- Email: [email protected]
Study on cross section calculations for (n,p) nuclear reactions of cadmium isotopes
Applied Radiation and Isotopes 154 (2019) 108868
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Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Study on cross section calculations for (n,p) nuclear reactions of cadmium isotopes
T
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Mustafa Yiğit , Sümeyye Nur Bostan Aksaray University, Department of Physics, Aksaray, Turkey
H I GH L IG H T S
cross sections are predicted using EMPIRE 3.2, ALICE/ASH and TALYS 1.8 codes. • The empirical systematics are used to calculate the (n,p) cross sections. • The • The equilibrium and pre-equilibrium model were used.
A R T I C LE I N FO
A B S T R A C T
Keywords: Cadmium Hauser-feshbach model ALICE code Excitation function
A theoretical study of the cross sections of (n, p) reactions on Cadmium isotopes has been performed using the Hauser-Feshbach model, Weisskopf-Ewing model, Exciton model, Two-component exciton model and Geometry dependent hybrid model for the incident energies from the reaction threshold to 20 MeV. Furthermore, the empirical formulae based on the different systematics at the energies near 14.5MeV were used for calculating the (n, p) nuclear cross sections. Finally, the results of the present paper are discussed and compared with the experimental cross sections and TENDL-2017 data found in the literature.
1. Introduction The generation of charged particle via the reactions of fast neutrons such as (n, α ) , (n, p) , (n, nα) and (n, np) is one of the problems faced in design and development of nuclear fusion reactors. Actually, these nuclear reactions can be produced with bombardment of fast neutron particles on the structural materials such as the first wall and the blanket components of nuclear devices. Thus, such reactions lead to the generation of helium and hydrogen gases in the walls of a nuclear reactor at different locations. Furthermore, some adverse effects such as atomic displacement damages and transmutation productions can also induce defects in the structural materials used in and around the nuclear devices. Accurate prediction of excitation curves of the hydrogen production through (n, p) nuclear reactions on reactor materials is particularly required up to the neutron energies up to 20 MeV (Lalremruata et al., 2009; Mulik et al., 2013). Cadmium is an important metal used in nuclear technology as plating element for corrosion resistance and as an alloying element in different structural materials. In addition, it has been also used to control neutron flux in fission devices. Besides, the activation of cadmium may be used for prediction of depth distribution and dose (Tarkanyi et al., 2006; Ditroi et al., 2016).
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Data on charged particle emission processes in neutron reactions are essential for understanding nuclear phenomena. These data have been also needed for determining the nuclear transmutation rates, nuclear heating, induced radioactivity and radiation damage (Yiğit et al., 2019; Korkmaz et al., 2017). Moreover, comparison between the theoretical and experimental cross section data based on the nuclear models is important in understanding of the experimental spectra (Yiğit and Kara, 2017). The empirical and semi-empirical systematics of cross sections based on the equilibrium and pre-equilibrium emission processes are very practical in calculating unknown cross section data (Yiğit, 2017a; 2017b). Particularly, the systematics of (n, p) nuclear reactions at incident neutron energies near 14.5MeV are required for better reproduction of the experimental cross section values. This paper is undertaken in attempt to calculate the excitation functions of the 110Cd(n, p) 110Ag, 111Cd(n, p) 111Ag, 112Cd(n, p) 112Ag, 113 Cd(n, p) 113Ag, 114Cd(n, p) 114Ag and 116Cd(n, p) 116Ag nuclear reactions using the equilibrium and pre–equilibrium emission models, and the empirical cross sections formulae. The present theoretical cross sections were compared with the literature data available in EXFOR (EXFOR, 2019) and TENDL-2017 (Koning et al., 2017).
Corresponding author. E-mail address: [email protected] (M. Yiğit).
https://doi.org/10.1016/j.apradiso.2019.108868 Received 4 June 2019; Received in revised form 29 July 2019; Accepted 15 August 2019 Available online 16 August 2019 0969-8043/ © 2019 Elsevier Ltd. All rights reserved.
