Theoretical calculation of neutron cross sections for ^{90,91,92,94,96}Zr in the incident energy range between 200 keV and 20 MeV

ABSTRACT

Neutron cross sections of ^{90,91,92,94,96}Zr were calculated in the incident energy (E_n) range from 200 keV to 20 MeV for the revision of the 4th version of the Japanese Evaluated Nuclear Data Library (JENDL-4.0). The calculation was carried out by using conventional nuclear reaction models such as the spherical optical model, the distorted wave Born approximation, preequilibrium models, and the multi-step statistical model. Parameter values of these nuclear models were adjusted with the aid of experimental cross sections which were published after the JENDL-4.0 evaluation. Cross sections were computed for total, elastic and inelastic scattering, (n, γ), (n, 2n), (n, p), (n, α), (n, nα), and (n, x) = (n, d) + (n, np) reactions, and they were almost consistent with the experimental data. The cross sections were also estimated for the metastable states with the half-life larger than 1 sec. The obtained results well reproduced measured cross sections for the reactions ⁹⁰Zr(n, 2n)^89mZr, ⁹¹Zr(n, x)^90mY and ⁹¹Zr(n, nα)^87mSr.

KEYWORDS:

• Neutron cross section
• Zr isotope
• nuclear data evaluation
• JENDL-4.0

1. Introduction

The fourth version of Japanese Evaluated Nuclear Data Library (JENDL-4.0 [1]) was released in 2010. In this library, neutron nuclear data are available for 406 nuclides in the incident energy range from 10${ }^{-5}$ eV to 20 MeV. In the JENDL-4.0 project, effort was made for improving the data of fission products and minor actinides, to contribute to innovative reactor designs and nuclear transmutation analyses. Moreover, covariance data were estimated mainly in actinides.

At present, we are engaged in the revision of JENDL-4.0. A part of our activity is being carried out for structural materials of nuclear power plants, and cross sections for metastable states will be supplemented substantially to the library for the evaluation of radioactivities.

Zirconium is an important structural material in nuclear reactors. Natural zirconium (${ }^{nat}$Zr) is composed of five isotopes ${ }^{90,91,92,94,96}$Zr with the abundances 51.45% (${ }^{90}$Zr), 11.22% (${ }^{91}$Zr), 17.15% (${ }^{92}$Zr), 17.38% (${ }^{94}$Zr), and 2.80% (${ }^{96}$Zr). Zirconium alloys are being employed in fuel rod cladding and channel boxes of fuel assemblies of light-water reactors, owing to the small absorption cross section for neutrons. Also, CuCrZr alloy is expected as a heat sink material of Tokamak fusion reactors because of its high thermal conductivity. Therefore, reliable nuclear data of Zr isotopes are required to investigate how these materials are affected in the reactors.

