Low-energy atomic displacement model of SRIM simulations

Low-energy atomic displacement model of SRIM simulations 1 corresp first-author chenshli23@mail.sysu.edu.cn chenshli23@mail.sysu.edu.cn chenshli23@mail.sysu.edu.cn https://orcid.org/0000-0003-0497-1579 Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Radiation-induced atomic displacement damage is a pressing issue for materials. The present work investigates the number of atomic displacements using the Primary Knock-on Atom (PKA) energy E PKA and threshold displacement energy E d as two major parameters via low-energy SRIM Binary Collision Approximation (BCA) full cascade simulations. It is found that the number of atomic displacements cannot be uniquely determined by E PKA / E d or E D / E d ( E D refers to the damage energy) when the energy is comparable with E d . The effective energy E D,eff proposed in the present work allows to describing the number of atomic displacements for most presently studied monatomic materials by the unique variable E D,eff / E d . Nevertheless, it is noteworthy that the BCA simulation damage energy depends on E d , whereas the currently used analytical method is independent of E d . A more accurate analytical damage energy function should be determined by including the dependence on E d . Atomic displacement Damage energy Effective energy SRIM neutron cascade simulations Introduction 1 Radiation damage is an important issue for materials because it changes the properties of materials (e.g., radiation-induced segregation, swelling, hardening, and variation of resistivity [ 1 1 ]). Primary radiation damage, widely known as atomic displacement damage, is fundamental for studying the irradiation effect on materials. The number of atomic Displacements per Atom (DPA) was proposed to unify the damage caused by different irradiation sources, such as neutrons, protons, photons, electrons, positrons, and ions. DPA is now used as an essential parameter for studying the secondary (or macroscopic) radiation damage of materials [ 2 2 ]. In the past decades, various methods and computational codes have been developed to calculate the number of stable atomic displacements (i.e., Frenkel pairs). The Stopping and Range of Ions in Matter (SRIM) Monte Carlo code [ 3 3 ] (or the former TRansport of Ions in Matter (TRIM) code [ 4 4 ] included in) is widely used for computing atomic vacancies because of its ease of use with a user-friendly graphical interface. Presently, it is almost a non-official standard step to use SRIM simulations for ion irradiation studies. Regardless of the discrepancy between the Quick Calculation (QC) and Full-Cascade simulations (FC) [ 5 5 , 6 6 ], SRIM is powerful for modeling ion irradiation in materials. However, it cannot be directly used to compute the displacement damage induced by other radiation sources. To unify the atomic vacancies induced by irradiation with different sources, simulation tools or models using the kinetic energy of the Primary Knock-on Atom (PKA) as a major parameter should be used. Therefore, the so-called neutron cascade simulation has been implemented in the SRIM code [ 3 3 ]. With the neutron cascade option, users identify the position and energy of PKA in a separate file (i.e., TRIM.DAT) to compute the atomic vacancies induced by a PKA rather than an external ion. For irradiation sources other than ions, the PKA spectra can be determined using specific calculation tools and combined with neutron cascade simulations to obtain the number of point defects. Therefore, consistent results can be obtained for the number of atomic vacancies induced by different irradiation sources with SRIM simulations. However, the numbers of atomic vacancies (from vacancies.txt file) obtained using the two basic options, QC and FC, of SRIM-like codes differ by a factor of about 2 [ 5 5 , 6 6 ] (1.0-2.7 depending on the incident ion and ta rget atom [ 7 7 ]). Stoller et al. [ 6 6 ] recommended the use of the QC option for obtaining results comparable with the Norgett-Robinson-Torrens (NRT) [ 8 8 ] model. Recently, Crocombette and Van Wambeke [ 9 9 ] and Weber and Zhang [ 10 10 ] recommended using the FC option, especially for compound materials, because it is physically more reasonable. Weber and Zhang [ 10 10 ] and Chen and Bernard [ 11 11 ] explained that the discrepancy between QC and FC is due to the displacements induced by low-energy atoms, which are considered unable to produce additional displacements