 1 Na 4 XeO 6 ·2H 2 O 1 1 GeLi 1 1 27 Al(n,α) 24 Na 1 1 14.6 1 1 655±60 1 1 906±83 a 1 1 [ 12 12 ] 1 1   1 1 T 1/2 =2.26 d, E γ=233 keV, I γ=13.55 % 1 1 Quinol clathrate 1 1 GeLi 1 1 136 Xe(n,2n) 135(m+g) Xe 1 1 14.4 1 1 665±80 1 1 890±107 a 1 1 [ 13 13 ] 1 1 134 Xe(n,2n) 133g Xe 1 1 T 1/2 =5.29 d, E γ=81.0 keV, I γ= 35% 1 1 Na 4 XeO 6 ·2H 2 O 1 1 GeLi 1 1 27 Al(n,α) 24 Na 1 1 14.614.4 1 1 805±901695±226 b 1 1 755±84 a 1366±202 b 1 1 [ 12 12 ][ 13 13 ] 1 1 134 Xe(n,2n) 133m+g Xe 1 1 T 1/2 =5.29 d, E γ=81.0 keV, I γ= 35% 1 1 Na 4 XeO 6 ·2H 2 O 1 1 GeLi 1 1 27 Al(n,α) 24 Na 1 1 14.6 1 1 1460±110 1 1 1661±118 a 1 1 [ 12 12 ] 1 1   1 1 T 1/2 =5.27 d, E γ=81 keV, I γ= 35.7% 1 1 Quinol clathrate 1 1 GeLi 1 1 128 Xe(n,2n) 127(m+g) Xe 1 1 14.4 1 1 2360±240 1 1 2256±229 a 1 1 [ 13 13 ] ii. There is a significant shortage of decay data. The 134 Xe(n,2n) 133m Xe reaction CSs [ 12 12 , 13 13 ] exhibited characteristic γ-ray intensities (14% and 13.55% at 233 keV, respectively ( Table 1 Table 1 )). Recently, 10.12% was also reported [ 9 9 ]. The 134 Xe(n,2n) 133g Xe reaction CSs were measured by Sigg and Kuroda (1976) [ 12 12 ], and Kondaiah et al . (1968) [ 13 13 ] using characteristic γ-ray intensities of 35% and 35.7%, respectively, for the same characteristic γ-ray of 81 keV ( Table 1 Table 1 ). However, 36.9% was obtained more recently [ 9 9 ]. As 79.61 keV and 80.998 keV γ-ray energies are so close that they cannot be distinguished by the detector, the intensity of the 81-keV gamma ray should be higher (37.34% = 0.44% + 36.9%). The CS of 133m Xe radionuclide product with a 2.26-d half-life was also determined [ 12 12 , 13 13 ], recently reported as 2.198 d [ 9 9 ]. iii. There are differences in the total CS measurement methods. Only two methods currently exist for calculating the total CS. 1) The excited-state and pure ground-state CSs can be added after measuring them separately [ 12 12 ]. 2) The ground state can be measured after full decay of the excited state via IT, and the result can be considered as the total CS [ 13 13 ]. The latter method satisfies the conditions that the excited-state half-life is less than that of the ground state, preferably much less, and IT = 100%. If these conditions are not satisfied, it is not advisable to replace the total CS with the obtained ground-state CS. Analysis of the existing data led us to conclude that the isomeric ratios (IRs) and CSs for the (n,2n) reaction of 134 Xe in the 13–15 MeV range must be further analyzed. We used residual nuclear decay analysis [ 19 19 - 21 21 19-21 ] and natural Xe with high-purity pressure as the target to measure the 134 Xe(n,2n) 133m Xe, 134 Xe(n,2n) 133g Xe, and 134 Xe(n,2n) 133 Xe CSs reactions and their corresponding isomeric-state CSs in the 13-15 MeV range. The results were compared with theoretical values obtained using TALYS-1.95 [ 22 22 ], evaluation data, and previous experimental results. Experimental Methods 1 Sample preparation 2 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 93 Nb(n,2n) 92m Nb (threshold energy 8.972 MeV) reaction was used to monitor the corresponding threshold energies of 134 Xe(n,2n) 133m,g Xe reactions ( E th = 8.853 MeV and 8.618 MeV). Low-threshold reactions such as (n,α) and (n,p) were monitored via a 27 Al(n,α) 24 Na reaction (threshold energy = 3.249 MeV). The CS of the 93 Nb(n,2n) 92m Nb reaction was obtained from the IRDFF-II data library [ 23 23 ]. To minimize the influence