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
2. Theoretical background
efficient of the 1–th partial wave, and Pv (l,ε) is decay probability at exit channel energy (Broeders et al., 2006). TALYS 1.8computer code (Koning et al., 2013) has been widely used for the analysis of nuclear structure and reaction experiments. The nuclear reactions induced by gammas, protons, neutrons, deuterons, tritons, 3He and 4He in this simulation code can be simulated in the bombarding energy region of 1keV–200 MeV. Nuclear models used at the code TALYS 1.8can be categorized as direct, optical, equilibrium, pre-equilibrium and fission models (Koning et al., 2013). The Hauser–Feshbach approach to accounting for the evaporation peak of the nuclear reaction spectrum has been employed for defining the compound emission, binary and multiple processes. Additionally, the nuclear state of the pre-equilibrium emission spectrum has been characterized with the Exciton model (Griffin, 1966) in this code. So, the reaction cross section for nuclear particle emission process in preequilibrium model of the TALYS code is given as follows,
The model-based calculation results on the prediction of new cross sections are at the forefront in researches of the nuclear physics today. Accordingly, the sophisticated simulation reaction codes, which contain different phenomenological nuclear theories, have been used successfully for evaluation of the new data of different reaction processes. So, contribution of various emission processes such as pre-equilibrium and equilibrium emissions on nuclear cross sections in these reaction codes have been widely studied. The nuclear reaction models such as the Exciton model (Griffin, 1966), Two–component exciton model (Kalbach, 1986) and Geometry dependent hybrid (GDH) model (Blann and Vonach, 1983) have been broadly and significantly used in the theoretical calculations for analysing the pre-equilibrium emissions. The pre-equilibrium emissions provide a sizeable part of the nuclear excitation curve of reaction at the projectile energies between 10 and 200 MeV. Calculations of cross sections on the equilibrium emission process of the compound nucleus are widely treated in terms of statistical models of nuclear reactions. The prediction of the equilibrium emission of the decay process going to continuum states have been mostly carried out using the Weisskopf-Ewing model and Hauser-Feshbach model. It is known that the equilibrium emission of nuclear reaction is dominant process at the bombarding energies up to 10 MeV (Yiğit, 2019; Yiğit and Tel, 2017; Yiğit and Korkmaz, 2018). The reaction model based on equilibrium emissions suggested by Weisskopf and Ewing (1940) does not include the angular momentum effect. Moreover, the statistical model of Hauser and Feshbach (1952) takes into account the conservation of angular momentum at the interactions between the target nucleus and the incident particle. EMPIRE 3.2 code (Herman et al., 2013), ALICE/ASH code (Broeders et al., 2006) and TALYS 1.8code (Koning et al., 2013) have widely been used for the prediction and analysis of (n, p) cross sections at low- and medium-energy region. The code EMPIRE 3.2 has be broadly used for the evaluation studies of the reaction cross section data. The projectile particle at the cross section calculations in this computer code can be a photon, neutron, proton, deuteron, Triton or heavier ion. The preequilibrium mechanism in the Exciton model (Griffin, 1966) is based on the solution of master equation in the form predicted by Cline (1972) and Ribansky et al. (1973) as follows,
p
π
r σa,b (Einc) is
the where the term Da,b (Einc) is the depletion factor and the (a, b) nuclear reaction cross section. Wb (E,n, ε b) Represents the particle emission probability of type b with energy ε b from a nuclear state with exciton n and excitation energy E of compound nucleus (Herman et al., 2013). ALICE/ASH computer code is an advanced version of the code ALICE used in the analytical computation of excitation functions. The pre-equilibrium differential emission spectrum of nucleons in the Geometry dependent hybrid model is calculated as follows,
3.1.
110
Cd(n , p) 110Ag reaction
The model results compared with the experimental cross sections measured by Yu and Gardner (1967) for 110Cd(n, p) 110Ag nuclear reaction are given in Fig. 1. Cross section value of 27 ± 5.4 mb reported by Yu and Gardner (1967) at the energy of 14.1MeV is observed to be very compatible with the Two-component exciton model and TENDL-2017 library data. Besides, the cross sections calculated using empirical cross section formulae of Yiğit (2018), Doczi et al. (1997), Luo et al. (2008) and Levkovskii (1964) give results in reasonably good agreement with the experimental data of Yu and Gardner (1967). However, it is seen in Fig. 1 that the calculation results of the equilibrium and pre-equilibrium model obtained using the simulation code ALICE/ASH are completely different from both experimental value and other calculation data.
∞
dσv (ε) = π 2 ∑ (2 l+ 1)Tl Pv (l,ε) dε l= 0
(4)
This study presents the results of theoretical model calculation of (n, p) cross sections resulting from interactions of neutron particles with Cadmium isotopes. The present simulation calculations for (n, p) cross sections were performed using the GDH and Weisskopf-Ewing models by the code ALICE/ASH, the Two–component exciton and HauserFeshbach models by the code TALYS 1.8, and the Exciton model by the code EMPIRE 3.2 . Furthermore, the Back-shifted fermi gas model (Dilg et al., 1973) in the TALYS 1.8code, the Enhanced generalized super-fluid model (D’Arrigo et al., 1994) in the EMPIRE 3.2 code and the Fermi gas model in the ALICE/ASH code are used for the level density calculations. Thus, the excitation functions of (n, p) nuclear reactions on 110-114,116 Cd isotopes considered in the present study have been shown in Figs. 1–6, and also the numerical values of data at the energies around 14.5MeV are given in Table 1.
(2)
n
Wk (pπ , h π , pυ , h υ, Ek) τ (pπ , h π , pυ , h υ)
3. Results and discussion
(1)
∑ Wb (E,n, εb) τ (n)
∑ pυ = p0υ
In the above equation, the terms pπ (pυ) and h π (h υ) represent the proton (neutron) particle number and the proton (neutron) hole number, respectively. The term σ CF represents the cross section for the compound nucleus formation predicted with the optical model. The term τ denotes the average lifetime of the exciton state. The terms Wk and Ek correspond to the emission rate and the emission energy of particle k , respectively. The quantity P is the part of the pre-compound stage for the nuclear emission to survive the previous states and now passes through the (pπ , h π , pυ , h υ) configuration, averaged over time. And also, the initial proton and neutron particle numbers are p0π = Z p and p0υ = Np , respectively with Zp (Np ) the proton (neutron) number of bombarding particle (Koning et al., 2013).
here the quantity L(n, E) denotes the total emission rate integrated over emission energy for γ -rays and particles. In addition to, qt (n) represents the initial occupation probability of composite nucleus in the state with n exciton number. And also, λ+ (E, n) and λ− (E, n) represent the transition rates of decay to neighboring states (Herman et al., 2013). The pre-equilibrium emission spectrum is given as follows,
dσa,b r (ε b) = σa,b (Einc)Da,b (Einc) × dε b
π
pmax υ
P(pπ , h π , pυ , h υ)
− qt= 0 (n) = λ− (E, n− 2) τ (n − 2) + λ+ (E, n+ 2) τ (n+2) − [λ+ (E, n) + λ− (E, n) + L(E, n)] τ (n)
max
π dσkPE = σ CF ∑ dEk p = p0
(3)
In the above equation, the term is the reduced de–Broglie wavelength of the incident particle. Also, Tl represents the transmission co2
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
Fig. 1. Cross section results for reaction.