in classical models [ 12 12 , 13 13 ] (details are given in Sect. 2.1). Nevertheless, the residual energy transfer effect [ 11 11 ] is not evident for a low initial energy. Consequently, further studies on the atomic displacement model for low PKA energy are crucial for unifying the displacement damage induced by different radiation sources, especially for light particles such as electrons, positrons, and photons. In addition, because the atomic displacements at low energies have an integrated effect on the model at high energies [ 10 10 ], studies on low-energy cascades can reveal the features of the primary radiation damage over the entire energy range. Therefore, the present work investigates the atomic displacements mainly based on SRIM-2013 FC simulations at low PKA energies. The simulation methods are described in Sect. 2.2. The simulated results and corresponding discussion are presented in Sect. 3. Detailed discussion and comments on the use of damage energy for quantifying displacement damage are presented in Sect. 4. The main conclusions of the present study are summarized in Sect. 5. Current atomic displacement models and SRIM simulations 1 Current atomic displacement models 2 The Threshold Displacement Energy (TDE) is defined as the minimum recoil energy required to create a stable point defect. The direction-averaged TDE, denoted by E d in the present study, is widely used in analytical atomic displacement models. Using E d , Kinchin and Pease (KP) got a formula for computing the number of atomic vacancies (denoted by ν in the present work) induced by a PKA with the kinetic energy of E PKA as [ 12 12 ] <math display="block" id="M1"><mrow><msub><mi>ν</mi><mrow><mtext>KP</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo><</mo><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mtext>d</mtext></msub><mo>≤</mo><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo><</mo><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mfrac><mo>,</mo></mrow></mtd><mtd><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub><mo>≤</mo><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo><</mo><msub><mi>E</mi><mtext>c</mtext></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><msub><mi>E</mi><mtext>c</mtext></msub></mrow><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mfrac><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo>></mo><msub><mi>E</mi><mtext>c</mtext></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> It is noteworthy that the cut-off energy E c is not used in more recent models nor in the present work. The main reason for this can be found in Ref. [ 14 14 ]. Considering electronic energy loss and a more realistic atomic collision cross section, Norgett, Robinson, and Torrens (NRT) proposed a modified KP formula based on several Binary Collision Approximation (BCA) calculations [ 8 8 , 15 15 ]: <math display="block" id="M2"><mrow><msub><mi>ν</mi><mrow><mtext>NRT</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>D</mtext></msub></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mo><</mo><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mtext>d</mtext></msub><mo>≤</mo><msub><mi>E</mi><mtext>D</mtext></msub><mo><</mo><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub><mo>/</mo><mn>0.8</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mn>0.8</mn><msub><mi>E</mi><mtext>D</mtext></msub></mrow><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></mfrac><mo>,</mo></mrow></mtd><mtd><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mo>></mo><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub><mo>/</mo><mn>0.8</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> where E d is the effective energy for atomic motion, also called damage energy, first proposed by Lindhard et al. [ 16 16 ]. Figure 1 Figure 1 shows Lindhard’s partition function (i.e., P = E D / E PKA ) with Robinson’s analytical fitting [ 17 17 ] for various monatomic materials. Because some displaced atoms are recombined before reaching thermal equilibrium, the Athermal Recombination-Corrected (ARC) model corrects the NRT model for E d > 2 E d /0.8-2.5 E d [ 13 13 , 18 18 , 19 19 ]. The athermal recombination of displaced atoms cannot be simulated by BCA codes. However, it has a quite limited influence for the low PKA energy, which is the case for the present work; thus, it is not considered here. In the KP and NRT (or NRT-based) models, one can conclude that the effective variables