Under the current JENDL project,we have evaluated resonance parameters of Zr isotopes [2] using recent experimental data. In the next version of JENDL-4.0, the resonance parameters of ${ }^{92,94,96}$Zr will be updated on the basis of the measurements by Tagliente et al. [3–5] Also, above the resonance region, there are experimental cross sections [6–11] which were published after the JENDL-4.0 evaluation. Especially, Semkova et al. provided activation cross sections for thirteen reactions ${ }^{90}$Zr$(n,\alpha )$${ }^{87m}$Sr, ${ }^{90}$Zr$(n,x)$${ }^{89m}$Y ($(n,x)=(n,d)+(n,np)$), ${ }^{90}$Zr $(n,p)$${ }^{90m}$Y, ${ }^{90}$Zr$(n,2n)$${ }^{89}$Zr, ${ }^{90}$Zr$(n,2n)$${ }^{89m}$Zr, ${ }^{91}$Zr $(n,n\alpha )$${ }^{87m}$Sr, ${ }^{91}$Zr$(n,x)$${ }^{90m}$Y, ${ }^{91}$Zr$(n,p)$${ }^{91m}$Y, ${ }^{92}$Zr $(n,x)$${ }^{91m}$Y, ${ }^{92}$Zr$(n,p)$${ }^{92}$Y, ${ }^{94}$Zr$(n,\alpha )$${ }^{91}$Sr, ${ }^{94}$Zr $(n,x)$${ }^{93}$Y, ${ }^{94}$Zr$(n,p)$${ }^{94}$Y, including metastable state production [6]. Frehaut revised the $(n,2n)$ cross section for natural zirconium ${ }^{nat}$Zr [10]. Filatenkov published activation cross sections in eight reactions ${ }^{90}$Zr$(n,\alpha )$${ }^{87m}$Sr, ${ }^{90}$Zr$(n,p)$${ }^{90m}$Y, ${ }^{90}$Zr$(n,x)$${ }^{89m}$Y, ${ }^{90}$Zr$(n,2n)$${ }^{89}$Zr, ${ }^{90}$Zr$(n,2n)$${ }^{89m}$Zr, ${ }^{92}$Zr$(n,p)$${ }^{92}$Y, ${ }^{94}$Zr$(n,\alpha )$${ }^{91}$Sr, ${ }^{96}$Zr$(n,2n)$${ }^{95}$Zr [11]. Regarding some reactions, theoretical approaches have been performed to explain the experimental data [12,13]. In the present study, we calculated cross sections in the incident energy $(E_n)$ range between 0.2 MeV (above the resolved resonance region) and 20 MeV in order to improve the JENDL-4.0 data, with the aid of these measured data. We also attempted to evaluate cross sections for the reactions ${ }^{90}$Zr$(n,x)$${ }^{89m}$Y, ${ }^{90}$Zr$(n,p)$${ }^{90m}$Y, ${ }^{91}$Zr$(n,n\alpha )$${ }^{87m}$Sr, ${ }^{91}$Zr$(n,x)$${ }^{90m}$Y, ${ }^{91}$Zr$(n,p)$${ }^{91m}$Y, ${ }^{92}$Zr$(n,x)$${ }^{91m}$Y, which were not considered in our previous work [14] and not included in JENDL-4.0.

This paper is structured as follows. The computational method and parameter values are mentioned in Section 2. Comparison of the obtained results with experimental data is presented in Section 3. Section 4 summarizes the conclusions of this study.

2. Computational method

Neutron cross sections were calculated by the CCONE code (Version 0.8.5.1) [15,16]. The cross sections were computed using (1) the spherical optical model, (2) the distorted-wave Born approximation (DWBA), (4) the two-component exciton preequilibrium model, and (3) the multi-step statistical model. The Kalbach particle-pickup and knockout processes [17] and the $(n,\gamma )$ reaction model of Akkermans and Gruppelaar [18] were added to the preequilibrium calculation. In the statistical model calculation, the effect of width fluctuation was taken into account in $E_n\le$ 5 MeV [19,20].

2.1. Parameter determination

Parameters values were determined similarly with our previous work [14]. Although some contents in [14] are repeated, we mention the definition of major parameters for this paper.

2.1.1. Optical model

The Zr isotopes, which appear near the double magic numbers, were assumed to be spherical. The global optical model potential (OMP) proposed by Kunieda et al. [21] were employed for neutrons and protons. In this study, radius parameters for incident neutrons have been tuned slightly from 1.21 to 1.215 fm in the ${ }^{90,91,92,94}$Zr target nuclei. Figures 1 and 2 show the total and elastic scattering cross sections, respectively. The compound elastic scattering component is included in the cross section. In Figures 1 and 2, the cross sections obtained from the original global spherical OMPs (Global) and the coupled-channels computations based on the rigid-rotor model (RRM-CC) by Kunieda et al. [21] are also shown. The OMPs of Kunieda et al. have been developed taking into account the measured neutron strength functions related to average neutron widths in the resolved resonance region. With the modified global OMP, experimental cross sections for ${ }^{nat}$Zr can be reproduced as with the RRM-CC computations by Kunieda et al. [21].

Figure 1. Total cross sections for ${ }^{90,91,92,94,nat}$Zr.

Figure 2. Elastic scattering cross sections for ${ }^{90,92,94,nat}$Zr.

Transmission coefficients for $\alpha$ particles were computed using the global OMP proposed by Avrigeanu et al. [22]. The global OMP of Lohr and Haeberli [23,24] was adopted for deuterons. The global OMPs of Becchetti and Greenlees [24,25] were used for tritons and ${ }^3$He.