of the 132 Xe(n,γ) 133m,g Xe reaction on the 134 Xe(n,2n) 133m,g Xe 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 17 , 18 18 ]. The neutron generator facility was provided by the China Academy of Engineering Physics (CAEP) for sample irradiation at (4-5)×10 10 n/s yield (for 2 h). The d+ 3 H→n+ 4 He+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/cm 2 . A beam of α-particles at 135° was used to correct the neutron flux. Neutron energy and detection efficiency 2 In the experiment, Zr and Nb were used to measure the CS ratios of 93 Nb(n,2n) 92m Nb and 90 Zr(n,2n) 89m+g Zr, 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 24 ]. Considering the sample distance from the target, size, and beam radius, the uncertainty in the neutron energy was 0.2 MeV [ 25 25 ]. The results obtained using the CS ratio method were consistent in terms of the experimental errors. The detector efficiency was calibrated using 152 Eu, 133 Ba, 137 Cs, and 226 Ra 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 17 , 18 18 ]. Radioactivity measurement 2 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 133m Xe, 133g Xe, and 92m Nb 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 Fig. 2 . A γ-spectrum analysis was performed using ORTEC® GammaVision® software to estimate the peak area [ 26 26 ]. Table 2 Table 2 displays the natural isotope abundance and product decay characteristics. Uncertainties are indicated by the lower index and italicized numbers (10.4357 21 % indicates (10.4357 ± 0.0021)%, 2.198 d 13 indicates (2.198 ± 0.013) d. 1 1 Isotope Abundance (%) 1 1 Reaction 1 1 E -threshold (MeV) 1 1 Mode decay (%) 1 1 Product half-life 1 1 E γ (keV) 1 1 I γ (%) 1 1 10.4357 21 1 1 134 Xe(n,2n) 133m Xe 1 1 8.853 1 1 IT(100) 1 1 2.198 d 13 1 1 233.22 1 1 10.12 15 1 1 10.4357 21   1 1 134 Xe(n,2n) 133g Xe  1 1 8.618  1 1 β - (100)  1 1 5.2475 d 5   1 1 80.99879.61 1 1 36.9 3 0.44 18 1 1 100 1 1 93 Nb(n,2n) 92m Nb 1 1 8.972 1 1 EC (100) 1 1 10.15 d 2 1 1 934.44 1 1 99.15 4 CS calculations and associated uncertainties 1 CS calculation 2 The reaction CSs can be expressed as [ 17 17 , 18 18 ] <math display="block" id="M1"><mrow><msub><mi>σ</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><msub><mrow><mrow><mo>[</mo><mrow><mi>S</mi><mi>ε</mi><msub><mi>I</mi><mi>γ</mi></msub><mi>η</mi><mtext>KMD</mtext></mrow><mo>]</mo></mrow></mrow><mrow><mtext>Nb</mtext></mrow></msub></mrow><mrow><msub><mrow><mrow><mo>[</mo><mrow><mi>S</mi><mi>ε</mi><msub><mi>I</mi><mi>γ</mi></msub><mi>η</mi><mtext>KMD</mtext></mrow><mo>]</mo></mrow></mrow><mi>x</mi></msub></mrow></mfrac><mfrac><mrow><msub><mrow><mrow><mo>[</mo><mrow><mi>λ</mi><mtext>AFC</mtext></mrow><mo>]</mo></mrow></mrow><mi>x</mi></msub></mrow><mrow><msub><mrow><mrow><mo>[</mo><mrow><mi>λ</mi><mtext>AFC</mtext></mrow><mo>]</mo></mrow></mrow><mrow><mtext>Nb</mtext></mrow></msub></mrow></mfrac><msub><mi>σ</mi><mrow><mtext>Nb</mtext></mrow></msub></mrow></math> 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; <math id="M2"><mrow><mtext>D</mtext><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mrow><mo>(</mo><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></msup></mrow></math> represents the counting collection factor; <math id="M3"><mrow><mi>S</mi><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>T</mi></mrow></msup></mrow></math> is the growth factor of the product nuclide; T denotes the irradiation time; t 1 is the cooling time; t 