Preamble

Monte Carlo Simulation of Spallation and Fission Fragment Distributions for ADS-Related Nuclear Reactions

Sun Wenming¹
¹(Graduate School of Science, The University of Tokyo, Tokyo, 113-0033)

Monte Carlo simulations with the CRISP code were conducted to study spallation and fission fragment distributions induced by intermediate- and high-energy protons and photons on actinide and pre-actinide nuclei. The model accounts for intranuclear cascade, pre-equilibrium, and evaporation-fission competition, enabling consistent treatment of both residues and fission products. Comparisons with experimental data show good agreement in mass and charge distributions, with minor deviations for light fragments. The results highlight the reliability of Monte Carlo approaches for predicting residual nuclei and fragment yields under accelerator-driven system (ADS) conditions. This work provides nuclear data relevant to ADS design, safety, and transmutation analysis.

Keywords: Monte Carlo simulation; Spallation and fission fragments; Accelerator-driven system (ADS)
PACS: 24.10.Lx; 25.85.-w; 28.50.Ft

1. Introduction

The accelerator-driven system (ADS) is an innovative reactor concept being developed as a dedicated burner in a double-strata fuel cycle to incinerate nuclear waste [1–4]. An ADS consists of a subcritical assembly driven by an accelerator that delivers a proton beam onto a target to produce neutrons via spallation reactions. The spallation target simultaneously constitutes the physical and functional interface between the accelerator and the subcritical reactor. For this reason, it is arguably the most innovative component of the ADS, and its design represents a key challenge in ADS development. The performance of the reactor is characterized by the number of neutrons emitted per incident proton, the mean energy deposited in the target per neutron produced, the neutron spectrum, and the spallation product distribution [5].

The detailed design of spallation neutron sources or accelerator-driven systems requires reliable computational tools to optimize their performance in terms of useful neutron production and to properly assess specific problems likely to occur in such systems. Among these problems are the radioactivity induced by spallation reactions and the associated shielding issues [6], radiation damage to target, window, or structural materials from energetic particles generated in the reaction [7], and induced radiotoxicity within the target [8] due to the production of various nuclides.

Radiation damage can arise from gas production that causes embrittlement of structural materials and from atomic displacements (DPA) which weaken the various components of the spallation source. Modifications to the chemical composition of these materials may lead to corrosion problems, loss of alloy cohesion, and changes in mechanical properties due to the formation of compounds not initially present in the materials.

It is important to emphasize that, at the current stage of ADS development, it is still necessary to study the optimal technological choices for variables such as target materials and incident particle energies. For this reason, the most important aspect in calculating nuclide yields is the confidence one has in the results obtained. At this stage, reliability of the calculation method may be more important than absolute accuracy. Methods that correctly incorporate the underlying reaction mechanisms should be favored over those with many free parameters fitted to experimental data. While parameter-rich models are valuable for interpolating available experimental data, their utility is limited when no data exist.

This paper is organized as follows: Section 2 describes the main mechanisms for fragment production in nuclear reactions and provides a brief description of the intranuclear cascade process, which is relevant for residual nucleus formation. Section 3 describes the evaporation mechanism, which is the most important process for spallation studies. Section 4 describes the fission process, a decay channel relevant for fission fragment formation. Section 5 presents and discusses results obtained with the CRISP code, and Section 6 presents our conclusions.

2. Production of Nuclides in Nuclear Reactions

The production of nuclides is associated with the different mechanisms through which nuclear reactions evolve. Depending on the target, projectile, and energy, different mechanisms become more or less relevant. The distribution of fragments generated when the reaction completes is strongly dependent on the channels available to the system and their respective branching ratios.

As the reaction energy increases, more channels become available and the complexity of the reaction grows. This high complexity is the primary motivation for adopting Monte Carlo (MC) methods in nuclear reaction calculations. Indeed, the large number of particles, the numerous available channels, and the fact that the reaction can be understood as a Markovian sequence of steps are features that perfectly match the characteristics of Monte Carlo calculations. In this work, we adopt the MC method and use the implementation for nuclear reaction calculations developed in the CRISP code [12], which has already been applied in ADS studies [13–15]. This implementation is described in detail below.