2.1.2. DWBA

Direct inelastic scattering cross sections were computed on the basis of the experimental data of Wang and Rapaport [26], the JENDL-3.2 [27] and JENDL-3.3 [28] evaluations. Table 1 shows the values of the deformation parameter ($\beta$) accompanied by the excitation energy ($E_X$), spin ($J$), parity ($\pi$), and the angular momentum transfer quantum number ($l$). In Table 1, the deformation parameters derived by Wang and Rapaport have been adjusted for reproducing the measured angular distributions with the present OMPs. Figure 3 shows several angular distributions for $E_n$ = 8 MeV. In Table 1, pseudo states have been introduced in ${ }^{92,94}$Zr. The parameter values were taken from JENDL-3.2 [14,27]. Cross sections for the pseudo states were added to the calculated neutron continuum spectra using the Gaussian distribution with the standard deviation of 0.6 MeV in order to reproduce the measured continuum spectra. The deformation parameters for ${ }^{96}$Zr were based on the JENDL-3.3 evaluation [14,28].

Table 1. DWBA parameters for ${ }^{90,91,92,94,96}$Zr

	$E_X$(MeV)	$J\pi$	$l$	$\beta$
${ }^{90}$Zr	2.186	2+	2	0.090
	2.319	5$-$	5	0.078
	2.748	3$-$	3	0.185
	3.309	2+	2	0.10
${ }^{91}$Zr	1.205	1/2+	2	0.040
	1.466	5/2+	2	0.017
	1.882	7/2+	2	0.045
	2.042	3/2+	2	0.028
	2.131	9/2+	2	0.038
	2.170	11/2$-$	3	0.094
${ }^{92}$Zr	0.9345	2+	2	0.113
	1.495	4+	4	0.044
	1.847	2+	2	0.048
	2.067	2+	2	0.039
	2.340	3$-$	3	0.16
	4.0	pseudo	2	0.08
	4.5	pseudo	2	0.08
${ }^{94}$Zr	0.919	2+	2	0.11
	1.470	4+	4	0.059
	1.671	2+	2	0.053
	2.058	3$-$	3	0.15
	2.151	2+	2	0.054
	4.0	pseudo	2	0.08
	4.5	pseudo	2	0.08
${ }^{96}$Zr	1.751	2+	2	0.12
	1.897	3$-$	3	0.17

Figure 3. Angular distributions of neutron inelastic scattering for ${ }^{90,91,92,94}$Zr in the center of mass (CM) system at $E_n$ = 8 MeV.

2.2. Preequilibrium model

In the two-component exciton model calculation, the single-particle state density parameter was initially set to $g_{\nu }$ = $N$/15 for neutrons and $g_{\pi }$ = $Z$/15 for protons, where $N$ and $Z$ are the neutron and proton numbers of the compound nucleus. Some density parameters were modified by employing the correction factor $C_0$, in order to improve the agreement with measured data. The $C_0$ values are shown in the eighth column of Table 2. Both the $g_{\nu }$ and $g_{\pi }$ parameters were multiplied by $C_0$ in the calculation.