are E PKA / E d and E D / E d , respectively. Therefore, the present study uses the energy normalized by E d as an essential parameter to reduce the number of variables and simplify the comparison among different materials as well as the analysis. To simplify the expressions in the following discussion, let <math id="M3"><mover accent="true"><mi>ν</mi><mo>^</mo></mover></math> denote the number of atomic vacancies using the normalized energy as a unique parameter, i.e., <math id="M4"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo>/</mo><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>ν</mi><mrow><mtext>KP</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow></math> and <math id="M5"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mo>/</mo><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>ν</mi><mrow><mtext>NRT</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>D</mtext></msub></mrow><mo>)</mo></mrow></mrow></math> . SRIM simulations 2 In SRIM-like codes, there are four methods to obtain the number of atomic displacements: number of vacancies directly from BCA simulations (vacancies.txt for SRIM) and the value calculated using the NRT formula with the damage energy from the BCA simulations for both the QC and FC options. Since the FC option is more physically reasonable, the present work is based on FC. Because the method of using damage energy is based on the NRT formula, the direct results from collision simulations should be more reliable. Conversely, Agarwal et al. [ 7 7 ] recently pointed out that the latter should be incorrect according to the details of collisions and recommended using the former method. Their reasoning is absolutely convincing. It is however surprising that the results obtained with the recently developed code Iradina are consistent with those of the SRIM FC [ 9 9 , 20 20 ]. In addition, because the first method using damage energy is slightly different from the NRT model calculations, and the theories behind it are well understood, the present work investigates the number of atomic vacancies from SRIM-2013 FC using the vacancies.txt file, simply referred to as SRIM FC hereinafter. In both the KP and NRT or most other models, it is a common conclusion or assumption that only one atomic vacancy is produced for PKA energy (or damage energy) larger than E d but smaller than ~2 E d . For a high initial energy, the atomic displacements induced by a PKA or an incident self-ion are almost identical [ 11 11 ]. For an initial energy comparable with E d , a PKA is very different from an incident self-ion. Therefore, PKAs, rather than externally incident ions in SRIM simulations, are used in the present work. The original position of the PKAs is set to the center of a 10×10×10 nm 3 (or larger for a few high PKA energies) cube. Because SRIM is a stochastic code, the convergence of the Monte Carlo simulations must be ensured. Figure 2 Figure 2 displays the number of atomic vacancies from the SRIM FC of the neutron cascade for 50 and 80 eV Fe PKAs in pure Fe. The study of the numerical convergence is performed on the grid of 2 n PKAs. One can conclude that 8192 (= 2 13 ) PKAs are reasonable to ensure the convergence of SRIM Monte Carlo simulations; thus, the following studies are based on 8192 PKAs simulations. Different from the assumption that ν =1 for E d ≤ E PKA < 2 E d , the SRIM FC gives ν > 1 for E d < E PKA < 2 E d , which is achievable for atomistic simulations because the TDE is direction-dependent. It is much less evident in SRIM simulations owing to the amorphism of the materials. Nevertheless, this is consistent with the case of Ni studied by Weber and Zhang [ 10 10 ]. Agarwal et al. [ 7 7 ] believed that this is due to the incorrect count of some replacements as displacements. Using SRIM FC, we again compare the displacement damage induced by a PKA and an externally incident ion. Figure 3 Figure 3 plots the number of atomic displacements in Si induced by PKAs and externally incident Si ions (coming from one side of the simulated cube) using the initial kinetic energy and corresponding damage energy as variables. It should be noted that the PKA-induced damage energies used in the present work are the PKA energies after subtracting the ionization energies stored in the IONIZ.TXT SRIM output file. SRIM FC confirms the non-negligible differences between the atomic displacements induced by a PKA and those induced by an incident self-ion when the energies are