2 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 Δ t i . This factor is calculated as <math id="M4"><mrow><mi>K</mi><mo>=</mo><mrow><mo>[</mo><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mi>ϕ</mi><mi>i</mi></msub></mrow></mstyle><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mtext>Δ</mtext><msub><mi>t</mi><mi>i</mi></msub></mrow></msup></mrow><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><msub><mi>T</mi><mi>i</mi></msub></mrow></msup></mrow><mo>]</mo></mrow><mo>/</mo><mo stretchy="false">(</mo><mi>S</mi><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></math> , 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; <math id="M5"><mrow><mi>ϕ</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mi>ϕ</mi><mi>i</mi></msub><mtext>Δ</mtext><msub><mi>t</mi><mi>i</mi></msub></mrow></mstyle><mo>/</mo><mi>T</mi></mrow></math> is the mean neutron fluence rate within the irradiation time. F is the total correction factor, expressed as <math display="block" id="M6"><mrow><mi>F</mi><mo>=</mo><msub><mi>F</mi><mi>Ω</mi></msub><mo>×</mo><msub><mi>F</mi><mtext>s</mtext></msub><mo>×</mo><msub><mi>F</mi><mtext>g</mtext></msub></mrow></math> 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 18 ]. Table 3 Table 3 presents the self-absorption correction factors. 1 2 Gamma-ray energy (keV) 2 1 μ / ρ (cm 2 /g) 3 1 μ (cm -1 ) 1 2 Sample density ρ (g/cm 3 ) 1 2 Correction factor f s 1 1 Fe 1 1 Xe 1 1 Fe 1 1 No. 1 1 Xe 1 1 80.998 1 1 0.5840 1 1 3.5521 1 1 4.5906 1 1 1 1 1 4.4117 1 1 1.2420 1 1 8.288 1 1   1 1   1 1   1 1   1 1 2 1 1 2.0804 1 1 0.5857 1 1 4.103 1 1   1 1   1 1   1 1   1 1 3 1 1 4.2326 1 1 1.1916 1 1 7.956 1 1   1 1   1 1   1 1   1 1 4 1 1 1.9529 1 1 0.5498 1 1 3.889 1 1   1 1   1 1   1 1   1 1 5 1 1 1.9369 1 1 0.5453 1 1 3.863 1 1 233.22 1 1 0.1340 1 1 0.3108 1 1 1.0533 1 1 1 1 1 0.3860 1 1 1.2420 1 1 1.390 1 1   1 1   1 1   1 1   1 1 2 1 1 0.1820 1 1 0.5857 1 1 1.204 1 1   1 1   1 1   1 1   1 1 3 1 1 0.3703 1 1 1.1916 1 1 1.375 1 1   1 1   1 1   1 1   1 1 4 1 1 0.1709 1 1 0.5498 1 1 1.195 1 1   1 1   1 1   1 1   1 1 5 1 1 0.1695 1 1 0.5453 1 1 1.194 To obtain the pure ground-state CSs for 134 Xe(n,2n) 133g Xe reactions, C x in Equation (1) was set to a peak area of 80.998 keV minus the contribution from 133m Xe via <math id="M7"><mrow><msup><mrow/><mrow><mtext>133m</mtext></mrow></msup><mtext>Xe</mtext><msup><mrow><mover><mo>→</mo><mrow><mtext>IT</mtext><mrow><mo>(</mo><mrow><mn>100</mn><mi>%</mi></mrow><mo>)</mo></mrow></mrow></mover></mrow><mrow><mtext>133g</mtext></mrow></msup><mtext>Xe</mtext></mrow></math> (counting C 1 ). C 1 can be expressed using the decay of an artificial nuclide, expressed as [ 18 18 ] <math display="block" id="M8"><mrow><msub><mi>C</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mtext>mg</mtext></mrow></msub><msub><mi>ε</mi><mtext>g</mtext></msub><msub><mi>I</mi><mtext>g</mtext></msub><msub><mi>C</mi><mtext>m</mtext></msub><msub><mi>F</mi><mtext>m</mtext></msub><mrow><mo>(</mo><mrow><msubsup><mi>λ</mi><mtext>g</mtext><mn>2</mn></msubsup><msub><mi>S</mi><mtext>m</mtext></msub><msub><mi>D</mi><mtext>m</mtext></msub><mo>−</mo><msubsup><mi>λ</mi><mtext>m</mtext><mn>2</mn></msubsup><msub><mi>S</mi><mtext>g</mtext></msub><msub><mi>D</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>λ</mi><mtext>g</mtext></msub><mo>−</mo><msub><mi>λ</mi><mtext>m</mtext></msub></mrow><mo>)</mo></mrow><msub><mi>S</mi><mtext>m</mtext></msub><msub><mi>D</mi><mtext>m</mtext></msub><msub><mi>I</mi><mtext>m</mtext></msub><msub><mi>ε</mi><mtext>m</mtext></msub><msub><mi>λ</mi><mtext>g</mtext></msub><msub><mi>K</mi><mtext>m</mtext></msub><msub><mi>F</mi><mtext>g</mtext></msub></mrow></mfrac><mo>,</mo></mrow></math> 