Evaluation of neutron nuclear data on tellurium isotopes

Abstract

Neutron nuclear data on 13 Te isotopes have been evaluated for the next release of Japanese Evaluated Nuclear Data Library (JENDL) general purpose file in the energy region from 10^{− 5} eV to 20 MeV. Thermal capture cross sections of ^{120, 121m, 127m, 129m, 131m, 132}Te were determined from the latest measurements or from a simplified formula, although the resolved resonance parameters, which describe the low-energy behavior of cross sections, remain unchanged from JENDL-4.0 for ^{122, 123, 124, 125, 126, 128, 130}Te. A statistical model code was applied to evaluate the cross sections above the resolved resonance region. Coupled-channel optical model parameters were employed for the interaction between neutrons and nuclei. Compound, pre-equilibrium, and direct-reaction processes were considered for cross-section calculation in high-energy region. The present results reproduce experimental data very well, and are found to be much better than the JENDL-4.0 data. The evaluated data are compiled into evaluated nuclear data file (ENDF)-formatted data files.

Keywords:

• neutron nuclear data
• evaluation
• tellurium isotopes
• cross section
• JENDL
• optical model
• statistical model

1. Introduction

The fourth version of the Japanese Evaluated Nuclear Data Library (JENDL-4.0 [1]) was released in May 2010. In this library, 215 nuclides are regarded as fission products in the region from Z = 30 to 68. Although the resolved resonance parameters (RRPs) were examined by reviewing the experimental data that were available then, the fast-neutron region above resolved resonance was re-evaluated for about 170 nuclides due to the deadline of release. Hence, the fast-neutron cross sections for the rest of fission-product nuclides may need further improvements for the next release of JENDL general purpose file. Tellurium isotopes are classified into such nuclides. The data of ¹²⁸Te were evaluated for JENDL-2 [2] in 1984 without those of other Te isotopes. In 1990, the data of ^{120, 122, 123, 124, 125, 126, 127m, 128, 129m, 130}Te were evaluated and compiled into JENDL-3.1 [3]. These data were essentially carried over to JENDL-3.2 [4], JENDL-3.3 [5], and JENDL-4.0, although the ¹³²Te data were newly evaluated and the RRPs of ^{122, 125, 126, 128, 130}Te were modified for JENDL-4.0. No gamma-ray production data have been evaluated except for ¹³²Te in JENDL-4.0. The direct reaction was not taken into account for the inelastic scattering in the JENDL-3.1 evaluation. Most of the Te data in JENDL-4.0 are based on the measurements and theoretical models which were available more than 20 years ago. Furthermore, neutron-induced activation cross sections of Te isotopes are needed [6] for the decommissioning of light water reactors. Therefore, it is necessary to re-examine the cross sections of Te isotopes.

The present work was undertaken to improve the evaluated data on Te isotopes by considering the latest knowledge on experimental and theoretical nuclear physics. Moreover, the data on ^{121m, 131m}Te were newly evaluated, since these isotopes could be produced through neutron capture by ¹²⁰Te and ¹³⁰Te, respectively. In addition, the independent yield of ^131mTe, which amounts to 7.11 × 10^{− 3} [7] for fast-neutron fission of ²³⁹Pu, is not small. Evaluated are the total, elastic and inelastic scattering, (n, γ), (n, p), (n, d), (n, t), $\left(n{,}^3\text{He}\right)$, (n, α), (n, np), (n, nd), (n, nα), (n, 2n), (n, 3n), (n, 2np) reaction cross sections, the angular distributions of elastically and inelastically scattered neutrons, and the energy distributions of emitted particles and γ-rays in the energy region from 10^{− 5} eV to 20 MeV. The Q-values of the reactions were calculated from the 2003 version of the mass table (AME2003) [8], and are listed in Table 1, together with isotopic abundances [9], half lives [10], and excitation energies [11] of target nuclei. The RRPs remain unchanged from JENDL-4.0. The unresolved resonance parameters (URPs) were obtained by fitting to the calculated total and (n, γ) cross sections.

Table 1. Isotopic abundances and reaction Q-values of tellurium isotopes.

	^120Te	^121mTe	^122Te	^123Te	^124Te	^125Te	^126Te	^127mTe	^128Te	^129mTe	^130Te	^131mTe	^132Te
$E_{\text{ex}}$ (keV)*	0.0	294.0	0.0	0.0	0.0	0.0	0.0	88.2	0.0	105.5	0.0	182.3	0.0
Abundances (%)	0.09	164.2d**	2.55	0.89	4.74	7.07	18.84	106.1d**	31.74	33.6d**	34.08	33.25h**	3.204d**
Q-values (MeV)
(n, γ)	7.218	10.128	6.929	9.424	6.569	9.114	6.288	8.871	6.082	8.525	5.929	8.226	5.834
(n, p)	−0.198	2.120	−1.202	0.835	−2.122	0.016	−2.884	−0.710	−3.601	−1.488	−4.277	−2.257	−4.726
(n, α)	6.642	9.045	5.401	7.580	4.326	6.570	3.402	5.690	2.553	4.769	1.794	3.954	1.058
(n, d)	−4.992	−4.897	−5.783	−5.906	−6.365	−6.466	−6.873	−6.859	−7.356	−7.354	−7.788	−7.800	−8.259
(n, t)	−8.284	−5.658	−8.768	−6.455	−9.073	−6.677	−9.323	−6.816	−9.472	−7.076	−9.621	−7.278	−9.769
(n, ^3He)	−4.608	−5.049	−6.069	−6.828	−7.438	−8.062	−8.688	−9.154	−9.832	−10.258	−10.877	−11.293	−11.903
(n, 2n)	−10.291	−6.924	−9.834	−6.929	−9.424	−6.569	−9.114	−6.200	−8.782	−5.977	−8.420	−5.747	−8.044
(n, np)	−7.216	−7.122	−8.008	−8.131	−8.589	−8.691	−9.098	−9.083	−9.581	−9.578	−10.013	−10.025	−10.483
(n, nα)	−0.301	−0.282	−1.083	−1.528	−1.844	−2.243	−2.544	−2.797	−3.180	−3.424	−3.756	−3.953	−4.272
(n, nd)	−14.541	−11.915	−15.025	−12.712	−15.330	−12.934	−15.580	−13.073	−15.729	−13.333	−15.879	−13.535	−16.026
(n, 3n)	−17.826	−17.215	−17.052	−16.763	−16.353	−15.993	−15.683	−15.313	−15.070	−14.759	−14.502	−14.167	−13.973
(n, 2np)	−16.765	−14.140	−17.250	−14.937	−17.555	−15.158	−17.805	−15.298	−17.954	−15.558	−18.103	−15.760	−18.251

*Excitation energy in a target nucleus.

^**Half-lives are given for ^{121m, 127m, 129m, 131m, 132}Te which are unstable.

This paper presents how the evaluation was performed. Section 2 deals with the resonance region. In Section 3, the computational methods and procedures above the resonance region are described. Comparisons of the evaluated results with experimental and existing evaluated data are made in Section 4. Finally, Section 5 summarizes the conclusion.

2. Resonance region

The resolved and unresolved resonance regions are given in Table 2 for each target nucleus. No RRPs are available for ^{120, 121m, 127m, 129m, 131m, 132}Te. The RRPs of ^{122, 123, 124, 125, 126, 128, 130}Te remain unchanged from JENDL-4.0, because no new measurements have been available since the JENDL-4.0 evaluation was performed. Concerning the nuclides for which RRPs are unavailable, the thermal elastic scattering cross sections were calculated as 4πR², where R stands for the scattering radius that can be estimated from nuclear model calculations in the higher energy region. On the other hand, the thermal capture cross sections of these nuclides except ¹²⁰Te were estimated from a simplified formula used in the previous work on antimony [12]. This formula gives rough estimates of thermal capture cross sections by using theoretical s-wave γ-ray strength functions. It is used only when no other options are available. The value measured by Eastman and Krane [13] was used to renormalize the 1/v capture cross sections of ¹²⁰Te at 0.0253 eV. The thermal capture and elastic scattering cross sections are listed in Table 3, where capture and elastic scattering cross sections are given at 300 and 0 K, respectively, in order to compare with the values recommended by Mughabghab [14]. There are large differences in the capture cross sections of ^{127m, 129m}Te between the present work and JENDL-4.0. The value of 3381 b for ^127mTe in JENDL-4.0 is based on the old recommendation [15], although the origin of 1601 b is unknown for ^129mTe. However, no data are found for ^127mTe in the new recommendation [14]. The new recommendation gives a value of 8.0 ±0.2 b for the scattering cross section of ¹²⁴Te, which seems larger than those of other nuclides in the same recommendation. Anyway, it is not possible to judge which one is reasonable, since no experimental data exist.

Table 2. Resolved and unresolved resonance regions.

Target		Unresolved
nuclei	Resolved resonance	resonance
^120Te	–	32 eV to 300 keV
^121mTe	–	10 eV to 100 keV
^122Te	10^−5 eV to 11 keV	11 to 300 keV
^123Te	10^−5 to 700 eV	700 eV to 200 keV
^124Te	10^−5 eV to 30 keV	30 to 300 keV
^125Te	10^−5 eV to 7 keV	7 to 300 keV
^126Te	10^−5 eV to 30 keV	30 to 300 keV
^127mTe	–	20 eV to 100 keV
^128Te	10^−5 eV to 8 keV	8 to 300 keV
^129mTe	–	80 eV to 100 keV
^130Te	10^−5 eV to 30.5 keV	30.5 to 300 keV
^131mTe	–	200 eV to 100 keV
^132Te	–	10 to 300 keV

Table 3. Thermal capture (C) and elastic scattering (E) cross sections in units of barns.

