1 Introduction

The cross section data are indispensable in a lot of applications in the history of nuclear physics. Especially, data on cross sections of nuclear reactions induced by deuterons are required to describe the nucleus–particle interaction. This interaction represents a good test for both nuclear models and evaluation of nuclear reaction data [1,2]. The deuteron particle has a weak binding energy of 2.224 MeV. So, it is responsible for the high complexity of the interaction process, which involves a variety of nuclear reactions produced by the proton and neutron following the deuteron breakup [1]. The available cross sections for the nuclear reactions induced by the deuterons in literature show large discrepancies. Actually, because the experimental studies for these reactions are quite poor, more measurements and theoretical studies must be carried out. Hence, the nuclear models for particle–induced reactions are very useful since they can be provided for estimating the cross sections quickly [3]. Particularly, both experimental and theoretical cross section data around 14 MeV are of considerable importance for verifying the accuracy of nuclear models [4,5]. On the other hand, the nuclear fusion will not contribute to acid rain and global warming because it will not produce SO₂ or CO₂. So, it offers the best hope as an energy source for the future generations. The success of the energy production from fusion requires the development of structural materials for fusion reactors [6-8]. In this framework, the selection of structural fusion materials is very important for nuclear fusion reactors. Chromium, nickel, zinc, scandium, titanium, and yttrium are important elements of structural fusion materials. In this paper, the deuteron reaction cross sections of these structural materials were calculated using different input parameters and nuclear models in the ALICE–ASH computer code [9]. And also, the (d,n) and (d,2n) cross section systematics were used at cross section calculations for the investigated deuteron reactions. In the calculations, the effect on cross sections of the variation of initial exciton number and level density parameters has been investigated by the Geometry dependent hybrid (GDH) model [10] in the ALICE code. Also, the initial exciton configuration is an important input parameter in the GDH model [11]. We used three particles–one hole and four particles–two holes were used as initial exciton configuration in the pre–equilibrium process for incident deuterons. Furthermore, the cross sections for the considered nuclear reactions were estimated using different level density parameters via Fermi gas model. Therefore, the exciton numbers (n₀=3p–1h to 4p–2h) and the nuclear level parameters (α=A/5; A/8; A/11) were varied obtaining a good agreement between the experimental and model calculations.

2 Theoretical Calculation Methods

Recently, a lot of theoretical calculation and experimental data analysis on the pre-equilibrium reaction mechanism have been reported because of the strong competition between the pre-equilibrium and equilibrium emissions of light particles [8,11,12,13]. Energetic light particles in the pre-equilibrium emissions have generally been emitted at the initial stage of interactions of target nucleus with incident particle. Actually, the features of excitation functions at low, medium, and high incident energies may reveal the characteristics of the nuclear reaction mechanism. The equilibrium decay process dominates at the portion with low energy of the excitation function, but the pre-equilibrium decay becomes important with increasing incident energies, and a slowly decreasing tail at the excitation function becomes apparent [12]. Hence, the excitation functions for nucleon emissions in a nuclear reaction may be measurable at incident energies where pure evaporative processes are greatly favored. In addition, the time scale at which the pre-equilibrium emissions occur is very short, ≈10⁻²¹ s, while the equilibrium nuclear processes take a longer time, 10⁻¹⁶ to 10⁻¹⁸s. Several models for explaining the pre-equilibrium and equilibrium reaction mechanisms have been proposed, including the Weisskopf-Ewing model, Hauser-Feshbach model, hybrid model, exciton model, and Geometry dependent hybrid model. The ALICE-ASH code has widely been used for the analysis of experimental values. This may be because of the fact that theoretical calculations with ALICE code are usually found to give reasonably good agreement with the measured values [12]. The ALICE–ASH code is a modified and advanced version of the ALICE code. This code can calculate the cross sections, angular distribution, and energy of particles in nuclear reactions produced by nuclei and nucleons with projectile energies up to 300 MeV [9]. The equilibrium particle emission can be calculated using the Weisskopf–Ewing (WE) model [14]. The basic nuclear input parameters in the WE model are the nuclear level density parameter, nuclear binding energy, inverse nuclear reaction cross section, and pairing energy. The probability of evaporation in the WE model is calculated as following,

where, the term “x” denotes the type of particle. The “Sx”, “μ_x”, and “ε_x” are the spin, reduced mass, and emitted particle energy, respectively. The term ${\text{σ}}_x^{\text{inv}}$ is inverse reaction cross section. The term “E” represents excitation energy for the emitting nucleus. ρ(Z,A,E) and ρ(Z',A'U) represent the nuclear level density for the x–particle emitted from the nucleus and the residual nucleus with “U” excitation energy, respectively [9].