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; <math id="M9"><mrow><msub><mi>S</mi><mtext>m</mtext></msub><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>m</mtext></msub><mi>T</mi></mrow></msup></mrow></math> and <math id="M10"><mrow><msub><mi>S</mi><mtext>g</mtext></msub><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>g</mtext></msub><mi>T</mi></mrow></msup></mrow></math> ; 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 <math id="M11"><mrow><msub><mi>D</mi><mtext>m</mtext></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>m</mtext></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>m</mtext></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup></mrow></math> and <math id="M12"><mrow><msub><mtext>D</mtext><mtext>g</mtext></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>g</mtext></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>g</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></msup></mrow></math> . Associated experimental uncertainties 2 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 133m Xe and 133g Xe, 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 27 - 30 30 27-30 ]. Table 4 Table 4 and Table 5 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 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 Tables 4 - 6 6 by adding 15 subset matrices in Equation 4 [ 29 29 ]: <math display="block" id="M13"><mrow><mtext>Cor</mtext><mo stretchy="false">(</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>j</mi></munder><mrow><mtext>Δ</mtext><msub><mi>x</mi><mi>i</mi></msub><mo>×</mo><mtext>Cor</mtext><mo stretchy="false">(</mo><mtext>Δ</mtext><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo><mtext>Δ</mtext><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>×</mo><mtext>Δ</mtext><msub><mi>x</mi><mi>j</mi></msub></mrow></mstyle></mrow></mstyle><mo>.</mo></mrow></math> 1 1 Attributes ( x ) 5 1 Fractional uncertainties (%) 1 1   1 1 13.5 MeV(Δ x i ) 1 1 13.8 MeV(Δ x j ) 1 1 14.1 MeV(Δ x k ) 1 1 14.4 MeV(Δ x l ) 1 1 14.8 MeV(Δ x n ) 1 1 C x 1 1 0.1236 1 1 0.1471 1 1 0.1349 1 1 0.1750 1 1 0.0726 1 1 C Nb 1 1 0.8523 1 1 0.8102 1 1 0.8083 1 1 0.8164 1 1 0.9315 1 1 I x 1 1 1.4822 1 1 1.4822 1 1 1.4822 1 1 1.4822 1 1 1.4822 1 1 I Nb 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 M x 1 1 0.0401 1 1 0.0869 1 1 0.0816 1 1 0.0876 1 1 0.0385 1 1 M Nb 1 1 0.0111 1 1 0.0109 1 1 0.0112 1 1 0.0110 1 1 0.0112 1 1 η x 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 σ Nb 1 1 0.6327 1 1 0.5615 1 1 0.5461 1 1 0.5427 1 1 0.5645 1 1 ε x 1 1 2.7201 1 1 2.7201 1 1 2.7201 1 1 2.7201 1 1 2.7201 1 1 ε Nb 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 S x 1 1 0.5836 1 1 0.5836 1 1 0.5836 1 1 0.5836 1 1 0.5836 1 1 S Nb 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 D x 1 1 0.4188 1 1 0.4197 1 1 0.2302 1 1 0.2335 1 1 0.0419 1 1 D Nb 1 1 0.1857 1 1 0.1857 1 1 0.1858 1 1 0.1858 1 1 0.1860 1 1 F x 1 1 2.0000 1 1 2.0000 1 1 2.0000 1 1 2.0000 1 1 2.0000 1 1 Total error (%) 1 1 4.65 1 1 4.63 1 1 4.62 1 1 4.62 1 1 4.63 1 1 Attributes ( x ) 5 1 Fractional uncertainties (%) 1 1   1 1 13.5 MeV(Δ x i ) 1 1 13.8 MeV(Δ x j ) 1 1 14.1 MeV(Δ x k ) 1 1 14.4 MeV(Δ x l ) 1 1 14.8 MeV(Δ x n ) 1 1 C x 1 1 0.1198 1 1 0.1224 1 1 0.1115 1 1 0.1174 1 1 0.0936 1 1 C Nb 1 1 0.8523 1 1 0.8102 1 1 0.8083 1 1 0.8164 1 1 0.9315 1 1 I x 1 1 1.2854 1 1 1.2854 1 1 1.2854 1 1 1.2854 1 1 1.2854 1 1 I Nb 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 0.0403 1 1 M x 1 1 0.0201 1 1 0.0869 1 1 0.0816 1 1 0.0876 1 1 0.0385 1 1 M Nb 1 1 0.0111 1 1 0.0109 1 1 0.0112 1 1 0.0110 1 1 0.0112 1 1 η x 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 0.0201 1 1 σ Nb 1 1 0.6327 1 1 0.5615 1 1 0.5461 1 1 0.5427 1 1 0.5645 1 1 ε x 1 1 3.9743 1 1 3.9743 1 1 3.9743 1 1 3.9743 1 1 3.9743 1 1 ε Nb 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 2.5038 1 1 S x 1 1 0.0094 1 1 0.0094 1 1 0.0094 1 1 0.0094 1 1 0.0094 1 1 S Nb 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 0.1965 1 1 D x 1 1 0.0027 1 1 0.0039 1 1 0.0039 1 1 0.0039 1 1 0.0052 1 1 D Nb 1 1 0.1857 1 1 0.1857 1 1 0.1858 1 1 0.1858 1 1 0.1860 1 1 F x 1 1 3.0000 1 1 3.0000 1 1 3.0000 1 1 3.0000 1 1 3.0000 1 1 Total error (%) 1 1 5.83 1 1 5.82 1 1 5.82 1 1 5.82 1 1 5.83 1 1 Cor(Δ x , Δ x ) 8 1 Correlation coefficient (Δ x , Δ x ) 1 1   1 1 C x 1 1 C Nb 1 1 M x 1 1 M Nb 1 1 ε x 1 1 S x 1 1 D x 1 1 F x 1 1 (Δ x i , Δ x i ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (Δ x i , Δ x j ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x i , Δ x k ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x i , Δ x l ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x i , Δ x n ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x j , Δ x j ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (Δ x j , Δ x k ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x j , Δ x l ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x j , Δ x n ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x k , Δ k ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (Δ x k , Δ x l ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x k , Δ x n ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x l , Δ x l ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (Δ x l , Δ x n ) 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.9987 1 1 0 1 1 0 1 1 0 1 1 (Δ x n , Δ x n ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The generated covariance matrix was [5×5]. The uncertainty in the experimental CS value was determined as (Equation 5) [ 29 29 ] <math display="block" id="M14"><mrow><mtext>Cor</mtext><mo stretchy="false">(</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mtext>Δ</mtext><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>.</mo></mrow></math> From the covariance matrix and total uncertainty between neutron energies, the correlation matrix [5×5] was derived using Eq. 6 [ 29 29 ]: <math display="block" id="M15"><mrow><mtext>Cor</mtext><mo stretchy="false">(</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mtext>Cor</mtext><mo stretchy="false">(</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mtext>Δ</mtext><msub><mi>σ</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mtext>Δ</mtext><msub><mi>σ</mi><mrow><mi>x</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> The experimentally obtained reaction CS values and their correlation matrices and uncertainties are presented in Table 7 Table 7 and Table 8 Table 8 . 