111
Cd(n, p) 110Ag
(2016) give close results with the cross section data obtained using the empirical formulae of Yiğit (2018), Doczi et al. (1997) and Luo et al. (2008). The experimental cross sections measured by Yu and Gardner (1967) and Levkovskii (1964) showed acceptable agreement with the Exciton model calculations with pre-equilibrium effects in the simulation code EMPIRE 3.2.
Especially the Weisskopf-Ewing model, which contains the equilibrium component of the reaction, is much lower than the other excitation function spectra.
3.2.
110
Cd(n , p) 111Ag reaction
The excitation functions of 111Cd(n, p) 111Ag nuclear reaction are presented at Fig. 2. It was observed that the data obtained using the Two-component exciton model showed similar results with the measured excitation function of Bayhurst and Prestwood (1961) in the incident energy range of 7 − 14.81MeV. Additionally, the cross section measurements obtained by Filatenkov (2016) and Grallert et al. (1993) have shown a very good fit with the excitation function calculated using the Two-component exciton model in the simulation code TALYS 1.8. It is observed that the results of experimental study made by Filatenkov
3.3.
112
Cd(n , p) 112Ag reaction
The cross section data for the 112Cd(n, p) 112Ag reaction are given in Fig. 3. The experimental studies performed for this reaction give similar cross section results in the energy region of 13 − 15MeV. The cross section measurements made by Filatenkov (2016), Grallert et al. (1993), Konno et al. (1993), Struwe and Winkler (1974), Charturverdi et al. (1977) and Mavaddat et al. (1974) are in excellent agreement Fig. 2. Cross section results for reaction.
3
111
Cd(n, p) 111Ag
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
Fig. 3. Cross section results for reaction.
113
Cd(n, p) 112Ag
lower than the measured data of the Konno et al. (1993) for this reaction. Moreover, the experimental cross section results of Konno et al. (1993) except for a few data points are compatible with the Twocomponent exciton model and TALS-based TENDL-2017 data. The highest cross section data are obtained by pre-equilibrium GDH model calculations made with the code ALICE/ASH. Moreover, the lowest results are obtained by the equilibrium emissions with Weisskopf-Ewing and Hauser-Feshbach models.
with the TENDL-2017 evaluation data. Furthermore, the agreement between the experimental results of Herman et al. (1978) and the TENDL data for the investigated reaction is acceptable except for a few data points. And also, the results for excitation function calculated with the empirical formulae are very compatible with the experimental measurements. The calculations including the equilibrium emissions with Weisskopf-Ewing and Hauser-Feshbach models give the lowest cross section results for this reaction.
3.4.
112
Cd(n , p) 113Ag reaction
3.5.
114
Cd(n , p) 114Ag reaction
The nuclear excitation functions of the 114Cd(n, p) 114Ag reaction produced by neutrons on the 114Cd target nucleus are shown in Fig. 5. The calculation results obtained using the empirical systematics of Yiğit (2018) and Doczi et al. (1997) appear to be in good agreement with the
In Fig. 4, the experimental and theoretical cross section values are shown for 113Cd(n, p) 113Ag nuclear reaction. It can be seen from Fig. 4 that the experimental data of Levkovskii (1964) and Yu and Gardner (1967) and also the calculation results of (n, p) empirical formulae are
Fig. 4. Cross section results for reaction.
4
113
Cd(n, p) 113Ag
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
Fig. 5. Cross section results for reaction.
114
Cd(n, p) 114Ag
compatible with each other for this reaction. In addition, the experimental data obtained by Anders vd. (1985) and Bjomstad and Alkstad (1973) have almost the same results. These experimental results show similar agreements with the results of Two-component exciton models, which include the pre-equilibrium effects of the nuclear reaction within the code TALYS 1.8. Fig. 6 shows that the cross section results calculated using Hauser-Feshbach and Weisskopf-Ewing models are too lower than the pre-equilibrium models and measured data.
experimental cross section of 12 ± 0.9mb reported by Anders et al. (1985) at 14.7 MeV energy. The cross section results measured by Struwe and Winkler (1974) and Prasad and Sarkar (1971) give lower values than the results of Anders et al. (1985) and empirical formulae for this reaction. The excitation function of Struwe and Winkler (1974) is consistent with cross section results of Two-component exciton model and TENDL-2017 . On the other hand, the data obtained by Prasad and Sarkar (1971) showed acceptable agreement with the Two-component exciton model and the Exciton model.
