Evaluation of neutron nuclear data on mercury isotopes

ABSTRACT

Neutron nuclear data on 10 isotopes of mercury have been evaluated for the next version of Japanese evaluated nuclear data library (JENDL) general-purpose file in the energy region from 10⁻⁵ eV to 20 MeV. Resolved resonance parameters (RRPs) for ^{200, 202}Hg were supplemented with the data which had not been considered in the previous library. Unresolved resonance parameters (URPs) were obtained by fitting to the total and capture cross sections calculated from nuclear models. A statistical model code was applied to evaluate cross sections above the resolved resonance region. Coupled-channel optical model parameters were employed for the interaction between neutrons and nuclei except ²⁰³Hg. Compound, pre-equilibrium, and direct-reaction processes were considered for cross-section calculation in the high energy region. The present results reproduce experimental data very well. The evaluated data are compiled into evaluated nuclear data file (ENDF) formatted data files.

KEYWORDS:

• Neutron nuclear data
• evaluation
• mercury isotopes
• cross section
• JENDL
• resonance parameter
• optical model
• statistical model

1. Introduction

Neutron nuclear data for mercury were evaluated [1] in 1997 for (Japanese evaluated nuclear data library)JENDL-3.3 [2]. The data were used to study neutron production rates, heat deposition, and shielding in a spallation-neutron target system for the J-PARC/MLF [3]. The latest library JENDL-4.0 [4] adopted almost the same data on mercury, although URPs were added. It is about 18 years since the data were evaluated. Recently, neutron-induced activation cross sections of mercury are required [5] for the decommissioning of light-water reactors, since mercury could be contained as an impurity in the structural materials of nuclear facility. Under such circumstances, it was decided to re-evaluate the data on mercury.

The present work was undertaken to evaluate the data on 10 mercury isotopes by considering the latest knowledge on experimental and theoretical nuclear physics. Evaluated are the total, elastic, and inelastic scattering, (n, γ), (n, p), (n, d), (n, t), (n, ³He), (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 mass table AME2012 [6], and are listed in Table 1, together with abundances [7] and half-lives [8].

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

	^195Hg	^196Hg	^197Hg	^198Hg	^199Hg	^200Hg	^201Hg	^202Hg	^203Hg	^204Hg
Abundances(%)	10.53 h^*	0.15	64.14 h^*	9.97	16.87	23.10	13.18	29.86	46.594 d^*	6.87
Q-values (MeV)
(n, γ)	8.898	6.786	8.485	6.663	8.029	6.230	7.754	5.995	7.493	5.669
(n, p)	2.352	0.096	1.382	−0.591	0.331	−1.481	−0.479	−2.210	−1.343	−3.258
(n, α)	10.939	8.301	9.868	7.488	8.747	6.565	7.889	5.691	6.978	4.697
(n, d)	−3.851	−4.322	−4.466	−4.878	−5.029	−5.473	−5.487	−6.009	−5.980	−6.611
(n, t)	−4.472	−6.492	−4.850	−6.693	−5.284	−6.800	−5.446	−6.984	−5.746	−7.216
(n, ^3He)	−3.378	−3.924	−4.605	−5.168	−5.984	−6.458	−7.132	−7.605	−8.388	−8.858
(n, 2n)	−6.887	−8.898	−6.786	−8.485	−6.663	−8.029	−6.230	−7.754	−5.995	−7.493
(n, np)	−6.075	−6.546	−6.690	−7.103	−7.253	−7.698	−7.712	−8.234	−8.205	−8.836
(n, nα)	2.277	2.040	1.516	1.383	0.825	0.718	0.334	0.135	−0.303	−0.514
(n, nd)	−10.729	−12.749	−11.107	−12.951	−11.541	−13.057	−11.704	−13.241	−12.004	−13.473
(n, 3n)	−16.079	−15.785	−15.684	−15.271	−15.148	−14.691	−14.259	−13.984	−13.749	−13.487
(n, 2np)	−12.954	−14.974	−13.332	−15.175	−13.766	−15.282	−13.928	−15.466	−14.228	−15.697

^*Half-lives are given for ^{195, 197, 203}Hg which are unstable.