The hybrid model [15] and GDH model [10] can be used to calculate the particle spectrum. The pre–equilibrium particle spectrum in GDH model is written as follows,

where, the “λ” and “T₁” represent the reduced de–Broglie wavelength of projectile particle and the transmission coefficient for l^-th partial wave, respectively. _nX_x symbolizes the number of x–nucleons for the n–exciton state. “ε_x” and “n₀” are the channel energy of nucleon and the initial exciton number, respectively. ω(p,h,E) represents the exciton state density at “E” excitation energy with “p” particles and “h” holes. “D_n” is a depletion factor determined by the work of Ref.[16], which takes into consideration the depletion of the n–exciton state due to nucleon emission. The final excitation energy is written as follows,

Here the term “Qx” is the separation energy of nucleon. In addition, the emission rate of nucleon is written as follows,

The “g_x” denotes single particle density. The density is equal to $\frac{Z}{14}$ for protons and $\frac{N}{14}$ for neutrons. The intra–nuclear transition rate is written as following,

where “ρ_l” denotes the average nuclear matter density at the distance from lλ to (1+l)λ. The term “V” represents the velocity of a nucleon in nucleus and also the term “σ₀” is the nucleon–nucleon scattering cross section corrected for the Pauli principle [9].

The TENDL based on TALYS model code is a nuclear data library, of which output contains in ENDF format nuclear data for use in both basic physics and applications. It has been updated annually since 2008. The TENDL database library includes data for stable and unstable target nuclei. The TENDL includes sub–libraries for bombarding photon, proton, neutron, deuteron, triton, ³He, and ⁴He particles from 10⁻⁵ eV up to 200 MeV [17].

The semi-empirical and empirical formulae based on the systematics of the measured excitation functions have been widely used to evaluate the cross sections at different energy ranges. These formulae work very well for a quick prediction of the cross sections. In general, the empirical and semi-empirical formulae of the nuclear reactions contain the exponential dependence of nuclear cross sections upon the proton and neutron numbers in nucleus. In recent years, the cross section systematics for different nuclear reaction channels were suggested by various studies [4,18-21]. In previous investigations we presented new empirical formulae to describe the cross sections of (d,n) and (d,2n) nuclear reactions produced by deuteron particles. The empirical nuclear cross section formula of (d,n) reactions induced by deuterons at energy of 8.6 MeV is given as follows [20],

On the other hand, for (d,2n) nuclear reactions at incident energies between 11.57 and 18.91 MeV, it is assumed that the cross section systematic including none-elastic and Coulomb effects given as follows,

where the term “S” is the asymmetry parameter [21].

3 Results and discussion

In this paper, the cross sections of the ⁵⁰Cr (d,n) ⁵¹Mn, ⁵⁸Ni(d,n)⁵⁹Cu, ⁶⁴Zn(d,n)⁶⁵Ga, ⁶⁶Zn(d,n)⁶⁷Ga, ⁴⁵Sc(d,2n)⁴⁵Ti, ⁴⁷Ti(d,2n)⁴⁷V, ⁴⁸Ti(d,2n)⁴⁸V, and ⁸⁹Y(d,2n)⁸⁹Zr nuclear reactions were calculated using different input parameters in the ALICE–ASH computer code, and also the cross section formulae. Furthermore, the calculated cross sections for these nuclear reactions are compared with the available experimental data [22] and TALYS-based TENDL library data [17]. The excitation functions are presented in Figs. 1, 2, 3, 4, 5, 6, 7, and 8 as a function of projectile particle energy. The WE model for equilibrium state and the GDH and hybrid models for pre–equilibrium state in these calculations were used. Changing the initial exciton number and nuclear level density parameter in pre–equilibrium GDH model calculations of the effects on the calculated cross sections was investigated. The Fermi gas model with α=A/8 parameter is used to calculate the nuclear level density in the equilibrium model. The initial exciton number and level density parameter in the hybrid model calculations are taken as n₀=4 and α=A/8, respectively. The initial exciton number in the GDH model calculations are taken as n₀=4 to 6. In addition, the nuclear level parameters for initial exciton numbers (n₀=3p–1h to 4p–2h) in the GDH model are taken as α=A/5; A/8; A/11. Thus, the effects of these input parameters on the cross sections are investigated.