4. Conclusions 3.6.
116
Cd(n , p) 116Ag reaction In this paper, the excitation function data of the (n, p) reactions produced by neutron particles on some cadmium isotopes were calculated from the threshold to 20 MeV energy. The data calculated using the empirical formulae were found to be consistent with experimental values except for 116Cd(n, p) 116Ag nuclear reaction. The excitation
Fig. 6 present a comparison of the calculated and measured excitation functions for 116Cd(n, p) 116Ag nuclear reaction. The cross section data predicted using the formulae of Yiğit (2018), Luo et al. (2008), Tel et al. (2003), Levkovskii (1964) and Doczi et al. (1997) are
Fig. 6. Cross section results for reaction.
5
116
Cd(n, p) 116Ag
6
Cross section systematics
Model calculations
Experimental data
– – – – – – 43.25 23.66 21.41 4.22 18.2 26.36 27.98 26.33 19.59 29.1 22.73
Chaturverdi et al. (1977) Mavaddat et al. (1974) Struwe and Winkler (1974) Anders vd. (1985) Prasad ve Sarkar (1971) Bjornstad and Alstad (1973) GDH model Two-component exciton model Exciton model Weisskopf-Ewing model Hauser-Feshbach model TENDL-2017 Yiğit (2018) Luo et al. (2008) Tel et al. (2003) Doczi et al. (1997) Levkovskii (1964) 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.7 14.5
– – – – – – 55 27.51 12.67 4.76 8.72 29.92 21.88 20.18 17.03 23.26 17.62
– –
Grallert et al. (1993) Levkovskii (1964) Bayhurst and Prestwood (1961)
Herman et al. (1978) Konno et al. (1993)
Cd(n, p) 111Ag
111
29± 2.1 (at 14.6 MeV) 15± 4 (at 14 MeV) 23.7± 1.2 (at 14.01 MeV) 22.5± 1.1 (at 14.09 MeV) 27.7± 1.4 (at 14.31 MeV) 28.7± 1.4 (at 14.5 MeV) 28.1± 1.4 (at 14.68 MeV) 30.7± 1.5 (at 14.81 MeV) 36.3± 1.8 (at 14.93 MeV) – –
Cd(n, p) 110Ag
– – –
110
(n, p) cross sections (mb)
14± 1.4 (at 14.1 MeV) 23.9± 1.87 (at 14.44 MeV) 26.6± 1.87 (at 14.64 MeV) 31± 2.18 (at 14.88 MeV)
14–15
Energy (MeV)
Cd isotopes at the energies around 14.5MeV.
27± 5.4 (at 14.1 MeV) –
110-114,116
Yu and Gardner (1967) Filatenkov (2016)
Table 1 Cross sections of (n, p) reactions on
19.1± 1.6 (at 14.5 MeV) 14.7± 1.9 (at 14.23 MeV) 16.3± 2.1 (at 14.44 MeV) 17.6± 2.3 (at 14.68 MeV) 18.1± 2.4 (at 14.95 MeV) 15± 1.3 (at 14 MeV) 18.7± 6.1 (at 14.8 MeV) 15.3± 1.9 (at 14.6 MeV) – – – 28.26 6.28 5.1 1.54 0.56 15.49 17.14 15.55 13.15 18.62 13.72
– 13.6± 1.3 (at 14.05 MeV) 14.7± 1.18 (at 14.28 MeV) 14.6± 1.18 (at 14.47 MeV) 16.9± 1.28 (at 14.68 MeV) 16.5± 1.44 (at 14.81 MeV) 15.4± 1.19 (at 14.86 MeV) 16± 1.2 (at 14.6 MeV) – –
Cd(n, p) 112Ag
112
– – – – – 41.86 16.61 4.12 1.80 2.82 19.02 13.67 12.03 12.31 14.93 10.74
– 18.2± 2.6 21.4± 2.6 25.9± 2.5 27.5± 2.8
(at (at (at (at
14.24 MeV) 14.45 MeV) 14.69 MeV) 14.96 MeV)
– 8± 2 (at 14 MeV) –
7.9±0.79 (at 14.1 MeV) –
Cd(n, p) 113Ag
113
Cd(n, p) 114Ag
– 5±1.3 (at 14.6 MeV) 12± 0.9 (14.7 MeV) 3±1.6 (14.8 MeV) – 18.86 3.40 1.70 0.42 0.43 5.04 10.68 9.35 8.96 11.98 8.44
– –
– – –
– –
114
Cd(n, p) 116Ag
– – 2.5±0.3 (14.7 MeV) – 2.2±0.5 (14.7 MeV) 12.00 1.43 0.37 0.11 0.09 3.07 6.77 5.72 6.19 7.75 5.27
– –
– – –
– –
116
M. Yiğit and S.N. Bostan
Applied Radiation and Isotopes 154 (2019) 108868
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
functions of Two-component exciton model and TENDL-2017 library except for the nuclear reaction performed on the 112Cd target nucleus showed similar spectra. We have also observed that the calculation results obtained using Two-component exciton model and TENDL-2017 for the investigated nuclear reactions were generally successful in estimating experimental values. On the other hand, the theoretical estimations with Exciton model using the code EMPIRE 3.2are in not good agreement with the measured values. It is also noted that the data obtained using GDH model with pre-equilibrium effects give the highest cross section values. The Hauser-Fescbah model and Weisskopf-Ewing model calculations have lowest cross section values at high incident neutron energies.