It is well established that intermediate- and high-energy reactions follow a two-step mechanism, as proposed by Bohr. In the first step, usually called the intranuclear cascade, the energy and momentum of the incident particle are distributed among a few nucleons through baryon-baryon reactions that are mainly elastic. However, at sufficiently high energies, nucleonic degrees of freedom can be excited. This step concludes when no nucleons possess kinetic energy high enough to escape from nuclear binding. Thereafter, collisions among nucleons lead only to system thermalization.

The second step depends on the excitation of the residual nucleus formed at the end of the intranuclear cascade. If the nuclear excitation energy is E/A ≤ 3 MeV, a statistical competition between evaporation and fission occurs. For heavy nuclei (A > 230), fission is the dominant channel, and in most cases the reaction ends with the formation of two fragments [16]. For nuclei with A < 230, fission is much less probable, and in most cases the nucleus evaporates until insufficient energy remains for particle emission (neutrons, protons, and alpha particles being the most frequent). A spallation product is then formed, characterized by its mass and atomic numbers.

If the nuclear excitation energy is E ≥ 3 MeV/A, an entirely different process may occur: nuclear multifragmentation [17–19]. This process is much faster than the evaporation/fission competition and is characterized by the simultaneous production of a large number of fragments with very different mass and atomic numbers.

Below we provide a brief description of each process mentioned above.

2.1 Intranuclear Cascade

We now describe the nuclear mechanisms relevant to nuclear reactions at intermediate and high energies, as implemented in the CRISP code. The first mechanism to consider is the intranuclear cascade (IC). As the incident particle enters the nuclear region, the IC is initiated by the collision of the projectile with one of the nucleons in an inelastic scattering event. This collision, called an elementary collision, always generates secondary particles, which may be two nucleons in the case of elastic scattering, mesons, or resonance states of the nucleon. These secondaries have relatively high energies compared to other nucleons in the Fermi sea and occupy high-energy single-particle states in the system [20–23]. We call them cascade particles.

The secondary particles generated in an elementary collision propagate inside the nucleus and may interact, increasing the number of cascade particles, or may reach the nuclear surface. In the latter case, if a particle has energy higher than the nuclear binding energy, it escapes from the nucleus; otherwise, it is reflected back and continues propagating inside the nucleus. In this way, as the intranuclear cascade continues, the number of cascade particles increases. The decision to stop the intranuclear cascade and initiate the second step of the reaction is based on energetic criteria: namely, when no bound particle remains in an excited state or possesses kinetic energy greater than its binding energy [21, 22, 24].

Several important aspects characterize intranuclear cascade calculations with the CRISP code. First, it is a multicollisional simulation of the cascade, with all nucleons moving simultaneously. The time-ordered sequence of elementary collisions considers the probability of interactions among all particles based on their respective cross sections. This represents an important difference from other Monte Carlo codes where only one cascade particle is considered while all others remain frozen in their initial states.

The multicollisional approach represents a significant improvement in intranuclear cascade simulation compared to the MC method used in well-known codes such as those from Barashenkov et al. or Bertini et al. [25, 26], since dynamical aspects such as nuclear density modifications or the evolution of occupancy numbers during the cascade are naturally accounted for in the former method but not in the latter [24].

Another important aspect is the Pauli blocking mechanism, which tracks possible violations of the Pauli Principle. With the multicollisional method, it is possible to adopt a very precise method for verifying the availability of single-particle states for fermions generated in elementary collisions, eliminating possible violations of this fundamental principle of quantum mechanics. It is because we use this method for Pauli blocking that we can employ energetic criteria to decide when to end the intranuclear cascade [27].

After the cascade concludes, no particles remain with sufficient energy to escape from the nucleus. A sequence of elementary collisions then distributes the excitation energy remaining in the nucleus among all nucleons. This is called the thermalization process. The main characteristics of the nucleus do not change during this step, and the mass number, atomic number, and excitation energy at the end of the thermalization step are the same as at the end of the intranuclear cascade.

In CRISP simulations, reactions can be initiated by intermediate- and high-energy protons [21] or photons [21, 22, 27, 28]. The code has been used for energies up to 3.5 GeV [24], where it was shown to provide good results for total photonuclear absorption cross sections from approximately 50 MeV (where the quasi-deuteron absorption mechanism dominates) up to 3.5 GeV (where the photon-hadronization mechanism dominates, leading to a shadowing effect in the cross section). Recently, the CRISP code has been used to study final-state interactions for the nonmesonic weak decay of hypernuclei [29]. The results demonstrate its applicability even for light nuclei such as ¹²C and relatively low energies, as in hypernucleus decay.