This paper presents how the evaluation was performed. Section 2 deals with the resonance region. In Section 3, the computational method 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 summarises the conclusion.

2. Resonance region

The resolved and unresolved resonance regions are listed in Table 2 for each target nucleus. In JENDL-4.0, the RRPs of mercury were taken from the work of Mughabghab [9], after which Beer and Macklin [10] reported the RRPs of ^{198, 199, 200, 201, 202, 204}Hg in the energy region above 2.8 keV. In the present work, the RRPs of ^{200, 202}Hg were supplemented with the data of Beer and Macklin. Figure 1 shows the capture cross section of ²⁰²Hg in the resolved resonance region. As for ²⁰⁴Hg where no RRPs were given for JENDL-4.0, the data were taken from the work of Beer and Macklin, and a negative resonance was placed so that the calculated thermal capture cross section could reproduce the value recommended by Mughabghab [11]. The RRPs of ^{196, 198, 199, 201}Hg remain unchanged from JENDL-4.0. The RRPs of ^{198, 199, 201}Hg obtained by Beer and Macklin were not considered, since there exists a large gap between the lowest resonance of Beer and Macklin and the highest one of JENDL-4.0. New measurements would be required to bridge the gap for further improvements of RRPs. Concerning ^{195, 197, 203}Hg for which RRPs are unavailable, constant and 1/v energy dependences are assumed for the elastic scattering and capture cross sections, respectively, up to the lower limit of unresolved resonance region. Normalisation of the cross sections is done at thermal energy. The thermal scattering cross sections were calculated as 4πR′², 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 ^{195, 197, 203}Hg were estimated by using the simplified formula [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.

Figure 1. Capture cross section of ²⁰²Hg in the resolved resonance region.

Table 2. Resolved and unresolved resonance regions.

Target nuclei	Resolved resonance	Unresolved resonance
^195Hg	–	3 eV– 100 keV
^196Hg	10^−5 eV– 102.85 eV	102.85 eV– 400 keV
^197Hg	–	15 eV– 100 keV
^198Hg	10^−5 eV– 459.14 eV	459.14 eV– 400 keV
^199Hg	10^−5 eV– 968.0 eV	968.0 eV– 100 keV
^200Hg	10^−5 eV– 10 keV	10 keV– 300 keV
^201Hg	10^−5 eV– 753.5 eV	753.5 eV– 100 keV
^202Hg	10^−5 eV– 20 keV	20 keV– 400 keV
^203Hg	–	400 eV– 100 keV
^204Hg	10^−5 eV– 10 keV	10 keV– 400 keV

The ASREP code [13] was used to determine the URPs by fitting to the total and capture cross sections calculated with the nuclear models which will be described in Section 3. The URPs obtained are used only for self-shielding calculations, since the pointwise cross sections are given in the evaluated data files. As an example, the cross sections of ¹⁹⁷Hg reconstructed from the URPs are compared with the total and capture cross sections calculated from the nuclear models in Figure 2. In the parameter fitting with the ASREP code, 10% uncertainties in the nuclear model calculations were assumed. The reconstructed cross sections reproduce the nuclear model calculations very well.

Figure 2. Unresolved resonance region for n + ¹⁹⁷Hg.

3. Computational methods and procedures above resonance region

3.1. Nuclear models

In the previous evaluation of Hg, several independent codes were used to calculate cross sections above the resonance region. After that, nuclear models have been sophisticated, and reliable model parameters could be obtained by many systematic studies. Such progress enables one to predict the cross sections more accurately than before.

In this work, the CCONE code (Version 0.8.4) [14,15] , which was developed for the JENDL projects, was used for calculating the neutron-induced reaction cross sections of Hg 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 semi-direct effects on the radiative capture reaction, the pre-equilibrium capture was considered by using the γ-ray emission rate of Akkermans and Gruppelaar [16] extended to the two-component exciton pre-equilibrium model. The CCONE code is a comprehensive system designed for nuclear data evaluation, and it is similar to the TALYS [17] and EMPIRE [18] codes.

