A study on the excitation functions of ^60,62Ni(α,n), ^60,61Ni(α,2n), ^58,64Ni(α,p), ^natNi(α,x) reactions

A study on the excitation functions of 60,62 Ni(α,n), 60,61 Ni(α,2n), 58,64 Ni(α,p), nat Ni(α,x) reactions 1 corresp first-author Corresponding Author. e-mail adress: rbaldik@gmail.com (R. Baldık) Corresponding Author. e-mail adress: rbaldik@gmail.com (R. Baldık) rbaldik@gmail.com 1 Bülent Ecevit University, Faculty of Arts and Science, Department of Physics, 67100, Zonguldak, Turkey Bülent Ecevit University, Faculty of Arts and Science, Department of Physics, 67100, Zonguldak, Turkey The prediction of nuclear cross-section data is crucial, especially in the absence of experimental data or in the difficulty of this experimental data. Nickel (Ni) is an important material in fusion and fission reactor technologies, the production of radionuclides in nuclear medicine, and many other fields. In this study, the excitation functions for 60,62 Ni(α,n), 60,61 Ni(α,2n), 58,64 Ni(α,p), and nat Ni(α,x) reactions have been investigated by using pre-equilibrium reaction models. The calculations of the excitation functions for the reactions are used with the geometry dependent hybrid model in ALICE/ASH code and the two-component exciton model in TALYS 1.8 code. The obtained results are compared to each other and the experimental data is taken from the EXFOR database. Nuclear reaction models and methods Level density Alpha induced reactions ALICE/ASH code TALYS 1.8 code 1 Introduction 1 The nuclear cross-section data for particle induced reaction benefits from the structural materials development in the fusion and fission reactors, medical radioisotope production, and astrophysical applications etc. Because the obtained nuclear reaction data via experiment has difficulties with regards to both time and financial considerations, the importance of theoretical data has increased. Therefore, theoretical cross-section data is used to determine the optimal energy ranges required for a reaction type and to calculate the radioisotope production yield [ 1 1 , 2 2 ]. The cross-section data can also be used to obtain the radioactive impurities that help to determine isotopically enriched target materials for medical radioisotopes production and in the calculations of astrophysical S-factors for nuclear astrophysics reactions etc. [ 1 1 , 2 2 ]. Therefore, research and development efforts on the nuclear cross-section data has gradually increased in the literature [ 1 1 , 3 3 , 12 12 – 19 19 12–19 , 4 4 – 11 11 4–11 ]. Nuclear reaction models are necessary in order to determine nuclear reaction data and make theoretical calculations. The models are typically required to provide the estimation of the nuclear reaction cross-sections. Pre-equilibrium reaction models are frequently used in cross-section calculations for estimating radiation damage, and producing radioisotopes, etc. The pre-equilibrium reaction mechanism, which was modeled classically and quantum mechanically, exists between direct and compound reaction types in terms of reaction time [ 20 20 , 21 21 ]. These reactions, which are on timescales of 10 -20 – 10 -18 s, are induced by light projectiles with incidences above ~10 MeV. Nuclear level density is one of the characteristic properties of all nuclei and the property is defined as the total number of energy levels per MeV at a certain excitation energy. The level density has an important role in the calculations of the nuclear cross-section. There are several models in literature for the determination of the nuclear level density. A brief summary of these models is given in section 2.3. Therefore, the investigations of different level densities effects on the nuclear reactions have been performed by some researchers [ 9 9 , 10 10 , 22 22 – 26 26 22–26 ]. In natural nickel (Ni) five isotopes exist ( 58 Ni % 68,077, 60 Ni % 26,223, 61 Ni % 1.140, 62 Ni % 3.635, and 64 Ni % 0.926)[ 27 27 ]. The Ni and its alloys are resistant to high temperatures. Therefore, they are used in structural mat erials for the fusion and fission appliances. In addition, the Ni is used in space and nuclear equipment working under radiation, radioisotopes production, astrophysical applications, etc. Nevertheless, the Ni activation data are interesting for the design of accelerator-driven systems and analyzed in thin film activation. The Ni targets are also used in the production