Table 2. Nuclear-level parameters and correction factors

	$E_{discrete}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$T$	$E_0$	$E_M$	$a^{\ast }$	$C_{sp}$	$C_0$
	MeV	MeV	MeV	MeV	MeV${ }^{-1}$
${ }^{97}$Zr	1.400	0.5882	0.4479	4.106	12.181	1.0	1.0
${ }^{96}$Zr	1.897	0.7059	0.6876	6.950	12.403	1.0	0.7
${ }^{95}$Zr	2.372	0.7279	$-$0.0395	5.469	11.637	1.4	1.0
${ }^{94}$Zr	2.151	0.7669	0.4687	7.586	12.185	1.0	1.2
${ }^{93}$Zr	2.458	0.7947	$-$0.6710	6.671	12.414	1.3	1.0
${ }^{92}$Zr	2.820	0.8508	0.5315	8.230	11.704	1.0	1.0
${ }^{91}$Zr	2.395	0.8232	0.0272	6.228	11.890	1.0	1.02
${ }^{90}$Zr	3.309	0.7907	1.946	6.578	11.748	1.0	0.9
${ }^{89}$Zr	2.151	0.8364	$-$0.5318	7.012	12.323	1.3	1.0
${ }^{96}$Y	0.932	0.4966	$-$0.3132	1.958	12.360	1.0	1.3
${ }^{95}$Y	2.021	0.4763	1.182	2.752	11.759	1.0	1.0
${ }^{94}$Y	0.724	0.3848	0.2067	0.774	13.1	1.0	0.95
${ }^{93}$Y	1.309	0.7505	$-$0.3510	5.778	11.549	1.0	1.4
${ }^{92}$Y	0.310	0.7319	$-$1.458	4.487	12.6	1.0	0.96
${ }^{91}$Y	1.547	0.8436	$-$0.5143	6.650	11.338	1.0	1.09
${ }^{90}$Y	1.417	0.7089	$-$0.3679	2.976	11.5	0.7	0.84
${ }^{89}$Y	3.248	0.6938	1.409	3.627	11.127	1.0	1.0
${ }^{88}$Y	0.766	0.8520	$-$1.643	5.422	11.495	1.0	1.0
${ }^{94}$Sr	3.047	0.6715	0.9663	6.494	12.185	1.0	1.0
${ }^{93}$Sr	1.238	0.6204	0.1804	4.623	12.762	1.0	1.0
${ }^{92}$Sr	2.088	0.7472	0.7359	7.218	11.967	1.0	1.0
${ }^{91}$Sr	1.368	0.6713	0.2433	4.893	12.543	1.0	1.0
${ }^{90}$Sr	2.674	0.8133	0.8175	7.712	11.748	1.0	1.0
${ }^{89}$Sr	2.079	0.8566	0.1184	6.194	10.953	1.0	1.0
${ }^{88}$Sr	3.585	0.8262	1.739	7.165	11.473	1.3	1.0
${ }^{87}$Sr	1.770	0.6797	0.4653	4.755	12.365	1.0	1.0
${ }^{86}$Sr	2.230	0.8537	0.4098	8.309	11.308	1.0	1.0

In evaluating the $\alpha$ particle emission, the pickup and knockout components were modified by multiplying the correction factor $C_{p,k}$. The same factor $C_{p,k}$ was applied to the pickup and knockout components. In this study, $C_{p,k}$ was selected to 3.0, 0.6, 2.7, 5.0 and 5.0 for ${ }^{90,91,92,94,96}$Zr, respectively.

2.3. Statistical model

2.3.1. Level data

Nuclear discrete level data were taken from the Reference Input Parameter Library, version 3, RIPL-3 [29]. Nuclear-level densities were applied above the highest discrete state to describe continuum states. On the basis of the Gilbert-Cameron formulation [30], the level density of the constant temperature form (${\rho }_T$) was used for lower excitation energies, and the Fermi-gas form (${\rho }_F$) was employed for higher energies. The constant temperature density ${\rho }_T$ is given in the form ${\rho }_T(U)$=exp$\left\{(U-E_0)/T\right\}/T$ as a function of excitation energy $U$, where the parameters $T$ and $E_0$ were determined to connect ${\rho }_T$ and ${\rho }_F$ smoothly at an appropriate matching energy $(E_M)$. In the Fermi-gas density, the $U$ dependent Mengoni-Nakajima density parameter [31] was employed together with their pairing energy and spin cutoff function. Table 2 shows the highest discrete energy $E_{discrete}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ applied in the calculation, the values of $T$, $E_0$, $E_M$ and the asymptotic level density parameter $a^{\ast }$ in the Mengoni-Nakajima density, for each nucleus. In Table 2, most $a^{\ast }$ values were taken from [31], while the values in ${ }^{90,92,94}$Y were altered to reproduce the experimental cross sections. Moreover, the correction factor $C_{sp}$ has been introduced to adjust the spin cutoff function by multiplying $C_{sp}$. The $C_{sp}$ values are shown in the seventh column of Table 2.

