2.1 ALICE/ASH code

The ALICE/ASH code, which is useful for pre-equilibrium calculations, was developed by Broeders et al.[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]. The pre-equilibrium models are the hybrid and GDH models in the ALICE/ASH code. The hybrid model was formulated by Blann [33] and combines the simplicity of the formulation of the exciton model [34] with the ability to predict the absolute cross-sections of the Harp-Miller-Berne model [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 [36–38].

2.2 TALYS 1.8 code

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, ³He, and alpha-particles, and for target nuclei from Li to Dy in the 1 keV–200 MeV incident particle energy range [20]. The code provides observables of the reaction and a complete description of all the reaction channels [20]. Also, the code can be used for the analysis of basic scientific experiments or generation of nuclear data for applications [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, 39].

2.3 Nuclear level densities

Two important models for level density approximation were firstly performed by Bethe [40, 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], Newton [43], and Cameron [44]. The excitation energy in the model was changed using the pairing energy parameter ($\Delta$) and the ($\Delta$) and (2$\Delta$) was represented by the shifts for odd mass and even-even nuclei, respectively [45]. The next step for nuclear level density by Gilbert and Cameron was to define the constant temperature (CT) model [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]. 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 [45–47]. 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]. 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, 47].

The other level density model is the generalized superfluid model (GSM) [48, 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]. 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].

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 $a=A/9$. 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:

Here, $\tilde{a}$ is the asymptotic level density, $\gamma$ is the damping parameter, and $\delta W$ is the shell correction energy. Also, the effective excitation energy ($U$) in Eq. (1) is determined by $U=E_x-\Delta$, where $E_x$ is the true excitation energy and an empirical parameter, $\Delta$, 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:

where $U^{\prime }$ is the effective excitation energy, $U_{\text{cr}}$ is the critical energy of the phase transition and $E_{\text{cond}}$ is condensation energy. The extensive reviews for nuclear level density models and the level density parameters can be seen in Refs. [20, 31].

3.1 The excitation functions of the ^60,62Ni(α,n), ^60,61Ni(α,2n), and ^58,64Ni(α,p) reactions

All theoretical results for the ⁶⁰Ni(α,n) reaction are close to each other and the experimental data [50–55] up to ~12 MeV incident alpha energy in 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 $\chi \frac{12}{\sqrt{A}}$ ($\chi$=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 ⁶²Ni(α,n), ^60,61Ni(α,2n), and ⁶⁴Ni(α,p) reactions in Figs. 2-4, 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. 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.

Fig. 2Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ⁶²Ni(α,n)⁶⁵Zn reaction.

For the ⁶²Ni(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] and Zyskind et al. [56] in 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]. 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 ⁶²Ni(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]. The obtained excitation function used for the CT+FGM model in the TALYS 1.8 code for the ⁶²Ni(α,n) reaction in this energy range is similar to those of Yıldız and Aydın.

In 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] and Levkovski [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]. 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 ⁶¹Ni(α,2n) reaction in 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].

Fig. 3Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ⁶⁰Ni(α,2n)⁶²Zn reaction.

Fig. 4Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ⁶¹Ni(α,2n)⁶³Zn reaction.

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 [53,57,58] up to ~ 12 MeV, except for one set of data by Yadav et al. [50] at 9 MeV (in Fig. 5 ). Also, the excitation functions of the ⁵⁸Ni(α,p) reaction to the production of ⁶¹Cu were analyzed by Aslam and Qaim using the EMPIRE and the BSFGM in the TALYS 1.4 codes [59]. When the Aslam and Qaim study and the present one for the ⁵⁸Ni (α,p) reaction are compared, the results are close to each other. For the ⁶⁴Ni(α,p) reaction, the calculated excitation functions using the GSM in the two–component exciton model are more compatible with experimental data [51,52,58,60] than those of other models in Fig. 6. Also, Levkovski’s experimental data for this reaction and ^60,62Ni(α,n), ⁶⁰Ni(α,2n), and ⁶⁴Ni(α,p) reactions in Figs. 1 -3,6 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].

Fig. 5Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ⁵⁸Ni(α,p)⁶¹Cu reaction.

Fig. 6Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ⁶⁴Ni(α,p)⁶⁷Cu reaction.

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 ^60,62Ni(α,n), ⁶⁰Ni(α,2n), ⁶¹Ni(α,2n), ⁵⁸Ni(α,p), and ⁶⁴Ni(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 ^60,62Ni(α,n) ^60,61Ni(α,2n) and ^58,64Ni(α,p) reactions are radioactive; ⁶²Zn (T_1/2= 9.19 h and β⁺ decay), ⁶³Zn (T_1/2= 38.47 m and β⁺ decay), ⁶⁵Zn (T_1/2= 243.93 d and β⁺ decay), ⁶¹Cu (T_1/2= 3.33 h and β⁺ decay) and ⁶⁷Cu (T_1/2= 61.83 h and β^- decay) [27].

3.2 The excitation functions of the ^natNi(α,x) reactions

The obtained excitation functions using the TALYS 1.8 code for the ^natNi(α,x) reactions are presented in Figs. 7, 8, 9, 10, 11, 12 and 13. The optimum energy ranges for the ^natNi(α,x) reactions in this study are obtained as E_α= 14 → 65 MeV. The different nuclear level density models’ effect on the ^natNi(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 ^natNi(α,x)⁵⁵Co, the ^natNi(α,x)⁵⁸Co, ^natNi(α,x)⁶¹Cu, and the GSM for the ^natNi(α,x)⁵⁶Co and the ^natNi(α,x)⁶²Zn, and the BSFGM and GSM together for the ^natNi(α,x)⁶⁵Zn are compatible with the experimental data [61–65]. For the ^natNi(α,x)⁵⁶Ni reaction, while the CT+FGM results are in good agreement with experimental data [61, 62, 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 ^natNi(α,x)⁶¹Cu reaction in the optimum energy range has the highest value among ^natNi(α,x) reactions. Also, the product nuclei of these reactions are radioactive (β⁺ decay); ⁵⁵Co (T_1/2= 17.53 h), ⁵⁶Co (T_1/2= 77.24 d), ⁵⁶Ni (T_1/2= 6.08 d), ⁵⁸Co (T_1/2= 70.86 d), ⁶¹Cu (T_1/2= 3.33 h), ⁶²Zn (T_1/2= 9.19 h), and ⁶⁵Zn (T_1/2= 243.93 d) [27].

Fig. 7Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁵⁵Co reaction.

Fig. 8Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁵⁶Co reaction.

Fig. 9Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁵⁶Ni reaction.

Fig. 10Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁵⁸Co reaction.

Fig. 11Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁶¹Cu reaction.

Fig. 12Full-screen View

The comparison of the theoretically calculated excitation functions with experimental data for the ^natNi(α,x)⁶²Zn reaction.