Target
nuclei	Reaction	Present	JENDL-4.0	Mughabghab [14]
^120Te	C	0.6563	2.341	2.340 ±0.306
	E	4.155	3.640	–
^121mTe	C	63.87	–	–
	E	4.324	–	–
^122Te	C	4.009	4.009	3.9±0.4
	E	2.508	2.508	2.03 ±0.20
^123Te	C	418.4	418.4	418 ±30
	E	0.5894	0.5894	–
^124Te	C	6.786	6.786	6.3±0.7
	E	3.653	3.653	8.0±0.2
^125Te	C	1.293	1.293	1.29± 0.16
	E	3.401	3.401	3.20±0.15
^126Te	C	0.4401	0.4401	0.44±0.06
	E	3.433	3.433	3.92±0.10
^127mTe	C	3.059	3381	–
	E	3.902	3.630	–
^128Te	C	0.1861	0.1861	0.200 ±0.008
	E	4.099	4.099	–
^129mTe	C	0.7698	1601	–
	E	3.649	3.400	–
^130Te	C	0.1862	0.1862	0.195 ±0.010
	E	3.836	3.836	–
^131mTe	C	0.1565	–	–
	E	3.138	–	–
^132Te	C	0.1394	0.1951	–
	E	5.308	5.373	–

The ASREP code [16] was used to determine the URPs by fitting to the total and capture cross sections calculated in the energy region above 10 eV. It should be noted that the URPs obtained are used only for self-shielding calculations, because the pointwise cross sections are given in the evaluated data files. As an example, the cross sections of ¹²³Te reconstructed from the URPs presently obtained are compared with the total and capture cross sections calculated from the nuclear models in Figure 1. In the parameter fitting, 10% uncertainties in the nuclear model calculations were assumed. It is found from the figure that the reconstructed cross sections reproduce the nuclear model calculations very well.

Figure 1. Total and capture cross sections of ¹²³Te in the unresolved resonance region.

3. Computational methods and procedures above resonance region

3.1. Nuclear models

The CCONE code (version 0.8.4) [17] was used for calculating the neutron-induced reaction cross sections of Te isotopes. The code is based on the spherical and coupled-channel optical models, the two-component exciton pre-equilibrium model, the distorted-wave Born approximation (DWBA), and the multi-step statistical model. In order to simulate the direct and semidirect effects on the radiative capture reaction, the pre-equilibrium capture was considered by using the γ-ray emission rate derived by Akkermans and Gruppelaar [18].

3.2. Parameter determination

3.2.1. Optical model potentials

As for neutrons, we employed the global optical model parameters obtained by Kunieda et al. [19] using the coupled-channel method based on the rigid rotor model (RRM-CC) [20]. The potential depths are given as follows: (1) $$\begin{array}{ccc}\hfill V_R& =& (V_R^0+V_R^1{E}^{†}+V_R^2{E}^{†2}+V_R^3{E}^{†3}\hfill \\ +V_R^{\text{DISP}}\text{exp}(-{\lambda }_R{E}^{†}))\hfill \\ \times \left[1-\frac{1}{V_R^0+V_R^{\text{DISP}}}{C}_{\text{viso}}\frac{N-Z}{A}\right],\hfill \end{array}$$(1) (2) $$\begin{array}{ccc}\hfill W_D& =& \left[W_D^{\text{DISP}}-C_{\text{wiso}}\frac{N-Z}{A}\right]\text{exp}(-{\lambda }_D{E}^{†})\hfill \\ \times \frac{E^{†2}}{E^{†2}+{\text{WID}}_D^2},\hfill \end{array}$$(2) (3) $$\begin{array}{ccc}\hfill W_V& =& W_V^{\text{DISP}}\frac{E^{†2}}{E^{†2}+{\text{WID}}_V^2},\hfill \end{array}$$(3) (4) $$\begin{array}{ccc}\hfill V_{\text{SO}}& =& V_{\text{SO}}^0\text{exp}(-{\lambda }_{\text{SO}}{E}^{†}),\hfill \end{array}$$(4) (5) $$\begin{array}{ccc}\hfill W_{\text{SO}}& =& W_{\text{SO}}^{\text{DISP}}\frac{E^{†2}}{E^{†2}+{\text{WID}}_{\text{SO}}^2},\hfill \end{array}$$(5) where Z, A, and N are the atomic number and mass number of a target and N = A − Z, respectively. The symbol E† is the incident-neutron energy (E) relative to the Fermi one (E_f), i.e., E† = E − E_f. The Fermi energy E_f is calculated from neutron-separation energies (S_n) of target and compound nuclei such as E_f = −{(S_n(Z, A) + S_n(Z, A + 1)}/2. The depths V_R, W_D, W_V, $V_{\text{SO}}$, and $W_{\text{SO}}$ correspond to real volume, imaginary surface, imaginary volume, real and imaginary spin–orbit terms, respectively. The form factor is of a Woods–Saxon shape, i.e., (6) $$f_i\left(r\right)=\frac{1}{1+\text{exp}\{(r-R_i)/a_i\}},i=R,D,V,\text{SO},$$(6) where R_i is written as (7) $$R_i=R_i^0\left[1+\sum _{l=2,4,...}{\beta }_l{Y}_l^0\left(\theta \right)\right].$$(7) The angle θ refers to the body-fixed system, and Y⁰_l and β_l denote spherical harmonics and a deformation parameter, respectively. From the neutron potentials, transmission coefficients are obtained together with total cross sections, shape elastic scattering cross sections, and the direct-reaction components of inelastic scattering cross sections for the CCONE calculation. It should be noted that renormalization would be done of the reaction cross section calculated from the transmission coefficients when additional DWBA calculation is performed for the inelastic scattering.