1 1 Neutron energy E n (MeV) 1 1 Cross-sectionσ x (mb) 1 1 Δσ x (%) 5 1 Correlation matrix 1 1 13.5±0.2 1 1 884±41 1 1 4.65 1 1 1.0000 1 1 13.8±0.2 1 1 888±41 1 1 4.63 1 1 0.7565 1 1 1.0000 1 1 14.1±0.2 1 1 909±42 1 1 4.62 1 1 0.7586 1 1 0.7593 1 1 1.0000 1 1 14.4±0.2 1 1 916±42 1 1 4.62 1 1 0.7581 1 1 0.7588 1 1 0.7610 1 1 1.0000 1 1 14.8±0.2 1 1 928±43 1 1 4.63 1 1 0.7562 1 1 0.7569 1 1 0.7591 1 1 0.7586 1 1 1.0000 1 1 Neutron energy E n (MeV) 1 1 Cross-sectionσ x (mb) 1 1 Δσ x (%) 5 1 Correlation matrix 1 1 13.5±0.2 1 1 824±48 1 1 5.83 1 1 1.0000 1 1 13.8±0.2 1 1 810±47 1 1 5.82 1 1 0.7125 1 1 1.0000 1 1 14.1±0.2 1 1 786±46 1 1 5.82 1 1 0.7125 1 1 0.7129 1 1 1.0000 1 1 14.4±0.2 1 1 767±45 1 1 5.82 1 1 0.7123 1 1 0.7127 1 1 0.7127 1 1 1.0000 1 1 14.8±0.2 1 1 747±44 1 1 5.83 1 1 0.7105 1 1 0.7108 1 1 0.7108 1 1 0.7106 1 1 1.0000 Table 9 Table 9 shows experimental values for 134 Xe(n,2n) 133m+g Xe reaction CSs and their IRs, and the monitor reaction CSs at different neutron energies. 1 1 E n (MeV) 2 1 This study 2 1 Monitor reaction 93 Nb(n,2n) 92m Nb 1 1   1 1 CS (mb) 1 1 IRs (σ m /σ g ) 1 1 CS (mb) 1 1 Reference 1 1 13.5±0.2 1 1 1708±63 1 1 1.07±0.08 1 1 453.0±2.9 1 1 [ 24 24 ] 1 1 13.8±0.2 1 1 1698±63 1 1 1.10±0.08 1 1 457.4±2.6 1 1 [ 24 24 ] 1 1 14.1±0.2 1 1 1696±62 1 1 1.16±0.09 1 1 459.7±2.5 1 1 [ 24 24 ] 1 1 14.4±0.2 1 1 1683±62 1 1 1.20±0.09 1 1 460.1±2.5 1 1 [ 24 24 ] 1 1 14.8±0.2 1 1 1675±61 1 1 1.24±0.09 1 1 460.2±2.6 1 1 [ 24 24 ] Theoretical calculations 1 TALYS software is widely used to analyze pre-equilibrium mechanisms, direct nuclear reactions, and compound nuclei (CN). It can evaluate nuclear reactions in the range of 0.001–200 MeV for target nuclei with masses of 12 and higher. TALYS also calculates the nuclear reaction CSs. Theoretical calculations for different ratios of 134 Xe(n,2n) 133m,g Xe reactions (σ m /σ g ) were performed for different ranges of neutron energies using TALYS-1.95 (ver. 1.95) [ 22 22 ]. We used the default parameters for our calculations, except for six distinct nuclear level density models (NLDs), which were divided into two groups: three microscopic levels and three phenomenological densities. The phenomenological level densities included: 1) ldmodel 1, with a constant temperature and Fermi gas model, and excitation energy divided into higher and lower energy regions in the Fermi gas model and with constant temperature laws, respectively; 2) ldmodel 2 is based on the back-shifted Fermi gas model, which uses the Fermi gas expression across the region; 3) ldmodel 3 is a generalized superfluid model that considers superconductive correlations based on Barden-Cooper-Schrieffer theory [ 31 31 - 32 32 31-32 ]. The microscopic level densities are i) ldmodel 4 is made up of Goriely's tables derived from the Skyrme force (microscopic level densities) [ 33 33 ]; ii) ldmodel 5 consists of Hilaire’s combinatorial tables derived from the Skyrme force [ 33 33 ]; iii) ldmodel 6 is the Gogny force, derived from Hilaire’s combinatorial tables [ 33 33 ]. Examining the spin distribution is feasible using relative feeding as a ‘probe’. <math id="M16"><mrow><msubsup><mi>σ</mi><mtext>F</mtext><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi><mi>x</mi></msub><mo stretchy="false">)</mo></mrow></math> indicates the angular momentum distribution of the level density, expressed as <math display="block" id="M17"><mrow><msubsup><mi>σ</mi><mtext>F</mtext><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.01389</mn><mfrac><mrow><msup><mi>A</mi><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mover