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 V(r) is defined as (1) $$\begin{array}{ccc}\hfill V\left(r\right)& =& -V_R{f}_R\left(r\right)-i\left\{W_V{f}_V\left(r\right)-4W_D{a}_D\frac{d}{dr}{f}_D\left(r\right)\right\}\hfill \\ & & +{\left(\frac{\hslash }{m_{\pi }c}\right)}^2(V_{\text{SO}}+i{W}_{\text{SO}})\frac{1}{r}\frac{d}{dr}{f}_{\text{SO}}\left(r\right)\mathit{L}·\mathit{\sigma },\hfill \end{array}$$(1) where L and σ are the orbital angular momentum and the Pauli matrix, respectively. The form factor f_i(r) is of a Woods–Saxon shape, i.e. (2) $$f_i\left(r\right)=\frac{1}{1+\exp \{(r-R_i)/a_i\}},i=R,D,V,SO,$$(2) where R_i is written as (3) $$R_i=R_i^0\left[1+\sum _{l=2,4,...}{\beta }_l{Y}_l^0\left(\theta \right)\right].$$(3) The angle θ refers to the body-fixed system, and Y⁰_l and β_l denote spherical harmonics and a deformation parameter, respectively. The potential depths are given as follows: (4) $$V_R=(V_R^0+V_R^1{E}^{†}+V_R^2{E}^{†2}+V_R^3{E}^{†3}+V_R^{\text{DISP}}\text{exp}(-{\lambda }_R{E}^{†}))\times \left[1-\frac{1}{V_R^0+V_R^{\text{DISP}}}{C}_{\text{viso}}\frac{N-Z}{A}\right],$$(4) (5) $$W_D=\left[W_D^{\text{DISP}}-C_{\text{wiso}}\frac{N-Z}{A}\right]\text{exp}(-{\lambda }_D{E}^{†})\times \frac{E^{†2}}{E^{†2}+{\text{WID}}_D^2},$$(5) (6) $$W_V=W_V^{\text{DISP}}\frac{E^{†2}}{E^{†2}+{\text{WID}}_V^2},$$(6) (7) $$V_{\text{SO}}=V_{\text{SO}}^0\text{exp}(-{\lambda }_{\text{SO}}{E}^{†}),$$(7) (8) $$W_{\text{SO}}=W_{\text{SO}}^{\text{DISP}}\frac{E^{†2}}{E^{†2}+{\text{WID}}_{\text{SO}}^2},$$(8) 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. 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.

The parameters required in Equatios (2(2) $$f_i\left(r\right)=\frac{1}{1+\exp \{(r-R_i)/a_i\}},i=R,D,V,SO,$$(2) )–(8(8) $$W_{\text{SO}}=W_{\text{SO}}^{\text{DISP}}\frac{E^{†2}}{E^{†2}+{\text{WID}}_{\text{SO}}^2},$$(8) ) are listed in Table 3, and the coupling schemes and deformation parameters are given in Table 4. The deformation parameters for even targets were taken from the work of Hogenbirk et al. [21], while those for odd targets were taken from the theoretical work of Möller et al. [22] The same neutron potentials were used as the spherical optical-model (SOM) ones for the ²⁰³Hg target, since no coupling levels were found within the framework of the rotational excitation. Vibrational excitation of the target nuclei was taken into account by DWBA using the deformation parameters given in Table 5. These β_L’s of DWBA for even nuclei were essentially based on the experimental work of Hogenbirk et al. [21], although those for odd nuclei were calculated using the weak coupling model [23]. The calculated total and non-elastic cross sections of elemental Hg are shown in Figures 3 and 4, respectively. In order to give better fits 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]. The measured angular distributions of neutrons elastically scattered from elemental Hg are satisfactorily reproduced by the present calculations, as seen in Figure 5. Concerning charged-particles, used were the spherical parameters of Koning and Delaroche [24] for protons, those of Lohr and Haeberli [25] for deuterons, those of Becchetti and Greenlees [26] for tritons and ³He, and those of Avrigeanu et al. [27] for α-particles. The transmission coefficients, which are required by CCONE to calculate charged-particle emission, are obtained from the charged-particle parameters.

