1 Introduction

Nuclear data are needed to explain the nature of the internal structure of nuclei. Nuclear physics researches are focused on understanding nuclear data, which are crucial for many applications such as fusion, fission, radiation therapy, Accelerator Driven System (ADS), radiobiology, nuclear wear measurement, astrophysics and cosmochemistry etc. [1-10]. Data relevant to radioisotopes can be grouped under two headings, the nuclear reaction data and decay data. Since the decay data have been well-established, the nuclear reaction data need to be further studied [2,5,11]. The nuclear reaction cross section data include a wide range of projectile energies from a few MeV up to the region of several GeV [2]. Obviously, the nuclear reaction data are indispensable items for radionuclide production. On the other hand, because cyclotrons generate very little radioactive waste, they are powerful sources with minimal environmental impact. Furthermore, the cyclotrons are very important in providing radioisotopes by different reaction mechanisms based on bombarding the target nuclei with charged particles for nuclear medicine [12]. The radioisotopes ^43,44Sc, ⁴⁵Ti, ⁵¹Cr, ⁵⁴Mn, and ⁵⁵Fe nuclei have been used in the various fields. The positron emitting radioisotope ⁴³Sc (with a half-life of 3.89 hours) could be used for an in vivo dosimetry [13]. The half-life (3.97 hours) of ⁴⁴Sc and its high positron branching of 94.27% may stimulate the practice of ⁴⁴Sc-labeled PET radiopharmaceuticals [14]. Because the ⁴⁴Sc has longer half-life than ⁶⁸Ga (67.71 minutes), it can be a useful alternative to ⁶⁸Ga as a positron emitter [12,15], and also it is the most interesting nuclear radioisotope for medical imaging using ${\beta }^{+}\gamma$ coincidences [13]. The radioisotope ⁵⁴Mn has a half-life of 312.3 days and decays by electron capture. It emits a single gamma ray at an energy of 834.8 keV. The ⁵⁴Mn is generally used as a standard source for $\gamma$-ray detectors [16]. The radioisotope ⁵⁵Fe (T_1/2=2.73 years) decays by electron capture and the major radiation emitted is the $K_{\alpha }$ X-ray with an energy of 5.89 keV. The ⁵⁵Fe is generally used as a standard source for X-ray detectors. The ⁵¹Cr radioisotope with a half-life of 27.7 days decays by electron capture. The decay results in the emission of a gamma ray of energy 320 keV. The ⁵¹Cr has been used to label red blood cells for the measurement of mass or volume of a living system and tracer investigations [16]. The radioisotope ⁴⁵Ti with a half-life of 3.08 hours decays 85% by positron emission and 15% by electron capture. The positron decay of the radioisotope ⁴⁵Ti proceeds mainly to the ground state of ⁴⁵Sc, so that almost no other concurrent γ-rays are emitted apart from the two annihilation photons. Thus, this radioisotope is suitable for PET [17]. In the present study, the excitation curves for the production of ^43,44Sc, ⁴⁵Ti, ⁵¹Cr, ⁵⁴Mn, and ⁵⁵Fe radioisotopes via nuclear reactions induced by proton particles were calculated using the ALICE/ASH nuclear reaction code [18] and compared to the measured values [19] and TENDL-2015 data [20]. In this context, there exist similar studies with different particle induced reactions in the literature [21-25]. In this paper, especially, the effects on the calculated cross section data of the variation of nuclear level densities have been investigated for the production of the considered radioisotopes.

2 Computational Method

The ALICE/ASH nuclear reaction code developed by Broeders et al. [18] is a modified version of the code ALICE/91. The ALICE/ASH is a reaction code describing the fast $\gamma$–emissions, the pre-compound composite particle emissions, the fission fragment yield calculations, and the nuclear level density calculations. Furthermore, this code is useful for cross section predictions using compound and pre-compound reaction mechanisms, angular and energy distributions of secondary particles at reactions induced by nuclei and nucleons with the incoming energy up to 300 MeV [18]. It is known that the reaction mechanism depends on the energies of incident particles [26-32]. Compound nuclear effects for a given reaction dominate in the incident energy range of about 0-10 MeV. The compound emission was described with the Weisskopf Ewing model [33] without angular momentum conservation. The basic physical parameters in this model are inverse reaction cross section, level density parameter, nuclear binding energy, and pairing. In the Weisskopf Ewing approximation, the evaporation cross sections for initial channel "a" and final channel "b" are given by