2.3.2. $\gamma$-ray strength function

The standard Lorentzian (SLO) form was adopted for the $\gamma$-ray strength functions in the E1, M1 and E2 radiations, since the $E_n$ dependence of measured capture cross sections was better reproduced by SLO than the generalized Lorentzian (GLO) [32]. The values of peak cross section, position and width parameters in the Lorentzian functions have been estimated using the systematic formulas built in the CCONE code [16] on the basis of RIPL-1 [33] and RIPL-2 [34]. The Lorentzian functions for ${ }^{90,91,92,94,96}$Zr targets have been normalized so as to obtain the $(n,\gamma )$ cross sections (${\sigma }_{\gamma }$) measured by Ohgama et al. [35,36] and Katabuchi et al. [7] by adopting ${\sigma }_{\gamma }$ = 6.6 mb at $E_n$=550 keV for ${ }^{90}$Zr [35], 32.3 mb at 70 keV for ${ }^{91}$Zr [36], 11.6 mb at 550 keV for ${ }^{92}$Zr [36], 9.3 mb at 550 keV for ${ }^{94}$Zr [35], and 19.2 mb at 21 keV for ${ }^{96}$Zr [7]. The contributions of M2 and E2 components relative to E1 were estimated by $M1/E1=0.45/(1.25A^{1/3}$)${ }^2$ [37] and $E2/E1=0.0008$ [38], where $A$ is the nuclear mass number. The intensities of $\gamma$-rays decaying to the ground and first excited states from the primary compound state were multiplied by factors of 2 and 3 in ${ }^{90}$Zr, and by 3 and 1.5 in ${ }^{91}$Zr. Figure 4 illustrates the $\gamma$-ray spectra from the $(n,\gamma )$ reaction of ${ }^{90,91,92,94}$ Zr at $E_n$ =550 keV. The total $\gamma$-ray emission strength has been normalized with the measured cross section after scaling the $\gamma$-ray emission spectra. In Figure 4 the $\gamma$-ray spectra measured by Ohgama et al. [36,39] are well reproduced by the SLOs applied.

Figure 4. $\gamma$-ray spectra from ${ }^{90,91,92,94}$Zr $(n,\gamma )$ reactions at $E_n$ = 550 keV.

3. Results and comparison with experimental and evaluated data

Calculated results were compared with experimental data, the JENDL-4.0 evaluations in which our previous results [14] were reflected, and the evaluated values in other major libraries ENDF/B-VIII.0 [40], JEFF-3.2 (for the metastable states) [41], and JEFF-3.3 [42]. The experimental data were obtained from the EXFOR database [43] that is compiled by the International Network of Nuclear Reaction Data Centers coordinated by the IAEA Nuclear Data Section.

Figures 5-10 show the $(n,\gamma )$ cross sections for ${ }^{90,91,92,94,96,nat}$Zr. In Figures 5-10 cross sections were calculated down to 10 keV in order to compare with the measured average cross sections, where the upper limits of resolved resonances are 171, 26, 71, 53.5 and 100 keV for ${ }^{90,91,92,94,96}$Zr in JENDL-4.0 [1]. To obtain the $E_n$ dependence of the measured cross sections by Ohgama et al. [35,36], the correction factor $C_{sp}$ of the spin cutoff function in the statistical model was adjusted in ${ }^{93,95}$Zr nuclei. It can be seen that the computed cross sections accord well with the experimental data. In the ${ }^{90}$Zr$(n,\gamma )$${ }^{91}$Zr reaction, the present result is similar to the ENDF/B-VIII.0 and JENDL-4.0 evaluations. In ${ }^{91}$Zr$(n,\gamma )$${ }^{92}$Zr, the $E_n$ dependence of the present cross section is close to the ENDF/B-VIII.0 and JEFF-3.3 evaluations. The JENDL-4.0 data adopts the JENDL-3.3 evaluation [28], whose energy dependence could not be obtained within our model calculation. In Figure 7, it is seen that the measured ${ }^{92}$Zr$(n,\gamma )$${ }^{93}$Zr data by Ohgama et al. [36] are well reproduced by our calculation. In Figure 8, we could not obtain the ${ }^{94}$Zr$(n,\gamma )$${ }^{95}$Zr cross section which reproduces the experimental data for 550 keV by Ohgama et al. [35] and for 2.45 MeV by Prajapati et al. [8] simultaneously. In the ${ }^{92,94}$Zr target nuclei, the cross sections have been normalized at 550 keV using the data measured by Ohgama et al. [35,36] With these cross sections, the capture cross section of ${ }^{nat}$Zr in Figure 10 measured by Poenitz [44] are well reproduced around 1 MeV. In Figures 9 and 10, the plausible cross sections were obtained for ${ }^{96,nat}$Zr.