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Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Study on cross section calculations for (n,p) nuclear reactions of cadmium isotopes
T
⁎
Mustafa Yiğit , Sümeyye Nur Bostan Aksaray University, Department of Physics, Aksaray, Turkey
H I GH L IG H T S
cross sections are predicted using EMPIRE 3.2, ALICE/ASH and TALYS 1.8 codes. • The empirical systematics are used to calculate the (n,p) cross sections. • The • The equilibrium and pre-equilibrium model were used.
A R T I C LE I N FO
A B S T R A C T
Keywords: Cadmium Hauser-feshbach model ALICE code Excitation function
A theoretical study of the cross sections of (n, p) reactions on Cadmium isotopes has been performed using the Hauser-Feshbach model, Weisskopf-Ewing model, Exciton model, Two-component exciton model and Geometry dependent hybrid model for the incident energies from the reaction threshold to 20 MeV. Furthermore, the empirical formulae based on the different systematics at the energies near 14.5MeV were used for calculating the (n, p) nuclear cross sections. Finally, the results of the present paper are discussed and compared with the experimental cross sections and TENDL-2017 data found in the literature.
1. Introduction The generation of charged particle via the reactions of fast neutrons such as (n, α ) , (n, p) , (n, nα) and (n, np) is one of the problems faced in design and development of nuclear fusion reactors. Actually, these nuclear reactions can be produced with bombardment of fast neutron particles on the structural materials such as the first wall and the blanket components of nuclear devices. Thus, such reactions lead to the generation of helium and hydrogen gases in the walls of a nuclear reactor at different locations. Furthermore, some adverse effects such as atomic displacement damages and transmutation productions can also induce defects in the structural materials used in and around the nuclear devices. Accurate prediction of excitation curves of the hydrogen production through (n, p) nuclear reactions on reactor materials is particularly required up to the neutron energies up to 20 MeV (Lalremruata et al., 2009; Mulik et al., 2013). Cadmium is an important metal used in nuclear technology as plating element for corrosion resistance and as an alloying element in different structural materials. In addition, it has been also used to control neutron flux in fission devices. Besides, the activation of cadmium may be used for prediction of depth distribution and dose (Tarkanyi et al., 2006; Ditroi et al., 2016).
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Data on charged particle emission processes in neutron reactions are essential for understanding nuclear phenomena. These data have been also needed for determining the nuclear transmutation rates, nuclear heating, induced radioactivity and radiation damage (Yiğit et al., 2019; Korkmaz et al., 2017). Moreover, comparison between the theoretical and experimental cross section data based on the nuclear models is important in understanding of the experimental spectra (Yiğit and Kara, 2017). The empirical and semi-empirical systematics of cross sections based on the equilibrium and pre-equilibrium emission processes are very practical in calculating unknown cross section data (Yiğit, 2017a; 2017b). Particularly, the systematics of (n, p) nuclear reactions at incident neutron energies near 14.5MeV are required for better reproduction of the experimental cross section values. This paper is undertaken in attempt to calculate the excitation functions of the 110Cd(n, p) 110Ag, 111Cd(n, p) 111Ag, 112Cd(n, p) 112Ag, 113 Cd(n, p) 113Ag, 114Cd(n, p) 114Ag and 116Cd(n, p) 116Ag nuclear reactions using the equilibrium and pre–equilibrium emission models, and the empirical cross sections formulae. The present theoretical cross sections were compared with the literature data available in EXFOR (EXFOR, 2019) and TENDL-2017 (Koning et al., 2017).
Corresponding author. E-mail address: [email protected] (M. Yiğit).
https://doi.org/10.1016/j.apradiso.2019.108868 Received 4 June 2019; Received in revised form 29 July 2019; Accepted 15 August 2019 Available online 16 August 2019 0969-8043/ © 2019 Elsevier Ltd. All rights reserved.