accent="true"><mi>a</mi><mo>˜</mo></mover></mfrac><msqrt><mrow><mi>α</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>x</mi></msub><mo>−</mo><mtext>Δ</mtext></mrow><mo>)</mo></mrow></mrow></msqrt><mo>,</mo></mrow></math> where A indicates the mass number; E x is the true excitation energy, Δ represents the empirical parameter and energy shift, which are nearly equal to the pairing energy of the observed nucleus odd–even effects; α is the energy-dependent level density parameter accounting for damping at higher excitation energies and shell effects at lower energies. Without shell effects, the asymptotic level density parameter ( <math id="M18"><mover accent="true"><mi>a</mi><mo>˜</mo></mover></math> ) is equal to α [ 34 34 ]. Results and discussion 1 Tables 7 Tables 7 , 8 8 and 9 9 show the CSs of the 134 Xe(n,2n) 133m Xe, 134 Xe(n,2n) 133g Xe, and 134 Xe(n,2n) 133 Xe reactions and the corresponding isomeric-state CS ratios obtained using the activation method. The data of the excited state and pure ground state were compared to data from the literature [ 12 12 , 13 13 ] and data obtained by TALYS-1.95 at six energy-level densities [ 22 22 ]. The 134 Xe(n,2n) 133 Xe CS was also compared to data from the ENDF/B-VIII.0 [ 35 35 ], JEFF-3.3 [ 36 36 ], JENDL-4.0 [ 37 37 ], ROSFOND [ 38 38 ], and CENDL-3 [ 39 39 ] databases. 134 Xe(n,2n) 133m Xe reaction 2 Two studies [ 12 12 , 13 13 ] have reported 134 Xe(n,2n) experiments for the 133m Xe 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 12 ] used I γ □ = 14%, and Kondaiah et al. [ 13 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 133m Xe was used to obtain the 134 Xe(n,2n) 133m Xe reaction CS. Thus, the CS values were modified according to the following equation: <math display="block" id="M19"><mrow><msub><mi>σ</mi><mrow><mtext>corrected</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>I</mi><mrow><mi>γ</mi><mo stretchy="false">(</mo><mtext>old)</mtext></mrow></msub></mrow><mrow><msub><mi>I</mi><mrow><mi>γ</mi><mo stretchy="false">(</mo><mtext>new</mtext><mo stretchy="false">)</mo></mrow></msub></mrow></mfrac><msub><mi>σ</mi><mrow><mtext>literature</mtext></mrow></msub><mo>,</mo></mrow></math> 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 12 , 13 13 ], as shown in Fig. 3 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. 134 Xe(n,2n) 133g Xe reaction 2 A thorough literature review yielded only one measurement of the 134 Xe(n,2n) 133g Xe reaction using γ-ray energy of 81 keV ( I γ □ = 35%) [ 12 12 ]. However, many investigations have used I γ □ = 36.9% for gamma-rays with the same energy. Production of the 133g Xe 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 12 ]. During the 133m,g Xe decay, the emitted 80.998-keV γ-ray was used to calculate the 134 Xe(n,2n) 133g Xe reaction CSs after subtracting the contribution of the 134 Xe(n,2n) 133m Xe reaction via the isomeric transition (IT). Almost 50% of the 80.998-keV γ-ray counts were from the isomeric transitions. Figure 4 Figure 4 shows the theoretical assessments of the excitation functions and the experimental results (Ref. [ 13 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 12 ]. At 14.4 MeV, the results in Ref. [ 13 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. 