Figure 5. ${ }^{90}$Zr$(n,\gamma )$${ }^{91}$Zr cross section.

Figure 6. ${ }^{91}$Zr$(n,\gamma )$${ }^{92}$Zr cross section.

Figure 7. ${ }^{92}$Zr$(n,\gamma )$${ }^{93}$Zr cross section.

Figure 8. ${ }^{94}$Zr$(n,\gamma )$${ }^{95}$Zr cross section.

Figure 9. ${ }^{96}$Zr$(n,\gamma )$${ }^{97}$Zr cross section.

Figure 10. ${ }^{nat}$Zr$(n,\gamma )$ cross section.

In Figure 11, the capture cross section of ${ }^{93}$Zr, which is a long-lived fission product (LLFP) with a half-life (T${ }_{1/2}$) of 1.61 $\times$ 10${ }^6$ years, was estimated by employing the global OMP of Kunieda et al. [21] and nuclear-level parameters in Table 2. The cross section was normalized at 30 keV with the recommended value 95 $\pm$ 10 mb of Mughabghab [45]. JEFF-3.3 adopts the ENDF/B-VIII.0 evaluation. All the evaluations and our computation have the similar $E_n$ dependence up to 1 MeV. Above 1 MeV the present cross section decreases more rapidly than other evaluations. This comes from the fact that the $E_X$ = 949.8 keV excited state with $J\pi$ =4.5+ [29,46,47] was included in the target ${ }^{93}$Zr. If this state is not included, the resulting cross section is close to the JEFF-3.2 evaluation.

Figure 11. ${ }^{93}$Zr$(n,\gamma )$${ }^{94}$Zr cross section.

Figures 12-14 show the $(n,2n)$ cross sections for ${ }^{90,96,nat}$Zr. In ${ }^{90}$Zr, the cross sections for the metastable (m.s.) state are also shown. To obtain the cross sections, the correction factor $C_0$ of the single-particle state density in the preequilibrium model was adjusted. Also, the $C_{sp}$ value was selected in ${ }^{89}$Zr to produce the ${ }^{90}$Zr$(n,2n)$${ }^{89m}$Zr cross section (${ }^{89m}$Zr: $E_X$ = 587.82 keV, $J\pi$ = 1/2$-$, T${ }_{1/2}$ = 4.161 m [47]). In Figure 12, the ${ }^{90}$Zr$(n,2n)$${ }^{89,89m}$Zr cross sections are reproduced quantitatively by our calculation. In Figures 13 and 14, the calculated cross sections accord with the experimental data for ${ }^{96,nat}$Zr.

Figure 12. ${ }^{90}$Zr$(n,2n)$${ }^{89,89m}$Zr cross sections. (m.s.) stands for the metastable state.

Figure 13. ${ }^{96}$Zr$(n,2n)$${ }^{95}$Zr cross section.

Figure 14. ${ }^{nat}$Zr$(n,2n)$Zr cross section.