Figure 3. Total cross section of elemental Hg.

Figure 4. Non-elastic cross section of elemental Hg.

Figure 5. Angular distributions of neutrons elastically scattered from elemental Hg.

Table 3. 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
Cviso	= 24.3 MeV
λR	= 1.05 × 10^− 2 + 1.63 × 10^− 4V^0R MeV^−1
$W_D^{\text{DISP}}$	= 11.0 MeV
WIDD	= 12.72 − 1.54 × 10^−2A + 7.14 × 10^−5A^2 MeV
Cwiso	= 18.0 MeV
λD	= 0.0140 MeV^−1
$W_V^{\text{DISP}}$	= 34.0 MeV
WIDV	= 105.05 − 14.13 × [1.0 + exp{(A − 100.21)/13.47}]^−1 MeV
$V_{\text{SO}}^0$	= 5.92 + 0.003A MeV
λSO	= 0.005 MeV^−1
$W_{\text{SO}}^{\text{DISP}}$	= −3.1 MeV
WIDSO	= 160.0 MeV
RR, 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
RSO	= (1.18 − 0.65A^−1/3)A^1/3 fm
aSO	= 0.59 fm

Table 4. Coupling schemes used in the interaction between neutron and target.

Target nuclei	Coupled levels^*	Deformation parameters
^195Hg	1/2^−(g.s.), 3/2^−(0.037), 5/2^−(0.053)	β2 = −0.130
^196Hg	0^+(g.s.), 2^+(0.426), 4^+(1.061), 6^+(1.785)	β2 = −0.110, β4 = −0.023
^197Hg	1/2^−(g.s.), 3/2^−(0.152), 5/2^−(0.308)	β2 = −0.130
^198Hg	0^+(g.s.), 2^+(0.412), 4^+(1.048), 6^+(1.816)	β2 = −0.110, β4 = −0.023
^199Hg	1/2^−(g.s.), 3/2^−(0.208), 5/2^−(0.414), 7/2^−(0.699)	β2 = −0.122
^200Hg	0^+(g.s.), 2^+(0.368), 4^+(0.947), 6^+(1.707)	β2 = −0.106, β4 = −0.024, β6 = −0.017
^201Hg	3/2^−(g.s.), 5/2^−(0.026), 7/2^−(0.414), 9/2^−(0.547)	β2=−0.105
^202Hg	0^+(g.s.), 2^+(0.440), 4^+(1.120), 6^+(1.989)	β2 = −0.088, β4 = −0.038
^203Hg	No coupling	–
^204Hg	0^+(g.s.), 2^+(0.437), 4^+(1.128), 6^+(2.191)	β2 = −0.069, β4 = −0.049, β6 = −0.013

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

Table 5. Deformation parameters for DWBA calculations of mercury isotopes.

Mass no.	Ex (MeV)	L	βL	Mass no.	Ex (MeV)	L	βL
195	0.844	3	0.020	201	0.002	2	0.028
	0.875	3	0.023		0.032	2	0.039
196	1.696	3	0.031		0.385	2	0.048
	1.757	5	0.027	202	0.960	2	0.012
	1.841	7	0.013		1.312	4	0.040
197	0.509	3	0.020		1.644	0	0.009
198	1.087	2	0.018		1.966	5	0.033
	1.401	0	0.013	203	0.007	2	0.018
	1.636	5	0.027		0.051	2	0.025
	1.683	7	0.013		0.369	2	0.031
	1.835	4	0.049		0.592	2	0.040
	1.930	3	0.031	204	2.263	5	0.033
199	0.158	2	0.082		2.301	7	0.021
	0.404	2	0.067		2.675	3	0.089
200	1.254	2	0.015
	1.515	0	0.010
	1.851	5	0.036
	2.151	3	0.030

3.2.2. Discrete levels and level density

In the calculation, it was necessary to input the discrete levels and level density parameters for 37 nuclei, i.e. ^193−205Hg, ^193−204Au, and ^{191 − 202}Pt. The discrete levels were taken from the reference input parameters library RIPL-3 [28].

