Reaction Cross-Section, Stopping Power and Penetrating Distance Calculations for the Structural Fusion Material ⁵⁴Fe in Different Reactions

Abstract

Nuclear reaction codes give us simplicity to investigate phenomena of nuclear physics. There have many computer programs such as TALYS, EMPIRE, ALICE/ASH, PCROSS, FLUKA and GEANT4. The stopping power of alpha, deuteron, proton and triton in ⁵⁴Fe materials is acquired as it has helpful applications of shielding and choosing the proper thickness of the target. Level density is very important to understanding nuclear reaction mechanism. The knowledge of level density for reaction cross-section calculations are required for various application such as astrophysics, accelerator driven sub-critical systems, nuclear medicine, fission and fusion reactor design and neutron capture. In this study, we calculated the cross-sections of ⁵⁴Fe using TALYS 1.6 and EMPIRE 3.1 codes for different reactions through the four level density models. Stopping powers and penetrating distances were calculated for the alpha, deuteron, proton and triton particles, taking into consideration all possible reactions in ⁵⁴Fe for incident energies of 1–45 MeV using GEANT4 calculation code. The obtained reaction cross-section results have been compared with the each other and against the experimental nuclear reaction data existing in EXFOR database.

Introduction

The most important problem of today’s world is to obtain clean and safe energy. Scientists have sought on this problem since middle of twentieth century. The fusion materials development for the safety of fusion reactors is very important. A good realizing on the mechanical properties and microstructure change of materials under effect of radiation is of great important for performing a safe design and operation of new nuclear systems and for developing structural materials in fusion power plants [1]. An extraordinary importance is being given for the structural materials development because the achievement of the fusion reactors is mainly dependent on the development of these materials. The development of suitable structural materials is an important stage towards fusion reactors becoming an effective energy source, especially if the promise of an environmentally free of risk machine is to be understood. The selection of structural materials for combined first-wall-breeding blanket parts depends not only on mechanical properties, suitability with other materials, nuclear properties and irradiation performance, but also on their radiological properties [2–5].

The process of simulation programs and nuclear reaction codes gave us the chance to design virtual experimental conditions to simulate phenomena of nuclear physics. The cross-section calculations, spectrum analyzes of outgoing particles and dosimeter calculations can be given example this situation. Estimating of reaction cross-section remains a major problem in applications such as data evaluations. There has been much progress in the development of nuclear reaction codes, which now include different reaction mechanisms to take care of [6].

The reaction cross-section data have a critical importance on fusion reactors and development of fusion reactor technology. In a fusion reactor design, neutron reaction cross-section data are needed and the evaluated values in nuclear data files are generally used for neutronic calculations [7–9]. The theoretical nuclear reaction models are usually required to get the prediction of the reaction cross-sections, specifically if the experimental data are unavailable or are improbably to be produced because of the experimental troubles [10–16].

Level density is a characteristic property of every nucleus and it is defined as the number of levels per unit energy at certain excitation energy. In other words it is the number of different ways in which individual nucleons can be placed in the various single particle orbitals such that the excitation energy lies in the range E to E + dE. It increases rapidly with excitation energy. Level density is crucial to understanding nuclear reaction mechanism. The knowledge of level density for reaction cross-section calculations are required for various application such as astrophysics, accelerator driven sub-critical systems, nuclear medicine, fission and fusion reactor design and neutron capture [17].

The stopping power of matter for charged particles such as proton, alpha, deuteron is very crucial in reactor applications and dosimeter calculations. It is also useful in understanding the interaction of particles with matter. The heavy charged particles like proton interact with matter primarily through coulomb forces between positive and negative charge of the orbital electrons within the absorber atoms [18].

In this paper, we calculated the cross-sections of ⁵⁴Fe using TALYS 1.6 [19] and EMPIRE 3.1 [20, 21] codes for different reactions through the four level density models. Also, stopping power and penetrating distances of ⁵⁴Fe for some charged particles have been calculated using GEANT4 [22] code. The obtained reaction cross-section results have been compared with the each other and against the National Nuclear Data Center (EXFOR) [23] database.

Calculation Methods

The reaction cross-sections of ⁵⁴Fe target nucleus have been calculated using Constant Temperature Fermi Gas Model (CTM), Back Shifted Fermi Gas Model (BSFM) and Generalized Superfluid Model (GSM) for the TALYS 1.6 and Default Model for the EMPIRE 3.1 computer codes.

