1 Introduction

Nuclear waste generated in a reactor contains a wide range of highly toxic products with long-lived, and the management of these LLFPs is one of the challenges of using nuclear energy. Due to the mobility of some of these LLFPs, their direct underground disposal is not acceptable because, in case of leakage, these radionuclides may contaminate underground water channels or food chains. Therefore, alternative methods are needed to reduce the hazard of these LLFPs. One such approach is the transmutation of these LLFPs by the accelerator-driven system (ADS) into some stable or short-lived radionuclides [1,2,3,4]. In the transmutation process, one isotope is converted to another isotope by receiving or losing  one (or multiple) nucleon(s). This process can reduce the activity, mass, volume, heat load or radiotoxicity of nuclear waste [5, 6]. Diverse studies have been performed on the transmutation of LLFPs by ADS [7,8,9,10,11]. In this work, by generating the transmutation response function, we make it possible for the computation of the transmutation rate for 129I at high energies in MCNPX code. 129I is the longest-lived radioactive fission product, with a half-life of \(1.57 \times 10^{7}\) years.

In this study, the MCNPX code was used due to its availability, transport of charged particles, robust physics models and the ability to run in analog mode. MCNPX code has two ways of calculating cross sections: it can use tabulated data when the data exists or it can use physics models to calculate the cross sections “on the fly” [12]. In the physics regime of interest here, MCNPX model interactions use a combination of the cascade excitation model (CEM), the nuclear optical model and the Los Alamos version of the quark gluon string model (LAQGSM) [13, 14].

In MCNPX, the Interaction tally (FM tally multiplier card) calculates the transmutation rate of nuclei for which tabulated data exist. The tabulated data generally only exist for a limited energy range. For 129I, the ENDF/B-VII.1 libraries are only evaluated up to 20 MeV. If the particle’s energy is in the physics model regime (i.e., above the energy cutoff for the tabulated data), the interaction tally will not score any contributions, and the warning “FM tallies do not score in model physics regime” is given. For example, in the case of 129I that neutrons cross-section library evaluated up to 20 MeV, neutrons with energies above 20 MeV will not contribute at all to the tally [12, 15, 16].

2 Response function generation

The number of transmutations that occur in a pure target is equal to the number of neutrons and protons lost in nuclear interactions, because in the bombardment of 129I with proton particles, any interaction leads to transmutation. To run the system in analog mode, the cutoff energy of the neutrons must be zero and for the protons less than the energy required to overcome the Coulomb barrier. In this mode, the MCNPX code tracks every particle until its death and prints a table of the number of neutrons and protons lost in the output file.

By proton bombardment of 129I, based on some reactions such as (p, n), (p, α), (p, np), (p, 2np), (p, d), (p, 3He), (p, 2p), (p, γ), (p, 2n), (p, 3n), (p, t) and (p, nα) radionuclides with shorter lives such as 128I (half-life 24.99 min), 129mXe (half-life 8.88 days), 127Xe (half-life 36.35 days), 127Te (half-life 9.35 h), 127mXe (half-life 62.9 s), 125mTe (half-life 57.40 days) and 127mTe (half-life 106.1 days), stable isotopes such as 129Xe, 125Te, 130Xe, 128Xe, 127I and 126Te are produced. As you can see, the longest half-live radionuclide 127mTe (half-life 106.1 days) is much shorter than 129I; so in the case of 129I, any nuclear interaction other than scattering will transform the nucleus into a shorter-lived nuclide [17].

The mentioned method of counting lost particles in the MCNPX output file can be used only for pure targets, if we use spallation materials such as lead or a reflector such as graphite to increase the transmutation rate causes a problem, because MCNPX does not distinguish between particle losses to different nuclei; so some particle losses have occurred in other material such as lead and graphite that would not result in transmutations of 129I.

In order to estimate the number of transmutations of iodine nuclei by particles whose energy is higher than the MCNPX’s tabulated data, the transmutation response function must be generated. To measure the average response function, for neutrons, 168 energy bins from 10 to 1500 MeV, and for protons, 120 energy bins from 1 to 1500 MeV, are considered. The energy considered in each bin is the upper limit of energy for that bin, and the lower limit is the energy considered in the previous bin. In order to regardless of the attenuation and the coulombic energy loss, a monodirectional point source of particles and a short and very thin cylindrical 129I target were modeled. Then, for each energy bin, we run a separate MCNPX deck. (Approximately 500 decks or more have been run for this study.)

The response value \(\sigma \left( E \right)\) is given by Equation [13]:

$$\sigma \left( E \right) = \frac{{n_{{{\text{intr}}}} }}{nps \cdot z \cdot N}$$
(1)

where \({n}_{\mathrm{intr}}\) is the number of interactions in the MCNPX output file, nps is the number of particles run, z is the target thickness is cm, and N is the atomic density in atoms/\({\mathrm{cm}}^{3}\).

At different energies, we need to choose different sizes of targets. A suitable target thickness is one in which the number of interactions with the target is less than 10% but more than 1% of the total incident particles. To check whether the attenuation is insignificant or not, you could try using two surface tallies, one at the entry and one at the exit of the target. The F1 and F11 tallies at the entrance and exit of your target should be close to the same. The F11 at the exit will be slightly lower, by the amount that has interacted, so if 1% of particles interact, the exit tally will be 99% of the entrance. If the exit is much lower, then the attenuation is significant and should be accounted for. To do so, many geometries have been studied, for example, in the energy bin of 1500 MeV, the target of 129I was considered a cylinder with a thickness of 0.001 cm. In addition, an energy cutoff card can be used to eliminate the interaction of particles whose energy is outside the range of each energy bin [13].