Concerning the level density, the composite formula of Gilbert and Cameron  [29] 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, (9) $${\rho }_T\left(E\right)=\frac{1}{T}\text{exp}\left(\frac{E-E_0}{T}\right).$$(9) On the other hand, the a parameter, which characterises the Fermi-gas part of level density ρ_F, is defined as (10) $$a\left(U\right)=a(*)\left[1+\frac{E_{sh}}{U}(1-e^{-\gamma U})\right],$$(10) where E_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. In the above equations, the values of a(*) and Δ were taken from the work of Mengoni and Nakajima  [30]. The shell correction energy E_sh is calculated as the difference between the experimental mass [6] and the theoretical mass [31]. 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.

Nuclei	a(^*)	Δ	Esh	T	E0	Em	$E_{\text{level}}^{\text{highest}}$
	1/MeV	MeV	MeV	MeV	MeV	MeV	MeV
^205Hg	23.809	0.838	−7.795	0.771	−0.981	10.703	1.395
^204Hg	23.713	1.680	−7.232	0.725	0.191	9.685	2.675
^203Hg	23.616	0.842	−6.529	0.712	−0.871	8.546	1.043
^202Hg	23.519	1.689	−5.927	0.734	−0.714	10.399	1.989
^201Hg	23.422	0.846	−5.263	0.685	−1.173	8.028	0.732
^200Hg	23.325	1.697	−4.720	0.629	0.176	7.391	2.274
^199Hg	23.228	0.851	−4.089	0.660	−1.381	7.625	0.822
^198Hg	23.131	1.706	−3.418	0.606	−0.077	7.236	2.110
^197Hg	23.033	0.855	−2.645	0.581	−0.901	6.006	0.904
^196Hg	22.936	1.714	−2.164	0.531	0.405	5.890	2.098
^195Hg	22.839	0.859	−1.303	0.517	−0.540	4.960	1.005
^194Hg	22.741	1.723	−1.041	0.427	1.165	4.247	2.374
^193Hg	22.644	0.864	−0.220	0.519	−0.866	5.258	0.375
^204Au	23.713	0.000	−6.873	0.717	−1.600	7.811	0.215
^203Au	23.616	0.842	−6.166	0.644	−0.215	6.466	1.087
^202Au	23.519	0.000	−5.862	0.688	−1.757	7.108	0.166
^201Au	23.422	0.846	−4.994	0.619	−0.457	6.218	0.810
^200Au	23.325	0.000	−4.616	0.611	−1.356	5.270	0.390
^199Au	23.228	0.851	−3.848	0.578	−0.431	5.576	1.159
^198Au	23.131	0.000	−3.422	0.604	−1.747	5.464	0.812
^197Au	23.033	0.855	−2.659	0.538	−0.378	5.039	1.231
^196Au	22.936	0.000	−2.059	0.591	−2.105	5.550	0.502
^195Au	22.839	0.859	−1.475	0.492	−0.211	4.422	1.560
^194Au	22.741	0.000	−0.842	0.356	−0.104	1.554	0.769
^193Au	22.644	0.864	−0.340	0.444	0.011	3.812	1.433
^202Pt	23.519	1.689	−5.152	0.641	0.170	7.639	0.535
^201Pt	23.422	0.846	−4.451	0.435	0.900	2.726	1.456
^200Pt	23.325	1.697	−4.153	0.611	0.132	7.147	2.258
^199Pt	23.228	0.851	−3.486	0.652	−1.577	7.628	0.474
^198Pt	23.131	1.706	−3.134	0.603	−0.154	7.247	1.944
^197Pt	23.033	0.855	−2.496	0.591	−1.093	6.292	0.965
^196Pt	22.936	1.714	−2.151	0.583	−0.239	7.037	1.992
^195Pt	22.839	0.859	−1.456	0.585	−1.366	6.400	0.875
^194Pt	22.741	1.723	−1.159	0.545	−0.056	6.437	1.992
^193Pt	22.644	0.864	−0.352	0.557	−1.327	6.020	0.692
^192Pt	22.546	1.732	−0.214	0.512	0.108	5.960	1.823
^191Pt	22.448	0.868	0.574	0.531	−1.227	5.657	0.613