where the term "$E_{\text{inc}}$" is incident particle energy, and the term "${\text{Γ}}_b$" represents emission probability per unit time of a particle of type "$b$"  by the compound nucleus. The emission probability is calculated by the formula ${\text{Γ}}_b=\frac{2s_b+1}{{\pi }^2{\hslash }^2}{\mu }_b\int d\epsilon {\sigma }_b^{\text{inv}}\left(\epsilon \right)\epsilon \frac{{\omega }_1\left(U\right)}{{\omega }_1\left(E\right)}$. The term "$U$" denotes excitation energy of residual nucleus, the term "$s_b$" is spin, the term "${\mu }_b$" is reduced mass, the term "${\sigma }_b^{\text{inv}}$" is inverse reaction cross section, and the term "${\omega }_1$" is total nuclear level density [33].

The pre-equilibrium nuclear effects dominate in the reactions produced by light particles with a projectile energy range above about 8-10 MeV. This process takes place in a number of steps, corresponding to the excitation of successive particle-hole pairs via the interaction of the target nucleus and projectile. For the pre-equilibrium nuclear reaction process, the Blann’s Hybrid model was written in the following form [34],

where the term "${\sigma }_{\text{R}}$" denotes the nuclear reaction cross section, the term "${}_n{}^{}\chi {}_{\nu }{}^{}$" is the particle number of the "ν" type (neutron or proton) with "n" exciton hierarchy, the term "$g_{\nu }$" represents the single particle level density for "ν" type, and the term "$P_{\upsilon }(\epsilon )d\epsilon$" corresponds to the particle number of the "ν" type emitted into the unbound continuum with the energy between "ε" and "ε + dε". The first set of square brackets of Eq. (3) represents the particle number to be found at a given energy "ε" for all scattering situations leading to an "n" exciton configuration. Moreover, the second set of square brackets in this equation corresponds to the fraction of particles with the "ν" type at an energy which should undergo emission into the continuum, rather than making an intranuclear transition. The terms "${\lambda }_{+}(\epsilon )$" and "${\lambda }_c(\epsilon )$" represent the intranuclear transition and emission rates, respectively. The terms "U" and "E" represent the excitation energies of the residual nucleus and composite system, respectively. Physically, the term "D_n" is the average fraction of the initial population surviving to the exciton number being treated [34]. Moreover, the Geometry Dependent Hybrid (GDH) model is a nuclear reaction model that takes into account nuclear geometry properties like the diffuseness of the nuclear surface. So, this model takes into consideration the reduced matter density and the shallow potential at the nuclear surface. The differential emission spectra in this model were described in the following form,

where the term "$T_{\mathcal{l}}$" denotes the transmission coefficient for the $\mathcal{l}$-th partial wave, and the term "$ƛ$" denotes the reduced de-Broglie wavelength of the incident particle. This model is made according to incoming orbital angular momentum "$\mathcal{l}$" in order to account for the nuclear effects of density distribution [34,35].

The combinatorics of a Fermi gas plus pairing have been widely suggested for calculating the nuclear level densities. The level density of the Fermi Gas model (FGM) with an energy-dependent nuclear level density parameter proposed by Ignatyuk, Smirenkin, and Tishin [36] is given in the following form,

Here, the level density parameter "$a$" is given by the phenomenological expression [36]

where the term $\tilde{a}$ corresponds to the asymptotic value of the nuclear level density parameter. The term "$\delta W$" represents shell correction [18]. The nuclear level density in the Superfluid nuclear model (SFM) [37] is presented by the expression

where the terms $K_{\text{rot}}\left(U^{ı}\right)$ and $K_{\text{vib}}\left(U^{ı}\right)$ represent rotational and vibrational enhancement factors at effective nuclear excitation energy "$U^{ı}$", respectively. The term ${\rho }_{\text{qp}}\left(U^{ı}\right)$ is the density of quasi-particle excitation [18]. The shell structure of the nucleus in the Kataria Ramamurthy Fermi gas model (KRM) [38] has quite important effects on level densities of the excited nucleus. The KRM is defined by the semi-empirical formula in terms of a Fourier expansion of single-particle level densities of nucleons in the nuclei. The nuclear level density parameter "$a$" in this model is given by the expression

The level density parameter is dependent upon the well parameters and also has been given by separation energies in the following form,

where the terms "$s_{\text{p}}$" and "$s_{\text{n}}$" denote the proton and neutron separation energies, respectively. The "${\beta }_0$", "${\beta }_1$", "${\beta }_2$", and "$\alpha$" are the fitting parameters [38].