3. Evaporation

Thermalization is followed by the evaporation process, during which nucleons or small clusters are emitted, carrying away part of the nuclear excitation energy. This process continues while sufficient energy remains in the nucleus to allow particle emission. It consists of a sequence of particle emissions, each governed by Weisskopf theory. The evaporation regime is determined by the relative probabilities of different particle emission channels [27, 28, 30].

These probabilities are obtained from the particle emission width, Γₖ, calculated according to the well-known Weisskopf evaporation model [31]. For proton emission we have:

[TABLE:1]

and for alpha particle emission:

[TABLE:2]

where Eₖ corresponds to the nuclear excitation energy after emission of a particle of type k (k = p, n, α), calculated as:

Eₙ = E - Bₙ,
Eₚ = E - Bₚ - Vₚ,
Eₐ = E - Bₐ - Vₐ,

where Bₙ, Bₚ, and Bₐ are the separation energies for neutrons, protons, and alpha particles, and Vₚ and Vₐ are the Coulomb potentials for protons and alpha particles, respectively.

These Coulomb potentials are given by the expressions:

B being the nuclear binding energy. The level density parameters aₖ are calculated from Dostrovsky's empirical formulas [32]:

[TABLE:3]

To evaluate the evaporation probability, we assume it is proportional to the corresponding width [15, 30]:

Pₖ = Γₖ / Σᵢ Γᵢ

While sufficient energy is available for particle evaporation, successive emissions are processed. The evaporation phase ends when the excitation energy of the nucleus becomes smaller than all separation energies Bₙ, Bₚ, and Bₐ. At this point, a nucleus that may be completely different from the initial one is formed. This nucleus is called a spallation product.

4. Fission

The CRISP code can also evaluate fission probability [36]. Fission is a process that competes with evaporation, as each nucleus in the evaporation sequence may undergo fission, forming two fragments with masses around one-half of the fissioning nucleus. This process can be easily incorporated into the evaporation framework by including the fission branching ratio, Γ_f, in the probability expression:

Pₖ = Γₖ / (Γ_f + Σᵢ Γᵢ)

The fission branching ratio is calculated according to the Bohr-Wheeler fission model [37]:

Γ_f = (1/2πρ(E)) ∫₀^(E-B_f) ρ*(E-B_f-K) dK

For the mass formula, we employ the semiempirical formulation proposed in [33] for nuclear binding energy, so that nuclear masses are calculated according to:

B(A,Z) = a_v A - a_s A^(2/3) - a_c Z(Z-1)/A^(1/3) - a_a (A-2Z)²/A + δ

where N is the number of neutrons. This formula was fitted to the compilation of Audi et al. [34] using the method of least squares with the MINUIT package [35]. The parameter values corresponding to the best fit are shown in Table 1.

5. Results and Discussions

We used the CRISP code to calculate cross sections for fragment formation in nuclear reactions. As explained above, the primary sources of fragments in the reactions studied here are spallation and fission processes. In this section, we present cross sections for both fission fragments and spallation products.

5.1 Fission Reactions

The relevant parameters in Dostrovsky's empirical formulas (see Eq. 6) for the evaporation process are shown in Table 2. This parameter set was used for all reactions studied in this work. The parameters for fission fragment calculations in Eq. 12 are shown in Table 3.

For the cases of ²⁴¹Am, ²³⁷Np, and ²³⁸U, the values for width and position (Table 3, first column) of the fission modes were taken from low-energy systematics [9]. The relative intensities of each fission mode for ²⁴¹Am and ²³⁷Np were considered fixed, with values given in Table 3. For ²³⁸U, the relative intensities were calculated according to Gaussian expressions depending on the mass number of the fissioning system.

No systematic study of multimodal parameters exists in the mass region of Pb. To obtain these parameters, we used a sigmoid fit for K_S and Gaussian fits for K₁ and K₂, all as functions of the fissioning system mass. For heavy fragment distributions, the peaks for the asymmetric modes are obtained by:

A_H¹ = b₁A_f + b₂,
A_H² = b₃A_f + b₄,

while the width for each mode is assumed constant:

Γ_S = b₅, Γ₁ = b₆, Γ₂ = b₇.