of 55-58 Co, 62 Zn, and 60-62,64 Cu medical radioisotopes [ 28 28 – 30 30 28–30 ]. In this study, the excitation functions of 60,62 Ni(α,n), 60,61 Ni(α,2n), 58,64 Ni(α,p), and nat Ni(α,x) reactions are calculated by using the pre-equilibrium reaction mechanism. The calculations are made by the ALICE/ASH code[ 31 31 ], which was used with the geometry dependent hybrid (GDH) models and by the TALYS 1.8 code[ 20 20 ] used with the two–component exciton model. However, in this study, for the first time, the reactions have been handled with the ALICE/ASH and the TALYS 1.8 codes. In addition, the different nuclear level density models effect on the GDH and the two–component exciton models have been investigated for the reactions in the present study. The obtained excitation functions have been discussed and compared with the experimental data [ 32 32 ]. 2 Calculations 1 2.1 ALICE/ASH code 2 The ALICE/ASH code, which is useful for pre-equilibrium calculations, was developed by Broeders et al.[ 31 31 ]. The ALICE/ASH code includes the calculations of excitation functions, energy and angular distributions of the particles emitted in nuclear reactions, residual nuclear yields, and total inelastic cross-sections for particles-nuclei reactions with up to 300 MeV [ 31 31 ]. The pre-equilibrium models are the hybrid and GDH models in the ALICE/ASH code. The hybrid model was formulated by Blann [ 33 33 ] and combines the simplicity of the formulation of the exciton model [ 34 34 ] with the ability to predict the absolute cross-sections of the Harp-Miller-Berne model [ 35 35 ]. The GDH model contains the effects of the interactions in the diffuse nuclear surface. The nuclear densities in the GDH model are calculated by averaging over the densities relative to the entrance channel of projectiles; at least for the first projectile and target interaction [ 36 36 – 38 38 36–38 ]. 2.2 TALYS 1.8 code 2 The TALYS 1.8 code is a computer code that contains the simulation of nuclear reactions for incident particles such as protons, neutrons, photons, deuterons, tritons, 3 He, and alpha-particles, and for target nuclei from Li to Dy in the 1 keV–200 MeV incident particle energy range [ 20 20 ]. The code provides observables of the reaction and a complete description of all the reaction channels [ 20 20 ]. Also, the code can be used for the analysis of basic scientific experiments or generation of nuclear data for applications [ 20 20 ]. In this study, the pre-equilibrium calculations in the TALYS 1.8 code are performed by using the two–component exciton model. The two–component exciton model is another version of the exciton model, where the temporal development of the system can be described by a master equation. The master equation is determined by the exciton numbers separating the neutron and proton excitons [ 21 21 , 39 39 ]. 2.3 Nuclear level densities 2 Two important models for level density approximation were firstly performed by Bethe [ 40 40 , 41 41 ]. The first model was based on the gas system, which consists of non-interacted particles called the Fermi gas model (FGM). The second was the liquid drop model. Hereupon, the conventional shifted Fermi gas was introduced by Erba et al. [ 42 42 ], Newton [ 43 43 ], and Cameron [ 44 44 ]. The excitation energy in the model was changed using the pairing energy parameter ( <math id="M1"><mi>Δ</mi></math> ) and the ( <math id="M2"><mi>Δ</mi></math> ) and (2 <math id="M3"><mi>Δ</mi></math> ) was represented by the shifts for odd mass and even-even nuclei, respectively [ 45 45 ]. The next step for nuclear level density by Gilbert and Cameron was to define the constant temperature (CT) model [ 46 46 ]. The important step includes the corrections of some parameterizations such as dependence on atomic mass effects, shell effects, and odd-even pairing effects. Their formula included a two-component level density formula with energy shifts [ 45 45 ]. To calculate the level density, they used a constant temperature formula for lower energies (for the first energy levels till ~10 MeV) and the conventional shifted Fermi gas formula at higher excitation [ 45 45 – 47 47 45–47 ]. The model is called the constant temperature plus Fermi gas model (CT+FGM), and the only free parameter for this model is the level density parameter. Another simpler and rather effective model is the back-shifted Fermi gas model (BSFGM) [ 47 47 ]. The model, in which all excitation energies are obtained by using the Fermi gas formula, has two free parameters: the level density and the back-shifted energy parameters [ 45 45 , 47 47 ]. The other level density model is the generalized superfluid model (GSM) [ 48 48 , 49 49 ], which is based upon the pairing correlations of the Bardeen-Cooper-Schrieffer (BCS) theory in the superconductor. Here, the pairing correlations represented to a phase transition from superfluid behavior, which strongly influenced on the nuclear level density at low energies. However, the model is described by the FGM at high energies [ 20 20 ]. The GSM at high energies is similar to CT+FGM, but at low energies, the GSM is independent of the specific discrete levels and follows the nature of the theory [ 20 20 ]. The level density models used to calculate the cross-sections in this study are as follows: the FGM with an energy independent level density parameter in the ALICE/ASH code is labeled as the FGM-1 in the present study (the model is the default model in the ALICE/ASH code). In this model, the level density parameter (a) is taken as the <math id="M4"><mrow><mi>a</mi><mo>=</mo><mi>A</mi><mo>/</mo><mn>9</mn></mrow></math> . Another option for the FGM in the ALICE/ASH code is the FGM with an energy dependent level density parameter and that option in this study is labeled as the FGM-2. The parameter is: <math display="block" id="M5"><mrow><mi>a</mi><mo>=</mo><mover accent="true"><mi>a</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>δ</mi><mi>W</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>γ</mi><mi>U</mi></mrow></msup></mrow><mi>U</mi></mfrac><mo stretchy="false">)</mo><mo>.</mo></mrow></math> Here, <math id="M6"><mover accent="true"><mi>a</mi><mo>˜</mo></mover></math> is the asymptotic level density, <math id="M7"><mi>γ</mi></math> is the damping parameter, and <math id="M8"><mrow><mi>δ</mi><mi>W</mi></mrow></math> is the shell correction energy. Also, the effective excitation energy ( <math id="M9"><mi>U</mi></math> ) in Eq. (1) is determined by <math id="M10"><mrow><mi>U</mi><mo>=</mo><msub><mi>E</mi><mi>x</mi></msub><mo>−</mo><mi>Δ</mi></mrow></math> , where <math id="M11"><mrow><msub><mi>E</mi><mi>x</mi></msub></mrow></math> is the true excitation energy and an empirical parameter, <math id="M12"><mi>Δ</mi></math> , is the energy shift. However, the level density parameter in Eq.(1) is used for the CT+FGM (default model) and the BSFGM in the TALYS 1.8 code. On the other hand, the level density parameter for the GSM in the ALICE/ASH and TALYS 1.8 codes is as follows: <math display="block" id="M13"><mrow><mi>a</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mover accent="true"><mi>a</mi><mo>˜</mo></mover><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>δ</mi><mi>W</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>γ</mi><mrow><mo>(</mo><mrow><msup><mi>U</mi><mo>′</mo></msup><mo>−</mo><msub><mi>E</mi><mrow><mtext>cond</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow></msup></mrow><mrow><msup><mi>U</mi><mo>′</mo></msup><mo>−</mo><msub><mi>E</mi><mrow><mtext>cond</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo><msup><mi>U</mi><mo>′</mo></msup><mo>></mo><msub><mi>U</mi><mrow><mtext>cr</mtext></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi>a</mi><mrow><mo>(</mo><mrow><msub><mi>U</mi><mrow><mtext>cr</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>,</mo><msup><mi>U</mi><mo>′</mo></msup><mo>≤</mo><msub><mi>U</mi><mrow><mtext>cr</mtext></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>,</mo></mrow></math> where <math id="M14"><msup><mi>U</mi><mo>′</mo></msup></math> is the effective excitation energy, <math id="M15"><mrow><msub><mi>U</mi><mrow><mtext>cr</mtext></mrow></msub></mrow></math> is the critical energy of the phase transition and <math id="M16"><mrow><msub><mi>E</mi><mrow><mtext>cond</mtext></mrow></msub></mrow></math> is condensation energy. The extensive reviews for nuclear level density models and the level density parameters can be seen in Refs. [ 20 20 , 31 31 ]. 3 Results and discussions 1 The comparison of the calculated excitation functions for 60,62 Ni(α,n), 60,61 Ni(α,2n), 58,64 Ni(α,p), and nat Ni(α,x) reactions using the pre-equilibrium reaction models and the available experimental data is presented in Figs. 1 Figs. 1 - 13 13 as a function of the incident alpha energy. The experimental data were taken from the EXFOR data files [ 32 32 ]. 