The parameters required in Equations (1(1) $$\begin{array}{ccc}\hfill V_R& =& (V_R^0+V_R^1{E}^{†}+V_R^2{E}^{†2}+V_R^3{E}^{†3}\hfill \\ +V_R^{\text{DISP}}\text{exp}(-{\lambda }_R{E}^{†}))\hfill \\ \times \left[1-\frac{1}{V_R^0+V_R^{\text{DISP}}}{C}_{\text{viso}}\frac{N-Z}{A}\right],\hfill \end{array}$$(1) )–(7(7) $$R_i=R_i^0\left[1+\sum _{l=2,4,...}{\beta }_l{Y}_l^0\left(\theta \right)\right].$$(7) ) are listed in Table 4, and the coupling schemes and deformation parameters are given in Table 5. The deformation parameters were determined by considering the compilation of Raman et al. [21], although their signs and magnitude were adjusted so as to reproduce measured total cross sections. The calculated total cross section is shown in Figure 2, together with experimental data and the spherical optical model calculations using the parameters of Koning and Delaroche [22]. In order to give a better fit to experimental data, the values of $W_D^{\text{DISP}}$ and $W_V^{\text{DISP}}$ were adjusted, while the rest of the parameters remain unchanged from the original ones [19]. Concerning charged particles, used were the spherical parameters of Koning and Delaroche [22] for protons, those of Lohr and Haeberli [23] for deuterons, those of Becchetti and Greenlees [24] for tritons and ³He, and those of Lemos [25] modified by Arthur and Young [26] for α-particles. The transmission coefficients, which are required by CCONE to calculate charged-particle emission, are obtained from the charged-particle parameters.

Figure 2. Total cross section of elemental Te.

Table 4. Optical model parameters for neutrons.

V^0R	=−34.40 − 6.18 × 10^−2A + 3.37 × 10^−4A^2 − 6.52 × 10^−7A^3 MeV
V^1R	=0.027
V^2R	=1.2 × 10^−4 MeV^−1
V^3R	=3.5 × 10^−7 MeV^−2
$V_R^{\text{DISP}}$	=94.88 MeV
$C_{\text{viso}}$	=24.3 MeV
λR	= 1.05 × 10^− 2 + 1.63 × 10^− 4V^0R MeV^−1
$W_D^{\text{DISP}}$	=9.8 MeV
${\text{WID}}_D$	=12.72 − 1.54 × 10^−2A + 7.14 × 10^−5A^2 MeV
$C_{\text{wiso}}$	=18.0 MeV
λD	=0.0140 MeV^−1
$W_V^{\text{DISP}}$	=23.0 MeV
${\text{WID}}_V$	$=105.05-14.13\times {[1.0+\text{exp}\{(A-100.21)/13.47\}]}^{-1}$ MeV
$V_{\text{SO}}^0$	=5.92 + 0.003A MeV
${\lambda }_{\text{SO}}$	=0.005 MeV^−1
$W_{\text{SO}}^{\text{DISP}}$	=−3.1 MeV
${\text{WID}}_{\text{SO}}$	=160.0 MeV
R^0R, D, V	=1.21A^1/3 fm
aR, D, V	=0.49 + 3.22 × 10^−3A − 2.07 × 10^−5A^2 + 4.47 × 10^−8A^3 fm
$R_{\text{SO}}^0$	=(1.18 − 0.65A^−1/3)A^1/3 fm
$a_{\text{SO}}$	=0.59 fm

Table 5. Coupling schemes used in the interaction between neutron and target.

Target nuclei	Coupled levels*	Deformation parameters
^120Te	0^+(g.s.), 2^+(0.056), 4^+(1.161), 6^+(1.776)	β2 = −0.1608
^121mTe	11/2^−(0.294), 13/2^−(0.975)	β2 = −0.1608
^122Te	0^+(g.s.), 2^+(0.564), 4^+(1.181), 6^+(1.751)	β2 = −0.1478
^123Te	1/2^+(g.s.), 3/2^+(0.440), 5/2^+(0.505)	β2 = −0.1478
^124Te	0^+(g.s.), 2^+(0.603), 4^+(1.249), 6^+(1.747)	β2 = −0.1356
^125Te	1/2^+(g.s.), 3/2^+(0.444), 5/2^+(0.463)	β2 = −0.1356
^126Te	0^+(g.s.), 2^+(0.666), 4^+(1.361), 6^+(1.776)	β2 = −0.1227
^127mTe	11/2^−(0.088), 13/2^−(1.545)	β2 = −0.1227
^128Te	0^+(g.s.), 2^+(0.743), 4^+(1.497), 6^+(1.811)	β2 = −0.1090
^129mTe	11/2^−(0.106), 13/2^−(1.110)	β2 = −0.1090
^130Te	0^+(g.s.), 2^+(0.840), 4^+(1.633), 6^+(1.815)	β2 = −0.1090
^131mTe	11/2^−(0.182), 13/2^−(1.015)	β2 = −0.0947
^132Te	0^+(g.s.), 2^+(0.974), 4^+(1.671), 6^+(1.775)	β2 = −0.0947

*The number in the parentheses indicates the excitation energy in MeV.