Applied Radiation and Isotopes 154 (2019) 108868
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2. Theoretical background
efficient of the 1–th partial wave, and Pv (l,ε) is decay probability at exit channel energy (Broeders et al., 2006). TALYS 1.8computer code (Koning et al., 2013) has been widely used for the analysis of nuclear structure and reaction experiments. The nuclear reactions induced by gammas, protons, neutrons, deuterons, tritons, 3He and 4He in this simulation code can be simulated in the bombarding energy region of 1keV–200 MeV. Nuclear models used at the code TALYS 1.8can be categorized as direct, optical, equilibrium, pre-equilibrium and fission models (Koning et al., 2013). The Hauser–Feshbach approach to accounting for the evaporation peak of the nuclear reaction spectrum has been employed for defining the compound emission, binary and multiple processes. Additionally, the nuclear state of the pre-equilibrium emission spectrum has been characterized with the Exciton model (Griffin, 1966) in this code. So, the reaction cross section for nuclear particle emission process in preequilibrium model of the TALYS code is given as follows,
The model-based calculation results on the prediction of new cross sections are at the forefront in researches of the nuclear physics today. Accordingly, the sophisticated simulation reaction codes, which contain different phenomenological nuclear theories, have been used successfully for evaluation of the new data of different reaction processes. So, contribution of various emission processes such as pre-equilibrium and equilibrium emissions on nuclear cross sections in these reaction codes have been widely studied. The nuclear reaction models such as the Exciton model (Griffin, 1966), Two–component exciton model (Kalbach, 1986) and Geometry dependent hybrid (GDH) model (Blann and Vonach, 1983) have been broadly and significantly used in the theoretical calculations for analysing the pre-equilibrium emissions. The pre-equilibrium emissions provide a sizeable part of the nuclear excitation curve of reaction at the projectile energies between 10 and 200 MeV. Calculations of cross sections on the equilibrium emission process of the compound nucleus are widely treated in terms of statistical models of nuclear reactions. The prediction of the equilibrium emission of the decay process going to continuum states have been mostly carried out using the Weisskopf-Ewing model and Hauser-Feshbach model. It is known that the equilibrium emission of nuclear reaction is dominant process at the bombarding energies up to 10 MeV (Yiğit, 2019; Yiğit and Tel, 2017; Yiğit and Korkmaz, 2018). The reaction model based on equilibrium emissions suggested by Weisskopf and Ewing (1940) does not include the angular momentum effect. Moreover, the statistical model of Hauser and Feshbach (1952) takes into account the conservation of angular momentum at the interactions between the target nucleus and the incident particle. EMPIRE 3.2 code (Herman et al., 2013), ALICE/ASH code (Broeders et al., 2006) and TALYS 1.8code (Koning et al., 2013) have widely been used for the prediction and analysis of (n, p) cross sections at low- and medium-energy region. The code EMPIRE 3.2 has be broadly used for the evaluation studies of the reaction cross section data. The projectile particle at the cross section calculations in this computer code can be a photon, neutron, proton, deuteron, Triton or heavier ion. The preequilibrium mechanism in the Exciton model (Griffin, 1966) is based on the solution of master equation in the form predicted by Cline (1972) and Ribansky et al. (1973) as follows,
p
π
r σa,b (Einc) is
the where the term Da,b (Einc) is the depletion factor and the (a, b) nuclear reaction cross section. Wb (E,n, ε b) Represents the particle emission probability of type b with energy ε b from a nuclear state with exciton n and excitation energy E of compound nucleus (Herman et al., 2013). ALICE/ASH computer code is an advanced version of the code ALICE used in the analytical computation of excitation functions. The pre-equilibrium differential emission spectrum of nucleons in the Geometry dependent hybrid model is calculated as follows,
3.1.
110
Cd(n , p) 110Ag reaction
The model results compared with the experimental cross sections measured by Yu and Gardner (1967) for 110Cd(n, p) 110Ag nuclear reaction are given in Fig. 1. Cross section value of 27 ± 5.4 mb reported by Yu and Gardner (1967) at the energy of 14.1MeV is observed to be very compatible with the Two-component exciton model and TENDL-2017 library data. Besides, the cross sections calculated using empirical cross section formulae of Yiğit (2018), Doczi et al. (1997), Luo et al. (2008) and Levkovskii (1964) give results in reasonably good agreement with the experimental data of Yu and Gardner (1967). However, it is seen in Fig. 1 that the calculation results of the equilibrium and pre-equilibrium model obtained using the simulation code ALICE/ASH are completely different from both experimental value and other calculation data.
∞
dσv (ε) = π 2 ∑ (2 l+ 1)Tl Pv (l,ε) dε l= 0
(4)
This study presents the results of theoretical model calculation of (n, p) cross sections resulting from interactions of neutron particles with Cadmium isotopes. The present simulation calculations for (n, p) cross sections were performed using the GDH and Weisskopf-Ewing models by the code ALICE/ASH, the Two–component exciton and HauserFeshbach models by the code TALYS 1.8, and the Exciton model by the code EMPIRE 3.2 . Furthermore, the Back-shifted fermi gas model (Dilg et al., 1973) in the TALYS 1.8code, the Enhanced generalized super-fluid model (D’Arrigo et al., 1994) in the EMPIRE 3.2 code and the Fermi gas model in the ALICE/ASH code are used for the level density calculations. Thus, the excitation functions of (n, p) nuclear reactions on 110-114,116 Cd isotopes considered in the present study have been shown in Figs. 1–6, and also the numerical values of data at the energies around 14.5MeV are given in Table 1.
(2)
n
Wk (pπ , h π , pυ , h υ, Ek) τ (pπ , h π , pυ , h υ)
3. Results and discussion
(1)
∑ Wb (E,n, εb) τ (n)
∑ pυ = p0υ
In the above equation, the terms pπ (pυ) and h π (h υ) represent the proton (neutron) particle number and the proton (neutron) hole number, respectively. The term σ CF represents the cross section for the compound nucleus formation predicted with the optical model. The term τ denotes the average lifetime of the exciton state. The terms Wk and Ek correspond to the emission rate and the emission energy of particle k , respectively. The quantity P is the part of the pre-compound stage for the nuclear emission to survive the previous states and now passes through the (pπ , h π , pυ , h υ) configuration, averaged over time. And also, the initial proton and neutron particle numbers are p0π = Z p and p0υ = Np , respectively with Zp (Np ) the proton (neutron) number of bombarding particle (Koning et al., 2013).
here the quantity L(n, E) denotes the total emission rate integrated over emission energy for γ -rays and particles. In addition to, qt (n) represents the initial occupation probability of composite nucleus in the state with n exciton number. And also, λ+ (E, n) and λ− (E, n) represent the transition rates of decay to neighboring states (Herman et al., 2013). The pre-equilibrium emission spectrum is given as follows,
dσa,b r (ε b) = σa,b (Einc)Da,b (Einc) × dε b
π
pmax υ
P(pπ , h π , pυ , h υ)
− qt= 0 (n) = λ− (E, n− 2) τ (n − 2) + λ+ (E, n+ 2) τ (n+2) − [λ+ (E, n) + λ− (E, n) + L(E, n)] τ (n)
max
π dσkPE = σ CF ∑ dEk p = p0
(3)
In the above equation, the term is the reduced de–Broglie wavelength of the incident particle. Also, Tl represents the transmission co2
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
Fig. 1. Cross section results for reaction.