The parameter values corresponding to the best fit are presented in Table 4.

[FIGURE:1]

We observe that the results obtained with the CRISP code provide a good description of the experimental data, although the data are spread over a large range.

Since CRISP can also be used for photon-induced reactions, we calculated mass distributions of fragments for fission induced in ²³⁸U by bremsstrahlung photons with endpoint energies of 50 MeV and 3500 MeV. The results are presented in Figure 2. Again, the data do not show good resolution, but we can observe that the CRISP results provide a good general description of the experimental data.

[FIGURE:2]

A more precise experiment was performed for proton-induced fission on ²⁰⁸Pb. In this case, the experimental results are very precise. Figure 3 shows the experimental data compared to the calculation results. We observe excellent agreement between experiment and CRISP calculation, particularly regarding the overall shape of the distribution. This agreement is even better when the result is normalized to the data.

[FIGURE:3]

For all results shown in Figures 1, 2, and 3, fragment evaporation was included. This was achieved by using Eq. 12 to obtain the mass and atomic numbers of heavy and light fragments. As a first approximation, the calculated excitation energy of the fissioning system was divided between the fragments, which continued the evaporation process until the stopping criterion was reached again, as explained in Section 3.

5.2 Spallation Reactions

Nuclei with mass number A < 220 exhibit low fission probabilities. In these cases, the most probable reaction channel is spallation, where evaporation continues until the residual nucleus becomes cold without undergoing fission. The nucleus at the end of evaporation is then called a spallation product.

Using the CRISP code with the parameters for evaporation and fission competition described above, we can also calculate spallation product distributions. Figure 4 shows our results compared to experimental data.

[FIGURE:4]

In general, we observe fair agreement with data, as the well-known spallation parabola calculated by our model shows position and width in good agreement with experimental results. For the most probable products, the absolute cross sections also agree well with data, although for some spallation products the agreement is not satisfactory. Similar behavior occurs for spallation on ¹⁹⁷Au, shown in Figure 5.

[FIGURE:5]

These results demonstrate that it is extremely difficult to obtain good results simultaneously for several different reactions across a wide range of target mass numbers and for quite different energies. The main difficulty arises because, at the end of the intranuclear cascade, we generally have excited residual nuclei that may be far from the line of stability. Since most nuclear models and their parameters are determined for cold, stable nuclei, we lack precise information on the structure of all nuclei formed during nuclear reactions at intermediate and high energies. Improvements to mass formulas and shell effects could lead to better agreement between calculations and experiments.

6. Conclusion

In this work, we addressed the problem of fragment production in nuclear reactions. This is a relevant issue in ADS development, as it is directly related to the study of damage induced in materials used in ADS.

We discussed the importance of calculation reliability and demonstrated that the model implemented in the CRISP code is developed with a reduced number of free parameters and careful attention to accurately reproducing the physical processes occurring during nuclear reactions. The parameters appearing in the mass formula were obtained by fitting to nuclear mass data. The parameters for evaporation and fission were fitted simultaneously to numerous results for fission and spallation cross sections on various nuclei with different projectiles at many different energies. For these reasons, the code can be used to calculate several observables in different reactions induced by photons, electrons, protons, and neutrons over a large energy range on nuclei with masses from A ≈ 12 to A ≈ 240.

We used the CRISP code to calculate nuclide production in nuclear reactions induced by intermediate- or high-energy projectiles. Proton and photon reactions on actinide and pre-actinide nuclei were considered. The most important fragment production mechanisms—spallation and fission—were studied in detail. We show that the results are in good qualitative agreement with available experimental data.