3 Results and Discussion

The calculated cross section values of the ⁴⁵Sc$\left(\text{p},\text{n}\right)$ ⁴⁵Ti, ⁴⁵Sc $\left(\text{p},\text{np}\right)$ ⁴⁴Sc, ⁴⁵Sc $\left(\text{p},2\text{np}\right)$ ⁴³Sc, ⁵⁵Mn $\left(\text{p},\text{np}\right)$ ⁵⁴Mn, ⁵⁵Mn $\left(\text{p},\text{n}\right)$ ⁵⁵Fe, and ⁵⁵Mn $\left(\text{p},\text{nα}\right)$ ⁵¹Cr nuclear reaction processes in comparison with the existing experimental values are graphically given in Figs. 1–6. In these calculations, the various level density models such as the FGM, SFM, and KRM in the ALICE/ASH nuclear reaction code were used. Additionally, the effects of the level density parameter on the calculated excitations functions are investigated via four level density parameters such as $a=A/6$, $A/12$, $A/18$, and $A/24$ values in the FGM.

Figure 1.Full-screen View

Nuclear model calculation results for the 45Sc (p, 2np) 43Sc reaction compared with the measured excitation functions

Figure 6.Full-screen View

Nuclear model calculation results for the ⁵⁵Mn (p, n) ⁵⁵Fe reaction compared with the measured excitation functions

3.2 Production of the ⁴⁴Sc radioisotope

The nuclear excitation curves for the ⁴⁵Sc$\left(\text{p},\text{np}\right)$⁴⁴Sc reaction are shown in Fig. 2 up to a proton energy of 85 MeV. Four data obtained by Ejnisman et al. [40] in the incident energy range of 16-22 MeV are very in good agreement with the predictions obtained using the KRM level density. Moreover, the cross section results of the level density parameter with $a=A/12$ using the FGM in the energy range of 12.8-24.8 MeV are in general agreement with the data obtained by Levkovskij [39] and are within the error bars. The cross section estimations obtained using the SFM and FGM (with $a\left(U\right)$) level densities using the code ALICE/ASH are mostly in good agreement with the measured cross sections of Mcgee et al. [42] in the proton energy region of 15-85 MeV. However, the excitation functions predicted using the code ALICE/ASH are not very consistent with the shape of the measured excitation function of Meadows et al. [41] for the considered reaction. The calculated excitation functions have maximum cross sections in the incident energy range 20-30 MeV. Also, in the peak portion of the excitation functions, we can observe important discrepancies between the calculations of different level density models.

Figure 2.Full-screen View

Nuclear model calculation results for the ⁴⁵Sc (p, np) ⁴⁴Sc reaction, comparison with the measured excitation functions

3.3 Production of ⁴⁵Ti radioisotope

The cross sections calculated in the present paper, and experimental values obtained by Levkovskij [39], Thomas and Bartolini [44], Howard et al. [43], Ejnisman et al. [40], and Mcgee et al. [42] are plotted in Fig. 3 for the ⁴⁵Sc $\left(\text{p},\text{n}\right)$ ⁴⁵Ti reaction. The cross section results predicted by the FGM with $a\left(U\right)$ show a good overall agreement with two experimental values of Ejnisman et al. [40] at the energies of 16 to 20 MeV. However, at the energies of 18 to 22 MeV, the data measured by Ejnisman et al. [40] are in good agreement with the cross section results calculated using the FGM with $a=A/24$ for the investigated reaction. The excitation curves for the ⁴⁵Sc$\left(\text{p},\text{n}\right)$⁴⁵Ti reaction reach maximum values in the proton energy region of 8-15 MeV and decreases above this incident energy region. Generally, the shape of the predicted cross sections at the maximum region of excitation functions is in agreement each other. The excitation function estimated by the KRM level density shows a maximum of 591.5 mb at 15 MeV. The cross section of 450$\pm$68 mb at an incident energy of 10 MeV obtained by Mcgee et al. [42] is in general agreement with the predictions of the FGM and SFM level densities. However, the cross sections of Mcgee et al. [42] at proton energies of 15, 20, and 25 MeV are quite a bit lower than the excitation functions estimated using the code ALICE/ASH. Besides, the data measured by Levkovskij [39] and Thomas and Bartolini [44], except for the value of 394 mb at a proton energy of 14.4 MeV, have lower results than the FGM, KRM, and SFM predictions by the code ALICE/ASH. At incident proton energies below 7 MeV, the excitation function obtained by Howard et al. [43] yields an acceptable harmony with the theoretical cross section values within the error bars.