3.2.2. Discrete levels and level density

In the calculation, it was necessary to input the discrete levels and level density parameters for 46 nuclei, i.e., ^{118 − 133}Te, ^{118 − 132}Sb, and ^{116 − 130}Sn. The discrete levels were taken from the reference input parameters library RIPL-3 [11].

Concerning the level density, the composite formula of Gilbert and Cameron [27] was used in the present work. In the region of low excitation energy E, the level density is described by the constant temperature formula ρ_T, namely, (8) $${\rho }_T\left(E\right)=\frac{1}{T}\text{exp}\left(\frac{E-E_0}{T}\right).$$(8) On the other hand, the a parameter, which characterizes the Fermi-gas part of level density ρ_F, is defined as (9) $$a\left(U\right)=a(*)\left[1+\frac{E_{\text{sh}}}{U}(1-e^{-\gamma U})\right],$$(9) where $E_{\text{sh}}$ is the shell correction energy and γ a damping factor. The damping factor is given by γ = 0.40A^{− 1/3} MeV^{− 1}. The energy U is expressed by E and the pairing energy Δ, i.e., U = E − Δ. The values of a(*) and Δ were taken from the work of Mengoni and Nakajima [28]. The shell correction energy $E_{\text{sh}}$ is calculated as the difference between the experimental mass [8] and the theoretical mass [29]. The two parameters T and E₀ were determined so as to connect ρ_F and ρ_T smoothly at an appropriate matching energy E_m. The parameters used in this work are listed in Table 6 for individual nuclei, together with the energies of the highest discrete levels $E_{\text{level}}^{\text{highest}}$.

Table 6. Level density parameters for each nucleus.

	a(*)	Δ	$E_{\text{sh}}$	T	E0	Em	$E_{\text{level}}^{\text{highest}}$
Nuclei	1/MeV	MeV	MeV	MeV	MeV	MeV	MeV
^133Te	16.646	1.041	−5.789	0.915	−0.848	10.932	2.024
^132Te	16.543	2.089	−4.646	0.879	−0.096	10.944	3.015
^131Te	16.441	1.048	−3.422	0.828	−1.216	8.719	1.952
^130Te	16.337	2.105	−2.610	0.813	−0.401	9.581	2.878
^129Te	16.234	1.057	−1.461	0.737	−1.056	7.086	1.110
^128Te	16.131	2.121	−0.941	0.739	−0.223	8.301	2.749
^127Te	16.028	1.065	0.102	0.681	−0.999	6.359	1.568
^126Te	15.924	2.138	0.364	0.662	0.232	7.099	2.812
^125Te	15.820	1.073	1.249	0.664	−1.161	6.275	1.322
^124Te	15.717	2.155	1.308	0.643	0.178	6.962	2.483
^123Te	15.613	1.082	1.993	0.629	−0.932	5.774	1.427
^122Te	15.509	2.173	1.907	0.589	0.655	6.145	2.204
^121Te	15.405	1.091	2.469	0.665	−1.463	6.440	1.173
^120Te	15.300	2.191	2.144	0.553	1.012	5.595	2.567
^119Te	15.196	1.100	2.607	0.641	−1.135	5.983	1.185
^118Te	15.091	2.209	2.046	0.660	−0.016	7.222	1.517
^132Sb	16.543	0.000	−7.202	1.031	−2.417	14.670	0.529
^131Sb	16.441	1.048	−5.794	0.992	−1.873	13.703	2.166
^130Sb	16.337	0.000	−4.508	0.871	−2.060	8.454	1.097
^129Sb	16.234	1.057	−3.542	0.810	−0.811	8.051	2.263
^128Sb	16.131	0.000	−2.378	0.581	−0.287	2.810	0.833
^127Sb	16.028	1.065	−1.606	0.695	−0.435	6.078	2.325
^126Sb	15.924	0.000	−0.622	0.726	−2.250	5.898	0.128
^125Sb	15.820	1.073	−0.073	0.472	0.909	2.998	2.299
^124Sb	15.717	0.000	0.761	0.590	−1.233	3.766	0.804
^123Sb	15.613	1.082	1.087	0.541	0.216	4.154	1.896
^122Sb	15.509	0.000	1.676	0.572	−1.283	3.652	0.748
^121Sb	15.405	1.091	1.842	0.540	0.082	4.271	2.048
^120Sb	15.300	0.000	2.181	0.525	−0.944	3.017	0.858
^119Sb	15.196	1.100	2.038	0.603	−0.521	5.218	1.547
^118Sb	15.091	0.000	2.134	0.578	−1.370	3.718	0.890
^130Sn	16.337	2.105	−7.042	1.041	−0.466	16.910	2.598
^129Sn	16.234	1.057	−5.562	0.907	−0.654	10.277	1.222
^128Sn	16.131	2.121	−4.644	0.904	−0.163	11.371	2.642
^127Sn	16.028	1.065	−3.311	0.834	−1.134	8.601	1.243
^126Sn	15.924	2.138	−2.605	0.784	0.189	8.613	2.795
^125Sn	15.820	1.073	−1.442	0.708	−0.554	6.273	1.259
^124Sn	15.717	2.155	−1.003	0.664	0.824	6.637	3.011
^123Sn	15.613	1.082	−0.022	0.693	−0.917	6.354	1.301
^122Sn	15.509	2.173	0.159	0.609	1.011	5.990	3.036
^121Sn	15.405	1.091	0.968	0.597	−0.188	4.903	1.571
^120Sn	15.300	2.191	0.882	0.621	0.725	6.337	2.931
^119Sn	15.196	1.100	1.467	0.611	−0.426	5.202	1.633
^118Sn	15.091	2.209	1.180	0.612	0.795	6.211	3.057
^117Sn	14.987	1.109	1.442	0.611	−0.344	5.115	1.710
^116Sn	14.882	2.228	1.077	0.624	0.794	6.301	3.315