comparable with E d . Accordingly, the neutron cascade option must be used to study atomic displacements versus PKA energy. The present work is based on selected important monatomic materials because the current analytical formula is valid only for monatomic materials. Moreover, the materials are chosen to cover a wide range of atomic numbers (from Z = 6 to 74). Fe and Ni are widely used in stainless steel, Al is used in many fission reactors [ 21 21 ], C, Cu, and W [ 22 22 ] are used for fusion applications, and Si is a necessary element for semiconductors in various applications [ 23 23 ]. The average TDEs of the studied elements are given in Table 1 Table 1 . All binding energies are set to be 0 to study the analytical atomic displacement models. 1 1 Element 1 1 C 1 1 Al 1 1 Si 1 1 Fe 1 1 Ni 1 1 Cu 1 1 W 1 1 Atomic number 1 1 6 1 1 13 1 1 14 1 1 26 1 1 28 1 1 29 1 1 74 1 1 E d (eV) 1 1 25 [ 24 24 ] 1 1 27 [ 25 25 ] 1 1 36 [ 26 26 ] 1 1 40 [ 27 27 ] 1 1 40 [ 27 27 ] 1 1 30 [ 27 27 ] 1 1 90 [ 27 27 ] Simulation results and discussion 1 Atomic displacements from SRIM FC 2 Figure 4 Figure 4 shows the number of atomic displacements for Fe PKA (i.e., using the neutron cascade option) up to 100 keV in pure Fe by SRIM FC. Both the values from the VACANCY.TXT file and those computed with the NRT formula using the damage energy from the SRM FC are illustrated. The NRT formula is also multiplied by a factor of 2 for an intuitive comparison. The results are quite similar to the case of Ni PKA in Ni shown by Weber and Zhang [ 10 10 ]. For damage energy above 2.5 E d , a typical discrepancy of a factor of about 2 is found between the SRIM FC and NRT calculation. Such discrepancy has been widely recognized and analyzed [ 5 5 , 6 6 , 9 9 – 11 11 9–11 ]; therefore, the present work does not emphasize this point. In the range of E d ≤ E D < 2.5 E d , SRIM FC show that v > 1 and is a strictly increasing function of damage energy, whereas the NRT model implies that v NRT =1. The ratio of SRIM FC to NRT increases from ~1 to ~ 2 when E D increases from ~ E d to 2.5 E d . This region has received much less attention because it is not crucial for the displacement damage induced by reactor neutrons and ions. However, it has a large influence on that induced by light particles (e.g., electrons, positrons, photons) (see Refs. [ 28 28 , 29 29 ] for example). Therefore, the atomic displacements for energy below 2.5 E d are yet to be studied. Atomic displacements for E D < E d 2 As the results shown in Fig. 3 Fig. 3 , atomic displacements are observed when E D < E d but E PKA ≥ E d . This is a direct consequence of the definition of E d : a PKA with E PKA ≥ E d is able to produce one atomic displacement (itself or a replacement) because the energy loss in inelastic collisions occurs after the displacement of PKA [ 29 29 ]. Therefore, for PKA energy comparable with E d , it is questionable to use the damage energy as the effective energy for computing the number of atomic displacements. The author has proposed a simple modification of the NRT (mNRT) model by assuming that v =1 when ( E PKA ≥ E d ) & ( E D < E d )[ 29 29 ]. This modification has a limited influence on the quantification of the primary damage induced by neutron and ions irradiations, of which the contribution of high-energy PKA is predominant. However, its influence on the displacement damage induced by light particles can be considerable [ 29 29 ]. Compared with the original NRT model, the mNRT is more consistent with the SRIM simulations for PKA energy in the range of ( E PKA ≥ E d ) & ( E D < E d ). Nevertheless, they are identical for E D > E d ; thus, the agreement with the SRIM BCA calculations is barely improved in general. Atomic displacements between E d and 2.5 E d 2 Once E PKA is larger than 2 times the minimum TDE, denoted by E d,min hereinafter, it is possible to produce two atomic displacements. Because the TDE is direction dependent, it is possible that ν > 1 for E d < E PKA < 2 E d . In fact, for some materials, E d > 2 E d,min . For iron, Table 2 Table 2 reveals that 9 of the 11 interatomic potentials used in Ref. [ 30 30 ] give E d > 2 E d,min . Because the NRT model and NRT-based models use 2.5 E d as a demarcation energy to ensure continuity, we investigate the point defects for both PKA energy and damage energy in the range of ［ E d , 2.5 E d ］. Because the present work and the SRIM code use only the average TDE, it is simply denoted by TDE hereinafter if without any other statement. 