When the transmutation response function was obtained for each energy bin, based on Eq. (2), En and EMn cards can be used to enter the values of the transmutation response function to MCNPX deck and create a custom tally to calculate the transmutation rate at high energies for 129I in the region containing target material:

$${\text{Transmutation}}\,{\text{rate}} = \sum\limits_{p}^{{{\text{particles}}}} {\sum\limits_{E}^{{{\text{bins}}}} {\Phi (p,E)} } \sigma (p,E) \times V \times N$$
(2)

where \(\sum\nolimits_{p}^{{{\text{particles}}}} {}\) is a sum over all particles, \(\sum\nolimits_{E}^{{{\text{bins}}}} {}\) is the sum over all energy bins tallied for particle type p, \(\Phi (p,E)\) is the flux of particle p within energy bin E, \(\sigma (p,E)\) is the response function value for particle p and energy bin E, V is the volume of material, and N is the atom density of 129I [13].

For 129I, radiative capture (n, γ) is the main mechanism for transmutation below 10 MeV, which leads to the production of 130I with 12.36 h half-life. MCNPX by the 102 capture tally will tally (n, γ) reactions for the nuclide specified in the range for which has tabulated cross-section data; therefore, for energies higher than 10 MeV, a custom tally can be used for all other non-scattering nuclear interactions. (Scattering does not lead to transmutation.) The final value of the response function can be found in Appendix, Tables 2 and 3.

Table 1 Number of proton-induced transmutations of 129I with 10,000 incident protons
Full size table

3 Results and discussion

Experiments performed on the 129I transmutation by proton beam mostly calculated the transmutation rate of 129I by neutrons produced in spallation reactions induced by protons in the target, not the transmutation rate of 129I by the proton itself. In the study presented by [18], the cross section is calculated for some of the proton-induced reactions leading to 129I transmutation. The response function presented in our study calculates the total transmutation rate, but in [18], the cross section of many of the reactions that lead to the transmutation of 129I is not calculated, such as the reactions that lead to the transmutation of iodine to 129mXe, 127Te, 127mXe, 125mTe, 127mTe, 129Xe, 125Te, 130Xe, 128Xe, 127I, 126Te, 127mTe, etc., so it is not possible to accurately compare the results. The total cross section calculated in 660 MeV for iodine is 0.496 b, which is certainly less than the value calculated for the total cross section in 660 MeV of this study, which includes all the reactions that lead to iodine transmutation. In addition, in experiments in which the transmutation rate of 129I is calculated by the produced neutrons, the transmutation rate is mainly calculated for limited reactions such as (n, γ), not for the total transmutation rate that the sum of all the reactions that lead to 129I transmutation. The transmutation response function by neutron presented in this study calculates the total 129I transmutation rate (other than radiative capture and scattering, scattering does not lead to transmutation), and it is not possible to compare it with the experimental results, which are mainly calculated for specific channels. With these interpretations, it is not possible to make an accurate comparison with the experimental results.

To validate, the results were tested by two methods for energies lower and higher than 200 MeV. Tabulated data for the interactions of protons with 129I up to 200 MeV are available in the JENDL/ImPACT-18 [19] and TENDL-2019 [20] libraries, but the format of these libraries is ENDF, to analyze the number of transmutations obtained from the custom tally and the FM tally of these libraries in MCNPX, was needed to convert their format to ACE by NJOY code. The NJOY nuclear data processing system is a modular computer code used for converting evaluated nuclear data in the ENDF format into libraries useful for applications calculations, and the ACER module prepares libraries in ACE format for the Los Alamos continuous-energy Monte Carlo MCNP codes [21, 22].

Figures 1 and 2 show the generated custom cross section in this study with the cross sections in the JENDL and TENDL libraries, and Table 1 contains the number of proton-induced transmutations caused by 10,000 protons in a 129I target at different energies using custom tally and FM tally of TENDL and JENDL libraries which both indicate a good agreement in the results.

Fig. 1
figure 1

Interaction cross section vs. energy using TENDL and JENDL and the response function generated in MCNPX code

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Fig. 2
figure 2

Total neutron absorption cross section vs. energy using TENDL and JENDL and the response function generated in MCNPX code

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For validation at energies higher than 200 MeV, the custom tally was used to estimate the number of transmutations in a pure 129I target, then the number of lost protons in the MCNPX output file was counted and the results were compared. The number of proton-induced transmutations obtained for energies 300 MeV, 900 MeV and 1500 MeV with 10,0000 incident protons are 44,672, 124,033 and 143,928, respectively, for the custom tally, and 43,776, 122,269 and 144,133 for counting the number of protons lost in transmutation in the output file. The results show that the error is about 2%, which is considered acceptable.

4 Conclusion

A light water reactor (LWR), 1 GWh (e), discharges about 21 tons of radioactive fuel every year, of which 760 kg are fission products [23]. One way to manage LLFPs is to transmute them to some stable or shorter-lived radionuclides by an accelerator-driven system (ADS). The MCNPX code can be used to simulate the transmutation rates of these LLFPs, but the tabulated data used in this code is not available for high-energy protons and neutrons. This study proposes a transmutation response function that is consistent with the physics models used in the MCNPX code. The generated custom tally can be input into MCNPX and used for the estimation of the transmutation rate of 129I by an accelerator-driven system with energies up to 1500 MeV.

For energies lower than 200 MeV, the results were compared with data from the JENDL and TENDL libraries. The difference in values is very small and can be caused by inherent errors in the simulation codes. As you can see, even the data from the JENDL and TENDL libraries are slightly different. As well as, using cross-section data from other libraries that are not always consistent with the models used in MCNPX can cause differences in the transmutation rates.