3.2.3. Gamma-ray transition

The γ-ray transmission coefficients are related to the photoabsorption cross sections by the detailed balance. The photoabsorption cross section is usually approximated by various Lorentzian shapes. In the present calculation, a modified Lorentzian proposed by Plujko et al. [30] was used for $\text{E}1$ radiation, since its use is recommended by RIPL-3 [11]. The resonance energy (E₀), resonance width (Γ₀), and peak cross section (σ₀) are given [31] by (10) $$\begin{array}{ccc}\hfill E_0& =& 31.2A^{-1/3}+20.6A^{-1/6}\mathrm{MeV},\hfill \end{array}$$(10) (11) $$\begin{array}{ccc}\hfill {\Gamma }_0& =& 0.026E_0^{1.91}\mathrm{MeV},\hfill \end{array}$$(11) (12) $$\begin{array}{ccc}\hfill {\sigma }_0& =& 1.2\times 120NZ/\left(A\pi {\Gamma }_0\right)\mathrm{mb}.\hfill \end{array}$$(12) The standard Lorentzian form was used to describe the M1 and E2 radiations for which the parameters were taken from the work of Kopecky and Uhl [32].

In cases where measured capture cross sections are available, the γ-ray strength functions are renormalized so that the calculated cross sections reproduce these data. The $\text{s}$-wave γ-ray strength functions are compared in Table 7 with the values recommended by Mughabghab [14] and those used in the JENDL-4.0 evaluation. The values of ^{121, 123, 124, 125, 126, 127, 129, 131}Te were obtained after renormalization using measurements.

Table 7. Gamma-ray strength functions for s-wave neutron in units of 10⁻⁴.

Compound	Present
nuclei	work	Mughabghab [14]	JENDL-4.0
^121Te	14.04	–	7.41
^122Te	146.6	–	–
^123Te	5.774	5.1 ±0.6	6.36
^124Te	61.29	48.2 ±4.9	79.3
^125Te	2.395	2.7±0.6	2.59
^126Te	26.85	24.8±2.2	32.8
^127Te	0.9730	0.83± 0.32*	1.26
^128Te	23.73	–	132.0
^129Te	0.3625	0.23±0.07*	0.485
^130Te	7.830	–	97.1
^131Te	0.1490	–	0.141
^132Te	2.078	–	–
^133Te	0.05886	–	0.0361

*Calculated from the s-wave average level-spacing and the s-wave average radiative width given in [14].

3.2.4. Pre-equilibrium parameters

The version 0.8.4 of CCONE incorporates pre-equilibrium processes for the (n, γ) and first-step binary reactions such as (n, n′), (n, p), (n, d), (n, t), $\left(n{,}^3\text{He}\right)$, and (n, α). In the two-exciton pre-equilibrium formalism, the single-particle state densities for protons and neutrons in a residual nucleus are described [33] by (13) $$\begin{array}{c}\hfill g_{\pi }=\frac{C_{0}Z}{15}{\mathrm{MeV}}^{-1},\end{array}$$(13) (14) $$\begin{array}{c}\hfill g_{\nu }=\frac{C_{0}N}{15}{\mathrm{MeV}}^{-1},\end{array}$$(14) respectively. The parameter C₀, of which the default value is unity, is changed so that the calculations reproduce measured neutron and proton emission data such as (n, 2n) and (n, p) in the present work. The (n, 2n) reaction is a ternary reaction, which proceeds from (n, n′) by sequential decay in the framework of the multi-step statistical model. Hence, the (n, 2n) cross section is sensitive to C₀ for the residual nucleus in the (n, n′) reaction. More experimental data are available for (n, 2n) than for (n, n′) in the nuclides presently concerned. In addition to the C₀'s, the parameters for knockout and pickup reactions [34] were adjusted for α emission. The pre-equilibrium parameters used are listed in Table 8, where C₀'s are given only for the residual nuclei in the (n, n′) and (n, p) reactions, and are assumed to be unity for those in the other first-step binary reactions.

Table 8. Pre-equilibrium parameters.