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 (MLO) proposed by Plujko et al.  [32] was used for E1 radiation, since its use is recommended by RIPL-3 [28]. The resonance energy (E₀), resonance width (Γ₀), and peak cross section (σ₀) are given [33] by (11) $$E_0=31.2A^{-1/3}+20.6A^{-1/6}\mathrm{MeV},$$(11) (12) $${\Gamma }_0=0.026E_0^{1.91}\mathrm{MeV},$$(12) (13) $${\sigma }_0=1.2\times 120NZ/\left(A\pi {\Gamma }_0\right)\mathrm{mb}.$$(13) The standard Lorentzian (SLO) form was used to describe the M1 and E2 radiations for which the parameters were taken from the work of Kopecky and Uhl  [34].

In cases where measured capture cross sections are available in the keV region, the γ-ray strength functions are renormalised so that the calculated cross sections reproduce these data. The s-wave γ-ray strength functions are compared in Table 7 with the values recommended by Mughabghab  [11]. The values of ^{197, 199, 200, 202, 203, 205}Hg were obtained after renormalisation using measurements.

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

Compound nuclei	Present work	Mughabghab  [11]
^196Hg	170.0	–
^197Hg	3.980	–
^198Hg	52.17	–
^199Hg	3.935	14.3 ± 4.9
^200Hg	23.63	28.9 ± 9.1
^201Hg	2.274	–
^202Hg	35.99	16.6 ± 4.2
^203Hg	1.490	–
^204Hg	8.546	–
^205Hg	0.6681	–

3.2.4. Pre-equilibrium parameters

The Version 0.8.4 of CCONE incorporates pre-equilibrium process for the (n, γ) and first-step binary reactions such as (n, n′), (n, p), (n, d), (n, t), (n, ³He), 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  [35] by (14) $$g_{\pi }=\frac{C_{0}Z}{15}{\mathrm{MeV}}^{-1},$$(14) (15) $$g_{\nu }=\frac{C_{0}N}{15}{\mathrm{MeV}}^{-1},$$(15) 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 pre-equilibrium parameters for complex-particle emission  [36] 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)
^195Hg	1.00	1.00	3.00	3.00
^196Hg	0.80	1.00	3.00	3.00
^197Hg	1.00	1.00	3.00	3.00
^198Hg	1.20	1.03	3.00	3.00
^199Hg	1.00	0.93	3.00	3.00
^200Hg	1.30	1.00	3.00	3.00
^201Hg	1.00	1.00	3.00	3.00
^202Hg	1.00	0.80	6.00	6.00
^203Hg	1.00	1.00	6.00	6.00
^204Hg	1.10	0.80	6.00	6.00

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

Comparisons are made with experimental data and other evaluated data. The experimental data are taken from the EXFOR database [37] that is compiled by the international data centres such as the IAEA Nuclear Data Section, the OECD NEA Data Bank, and the National Nuclear Data Center at the Brookhaven National Laboratory. As for the general-purpose evaluated nuclear data, JENDL-4.0  [4], ENDF/B-VII.1  [38] and JEFF-3.2  [39] contain the data for ^{196, 198, 199, 200, 201, 202, 204}Hg. The libraries ENDF/B-VII.1 and JEFF-3.2 took cross sections from JENDL-4.0 below 20 MeV. Therefore, they are not independent of one another. To avoid duplication, the JENDL-4.0 data are solely used to compare with the present data. These general-purpose libraries do not contain activation cross sections leading to the ground or isomeric state. The activation files JENDL/A-96  [40] and JEFF-3.1/A  [41] are used for comparison with such activation data. Concerning the ¹⁹⁹Hg(n, n′)^199mHg reaction, the dosimetry file IRDFF-1.05  [42] is included in the comparison. 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.