The TALYS is a nuclear reaction simulation code for the analysis and estimation of nuclear reactions that include neutrons, protons, photons, tritons, deuterons, ³He and alpha particles in the energy range of 1 keV–1 GeV. For this, TALYS integrates the optical model, pre-equilibrium, direct, fission and statistical nuclear reaction models in one calculation plan and in that connection gives a prediction for all the possible reaction. In TALYS, various options are contained for the select of various parameters such as level density, gamma strength function and compound nuclear model parameters [24]. The default cross-section calculations were considered by the two-component exciton model [25]. This model is based on the Kalbach’s theory [26]. The proton and neutron type of the produced particles and holes is clearly followed during the reaction in the two-component exciton model.

The EMPIRE 3.1 contains the mechanism of pre-equilibrium as explained in the exciton model [27], as dependent upon the master equation solution [28] in the form recommended by Cline [29] and Ribansky [30]

$$- q_{t = 0} (n) = \lambda_{ + } \left( {E,n + 2} \right)\tau \left( {n + 2} \right) + \lambda_{ - } \left( {E,n - 2} \right)\tau \left( {n - 2} \right) - \left[ {\lambda_{ + } \left( {E,n} \right) + \lambda_{ + } \left( {E,n} \right) + L\left( {E,n} \right)} \right]\tau \left( n \right)$$

(1)

where q _t (n) is the initial occupation probability of the composite nucleus in the state with the exciton number n, $\lambda_{ + } (E,n)$ and $\lambda_{ - } (E,n)$ are the transition rates for decay to neighboring states, and L(E,n) is the total emission rate integrated over emission energy for particles (neutrons υ, protons π and clusters) and γ-rays. The pre-equilibrium spectra can be calculated as.

$$\frac{{d\sigma_{a,b} }}{{d\varepsilon_{b} }}\left( {\varepsilon_{b} } \right) = \sigma_{a,b}^{r} \left( {E_{inc} } \right)D_{a,b} \left( {E_{inc} } \right) \times \sum\limits_{n}^{{}} {W_{b} \left( {E,n,\varepsilon_{b} } \right)\tau \left( n \right)}$$

(2)

where $\sigma_{a,b}^{r} \left( {E_{inc} } \right)$ is the cross-section of the reaction (a, b), W _b (E, n, ɛ _b ) is the probability of emission of a particle of type b (or γ-ray) with energy ɛ _b from a state with n excitons and excitation energy E of the compound nucleus, and $D_{a,b} \left( {E_{inc} } \right)$ is the depletion factor, which takes into account the flux loss as a result of the direct reaction processes.

In the BSFM [31], the pairing energy is treated as an adjustable parameter and the Fermi gas expression is used all the way down to 0 MeV. However, it should be noted that standard version of the Fermi gas expression at low excitation energies leads to numerical divergence.

The CTM [32] requires solving the problem of matching the constant temperature part at low energies and the Fermi gas expression at high energies. Since this matching depends on experimental information (discrete levels, mean resonance spacings, shell corrections, separation energies, etc.) it may occur that the solution of the matching problem yields hard to believe, and sometimes unphysical, values for the matching parameters.

The GSM takes superconductive pairing correlations into account according to the Bardeen–Cooper–Schrieffer theory. The phenomenological version of the model [33, 34] is characterized by a phase transition from a superfluid behavior at low energy, where pairing correlations strongly influence the level density, to a high-energy region that is described by the FGM. The GSM thus resembles the CTM to the extent that it distinguishes between a low energy and a high-energy region, although for the GSM this distinction follows naturally from the theory and does not depend on specific discrete levels that determine a matching energy. Instead, the model automatically provides a constant temperature-like behavior at low energies.

GEANT4 is a free simulation and calculation code that can be used to investigation of high-energy physics, medical physics, space and radiation physics. At the heart of GEANT4 is an abundant set of physics models to handle the interactions of particles with matter across a very wide energy range. Data and expertise have been drawn from many sources around the world and in this respect, GEANT4 acts as a repository that incorporates a large part of all that is known about particle interactions [22].

The GEANT4 processes are dedicated to stopping power into ⁵⁴Fe. Stopping powers and penetrating distances in material for different reactions were calculated for the incident alpha, deuteron, proton and triton particles, taking into consideration all possible reactions such as ionization, scattering, absorption, transmission and charge transfer in ⁵⁴Fe for incident energies of 1–45 MeV using by Bethe Bloch’s and Niels Bohr’s approximation [35, 36].