	C0		Knockout	Pickup
Target nuclei	(n, n′)	(n, p)	for (n, α)	for (n, α)
^120Te	1.03	1.40	1.0	0.0
^121mTe	1.00	1.00	1.0	0.0
^122Te	1.05	0.94	1.0	0.0
^123Te	1.50	0.80	1.0	0.0
^124Te	1.00	0.97	1.5	1.5
^125Te	1.00	1.00	1.5	1.5
^126Te	0.80	0.70	3.0	3.0
^127mTe	1.00	1.00	3.0	3.0
^128Te	1.30	0.70	8.0	8.0
^129mTe	1.00	1.00	8.0	8.0
^130Te	0.90	0.75	10.0	10.0
^131mTe	1.00	1.00	10.0	10.0
^132Te	1.00	1.00	10.0	10.0

4. Comparison with experimental data and other evaluated data in the fast-neutron energy region

The presently evaluated data are compared with experimental data and other evaluated data. The experimental data are taken from the EXFOR database [35], which is internationally maintained. Concerning the general purpose evaluated nuclear data, there are three major libraries, i.e., JENDL-4.0 [1], evaluated nuclear data file (ENDF)/B-VII.1 [36], and joint evaluated fission and fusion (JEFF)-3.2 [37]. However, the fast-neutron cross sections of ENDF/B-VII.1 except for ^{130, 132}Te were taken from JENDL-3.3 [5], which means that those of ENDF/V-II.1 data are equivalent to the JENDL-4.0 data. To avoid duplication, identical evaluated data are not drawn in the following figures. Except ^{122, 124, 131m}Te in JEFF-3.2, the general purpose libraries do not contain activation cross sections leading to the ground or isomeric state. The activation cross section file JEFF-3.1/A [38] is used for comparison with such activation data. We do not discuss most of the reactions where experimental data are unavailable, since it is difficult to judge whether the present evaluation is reasonable without measurements.

Figures 3–10 show the evaluated capture cross sections together with experimental data and other evaluated data. It is found from the figures that the present evaluations reproduce measured cross sections except for ¹²⁰Te. The JEFF-3.2 evaluations deviate considerably from the measurements for ^{122, 123, 124, 126}Te. In JEFF-3.2, the RRPs, which are given below 26 and 54 keV for ¹²²Te and ¹²⁴Te, respectively, cannot yield proper energy-average cross section below several tens of keV. An explanation is required for ¹²⁰Te. There exists only one measurement of Trofimov [39] using mono-energetic neutrons in the keV region. Recently, Dillmann et al. [40] measured spectrum-averaged cross sections using the p-⁷Li continuous neutron source. The Karlsruhe Astrophysics Database of Nucleosynthesis in Stars (KADoNIS) [41] adopted their data and provides a 30-keV Maxwellian-average cross section (MACS30). The MACS30 is considered as more reliable than the data of Trofimov in the keV region. Therefore, the MACS30 value was used to renormalize the γ-ray strength function of the compound nucleus ¹²¹Te, as seen in Figure 11. Although experimental data are unavailable for the capture cross sections of the remaining isotopes, they are shown in Figure 12, where the JENDL-4.0 data are also depicted for ^{127m, 129m, 132}Te.

Figure 3. Capture cross section of ¹²⁰Te.

Figure 4. Capture cross section of ¹²²Te.

Figure 5. Capture cross section of ¹²³Te.

Figure 6. Capture cross section of ¹²⁴Te.

Figure 7. Capture cross section of ¹²⁵Te.

Figure 8. Capture cross section of ¹²⁶Te.

Figure 9. Capture cross section of ¹²⁸Te.

Figure 10. Capture cross section of ¹³⁰Te.

Figure 11. Maxwellian-averaged capture cross section of ¹²⁰Te.

Figure 12. Capture cross sections of ^{121m, 127m, 129m, 131m, 132}Te.

Figure 13. ¹²⁰Te(n, 2n)¹¹⁹Te reaction cross section.

Figure 14. ¹²²Te(n, 2n)¹²¹Te reaction cross section.

Figure 15. ¹²⁴Te(n, 2n)^123mTe (T_1/2 = 119.2 d) reaction cross section.

Figure 16. ¹²⁶Te(n, 2n)^125mTe (T_1/2 = 57.40 d) reaction cross section.

Figure 17. ¹²⁸Te(n, 2n)^127mTe (T_1/2 = 106.1 d) reaction cross section.

The total (n, 2n) cross sections (the sum of the ground and isomeric state production) of ^{120, 122}Te and isomeric state productions of ^{124, 126, 128, 130}Te are illustrated in Figures 13 and 14 and in Figures 15–18, respectively. On the whole, the presently evaluated cross sections are in good agreement with available experimental data. The JENDL-4.0 evaluations of ^{120, 122}Te drastically decrease above 18 MeV, although no measurements are available in that energy region. The isomeric state production in JEFF-3.1/A and JEFF-3.2 is also in fair agreement with the measurements within experimental uncertainties around 14 MeV.

Figure 18. ¹³⁰Te(n, 2n)^129mTe (T_1/2 = 33.6 d) reaction cross section.

