Optimizing the primary knock-on atom for simulating particle radiation-induced structural damage in Zr-based metals

Neutron radiation damaging nuclear materials spans from the atomic (nanometer, picosecond) to the macroscopic (meter, year) length and time scales. During the MD simulation, the structural model usually has a small volume not greater than 1 cubic micron [14]. In addition, the structural change caused by ECC usually lasts a short time of the order of the picosecond to nanosecond scale. Considering such small size- and time-scale limitations, it is almost impossible for all of the incident neutrons to collide two PKAs in the structural model during one MD simulation. Therefore, to cause the ECC in structural models one PKA is used. This implies that the kinetic energy value of PKA should be given properly.

Figure 1 shows the energy spectra of PKAs excited by incident neutrons with different kinetic energies in pure Zr metal. The tendency could be observed in all of the curves that PKA counts decrease with the increase of their kinetic energies. It is found that there is an energy cut-off for all of the PKA spectra. (the energy spectra of Zr₂Cu and Zr₂Ni are not shown here, because their energy cut-off values are very similar with those shown in fig. 1). PKA energy also can be estimated from a binary collision approximation based on momentum and energy conservation [31]:

$$\begin{equation} E=\frac{4M_{n} M}{(M_{n} +M)^{2}}E_{n} \cos^{2}\theta, \end{equation} \tag{ 1 }$$

where M_n and M are the masses of the incident neutron and the target atom, respectively. E_n is the energy of the incident neutron, θ is scattering angle of the recoil atom. The maximal kinetic energy of PKA can be obtained when there is a central collision between the neutron and the target atom. Both the maximal energies of PKAs calculated from eq. (1) and those shown in fig. 1 are listed in table 1. It is found that the energy cut-off values in this simulation are 21, 45, 87, 175, 305, 435, and 609 keV, corresponding to incident neutrons with energies of 0.5, 1, 2, 4, 7, 10, and 14 MeV, respectively. These values are similar with their counterparts obtained from eq. (1), indicating the success of the Geant4 simulation.

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In addition, in many previous MD simulations, choosing a proper PKA energy has a great physical significance [14,32,33]. In this work, as shown in fig. 1, compared with PKAs with small energies, those whose energies are relatively large (about hundreds of keV) have a small probability to be excited, while they can cause much greater atomic level structural damages in materials. Therefore, studying structural damages of materials caused by a PKA with relatively large energy via MD simulation also has physical significance. Especially, if we want to probe the 14 MeV neutron-induced structural damage which is a key issue for designing fusion reactors, a relatively large input value of PKA energy in MD simulation should be set.

The energy spectrum of PKAs also can be obtained by using the SPECTER program. However, such information is lacking when the kinetic energy of neutrons is between 6 to 13 MeV or larger than 15 MeV [27]. Fortunately, the program Geant4 can track all the PKAs including those caused by less important reactions when the energy of neutrons ranges from 6 to 13 MeV. As shown in fig. 1, we can provide the energy spectra of PKAs corresponding to the incident neutrons with kinetic energies ranging from 6 to 10 MeV.

Some metallic materials such as Fe-based and Zr-based alloys are widely applied in fission reactors [5,21]. Because these alloys usually contain many constituent elements and crystal phases, some simplified structural models usually are used to investigate the radiation damage via MD simulation. Whether it can approximate the structural damage is a long-standing issue [34]. In this work, the energy spectra of PKAs for pure Zr, Zr₂Cu, Zr₂Ni, and pure Ni are compared with each other, as shown in fig. 2. It is found that the energy spectrum of Zr is similar with those of Zr₂Cu and Zr₂Ni binary alloys. The Zr concentration in both Zr₂Cu and Zr₂Ni is 67 at.%, while Zr-2 and Zr-4 alloys (two typical Zr-based nuclear materials) can be regarded as pure Zr, considering their extremely high Zr concentration (> 98 at.%) [30]. Therefore, we conclude that the energy spectra of Zr-2 and Zr-4 should be very similar to that of pure Zr. In this sense, it is reasonable to study the neutron-induced structural damage in Zr-2 and Zr-4 via the MD simulation by using a structural model of pure Zr.

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In a MD simulation, the PKA's velocity direction usually is defined to be perpendicularly toward the grain boundary, which is simplified as one plane paralleling to a certain crystallographic plane or even to the surface of the structural model [14]. Recently, it was suggested that the distance and the direction between a PKA and a grain boundary obviously affect the damage production [14]. In detail, it is known that the grain boundary is the absorber of interstitials which are moving atoms excited after the ECC initiated by a PKA. The distance and the direction of a PKA to the grain boundary affect the interstitials during a collision cascade, in terms of collision numbers, kinetic energies, and moving paths. In this sense, finding the possible direction of PKA's velocity has important physical significance for the MD simulation.

In the SPECTER program, the angle distribution of PKAs caused by the elastic scattering mode can be obtained, while some other modes such as inelastic collision or nuclear reactions cannot be considered, because in that case the angle distribution of PKA is assumed to be isotropic in the center-of-mass system [27]. Such shortcomings can be avoided by using Geant4. In other words, PKAs caused by all the particle collisions and other nuclear reactions can be tracked in Geant4, and the PKAs angle distribution can be obtained accordingly. In detail, the velocity of the incident neutron is defined to be perpendicularly toward the surface of a structural model, and the scattering angle of PKA is defined to be the angle between the velocity directions of the PKA and the incident neutron. If the value of scattering angle is known, the angle between the PKA's moving direction and the surface of a structural model (and even the grain boundary) is known.

The distributions of scattering angles for PKAs excited by incident neutrons with different kinetic energies are plotted in figs. 3(a)–(h). When the kinetic energy of neutrons is lower than about 2 MeV, only a broad peak could be observed. We notice that the angle distribution ranging from 0° to 90° has higher intensity than that from 90° to 180°. This is because that the former is mainly attributed to relatively strong primary collision between a neutron and a PKA, while the latter is caused by the relatively weak secondary and even tertiary collisions. In particular, the peak value about 70° denotes the most probable scattering angle of a PKA. Furthermore, it is worth noting that the position of this main peak slightly increases with the kinetic energy of neutrons, and when the kinetic energy of neutrons varies from 2 to 11 MeV, other two peaks with angle values about 18° and 52° appear. These two peaks probably are attributed to the PKAs generated from some nuclear reactions such as (n, n') or (n, 2n). This indicates that there are three possible scattering angles of PKAs. However, if the kinetic energy of neutrons is larger than 12 MeV, the peaks in the low-angle region have relatively weak intensities, while there is a strong peak locating at 82°, representing the most probable scattering angle. According to the above analysis, the preferred scattering angles of PKAs can be estimated. This is helpful for optimizing the moving direction of a PKA used in the MD simulation.

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