134 Xe(n,2n) 133 Xe reaction 2 There are two published reports on the 134 Xe(n,2n) 133 Xe 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 35 ], (JEFF-3.3 [ 36 36 ], CENDL-3 [ 39 39 ]), JENDL-4.0 [ 37 37 ], and ROSFOND [ 38 38 ] databases were almost identical to those obtained by TALYS-1.95 [ 22 22 ] at ~15 MeV ( Fig. 5 Fig. 5 ). We conclude that the experimental data agree with the ROSFOND [ 38 38 ] results, TALYS-1.95 with ldmodel 1, and the corrected experimental results of Sigg and Kuroda [ 12 12 ]. At 14.4 MeV, the values [ 13 13 ] were 570 mb higher than ours. The evaluation excitation curve taken from JENDL-4.0 [ 37 37 ] is above the corrected results of Sigg and Kuroda [ 12 12 ], our experimental results, and results from the other databases [ 35 35 , 36 36 , 38 38 , 39 39 ]. Data for the 13.5-14.1 MeV neutron energy range are reported for the first time. Isomeric CS ratio 2 The ratios of the isomeric CSs were calculated using the measured 134 Xe(n,2n) 133m.g Xe CSs. The CS ratios (σ m /σ g ) for the isomeric 133m,g Xe pair created from the 134 Xe(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 Table 9 ). Figure 6 Figure 6 shows the corrected isomeric CS ratios from the literature [ 12 12 , 13 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 40 - 43 43 40-43 ]. The obtained σ m /σ g values were consistent with the modified values from Sigg and Kuroda [ 12 12 ], but slightly lower than the values obtained using ldmodels 1-6 and higher than the results of Ref. [ 13 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. Summary and Conclusions 1 The σ m , σ g , and σ m+g for the 134 Xe(n,2n) 133m Xe, 134 Xe(n,2n) 133g Xe, and 134 Xe(n,2n) 133 Xe reactions and the corresponding isomeric CS ratios (σ m /σ g ) were determined from the neutron activation and offline γ–ray spectra of the residual products at 13.5–14.8 MeV using the 93 Nb(n,2n) 92m Nb reaction as a reference. We thoroughly analyzed the uncertainties of the experimentally obtained CSs using covariance analysis. Experimental data were compared to data from the literature and CS values obtained using TALYS-1.95 software and from the ENDF/B-VIII.0, JEFF-3.3, JENDL-4.0, ROSFOND, and CENDL-3 databases. The theoretically obtained CS values for the 134 Xe(n,2n) reaction at six different NLDs (to generate nuclear excited and ground states) disagreed with the experimental results. However, the total CSs values were in agreement. The magnitude of the calculated CSs differed due to variations in the nuclear inputs and reaction models used for the simulations. The reaction mechanism involves photon and particle emissions in the CN. The CN CS is determined by the OPs, NLDs, and γSFs. Significant discrepancies were found in the experimental values; however, these were observed only for the pure ground state and total CSs as they are the most affected by the excited states. The CSs at neutron energies of 13.5–14.1 MeV are presented for the first time for the 134 Xe(n,2n) 133m Xe, 134 Xe(n,2n) 133g Xe, and 134 Xe(n,2n) 133 Xe reactions, significantly extending the currently published energy ranges. 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We acknowledge the Intense Neutron Generator group at the China Academy of Engineering Physics for providing the neutron generator facility. 10.1007/s41365-022-01158-z This work was supported by the National Natural Science Foundation of China (Nos. 11875016 and 12165006). 2022-08-31 2022-11-17 2022-11-19 2023-01-09 2023-02-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01158-z http://dx.doi.org/10.1007/s41365-022-01158-z . 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