The charged-particle emission cross sections are shown in Figures 19–25. As for ^{122, 126}Te(n, p), the JENDL-4.0 evaluations deviate from the measurements at 14 MeV, while the present and JEFF-3.2 ones agree with them. Concerning ¹²⁴Te(n, p), all the evaluations are almost consistent with one another and with the data measured by Zhou et al. [42], although a value of 149 ±40 mb measured by Brzosko et al. [43], which would be located outside Figure 20, is considerably larger than the evaluations. The presently evaluated ¹²⁸Te(n, p)^128gSb and ¹³⁰Te(n, p)^130mSb cross sections almost reproduce the measurements around 14 MeV. The pre-equilibrium α emission was enhanced for heavier targets in the present evaluation, as seen from Table 8. This increases the cross sections for ¹²⁸Te(n, α)^125mSn and ¹³⁰Te(n, α)^127mSn above 14 MeV, as compared with the JEFF-3.1/A evaluations.

Figure 19. ¹²²Te(n, p)¹²²Sb reaction cross section.

Figure 20. ¹²⁴Te(n, p)¹²⁴Sb reaction cross section.

Figure 21. ¹²⁶Te(n, p)¹²⁶Sb reaction cross section.

Figure 22. ¹²⁸Te(n, p)^128gSb (T_1/2 = 9.01 h) reaction cross section.

Figure 23. ¹³⁰Te(n, p)^130mSb (T_1/2 = 6.3 m) reaction cross section.

Figure 24. ¹²⁸Te(n, α)^125mSn (T_1/2 = 9.52 m) reaction cross section.

Figure 25. ¹³⁰Te(n, α)^127mSn (T_1/2 = 4.13 m) reaction cross section.

The angular distributions of neutrons elastically scattered from elemental Te are illustrated in Figure 26. The present evaluation agrees with available experimental data, whereas the JENDL-4.0 and ENDF/B-VII.1 ones overestimate the measurements at 14 MeV in the angular region from 40° to 130°. This fact proves the reliability of the neutron optical model parameters presently used. The JEFF-3.2 evaluation cannot reproduce the measurements at all incident energies, since isotropic distributions are given for the naturally occurring isotopes ^{120, 123, 125, 126, 130}Te.

Figure 26. Angular distributions of neutrons elastically scattered from elemental Te.

The double-differential neutron emission spectra from elemental Te are illustrated in Figure 27 at 70° at an incident energy of 14.1 MeV. As seen from Table 5, the present coupled-channel calculation took account of only rotational bands in target nuclei. In order to enhance the inelastic scattering to 3⁻ vibrational states in even Z–even N targets, DWBA calculations were performed using the deformation parameters given in Table 9. The β₃ values were determined by considering the compilation of Kibédi and Spear [44]. In the calculations, the neutron potentials mentioned in Section 3 were used as spherical ones. The presently evaluated spectra reproduce the data measured by Watanabe et al. [45] very well. The other evaluated data obviously underestimate the measurements in the energy region from 6 to 13 MeV. The JEFF-3.2 evaluation overestimates even the elastic peak at 14 MeV, which comes from the fact that isotropic elastic angular distributions were adopted for many Te isotopes by JEFF-3.2.

Figure 27. Neutron emission spectra for elemental Te at 14.1 MeV.

Table 9. Deformation parameters for DWBA calculations of 3⁻ states.

Target	Excitation	Deformation
nuclei	energy (MeV)	parameters
^120Te	2.461	β3 = 0.132
^122Te	2.197	β3 = 0.132
^124Te	2.294	β3 = 0.130
^126Te	2.386	β3 = 0.131
^128Te	2.494	β3 = 0.110
^130Te	2.730	β3 = 0.100
^132Te*	2.488	β3 = 0.100

*¹³²Te is not a naturally occurring isotope.

5. Concluding remarks

The neutron nuclear data on Te isotopes were evaluated in the energy region from 10⁻⁵ eV to 20 MeV. The RRPs remain unchanged from JENDL-4.0. The thermal capture cross section of ¹²⁰Te was modified by taking account of the latest measurement, while those of several isotopes were determined from the simplified formula. The URPs were obtained for self-shielding calculations by fitting to the total and capture cross sections calculated from the nuclear models.

Above the resolved resonance region, the present evaluation is based on the statistical model using the CCONE code. The neutron transmission coefficients were obtained by the RRM-CC, together with the total cross sections, the shape elastic scattering cross sections, and the direct-reaction components of the inelastic scattering cross sections. The neutron potentials were found to be reliable by comparing with measured total cross sections and elastic angular distributions for elemental Te. DWBA calculations were made to enhance the inelastic scattering to 3⁻ vibrational levels in even Z–even N targets, of which contributions were remarkable in neutron emission spectra at 14 MeV. On the whole, the presently evaluated cross sections are in good agreement with measurements, and better than other evaluated data. The data on ^{121m, 131m}Te, which were missing in JENDL-4.0, were newly evaluated. Moreover, γ-ray production data were evaluated for all nuclides, while these data were given only for ¹³²Te in JENDL-4.0. The evaluated data are compiled into ENDF-formatted data files for the next release of JENDL general purpose file.

Acknowledgements

The author would like to thank the members of the Nuclear Data Center, Japan Atomic Energy Agency, for their helpful comments on this work. He is also grateful to Dr. I. Dillmann for providing information on KADoNIS.