111
Cd(n, p) 110Ag
(2016) give close results with the cross section data obtained using the empirical formulae of Yiğit (2018), Doczi et al. (1997) and Luo et al. (2008). The experimental cross sections measured by Yu and Gardner (1967) and Levkovskii (1964) showed acceptable agreement with the Exciton model calculations with pre-equilibrium effects in the simulation code EMPIRE 3.2.
Especially the Weisskopf-Ewing model, which contains the equilibrium component of the reaction, is much lower than the other excitation function spectra.
3.2.
110
Cd(n , p) 111Ag reaction
The excitation functions of 111Cd(n, p) 111Ag nuclear reaction are presented at Fig. 2. It was observed that the data obtained using the Two-component exciton model showed similar results with the measured excitation function of Bayhurst and Prestwood (1961) in the incident energy range of 7 − 14.81MeV. Additionally, the cross section measurements obtained by Filatenkov (2016) and Grallert et al. (1993) have shown a very good fit with the excitation function calculated using the Two-component exciton model in the simulation code TALYS 1.8. It is observed that the results of experimental study made by Filatenkov
3.3.
112
Cd(n , p) 112Ag reaction
The cross section data for the 112Cd(n, p) 112Ag reaction are given in Fig. 3. The experimental studies performed for this reaction give similar cross section results in the energy region of 13 − 15MeV. The cross section measurements made by Filatenkov (2016), Grallert et al. (1993), Konno et al. (1993), Struwe and Winkler (1974), Charturverdi et al. (1977) and Mavaddat et al. (1974) are in excellent agreement Fig. 2. Cross section results for reaction.
3
111
Cd(n, p) 111Ag
Applied Radiation and Isotopes 154 (2019) 108868
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Fig. 3. Cross section results for reaction.
113
Cd(n, p) 112Ag
lower than the measured data of the Konno et al. (1993) for this reaction. Moreover, the experimental cross section results of Konno et al. (1993) except for a few data points are compatible with the Twocomponent exciton model and TALS-based TENDL-2017 data. The highest cross section data are obtained by pre-equilibrium GDH model calculations made with the code ALICE/ASH. Moreover, the lowest results are obtained by the equilibrium emissions with Weisskopf-Ewing and Hauser-Feshbach models.
with the TENDL-2017 evaluation data. Furthermore, the agreement between the experimental results of Herman et al. (1978) and the TENDL data for the investigated reaction is acceptable except for a few data points. And also, the results for excitation function calculated with the empirical formulae are very compatible with the experimental measurements. The calculations including the equilibrium emissions with Weisskopf-Ewing and Hauser-Feshbach models give the lowest cross section results for this reaction.
3.4.
112
Cd(n , p) 113Ag reaction
3.5.
114
Cd(n , p) 114Ag reaction
The nuclear excitation functions of the 114Cd(n, p) 114Ag reaction produced by neutrons on the 114Cd target nucleus are shown in Fig. 5. The calculation results obtained using the empirical systematics of Yiğit (2018) and Doczi et al. (1997) appear to be in good agreement with the
In Fig. 4, the experimental and theoretical cross section values are shown for 113Cd(n, p) 113Ag nuclear reaction. It can be seen from Fig. 4 that the experimental data of Levkovskii (1964) and Yu and Gardner (1967) and also the calculation results of (n, p) empirical formulae are
Fig. 4. Cross section results for reaction.
4
113
Cd(n, p) 113Ag
Applied Radiation and Isotopes 154 (2019) 108868
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Fig. 5. Cross section results for reaction.
114
Cd(n, p) 114Ag
compatible with each other for this reaction. In addition, the experimental data obtained by Anders vd. (1985) and Bjomstad and Alkstad (1973) have almost the same results. These experimental results show similar agreements with the results of Two-component exciton models, which include the pre-equilibrium effects of the nuclear reaction within the code TALYS 1.8. Fig. 6 shows that the cross section results calculated using Hauser-Feshbach and Weisskopf-Ewing models are too lower than the pre-equilibrium models and measured data.
experimental cross section of 12 ± 0.9mb reported by Anders et al. (1985) at 14.7 MeV energy. The cross section results measured by Struwe and Winkler (1974) and Prasad and Sarkar (1971) give lower values than the results of Anders et al. (1985) and empirical formulae for this reaction. The excitation function of Struwe and Winkler (1974) is consistent with cross section results of Two-component exciton model and TENDL-2017 . On the other hand, the data obtained by Prasad and Sarkar (1971) showed acceptable agreement with the Two-component exciton model and the Exciton model.