References

[1] W. Gudowski, "Accelerator-driven transmutation projects. The importance of nuclear physics research for waste transmutation," Nuclear Physics A, vol. 654, no. 1-2, pp. 436–457, 1999.
[2] E. Birgersson, S. Oberstedt, A. Oberstedt et al., "Light fission-fragment mass distribution from the reaction ²⁵¹Cf(n_th, f)," Nuclear Physics A, vol. 791, no. 1-2, pp. 1–23, 2007.
[3] Department of Nuclear Energy Research Advisory Committee and the Generation IV International Forum, A Technology Roadmap for Generation IV Nuclear Energy Systems, GIF-002-00, 2002.
[4] C. Rubbia et al., "Conceptual design of a fast neutron operated high power energy amplifier," CERN, Report CERN/AT/95-44(ET), 1995.
[5] S. T. Mongelli, J. R. Maiorino, S. Anefalos, A. Deppman, and T. Carluccio, "Spallation physics and the ADS target design," Brazilian Journal of Physics, vol. 35, no. 3B, pp. 894–897, 2005.
[6] S. Leray, "Long term needs for nuclear data development," IAEA report INDC-NDS-0428, 2001.
[7] Villagrasa et al., "AccApp'03 Accelerator Applications in a Nuclear Renaissance," San Diego, Calif, USA.
[8] T. Aoust, E. M. Malambu, and H. A. Abderrahim, "Importance of nuclear data for windowless spallation target design," in Proceedings of the International Conference on Nuclear Data for Science and Technology, vol. 769, pp. 1572–1575, October 2004.
[9] C. Böckstiegel, S. Steinhauser, K.-H. Schmidt et al., "Nuclear-fission studies with relativistic secondary beams: analysis of fission channels," Nuclear Physics A, vol. 802, pp. 12–25, 2008.
[10] G. S. Karapetyan, A. R. Balabekyan, N. A. Demekhina, and J. Adam, "Multimode approach to ²⁴¹Am and ²³⁷Np fission induced by 660-MeV protons," Physics of Atomic Nuclei, vol. 72, no. 6, pp. 911–916, 2009.
[11] N. A. Demekhina and G. S. Karapetyan, "Multimode approximation for ²³⁸U photofission at intermediate energies," Physics of Atomic Nuclei, vol. 71, no. 1, pp. 27–35, 2008.
[12] A. Deppman, S. B. Duarte, G. Silva et al., "The CRISP package for intermediate- and high-energy photonuclear reactions," Journal of Physics G, vol. 30, no. 12, pp. 1991–2002, 2004.
[13] S. Anefalos, A. Deppman, G. Da Silva, J. R. Maiorino, A. Dos Santos, and F. Garcia, "The utilization of CRISP code in hybrid reactor studies," Brazilian Journal of Physics, vol. 35, pp. 912–914, 2005.
[14] S. Anefalos, A. Deppman, J. D. T. Arruda-Neto et al., "The CRISP code for nuclear reactions," in Proceedings of the International Conference on Nuclear Data for Science and Technology, vol. 769, pp. 1299–1302, 2004.
[15] S. A. Pereira, "Spallation product distributions and neutron multiplicities for accelerator-driven system using the CRISP code," Nuclear Science and Engineering, vol. 159, pp. 102–105, 2008.
[16] M. L. Terranova and O. A. P. Tavares, "Total nuclear photoabsorption cross section in the range 0.2-1.0 GeV for nuclei throughout the periodic table," Physica Scripta, vol. 49, pp. 267–279, 1994.
[17] J. P. Bondorf, R. Donangelo, I. N. Mishustin, C. J. Pethick, H. Schulz, and K. Sneppen, "Statistical multifragmentation of nuclei. (I). Formulation of the model," Nuclear Physics, Section A, vol. 443, no. 2, pp. 321–347, 1985.
[18] J. Bondorf, R. Donangelo, I. N. Mishustin, and H. Schulz, "Statistical multifragmentation of nuclei. (II). Application of the model to finite nuclei disassembly," Nuclear Physics, Section A, vol. 444, no. 3, pp. 460–476, 1985.
[19] C. E. Aguiar, R. Donangelo, and S. R. Souza, "Nuclear multifragmentation within the framework of different statistical ensembles," Physical Review C, vol. 73, no. 2, Article ID 024613, pp. 1–8, 2006.
[20] J. Cugnon, "Proton-nucleus interactions at high energy," Nuclear Physics A, vol. 462, p. 751, 1987.
