Introduction

Current research and development activities on materials for fusion power reactors are mainly focused on plasma facing, tritium breeding and structural materials [1]. The current approach to developing structural materials for fusion reactors requires a heavy reliance on the use of small irradiation specimens largely because of limitations in available irradiation volumes [2]. The selection of fusion structural materials are an indispensable component for fusion reactor technology [3]. Intense neutron fluxes within fusion reactors that are currently being designed will lead to the activation of structural components. To evaluate and reduce this radiation, nuclear cross-section data are required for neutrons [4].

The reaction cross-section data have a critical importance on fusion reactors and development for fusion reactor technology. In a fusion reactor design, neutron reaction cross-section data are required and the evaluated values in nuclear data files are usually used for neutronic calculations [5–7]. Besides, photonuclear cross-sections are also important for some applications as; analysis of radiation transport and shielding, absorbed dose calculations in the human body throughout photon-radiotherapy, fission and fusion reactor technology, activation analysis including protections and material analysis studies for photon-induced reactions and transmutation of nuclear waste [8, 9].

The nuclear reaction models are generally required to get the prediction of the reaction cross-sections, especially if the no experimental data obtained or in cases where it is difficult to carry out the experimental measurements [10–15].

In our previous studies [7, 12], we examined (γ,n) and (γ,2n) photo-neutron cross-sections of several structural fusion materials. In this study, the theoretical (γ,3n) reaction cross-sections of several structural fusion materials such as 55Mn, 65Cu, 94Zr, 98,100Mo, 181Ta and 186W in photon-induced reactions have been investigated. The photo-neutron cross-sections of 55Mn(γ,3n)52Mn, 65Cu(γ,3n)62Cu, 94Zr(γ,3n)91Zr, 98Mo(γ,3n)95Mo, 100Mo(γ,3n)97Mo, 181Ta(γ,3n)178Ta and 186W(γ,3n)183W reactions have been calculated using PCROSS [16], EMPIRE 3.1 [17, 18] and TALYS 1.6 [19] computer codes in the photon energy range of 20–110 MeV. EMPIRE 3.1 exciton, TALYS 1.6 two component exciton and TALYS 1.6 pre-equilibrium models have been used to calculate the pre-equilibrium photo-neutron cross-sections. PCROSS Weisskopf–Ewing (WE) [20] model have been used for the reaction equilibrium component. The calculated results have been compared with each other and available experimental data existing in the EXFOR [21] database.

Calculation Methods

Photo-neutron cross-sections as a function of photon energy have been calculated using PCROSS code for the WE model, EMPIRE 3.1 code for the exciton model and TALYS 1.6 code for the two component exciton and pre-equilibrium models.

The equilibrium particle emission is given by the WE model in which angular momentum conservation is neglected. In the process, the basic parameters are inverse reaction cross-section, binding energies, the pairing and the level density parameters. The reaction cross-section for incident channel a and exit channel b can be written as

$$ \sigma_{ab}^{WE} \, = \,\sigma_{ab} \left( {E_{inc} } \right)\,\frac{{\Upgamma_{b} }}{{\sum\nolimits_{{b^{{\prime }} }} {\Upgamma_{{b^{{\prime }} }} } }} $$
(1)

where E inc is the incident energy. In Eq. (1), Γ b can be also expressed as

$$ \Upgamma_{b} \, = \,\frac{{2{\text{s}}_{\text{b}} \, + \,1}}{{\pi^{2} \hbar^{2} }}\mu_{\text{b}} \int {{\text{d}}\varepsilon } \,\sigma_{b}^{inv} \left(\varepsilon\right)\,\varepsilon \frac{{\omega_{1} (U)}}{{\omega_{1} (E)}} $$
(2)

where U, μ b, s b are the excitation energy of the residual nucleus, the reduced mass and the spin, respectively. The total single-particle level density is taken as

$$ \omega_{1} (E)\, = \,\frac{1}{{\sqrt {48} }}\frac{{\exp \left[ {2\sqrt {\alpha \left( {E - D} \right)} } \right]}}{E - D};\,\alpha \, = \,\frac{6}{{\pi^{2} }}g $$
(3)

where \( \sigma_{b}^{inv} , \) E, D and g are the inverse reaction cross-section, the excitation energy of the compound nucleus, the pairing energy and the single-particle level density, respectively.