4. Conclusions 3.6.
116
Cd(n , p) 116Ag reaction In this paper, the excitation function data of the (n, p) reactions produced by neutron particles on some cadmium isotopes were calculated from the threshold to 20 MeV energy. The data calculated using the empirical formulae were found to be consistent with experimental values except for 116Cd(n, p) 116Ag nuclear reaction. The excitation
Fig. 6 present a comparison of the calculated and measured excitation functions for 116Cd(n, p) 116Ag nuclear reaction. The cross section data predicted using the formulae of Yiğit (2018), Luo et al. (2008), Tel et al. (2003), Levkovskii (1964) and Doczi et al. (1997) are
Fig. 6. Cross section results for reaction.
5
116
Cd(n, p) 116Ag
6
Cross section systematics
Model calculations
Experimental data
– – – – – – 43.25 23.66 21.41 4.22 18.2 26.36 27.98 26.33 19.59 29.1 22.73
Chaturverdi et al. (1977) Mavaddat et al. (1974) Struwe and Winkler (1974) Anders vd. (1985) Prasad ve Sarkar (1971) Bjornstad and Alstad (1973) GDH model Two-component exciton model Exciton model Weisskopf-Ewing model Hauser-Feshbach model TENDL-2017 Yiğit (2018) Luo et al. (2008) Tel et al. (2003) Doczi et al. (1997) Levkovskii (1964) 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.7 14.5
– – – – – – 55 27.51 12.67 4.76 8.72 29.92 21.88 20.18 17.03 23.26 17.62
– –
Grallert et al. (1993) Levkovskii (1964) Bayhurst and Prestwood (1961)
Herman et al. (1978) Konno et al. (1993)
Cd(n, p) 111Ag
111
29± 2.1 (at 14.6 MeV) 15± 4 (at 14 MeV) 23.7± 1.2 (at 14.01 MeV) 22.5± 1.1 (at 14.09 MeV) 27.7± 1.4 (at 14.31 MeV) 28.7± 1.4 (at 14.5 MeV) 28.1± 1.4 (at 14.68 MeV) 30.7± 1.5 (at 14.81 MeV) 36.3± 1.8 (at 14.93 MeV) – –
Cd(n, p) 110Ag
– – –
110
(n, p) cross sections (mb)
14± 1.4 (at 14.1 MeV) 23.9± 1.87 (at 14.44 MeV) 26.6± 1.87 (at 14.64 MeV) 31± 2.18 (at 14.88 MeV)
14–15
Energy (MeV)
Cd isotopes at the energies around 14.5MeV.
27± 5.4 (at 14.1 MeV) –
110-114,116
Yu and Gardner (1967) Filatenkov (2016)
Table 1 Cross sections of (n, p) reactions on
19.1± 1.6 (at 14.5 MeV) 14.7± 1.9 (at 14.23 MeV) 16.3± 2.1 (at 14.44 MeV) 17.6± 2.3 (at 14.68 MeV) 18.1± 2.4 (at 14.95 MeV) 15± 1.3 (at 14 MeV) 18.7± 6.1 (at 14.8 MeV) 15.3± 1.9 (at 14.6 MeV) – – – 28.26 6.28 5.1 1.54 0.56 15.49 17.14 15.55 13.15 18.62 13.72
– 13.6± 1.3 (at 14.05 MeV) 14.7± 1.18 (at 14.28 MeV) 14.6± 1.18 (at 14.47 MeV) 16.9± 1.28 (at 14.68 MeV) 16.5± 1.44 (at 14.81 MeV) 15.4± 1.19 (at 14.86 MeV) 16± 1.2 (at 14.6 MeV) – –
Cd(n, p) 112Ag
112
– – – – – 41.86 16.61 4.12 1.80 2.82 19.02 13.67 12.03 12.31 14.93 10.74
– 18.2± 2.6 21.4± 2.6 25.9± 2.5 27.5± 2.8
(at (at (at (at
14.24 MeV) 14.45 MeV) 14.69 MeV) 14.96 MeV)
– 8± 2 (at 14 MeV) –
7.9±0.79 (at 14.1 MeV) –
Cd(n, p) 113Ag
113
Cd(n, p) 114Ag
– 5±1.3 (at 14.6 MeV) 12± 0.9 (14.7 MeV) 3±1.6 (14.8 MeV) – 18.86 3.40 1.70 0.42 0.43 5.04 10.68 9.35 8.96 11.98 8.44
– –
– – –
– –
114
Cd(n, p) 116Ag
– – 2.5±0.3 (14.7 MeV) – 2.2±0.5 (14.7 MeV) 12.00 1.43 0.37 0.11 0.09 3.07 6.77 5.72 6.19 7.75 5.27
– –
– – –
– –
116
M. Yiğit and S.N. Bostan
Applied Radiation and Isotopes 154 (2019) 108868
Applied Radiation and Isotopes 154 (2019) 108868
M. Yiğit and S.N. Bostan
functions of Two-component exciton model and TENDL-2017 library except for the nuclear reaction performed on the 112Cd target nucleus showed similar spectra. We have also observed that the calculation results obtained using Two-component exciton model and TENDL-2017 for the investigated nuclear reactions were generally successful in estimating experimental values. On the other hand, the theoretical estimations with Exciton model using the code EMPIRE 3.2are in not good agreement with the measured values. It is also noted that the data obtained using GDH model with pre-equilibrium effects give the highest cross section values. The Hauser-Fescbah model and Weisskopf-Ewing model calculations have lowest cross section values at high incident neutron energies.
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