Results and Discussion

In the present study, the reaction cross-sections of ⁵⁴Fe(α,n)⁵⁷Ni, ⁵⁴Fe(α,2n)⁵⁶Ni,⁵⁴Fe(d,n)⁵⁵Co, ⁵⁴Fe(d,α)⁵²Mn, ⁵⁴Fe(n,2n)⁵³Fe, ⁵⁴Fe(n,α)⁵¹Cr, ⁵⁴Fe(n,p)⁵⁴Mn and ⁵⁴Fe(p,α)⁵¹Mn reactions have been calculated for different level density models using TALYS 1.6 and EMPIRE 3.1 computer codes. The comparison of the calculated and experimental reaction cross-sections have been shown in Figs. 1, 2, 3, 4, 5, 6, 7 and 8. All experimental data used in this study have been taken from the Ref. [23].

Fig. 1

The comparison of calculated cross-sections of ⁵⁴Fe(α,n)⁵⁷Ni reaction with the experimental values reported in Ref. [23]

Fig. 2

The comparison of calculated cross-sections of ⁵⁴Fe(α,2n)⁵⁶Ni reaction with the experimental values reported in Ref. [23]

Fig. 3

The comparison of calculated cross-sections of ⁵⁴Fe(d,n)⁵⁵Co reaction with the experimental values reported in Ref. [23]

Fig. 4

The comparison of calculated cross-sections of ⁵⁴Fe(d,α)⁵²Mn reaction with the experimental values reported in Ref. [23]

Fig. 5

The comparison of calculated cross-sections of ⁵⁴Fe(n,2n)⁵³Fe reaction with the experimental values reported in Ref. [23]

Fig. 6

The comparison of calculated cross-sections of ⁵⁴Fe(n,α)⁵¹Cr reaction with the experimental values reported in Ref. [23]

Fig. 7

The comparison of calculated cross-sections of ⁵⁴Fe(n,p)⁵⁴Mn reaction with the experimental values reported in Ref. [23]

Fig. 8

The comparison of calculated cross-sections of ⁵⁴Fe(p,α)⁵¹Mn reaction with the experimental values reported in Ref. [23]

The calculated cross-sections of ⁵⁴Fe(α,n)⁵⁷Ni reaction have been compared with the experimental data in Fig. 1. The TALYS 1.6 CTM and BSFM model calculations are in good agreement the experimental results up to 17 MeV. The TALYS 1.6 GSM model calculations are in good harmony with the experimental data except in the 17–27 MeV alpha energy region. EMPIRE 3.1 Default Model calculations exhibit a little discrepancy with the experimental data up to 27 MeV. The comparison of calculated cross-sections of ⁵⁴Fe(α,2n)⁵⁶Ni reaction with the experimental values shown in Fig. 2. The TALYS 1.6 CTM and BSFM model results are in good harmony with the experimental results but they follow experimental data from above in the alpha energy region of 36–41 MeV. EMPIRE 3.1 Default and TALYS 1.6 GSM Model calculations has similar structure with EXFOR data up to 28 MeV. The ⁵⁴Fe(d,n)⁵⁵Co reaction cross-section calculations have been compared with the experimental results in Fig. 3. TALYS 1.6 level density models are in good agreement with the experimental values up to 7 MeV deuteron incident energy. After this energy they follow experimental data from below. EMPIRE 3.1 Default Model give similar results with the TALYS 1.6 model calculations up to 7 MeV but this model goes to zero after 8 MeV. The experimental and theoretical results of ⁵⁴Fe(d,α)⁵²Mn reaction have been given in Fig. 4. The cross-section results of the EMPIRE 3.1 Default Model calculations are not in good agreement with the experimental values. They follow the experimental results from above in the 5–15 MeV incident deuteron energy region. The TALYS 1.6 CTM and BSFM model calculations are in good harmony with the experimental values in the region of all reaction incident energies. The TALYS 1.6 GSM model results give the same geometry with the EXFOR data but they gave different cross-section values in the energy range of 8–14 MeV. The comparison of calculated neutron cross-sections of ⁵⁴Fe(n,2n)⁵³Fe reaction with the experimental values can be seen in Fig. 5. All level density model calculations are in good harmony with the experimental data. The calculated reaction cross-sections of ⁵⁴Fe(n,α)⁵¹Cr reaction against the EXFOR data have been given in Fig. 6. All the TALYS 1.6 code calculations follow experimental values from below except BSFM model for 15–20 MeV neutron energy region. The EMPIRE 3.1 Default Model calculations are the nearest to the experimental results up to 9 MeV. The comparison of calculated neutron-production cross-sections of ⁵⁴Fe(n,p)⁵⁴Mn reaction with the experimental values have been given in Fig. 7. The TALYS 1.6 BSFM and CTM model are in good harmony with the EXFOR values up to 15 MeV but CTM model follow experimental data from below in the neutron energy region of 10–13 MeV. The GSM model of the TALYS 1.6 calculations exhibit a little discrepancy with experimental values up to 7 MeV. The EMPIRE 3.1 Default Model calculations give higher results than the EXFOR data. The reaction cross-section calculations of ⁵⁴Fe(p,α)⁵¹Mn reaction have been shown in Fig. 8. All the TALYS 1.6 level density model calculations follow the experimental results from below while EMPIRE 3.1 Default Model level density calculations follow the experimental data from above after 12 MeV incident proton energy.