Figure 3.Full-screen View

Nuclear model calculation results for the ⁴⁵Sc (p, n) ⁴⁵Ti reaction compared with the measured excitation functions

3.4 Production of ⁵¹Cr radioisotope

In Fig. 4, the excitation function data for production of the ⁵¹Cr radioisotope via the ⁵⁵Mn$\left(\text{p},\text{nα}\right)$⁵¹Cr reaction are plotted for comparison with the measured data in the literature up to the incident energy of 50 MeV. The cross section data obtained using level density parameters $a=A/18$ to $A/24$ in the FGM agree with the cross sections measured by Levkovskij [39] and Al-Abyad et al. [16] within the error bars. The excitation functions for the ⁵⁵Mn$\left(\text{p},\text{nα}\right)$⁵¹Cr nuclear reaction have maximum values in the proton energy range of 23-35 MeV. The calculated excitation functions via different level density parameters in the FGM using ALICE/ASH code differ greatly with each other at the maximum cross section region. For the instance, at proton energy of 28 MeV, the cross section value calculated by the FGM level density with $a=A/24$ is 110.2 mb, whereas the cross section value predicted using the level density parameter $a=A/6$ is 32.5 mb.

Figure 4.Full-screen View

Nuclear model calculation results for the ⁵⁵Mn (p, nα) ⁵¹Cr reaction compared with the measured excitation functions

3.5 Production of ⁵⁴Mn radioisotope

The comparison between the experimental and model results for cross sections of the ⁵⁵Mn$\left(\text{p},\text{np}\right)$⁵⁴Mn reaction is illustrated in Fig. 5. Excitation curves for this reaction have maximum position in the proton energy range of 20-30 MeV. The data calculated using the SFM, FGM and KRM level densities via the code ALICE/ASH give lower results than the experimental data reported by Ditroi et al. [45], Levkovskij [39], and Michel and Brinkmann [46] at the maximum position of nuclear cross sections. However, these experimental data in the low energy ranges have an acceptable fit with theoretical cross section data predicted by the FGM with $a\left(U\right)$. In addition, the excitation function estimated using KRM level density by the ALICE/ASH code for this reaction gives good agreement within the statistical errors of the measured data of Ditroi et al. [45] and Michel and Brinkmann [46] in the proton energy range of 27-44 MeV except for the data of Ditroi et al. [45] at 35.5 to 40 MeV. However, the obtained cross sections using the level density parameter $a=A/6$ by the FGM at 35.5 to 40 MeV are in good agreement with two cross section data points measured by Ditroi et al. [45]. Furthermore, the FGM predictions with $a\left(U\right)$ have a good agreement with the experimental cross section values of Al-Abyad et al. [16] in the incident energy region of 11.4-44.8 MeV. The results of the KRM level density in the energy range of 22.2-38 MeV show closer agreement with the experimental values by Gusakow et al. [47] within the error bars.

Figure 5.Full-screen View

Nuclear model calculation results for the ⁵⁵Mn (p, np) ⁵⁴Mn reaction compared with the measured excitation functions

4 Conclusion

In this paper, the excitation functions for the nuclear reactions ⁴⁵Sc$\left(\text{p},\text{n}\right)$ ⁴⁵Ti, ⁴⁵Sc $\left(\text{p},\text{np}\right)$ ⁴⁴Sc, ⁴⁵Sc $\left(\text{p},2\text{np}\right)$ ⁴³Sc, ⁵⁵Mn $\left(\text{p},\text{np}\right)$ ⁵⁴Mn, ⁵⁵Mn $\left(\text{p},\text{n}\right)$ ⁵⁵Fe, and ⁵⁵Mn $\left(\text{p},\text{nα}\right)$ ⁵¹Cr have been investigated using the FGM, KRM, and SFM level densities by the code ALICE/ASH. The experimental values and the obtained model data are graphically given in Figs. 1–6. Generally, the cross sections predicted with the ALICE/ASH nuclear reaction code, except for the calculated cross sections for the ⁴⁵Sc $\left(\text{p},\text{n}\right)$ ⁴⁵Ti and ⁵⁵Mn$\left(\text{p},\text{np}\right)$ ⁵⁴Mn reactions at the maximum of excitation functions, are in fair agreement with the existing experimental results in literature. We have observed that the cross sections can strongly vary with choice of the nuclear model parameters in the code. So, the size discrepancies between the calculated and the measured excitation functions can be reduced with the different level density models and level density parameters. Particularly, the shape of the excitation functions obtained with the SFM and FGM level densities, except for the maximum cross section region, follows the trend of the experimental cross sections. The obtained excitation functions contribute to the new investigations for radionuclide production.