[21] M. Gonçalves, S. De Pina, D. A. Lima, W. Milomen, E. L. Medeiros, and S. B. Duarte, "Many-body cascade calculation for photonuclear reactions," Physics Letters Section B, vol. 406, no. 1-2, pp. 1–6, 1997.
[22] S. De Pina, E. C. De Oliveira, E. L. Medeiros, S. B. Duarte, and M. Gonçalves, "Photonuclear K⁺ production calculation near threshold," Physics Letters Section B, vol. 434, no. 1-2, pp. 1–6, 1998.
[23] J. Cugnon, C. Volant, and S. Vuillier, "Improved intra-nuclear cascade model for nucleon-nucleus interactions," Nuclear Physics A, vol. 620, p. 475, 1997.
[24] A. Deppman, G. Silva, S. Anefalos et al., "Photofission and total photoabsorption cross sections in the energy range of shadowing effects," Physical Review C, vol. 73, no. 6, Article ID 064607, 2006.
[25] V. S. Barashenkov, F. G. Gereghi, A. S. Iljinov, G. G. Jonsson, and V. D. Toneev, "A cascade-evaporation model for photonuclear reactions," Nuclear Physics, Section A, vol. 231, no. 3, pp. 462–476, 1974.
[26] H. W. Bertini, "Low-energy intranuclear cascade calculation," Physical Review, vol. 131, no. 4, pp. 1801–1821, 1963.
[27] A. Deppman, O. A. P. Tavares, S. B. Duarte et al., "The MCEF code for nuclear evaporation and fission calculations," Computer Physics Communications, vol. 145, no. 3, pp. 385–394, 2002.
[28] A. Deppman, O. A. P. Tavares, S. B. Duarte et al., "Photofissility of heavy nuclei at intermediate energies," Physical Review C, vol. 66, no. 6, pp. 676011–676014, 2002.
[29] I. Gonzales, "Many-body cascade calculation of final state interactions in ¹²C nonmesonic weak decay," Journal of Physics G, vol. 38, no. 11, Article ID 115105, 2011.
[30] A. Deppman, O. A. P. Tavares, S. B. Duarte et al., "A Monte Carlo method for nuclear evaporation and fission at intermediate energies," Nuclear Instruments and Methods in Physics Research Section B, vol. 211, no. 1, pp. 15–21, 2003.
[31] V. Weisskopf, "Statistics and nuclear reactions," Physical Review, vol. 52, no. 4, pp. 295–303, 1937.
[32] I. Dostrovsky, P. Rabinowitz, and R. Bivins, "Monte Carlo calculations of high-energy nuclear interactions. I. Systematics of nuclear evaporation," Physical Review, vol. 111, no. 6, pp. 1659–1676, 1958.
[33] J. M. Pearson, "The quest for a microscopic nuclear mass formula," Hyperfine Interactions, vol. 132, no. 1–4, pp. 59–74, 2001.
[34] G. Audi, A. H. Wapstra, and C. Thibault, "The Ame2003 atomic mass evaluation—(II). Tables, graphs and references," Nuclear Physics A, vol. 729, no. 1, pp. 337–676, 2003.
[35] F. James, "Minuit: Function Minimization and Error Analysis-Reference Manual Version 94.1," CERN Program Library Long Writeup D506.
[36] A. Deppman, O. A. P. Tavares, S. B. Duarte et al., "Photofissility of actinide nuclei at intermediate energies," Physical Review Letters, vol. 87, no. 18, Article ID 182701, pp. 1827011–1827014, 2001.
[37] N. Bohr and J. A. Wheeler, "The mechanism of nuclear fission," Physical Review, vol. 56, no. 5, pp. 426–450, 1939.
[38] J. R. Nix, "Calculation of fission Barriers for heavy and superheavy nuclei," Annual Review of Nuclear Science, vol. 22, pp. 65–120, 1972.
[39] E. Andrade II, E. Freitas, O. A. P. Tavares, S. B. Duarte, A. Deppman, and F. Garcia, "Monte Carlo simulation of heavy nuclei photofission at intermediate energies," in Proceedings of the 31st Workshop on Nuclear Physics, vol. 1139, pp. 64–69, American Institute of Physics, 2009.
[40] E. Andrade II, J. C. M. Menezes, S. B. Duarte et al., "Fragment mass distributions in the fission of heavy nuclei by intermediate- and high-energy probes," Journal of Physics G, vol. 38, no. 8, Article ID 085104, 2011.
[41] U. Brosa, S. Grossmann, and A. Müller, "Nuclear scission," Physics Report, vol. 197, no. 4, pp. 167–262, 1990.