3.1 The excitation functions of the 60,62 Ni(α,n), 60,61 Ni(α,2n), and 58,64 Ni(α,p) reactions 2 All theoretical results for the 60 Ni(α,n) reaction are close to each other and the experimental data [ 50 50 – 55 55 50–55 ] up to ~12 MeV incident alpha energy in Fig. 1 Fig. 1 , except for the FGM-2. The reason for this, can be arised from using the CT formula for low energies in the FGM-2. While the CT approach is used at the excitation energies < 2 MeV in the FGM-2, this approach for the CT+FGM in the two-component exciton model is applied into a low energy range from 0 MeV up to a matching energy, E M . The matching energy is determined by E M = 2.33 + 253/ A + ∆ CTM , where the ∆ CTM is the energy shift which is given by <math id="M17"><mrow><mi>χ</mi><mfrac><mrow><mn>12</mn></mrow><mrow><msqrt><mi>A</mi></msqrt></mrow></mfrac></mrow></math> ( <math id="M18"><mi>χ</mi></math> =0 for odd-odd nuclei, 1 for odd-even nuclei, 2 for even-even nuclei). The same effect (the CT approach) on the excitation functions of 62 Ni(α,n), 60,61 Ni(α,2n), and 64 Ni(α,p) reactions in Figs. 2 Figs. 2 - 4 4 , 6 6 is seen more or less. In the range of ~12-20 MeV incident energy, the obtained excitation functions using the different level density models have different values in Fig.1 Fig.1 . Nevertheless, it can be said that the results for different level density models in the GDH model are generally compatible with the experimental datasets. According to the obtained results above 20 MeV, the closest theoretical results to the experimental datasets are the FGM-2 in the GDH model. For the 62 Ni(α,n) reaction up to 10.8 MeV, the calculated excitation functions using the three level density models in the two-component exciton model, are close to each other and in good agreement with the experimental data of Stelson and McGowan [ 53 53 ] and Zyskind et al. [ 56 56 ] in Fig. 2 Fig. 2 . The excitation functions for the FGM-2 in the GDH model for all incident alpha energies are also in good agreement with the experimental data of Tanaka [ 52 52 ]. Furthermore, the most compatible model with the experimental data in the high incident energy range for this reaction is the FGM-2 in the GDH model. Also, in the literature, the calculations of the excitation function for the 62 Ni(α,n) reaction in the incident alpha energy of 6-10 MeV for the CT+FGM model in TALYS 1.6 were conducted by Yıldız and Aydın [ 1 1 ]. The obtained excitation function used for the CT+FGM model in the TALYS 1.8 code for the 62 Ni(α,n) reaction in this energy range is similar to those of Yıldız and Aydın. In Fig. 3 Fig. 3 , the different level density models for the two-component exciton model are not effective in the excitation functions for the 60 Ni(α,2n) reaction up to ~25 MeV. The results of the two-component exciton model for this reaction up to ~29 MeV are in agreement with the experimental data of Tanaka [ 52 52 ] and Levkovski [ 51 51 ]. In the range of 25-29 MeV, the results of the GSM in the two-component exciton model and the FGM-2 in the GDH model are compatible with the data of Tanaka [ 52 52 ]. After this incident alpha energy, i.e. in the high-energy range, the most compatible model with the experimental data is the FGM-1 in the GDH model. Also, for the 61 Ni(α,2n) reaction in Fig. 4 Fig. 4 , the obtained excitation functions used for the GSM and the FGM-2 in the GDH model are close to each other and have higher values than those of the FGM-1. On the other hand, the results of the BSFGM and the GSM in the two-component exciton model are higher than the CT+FGM. For this reaction, the results for the FGM-1 in the GDH model at the 36 MeV and the BSFGM in the two-component exciton model at the 40 MeV are in good agreement with the Yadav et al. data [ 50 50 ]. The calculated excitation functions used in the different level density models in the two-component exciton model are close to each other and are in good agreement with the experimental data [ 53 53 , 57 57 , 58 58 ] up to ~ 12 MeV, except for one set of data by Yadav et al. [ 50 50 ] at 9 MeV (in Fig. 5 Fig. 5 ). Also, the excitation functions of the 58 Ni(α,p) reaction to the production of 61 Cu were analyzed by Aslam and Qaim using the EMPIRE and the BSFGM in the TALYS 1.4 codes [ 59 59 ]. When the Aslam and Qaim study and the present one for the 58 Ni (α,p) reaction