Figures 15-18 show the $(n,p)$ cross sections for ${ }^{90,91,92,94}$Zr. In this study the nuclear level parameters and correction factors for ${ }^{90}$Y have been adjusted by referring to the experimental ${ }^{90}$Zr$(n,p)$${ }^{90,90m}$Y and ${ }^{90}$Zr$(n,x)$${ }^{89,89m}$Y cross sections (${ }^{90m}$Y: $E_X$ = 681.67 keV, $J\pi$=7+, T${ }_{1/2}$ = 3.19 h. ${ }^{89m}$Y: $E_X$ = 908.97 keV, $J\pi$ = 9/2+, T${ }_{1/2}$ = 15.663 s [47]). Similarly, the parameters for ${ }^{91,92}$Y were selected when evaluating the ${ }^{91}$Zr$(n,p)$${ }^{91,91m}$Y, ${ }^{92}$Zr$(n,p)$${ }^{92}$Y and ${ }^{92}$Zr$(n,x)$${ }^{91m}$Y cross sections (${ }^{91m}$Y: $E_X$ = 555.58keV, $J\pi$ = 9/2+, T${ }_{1/2}$ = 49.71 m [47]). In Figure 15, the calculated ${ }^{90}$Zr$(n,p)$${ }^{90}$Y cross section is close to the ENDF/B-VIII.0 evaluation for $E_n\ge$ 14 MeV. In the ${ }^{90}$Zr$(n,p)$${ }^{90m}$Y reaction, the reproducibility of the measured data is slightly improved compared with the JEFF-3.2 evaluation. In Figure 16, we could not reproduce the measured ${ }^{91}$Zr$(n,p)$${ }^{91}$Y cross section in $E_n\le$ 10 MeV. To increase the cross section in this energy region as with ENDF/B-VIII.0 and JEFF-3.3, the level density of ${ }^{91}$Y in the statistical model could be increased, but that led to the unacceptably large cross section for $E_n\ge$ 10 MeV and overestimated the ${ }^{91}$Zr$(n,p)$${ }^{91m}$Y and ${ }^{91}$Zr$(n,x)$${ }^{90,90m}$Y (Figure 20) cross sections. In Figure 17, we could obtain the ${ }^{92}$Zr$(n,p)$${ }^{92}$Y cross section which is almost consistent with the experimental data. In Figure 18, the parameter values for the proton emission were selected to reproduce the experimental ${ }^{94}$Zr$(n,p)$${ }^{94}$Y and ${ }^{94}$Zr$(n,x)$${ }^{93}$Y (Figure 22) cross sections. The present ${ }^{94}$Zr$(n,p)$${ }^{94}$Y cross section is slightly larger than the measured cross section for $E_n\ge$ 17 MeV.

Figure 15. ${ }^{90}$Zr$(n,p)$${ }^{90,90m}$Y cross sections (m.s.) stands for the metastable state.

Figure 16. ${ }^{91}$Zr$(n,p)$${ }^{91,91m}$Y cross sections (m.s.) stands for the metastable state.

Figure 17. ${ }^{92}$Zr$(n,p)$${ }^{92}$Y cross section.

Figure 18. ${ }^{94}$Zr$(n,p)$${ }^{94}$Y cross section.

Figure 19. ${ }^{90}$Zr$(n,x)$${ }^{89,89m}$Y cross sections (m.s.) stands for the metastable state.

Figure 20. ${ }^{91}$Zr$(n,x)$${ }^{90,90m}$Y cross sections (m.s.) stands for the metastable state.

Figure 21. ${ }^{92}$Zr$(n,x)$${ }^{91,91m}$Y cross sections (m.s.) stands for the metastable state.

Figure 22. ${ }^{94}$Zr$(n,x)$${ }^{93}$Y cross section.

Figures 19-22 show the $(n,x)$ cross sections for ${ }^{90,91,92,94}$Zr. In each figure, the obtained $E_n$ dependence is almost consistent with the experimental data. In Figures 20 and 21, the present ${ }^{91}$Zr$(n,x)$${ }^{90}$Y and ${ }^{92}$Zr$(n,x)$${ }^{91}$Y cross sections are close to the JENDL-4.0 evaluations. In Figure 22, our result overlaps with the JEFF-3.3 evaluation. The $E_n$ dependence of the measured ${ }^{94}$Zr$(n,x)$${ }^{93}$Y cross section could be reproduced by JEFF-3.3 and the present computation. The JENDL-4.0 evaluation underestimated the cross section.

Figures 23-26 show the $(n,\alpha )$ cross sections for ${ }^{90,92,94,96}$Zr, and Figure 27 gives the ${ }^{91}$Zr$(n,n\alpha )$${ }^{87}$Sr cross section. In Figure 23, our ${ }^{90}$Zr$(n,\alpha )$${ }^{87m}$Sr (${ }^{87m}$Sr: $E_X$ = 388.53 keV, $J\pi$ =1/2$-$, T${ }_{1/2}$ = 2.815 h [47]) cross section is smaller than the experimental data for $E_n\le$ 15 MeV. We could not obtain the cross section which is almost flat for $E_n\ge$ 12 MeV by the calculation. In Figures 24 and 25, the computed cross sections for ${ }^{92,94}$Zr target nuclei accord well with the experimental cross sections. In Figure 26, the ${ }^{96}$Zr$(n,\alpha )$${ }^{93}$Sr cross section has been evaluated so as to be consistent with the experimental data by Ikeda et al. [48] as with ENDF/B-VIII.0. It is noted that the present result was obtained by multiplying the preequiriblium pickup and knockout components by the factor ($C_{p,k}$) of 5. In Figure 27, the computed ${ }^{91}$Zr$(n,n\alpha )$${ }^{87m}$Sr cross section agrees with the measured data. The JENDL-4.0 evaluation underestimated the cross section.