The EMPIRE 3.1 includes the pre-equilibrium mechanism as defined in the exciton model [22], as based on the solution of the master equation [23] in the form proposed by Cline [24] and Ribansky [25]

$$ - q_{t = 0} (n) = \lambda_{ + } \left( {E,n + 2} \right)\tau \left( {n + 2} \right) + \lambda_{ - } \left( {E,n - 2} \right)\tau \left( {n - 2} \right) - \left[ {\lambda_{ + } \left( {E,n} \right) + \lambda_{ + } \left( {E,n} \right) + L\left( {E,n} \right)} \right]\tau \left( n \right) $$
(4)

where qt(n) is the initial occupation probability of the composite nucleus in the state with the exciton number n, λ +(E,n) and λ −(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 (protons π, neutrons υ and clusters) and γ-rays.

The pre-equilibrium spectra can be calculated as

$$ \frac{{d\sigma_{a,b} }}{{d\varepsilon_{b} }}\left( {\varepsilon_{b} } \right) = \sigma_{a,b}^{r} \left( {E_{inc} } \right)D_{a,b} \left( {E_{inc} } \right) \times \sum\limits_{n} {W_{b} \left( {E,n,\varepsilon_{b} } \right)\tau \left( n \right)} $$
(5)

where σ r a,b (E inc ) is the cross-section of the reaction (a, b), W b (E, n, ε b ) is the probability of emission of a particle of type b (or γ ray) with energy ε b from a state with n excitons and excitation energy E of the compound nucleus, and D a,b (E inc ) is the depletion factor, which takes into account the flux loss as a result of the direct reaction processes.

TALYS [19, 26] is a nuclear reaction simulation code for the estimation and analysis of nuclear reactions that include protons, neutrons, photons, tritons, deuterons, 3He and alpha particles in the energy range of 1 keV–200 MeV. For this, TALYS integrates the optical model, direct, pre-equilibrium, fission and statistical nuclear reaction models in one calculation scheme and thereby gives a prediction for all the open reaction channels. In TALYS, several options are included for the choice of different parameters such as γ-strength functions, nuclear level densities and nuclear model parameters [27]. The pre-equilibrium reactions were considered by the two component exciton model [28]. The pre-equilibrium model of TALYS is the two-component exciton model of Kalbach [29]. In the two component model, the neutron and proton type of the created particles and holes is explicitly followed throughout the reaction.

The details of the other code model parameters and options of TALYS can be found in Refs. [19, 26].

Results and Discussion

In the present study, (γ,3n) reaction cross-sections of 55Mn(γ,3n)52Mn, 65Cu(γ,3n)62Cu, 94Zr(γ,3n)91Zr, 98Mo(γ,3n)95Mo, 100Mo(γ,3n)97Mo, 181Ta(γ,3n)178Ta and 186W(γ,3n)183W reactions have been performed in the photon energy range of 20–110 MeV using PCROSS, EMPIRE 3.1 and TALYS 1.6 computer codes. The photo-neutron cross-sections shown by (γ,3n) reactions for 55Mn, 65Cu, 94Zr, 98,100Mo, 181Ta and 186W target nuclei have been plotted as a function of photon energy in Figs. 1, 2, 3, 4, 5, 6 and 7. All experimental values used in this study have been taken from the EXFOR database. In the EXFOR library, the experimental cross-section data concerning (γ,3n) reactions are scarce for the structural fusion materials worked in this study.

Fig. 1
figure 1

Calculated values for the 55Mn(γ,3n)52Mn reaction with experimental data taken from EXFOR