are compared, the results are close to each other. For the 64 Ni(α,p) reaction, the calculated excitation functions using the GSM in the two–component exciton model are more compatible with experimental data [ 51 51 , 52 52 , 58 58 , 60 60 ] than those of other models in Fig. 6 Fig. 6 . Also, Levkovski’s experimental data for this reaction and 60,62 Ni(α,n), 60 Ni(α,2n), and 64 Ni(α,p) reactions in Figs. 1 Figs. 1 - 3 3 , 6 6 are considerably different in behavior than those of other experimental data. The reason of this may be due to Levkovski’s use of a different target holder, which was a rotating target for the simultaneous irradiation of a few foils [ 51 51 ]. The optimum energy range for a nuclear reaction is the incident particle energy range corresponding to the highest cross-sections. In this study, the optimum energy ranges for the 60,62 Ni(α,n), 60 Ni(α,2n), 61 Ni(α,2n), 58 Ni(α,p), and 64 Ni(α,p) reactions was obtained as E α = 14 → 19 MeV, E α = 15 → 18 MeV, E α = 26 → 32 MeV, E α = 14 → 17 MeV, and E α = 18 → 21 MeV, respectively. Also, the product nuclei of the 60,62 Ni(α,n) 60,61 Ni(α,2n) and 58,64 Ni(α,p) reactions are radioactive; 62 Zn ( T 1/2 = 9.19 h and β + decay), 63 Zn ( T 1/2 = 38.47 m and β + decay), 65 Zn ( T 1/2 = 243.93 d and β + decay), 61 Cu ( T 1/2 = 3.33 h and β + decay) and 67 Cu ( T 1/2 = 61.83 h and β - decay) [ 27 27 ]. 3.2 The excitation functions of the nat Ni(α,x) reactions 2 The obtained excitation functions using the TALYS 1.8 code for the nat Ni(α,x) reactions are presented in Figs. 7 Figs. 7 , 8 8 , 9 9 , 10 10 , 11 11 , 12 12 and 13 13 . The optimum energy ranges for the nat Ni(α,x) reactions in this study are obtained as E α = 14 → 65 MeV. The different nuclear level density models’ effect on the nat Ni(α,x) reactions prominently arise after ~20-30 MeV incident alpha energies. In addition, it is difficult to state that a single model could substitute the experimental data. Therefore, the calculated excitation functions using the CT+FGM for the nat Ni(α,x) 55 Co, the nat Ni(α,x) 58 Co, nat Ni(α,x) 61 Cu, and the GSM for the nat Ni(α,x) 56 Co and the nat Ni(α,x) 62 Zn, and the BSFGM and GSM together for the nat Ni(α,x) 65 Zn are compatible with the experimental data [ 61 61 – 65 65 61–65 ]. For the nat Ni(α,x) 56 Ni reaction, while the CT+FGM results are in good agreement with experimental data [ 61 61 , 62 62 , 64 64 ] in the low energy range, the GSM results are compatible with the experimental data in the high energy range. The cross-section for the nat Ni(α,x) 61 Cu reaction in the optimum energy range has the highest value among nat Ni(α,x) reactions. Also, the product nuclei of these reactions are radioactive (β + decay); 55 Co ( T 1/2 = 17.53 h), 56 Co ( T 1/2 = 77.24 d), 56 Ni ( T 1/2 = 6.08 d), 58 Co ( T 1/2 = 70.86 d), 61 Cu ( T 1/2 = 3.33 h), 62 Zn ( T 1/2 = 9.19 h), and 65 Zn ( T 1/2 = 243.93 d) [ 27 27 ]. 4 Conclusion 1 The excitation functions of 60,62 Ni(α,n), 60,61 Ni(α,2n), 58,64 Ni(α,p), and nat Ni(α,x) reactions are obtained by using the different nuclear level density models effects on the GDH and the two–component exciton models. According to the results, while nuclear level density models have an effect on the excitation functions of the handled reaction in the low incident energy range, it is relatively small, and this influence is more visible in the high incident energy range. Therefore, in this study, it is hard to state that a single model can explain the pre-equilibrium cross-sections of the alpha incident reactions. Furthermore, the results of the two models are different although the GDH and the two–component exciton models are based on the exciton model. The reason for this originates in the difference between the GDH model and the exciton model. Namely, while the GDH model is an independent particle model and also an inclusive model, the exciton model is a system approximation and exclusive model [ 66 66 ]. However, mixing a complete particle-hole configuration within exciton states is assumed in the exciton model, whereas the GDH model assumes no configuration mixing at all [ 66 66 , 67 67 ]. In addition, the initial exciton number and the internal decay rates in two models are crucial and adjustable parameters, so it is difficult to make an experimental verification as to which of the models is correct [ 68 68 ]. 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