The capture cross sections of ^{198, 199, 200, 201}Hg and ^{202, 204}Hg are illustrated in Figure 6 and 7, respectively. It is found from the figures that the present evaluation is in good agreement with the data measured by Beer and Macklin [10]. In the MeV region, the JENDL-4.0 evaluation is considerably smaller than the present data, since the pre-equilibrium capture is not taken into account in JENDL-4.0. There exist no experimental data on ¹⁹⁶Hg with mono-energetic neutrons in the keV region. Alternatively, we used the measured Maxwellian-averaged cross section [43] at kT = 30 keV to renormalise the γ-ray strength function, as shown in Figure 8. The calculated elemental cross section is compared with measured data and the JENDL-4.0 evaluation in Figure 9, where the open and closed symbols stand for neutron and γ-ray measurements, respectively. The present results are almost consistent with the γ-ray measurements, whereas the data obtained by the shell transmission method [44], i.e. the neutron measurements, are systematically larger than the present and JENDL-4.0 evaluations.

Figure 6. Capture cross sections of ^{198, 199, 200, 201}Hg.

Figure 7. Capture cross sections of ^{202, 204}Hg. The open symbols belong to ²⁰⁴Hg.

Figure 8. Maxwellian-averaged capture cross section of ¹⁹⁶Hg.

Figure 9. Capture cross section of ^natHg. The open and closed symbols stand for neutron and γ-ray measurements, respectively.

Figure 10 shows the ¹⁹⁹Hg(n, n′)^199mHg cross section, which is regarded as one of the dosimetry cross sections. In the figure, the data of Grudzevich et al. [45] were corrected for the ²⁰⁰Hg(n, 2n)^199mHg contribution estimated in this work. The present evaluation is almost consistent with the data measured by Sakurai et al. [46] above 3 MeV. However, it cannot reproduce the data of Sakurai et al. obtained below 3 MeV, and is rather close to the data of Swann and Metzger [47]. Two 2.8-MeV measurements [48,49] are obviously much lower than the present and JEFF-3.1/A evaluations. The IRDFF-1.05 data are based on a least-squares fit to the renormalised experimental data below 10 MeV, although the theoretical calculations were taken into account above 10 MeV. The present evaluation yields the dosimetry cross sections by placing much emphasis on the theoretical understanding of the reaction.

Figure 10. ¹⁹⁹Hg(n, n′)^199mHg reaction cross section. The data of Grudzevich et al. [45] were corrected for the ²⁰⁰Hg(n, 2n)^199mHg cross section estimated in the present work.

Figures 11–18 show the (n, 2n) reaction cross sections. As for ¹⁹⁶Hg(n, 2n), the present evaluation is consistent with the measured data on the ground- and isomeric-state production. The activation files JENDL/A-96 and JEFF-3.1/A are systematically smaller than the measured data on the ground-state production. In the present parametrisation, it is not possible to simultaneously fit the data measured by Al-Abyad et al. [50] below 12.5 MeV and the other data around 14 MeV for the ^{198, 204}Hg(n, 2n) reactions. We placed emphasis on the 14-MeV measurements obtained by several experiments. In the experiment of Al-Abyad et al., quasi-monoenergetic neutrons produced via the ²H(d, n)³He were incident on a natural Hg target, and so ¹⁹⁷Hg and ²⁰³Hg could be formed by the ¹⁹⁶Hg(n, γ) and ²⁰²Hg(n, γ) reactions, respectively, due to the low-energy breakup neutrons having continuous energies. These contributions were corrected in the experiment, but it is not certain whether the corrections were appropriate. It is suggested that more experiments should be performed in the energy region below 14 MeV.

Figure 11. ¹⁹⁶Hg(n, 2n)¹⁹⁵Hg reaction cross section.

Figure 12. ¹⁹⁶Hg(n, 2n)^195gHg reaction cross section.

Figure 13. ¹⁹⁶Hg(n, 2n)^195mHg reaction cross section.

Figure 14. ¹⁹⁸Hg(n, 2n)¹⁹⁷Hg reaction cross section.

Figure 15. ¹⁹⁸Hg(n, 2n)^197gHg reaction cross section.

Figure 16. ¹⁹⁸Hg(n, 2n)^197mHg reaction cross section.