Full size image
Fig. 2
figure 2

The same as Fig. 1 but for 65Cu(γ,3n)62Cu

Full size image
Fig. 3
figure 3

The same as Fig. 1 but for 94Zr(γ,3n)91Zr

Full size image
Fig. 4
figure 4

The same as Fig. 1 but for 98Mo(γ,3n)95Mo

Full size image
Fig. 5
figure 5

The same as Fig. 1 but for 100Mo(γ,3n)97Mo

Full size image
Fig. 6
figure 6

The same as Fig. 1 but for 181Ta(γ,3n)178Ta

Full size image
Fig. 7
figure 7

The same as Fig. 1 but for 186W(γ,3n)183W

Full size image

The calculated photo-neutron cross-sections of 55Mn(γ,3n)52Mn reaction have been compared with the experimental values in Fig. 1. All theoretical model calculations are in good agreement with the experimental data in the photon energy region 29–32.5 MeV. Although PCROSS, EMPIRE 3.1 and TALYS 1.6 model calculations exhibit a little discrepancy with each other in the photon energy region 32.5–36.5 MeV, in general, they are in agreement with the experimental data including error bars. 65Cu(γ,3n)62Cu reaction cross-section calculations have been compared with the experimental data in Fig. 2. The PCROSS-WE calculations are not in good agreement with the experimental values. The PCROSS code can work legally up to 69 MeV incident photon energy. Therefore, the PCROSS-WE calculations have been shown up to 69 MeV. EMPIRE 3.1 exciton, TALYS 1.6 two component exciton and pre-equilibrium model calculations are in harmony with each other. The TALYS 1.6 pre-equilibrium curves fit the experimental values the best in the energy range of 60–110 MeV. The theoretical model calculations of 94Zr(γ,3n)91Zr reaction have been compared with the experimental results in Fig. 3. All photo-neutron cross-sections are in agreement with the experimental values including error bars in the photon energy region 24–31 MeV. The calculated values of (γ,3n) reactions for 98,100Mo target nuclei have been compared with the experimental data in Figs. 4, 5. All theoretical model calculations are in agreement with the experimental data including error bars at the photon energy regions 24.5–29 MeV for the 98Mo(γ,3n)95Mo and 22.5–28.5 MeV for the 100Mo(γ,3n)97Mo reactions. The TALYS 1.6 pre-equilibrium model calculations are higher than the other theoretical calculations and experimental values in Fig. 5. The TALYS 1.6 two component exciton model cross-sections are shown to have a good agreement with the experimental data. The comparison of theoretical and experimental results of 181Ta(γ,3n)178Ta reaction has been given in Fig. 6. PCROSS-WE and EMPIRE 3.1 model calculations are in agreement with the measurements in the 22–28 MeV photon energy region. The TALYS 1.6 pre-equilibrium model calculations follow the experimental results from above up to 32 MeV but the TALYS 1.6 two component exciton model cross-sections are the best agreement with the measurements including error bars. The theoretical cross-section calculations and experimental values of 186W(γ,3n)183W reaction have been given in Fig. 7. Generally the PCROSS-WE and EMPIRE 3.1 model calculations are in harmony with the measurements in the energy region of 20–29 MeV. The PCROSS-WE model results are in good agreement with the experimental data. The TALYS 1.6 two component exciton model cross-sections are in agreement with the experimental data in the 24–28 MeV but the TALYS 1.6 pre-equilibrium model results are the best agreement with the experimental values in the 26.5–29 MeV photon energy region.

In general Figs. 1, 2, 3, 4, 5, 6 and 7 show that PCROSS, EMPIRE 3.1 and TALYS 1.6 model calculations exhibit a similar structure with experimental data except for 65Cu(γ,3n)62Cu reaction. As we can see from these results, the good agreement between photo-neutron cross-section calculations and experimental values of (γ,3n) reactions shows that PCROSS, EMPIRE 3.1 and TALYS 1.6 models are able to reproduce the cross-sections with reasonable accuracy in this case even without any tuning of the parameters. Besides, the agreement with the theoretical values obtained with TALYS 1.6 using two component exciton model parameters proves again the prediction strength of the code.

Summary and Conclusions

In this study, theoretical photo-neutron cross-sections of (γ,3n) reactions for several structural fusion materials such as 55Mn, 65Cu, 94Zr, 98,100Mo, 181Ta and 186W have calculated in the photon energy range of 20–110 MeV using PCROSS, EMPIRE 3.1 and TALYS 1.6 computer codes. The calculated results have been also compared with the available experimental values in the EXFOR library. The results can be summarized and concluded as follows:

  1. 1.

    The photo-neutron cross-section results calculated with PCROSS, EMPIRE 3.1 and TALYS 1.6 computer codes for (γ,3n) reaction are mostly in agreement with the experimental data except for 65Cu(γ,3n)62Cu reaction.

  2. 2.

    All model cross-section calculations are in good harmony with each other for all reactions in this study.

  3. 3.

    The good agreement between the calculations and experimental values shows that PCROSS, EMPIRE 3.1 and TALYS 1.6 models are able to reproduce the cross-sections with reasonable accuracy in this case even without any tuning of the parameters.

  4. 4.

    The agreement with the theoretical results obtained with TALYS 1.6 using two component exciton model parameters proves again the prediction strength of the code.

  5. 5.

    The TALYS 1.6 two component exciton option for (γ,3n) reaction cross-section calculations can be chosen, if the experimental data are not available or are improbable to be produced due to the experimental difficulty.