Fig. 1.Full-screen View

The cross section values of the ⁵⁰Cr(d,n) ⁵¹Mn nuclear reaction with the data reported by Klein et al. [23], Cogneau et al. [24], and Yiğit [20].

Fig. 2.Full-screen View

The cross section values of the ⁵⁸Ni(d,n) ⁵⁹Cu nuclear reaction with the data reported by Coetzee and Peisach [25], Carver and Jones [26], Cogneau et al. [27], and Yiğit [20].

Fig. 3.Full-screen View

The cross section values of the ⁶⁴Zn(d,n) ⁶⁵Ga nuclear reaction with the data reported by Coetzee and Peisach [25], Bissem et al. [28], and Yiğit [20].

Fig. 4.Full-screen View

The cross section values of the ⁶⁶Zn(d,n) ⁶⁷Ga nuclear reaction with the data reported Williams and IrvineJr [29], Bissem et al. [28], Nassiff and Munzel [30], and Yiğit [20].

Fig. 5.Full-screen View

The cross section values of the 45Sc(d,2n)45Ti nuclear reaction with the data reported by Hermanne et al. [31] and Yiğit [21].

Fig. 6.Full-screen View

The cross section values of the ⁴⁷Ti(d,2n) ⁴⁷V nuclear reaction with the data reported by Chen and Miller [32] and Yiğit [21].

Fig. 7.Full-screen View

The cross section values of the 48Ti(d,2n) 48V nuclear reaction with the data reported by WestJr et al. [33], Chen and Miller [32], Burgus et al. [34], and Yiğit [21].

Fig. 8.Full-screen View

The cross section values of the ⁸⁹Y(d,2n) ⁸⁹Zr nuclear reaction with the data reported by LaGamma and Nassiff [35], Bissem et al. [28], Lebeda et al. [36], Uddin et al. [37], and Yiğit [21].

4 Conclusion

In this study, the excitation functions of some (d,n) and (d,2n) nuclear reactions produced by the deuteron particle on the fusion structural materials such as chromium, nickel, zinc, scandium, titanium, and yttrium were investigated by using the different initial exciton numbers and level density parameters via the pre-equilibrium and equilibrium models. The obtained excitation functions were also compared with the experimental results and TENDL library data, and the cross section values where calculated using the formulae of Yiğit [20,21]. The excitation functions for the (d,n) nuclear reactions have maximum position in the deuteron energy of 4-9 MeV, whereas the maximum cross section values for the (d,2n) nuclear reactions are reached in the energy range of 14-20 MeV. The cross section values predicted by the empirical formula of Yiğit [21] for the (d,2n) nuclear reactions give an acceptable agreement with the measured data. On the other hand, the calculated cross section results of Yiğit [20] are not very consistent with the measured data for the ⁵⁰Cr (d,n) ⁵¹Mn and ⁵⁸Ni(d,n)⁵⁹Cu nuclear reactions. It seemed that the calculated excitation functions change very little with the variation of the nuclear level density parameters. Generally, the calculated cross section data via the pre–equilibrium model (initial exciton number n₀=4) for the investigated nuclear reactions have an acceptable harmony with the available experimental data. Especially, selecting the initial exciton number of n₀=4 in the pre-equilibrium models for cross section calculations of (d,n) nuclear reactions appears to be good. So, the initial exciton number is an important parameter of the pre-equilibrium process. We hope that the obtained results in this paper will stimulate future experimental cross section investigations of deuteron–nuclei interaction.