Figure 17. ²⁰⁰Hg(n, 2n)^199mHg reaction cross section.

Figure 18. ²⁰⁴Hg(n, 2n)²⁰³Hg reaction cross section.

Charged-particle emission cross sections are illustrated in Figures 19–26. As seen from Table 8, the pre-equilibrium parameters for (n, p) are not drastically changed from the default value, i.e. unity. The present evaluation agrees with the 14-MeV data measured by Kasugai et al. [51] On the other hand, the pre-equilibrium effects were much enhanced for (n, α). We could reproduce the data of Temperley [52] for ²⁰⁰Hg, although the data of Coleman et al. [53] that were obtained by the β counting are too high. However, the ²⁰²Hg case was obviously unsuccessful. As a matter of fact, Temperley could not observe 544-keV γ-rays due to the decay of ¹⁹⁹Pt. Then, the author set an upper limit of 1 mb for ²⁰²Hg(n, α)¹⁹⁹Pt in order to deduce the ¹⁹⁹Hg(n, p)¹⁹⁹Au cross section, since the residual nucleus ¹⁹⁹Pt decays to ¹⁹⁹Au. It is unlikely that the 14-MeV cross section could reach 1 mb by comparing with the ²⁰⁰Hg(n, α) cross section. Therefore, the data of Coleman et al. for ²⁰²Hg(n, α) might be overestimated as in the case of ²⁰⁰Hg.

Figure 19. ¹⁹⁸Hg(n, p)^198gAu reaction cross section.

Figure 20. ¹⁹⁹Hg(n, p)¹⁹⁹Au reaction cross section.

Figure 21. ²⁰⁰Hg(n, p)²⁰⁰Au reaction cross section.

Figure 22. ²⁰¹Hg(n, p)²⁰¹Au reaction cross section.

Figure 23. ²⁰²Hg(n, p)²⁰²Au reaction cross section.

Figure 24. ²⁰⁴Hg(n, p)²⁰⁴Au reaction cross section.

Figure 25. ²⁰⁰Hg(n, α)¹⁹⁷Pt reaction cross section.

Figure 26. ²⁰²Hg(n, α)¹⁹⁹Pt reaction cross section.

The double-differential neutron emission spectra from elemental Hg are depicted in Figure 27. The presently calculated cross sections are in good agreement with the data measured by Maekawa et al. [54] at 14 MeV. The JENDL-4.0 evaluation underestimates the spectra at forward angles, since symmetric angular distributions are given for neutrons inelastically scattered from the overlapping levels. In the present work, forward peaking is assumed by using the systematics of Kalbach [55].

Figure 27. Double-differential neutron emission spectra from elemental Hg at 14 MeV.

5. Concluding remarks

The neutron nuclear data on Hg isotopes were evaluated in the energy region from 10^{− 5} eV to 20 MeV. The RRPs of ^{200, 202, 204}Hg were revised by considering the latest measurements, while those of ^{196, 198, 199, 201}Hg remain unchanged from JENDL-4.0. However, more resonance experiments would be needed to improve the low-energy cross section. As for ^{195, 197, 203}Hg whose RRPs are unavailable, constant and 1/v energy dependences are assumed for the low-energy elastic scattering and capture cross sections, respectively. 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 or the SOM, together with the total and shape-elastic scattering cross sections. The neutron potentials were found to be reliable by comparing with measured total and non-elastic cross sections of elemental Hg. On the whole, the presently evaluated cross sections are in good agreement with available experimental data, and better than the JENDL-4.0 data. However, it was not possible to reproduce the measured ^{198, 204}Hg(n, 2n) cross sections below 12.5 MeV. More experimental data would be desired to resolve the difference in this energy region.

The evaluated data are compiled into ENDF-formatted data files and will be available in the next release of JENDL general purpose file. Part of the cross sections are used as activation data for the decommissioning of light-water reactors.

Acknowledgments

The author would like to thank the members of the Nuclear Data Center, JAEA, for their helpful comments on this work.

Disclosure statement

No potential conflict of interest was reported by the author.