Although there are some disagreements between the calculated values and the experimental results, in generally, the TALYS 1.6 CTM and BSFM model cross-section calculations are close to the experimental results except for the Fig. 8. These models can be suggested, if the experimental data are unavailable or are improbably to be produced due to experimental problems.

The penetrating distance calculations of proton, triton, deuteron and alpha particles in the incident energy range of 1–45 MeV for ⁵⁴Fe structural fusion material shown in Fig. 9. According to calculated results penetrating of alpha particles is the poorest. So this particles cannot be managed to enter into ⁵⁴Fe. On the contrary alpha particles, proton has the most penetrating in the ⁵⁴Fe target. As can be seen in Fig. 9, if the mass number of the projectile particles is increased, the penetrating in ⁵⁴Fe is decreased.

Fig. 9

Alpha, deuteron, proton and triton penetrating distance calculations in ⁵⁴Fe for incident energies of 1–45 MeV using GEANT4 code

The calculated stopping power values of alpha, deuteron, proton and triton projectile particles in ⁵⁴Fe target for incident energies of 1–45 MeV have been exhibited in Fig. 10. Bethe Bloch suggested that for high energies above approximately 1 MeV region, the stopping power decreases as the incident particle’s energy. All the projectile particles give similar results except for alpha. We note that in the incident energy range of 1–45 MeV proton, deuteron and triton have a far going that exceeds the penetration in ⁵⁴Fe fusion structural material. All calculated stopping power and penetrating distance results used by GEANT4 have been given in Tables 1 and 2.

Fig. 10

Alpha, deuteron, proton and triton stopping power calculations in ⁵⁴Fe for incident energies of 1–45 MeV using GEANT4 code

Table 1 Alpha, deuteron, proton and triton distances in ⁵⁴Fe for incident energies of 1–45 MeV

Table 2 Alpha, deuteron, proton and triton stopping power in ⁵⁴Fe for incident energies of 1–45 MeV

Summary and Conclusions

In this study, the reaction cross-section of ⁵⁴Fe reactions have been calculated for the different level density models using the TALYS 1.6 and EMPIRE 3.1 codes. The theoretical calculation results have been also compared with the obtainable experimental data in the EXFOR database. Also, stopping power and distance of penetrating in ⁵⁴Fe target using by GEANT4 have been simulated. The results can be summarized and concluded as follows:

• 1. Generally the calculated reaction cross-sections of ⁵⁴Fe reactions are in agreement with the experimental data for the TALYS 1.6 CTM and BSFM model results.

• 2. The TALYS 1.6 GSM and EMPIRE 3.1 Default model calculations show some disparateness with the EXFOR values for all reactions investigated in this study.

• 3. The TALYS 1.6 CTM and BSFM option for ⁵⁴Fe reaction cross-section calculations can be chosen, if the experimental data are unavailable or are improbably to be produced because of the experimental troubles.

• 4. In the incident energy range of 1–45 MeV, required target thickness of ⁵⁴Fe could be stopped alpha, triton, deuteron and proton, approximately 328, 1,603, 2,155, 3,619 µm respectively.

• 5. The obtained ⁵⁴Fe stopping power results for the projectile charged particles can be used in several applications such as fusion reactor design and shielding.