Figure 23. ${ }^{90}$Zr$(n,\alpha )$${ }^{87,87m}$Sr cross sections (m.s.) stands for the metastable state.

Figure 24. ${ }^{92}$Zr$(n,\alpha )$${ }^{89}$Sr cross section.

Figure 25. ${ }^{94}$Zr$(n,\alpha )$${ }^{91}$Sr cross section.

Figure 26. ${ }^{96}$Zr$(n,\alpha )$${ }^{93}$Sr cross section.

Figure 27. ${ }^{91}$Zr$(n,n\alpha )$${ }^{87,87m}$Sr cross sections (m.s.) stands for the metastable state.

Finally, we calculated the energy-differential and angle-energy double-differential cross sections (EDX and DDX) for emitted neutrons to test the validity of present work. The experimental data were taken from the EXFOR [43] data base. Figure 28 shows the neutron EDXs. The EDXs (and DDXs) were derived from ENDF/B-VIII.0, JEFF-3.3 and JENDL-4.0 by using the modified version of the PLDDX code [49]. In Figure 28, elastic and inelastic scattering cross sections for discrete states were added to the spectra using the Gaussian distribution with the standard deviation of 0.3 MeV. It can be seen that the experimental EDXs are reproduced by the present calculations as well as other evaluations. Figure 29 shows the neutron DDXs for $E_n=$ 14.1 MeV. The measured DDXs are reasonably reproduced by our calculation.

Figure 28. Neutron energy-differential cross sections for ${ }^{nat}$Zr.

Figure 29. Neutron double-differential cross sections for ${ }^{nat}$Zr at $E_n$=14.1 MeV.

4. Summary and concluding remarks

Toward the revision of JENDL-4.0, neutron cross sections of ${ }^{90,91,92,94,96}$Zr were computed in the energy range $E_n$ between 200 keV and 20 MeV. The calculation was performed on the basis of nuclear reaction models which deal with the compound, preequilibrium, and direct processes. The cross sections were computed for total, elastic and inelastic scattering $(n,\gamma )$, $(n,2n)$, $(n,p)$, $(n,\alpha )$, $(n,n\alpha )$, and $(n,x)=(n,d)+(n,np)$ reactions. The results accord well with the experimental data in most reactions. The cross sections for the reactions ${ }^{94}$Zr$(n,x)$${ }^{93}$Y and ${ }^{91}$Zr$(n,n\alpha )$${ }^{87}$Sr underestimated in JENDL-4.0 were improved by the present calculation. The results better than our previous work were derived by adjusting parameter values more precisely with correction factors and by referring to more experimental data published after the JENDL-4.0 evaluation. Also, cross sections were estimated for the metastable states which had not been considered in the JENDL-4.0 evaluation. In the consistent computation, we could obtained the results which reproduce the experimental data in the reactions ${ }^{90}$Zr$(n,2n)$${ }^{89m}$Zr, ${ }^{91}$Zr$(n,x)$${ }^{90m}$Y and ${ }^{91}$Zr$(n,n\alpha )$${ }^{87m}$Sr. Moreover, we estimated the cross section for ${ }^{93}$Zr$(n,\gamma )$${ }^{94}$Zr, which is important for the transmutation of LLFP, by using the parameter values obtained in the natural isotopes. These data will be applied for the revision of JENDL-4.0.

Acknowledgments

The author would like to thank Dr. K. Ohgama for kindly sending their numerical data of capture $\gamma$-ray spectra to us, and thank Dr. O. Iwamoto for providing CCONE version 0.8.5.1. The author is grateful to all members of JAEA/NDC for their helpful advice and encouragement during this work.

Disclosure statement

No potential conflict of interest was reported by the author.