1 1 Potential 1 1 ERG 1 1 ABC 1 1 WOL 1 1 COWP 1 1 MHS 1 1 AMS 1 1 SP-RB 1 1 HA-VD 1 1 HV-TB 1 1 JO-GA 1 1 FS-CB 1 1 E d,min (eV) 1 1 15 1 1 17 1 1 21 1 1 19 1 1 15 1 1 15 1 1 15 1 1 19 1 1 17 1 1 15 1 1 15 1 1 E d (eV) 1 1 40 1 1 45 1 1 41 1 1 54 1 1 37 1 1 39 1 1 42 1 1 33 1 1 46 1 1 37 1 1 39 1 1 E d / E d,min 1 1 2.7 1 1 2.6 1 1 1.9 1 1 2.8 1 1 2.5 1 1 2.6 1 1 2.8 1 1 1.8 1 1 2.7 1 1 2.4 1 1 2.6 Figure 5 Figure 5 plots the numbers of atomic displacements for various monatomic materials with PKA and damage energies between E d and 2 E d . For the sake of simplification, they are respectively denoted by <math id="M6"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> (left plot) and <math id="M7"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub></mrow></math> (right plot) with the variables E PKA / E d and E D / E d . It is obviously confirmed that ν > 1 for E PKA > E d for all monatomic materials. Excluding the material dependence already included in E d (and E D ), <math id="M8"><mover accent="true"><mi>ν</mi><mo>^</mo></mover></math> is additionally dependent on the material. For the seven monatomic materials studied in the present work, Fe, Ni, and Cu follow almost the same law. <math id="M9"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> of C and W are quasi-identical but smaller than that of the other five. <math id="M10"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> of Si is between those of Al, Fe, Ni, and Cu and the ones of C and W. However, <math id="M11"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub></mrow></math> seems to be decreasing with the increasing atomic number of the target. The main reason is that the partition function is larger for heavier atoms [ 14 14 , 16 16 ] ( Fig. 1 Fig. 1 ). It is noticeable that <math id="M12"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub></mrow></math> of C and W are quite different, whereas their <math id="M13"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> are quasi-identical. The discrepancies shown in Fig. 5 Fig. 5 can be attributed to the different materials and different TDEs. Therefore, Figure 6 Figure 6 compares the results of Si with two different values of E d . It is noted that 24 eV is the average threshold energy for a bond defect or a Frenkel pair [ 26 26 ]. This value is comparable with E d =21 eV obtained by Bourgoin et al. [ 31 31 ]. It can be concluded that the value of E d influences both <math id="M16"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> and <math id="M17"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub></mrow></math> , even though they are independent of E d in typical models (cf. Sect. 2.1). In addition, comparing the results of C ( Fig. 5 Fig. 5 ) and Si ( Figs. 5 Figs. 5 and 6 6 ), <math id="M18"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>KP</mtext></mrow></msub></mrow></math> and <math id="M19"><mrow><msub><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mrow><mtext>NRT</mtext></mrow></msub></mrow></math> of Si with E d =24 eV are larger than those of C, of which E d =25 eV. Consequently, <math id="M20"><mrow><mover accent="true"><mi>ν</mi><mo>^</mo></mover><mo>></mo><mn>1</mn></mrow></math> for E PKA > E d and <math id="M21"><mover accent="true"><mi>ν</mi><mo>^</mo></mover></math> depends on both the material and value of E d . Therefore, we cannot obtain a unique simple function of E PKA / E d or E D / E d to describe the number of atomic displacements from the SRIM FC in the range of ( E PKA ≥ E d ) & ( E D < 2.5 E d ). Nevertheless, as the results shown in Figs. 5 Figs. 5 and 6 6 , a linear fitting of the number of atomic displacements vs. PKA energy or damage energy is suitable for each monatomic material. Comments on the effective energy in displacement models 1 Comparing the two plots with the PKA energy shown in Fig. 6 Fig. 6 , E d =24 eV has large number of atomic displacements than E d =36 eV for a given E PKA / E d . For a specific material with a smaller E d , the same E PKA / E d implies a smaller E PKA , so that the partition function is larger, which further implies a larger E D / E d . Therefore, the two different plots versus PKA energy in Fig. 6 Fig. 6 confirm that the damage energy better describes the number of atomic displacements than the PKA energy. However, the results versus damage energy illustrated in Fig. 6 Fig. 6 show that the damage energy is not necessarily better than the PKA energy for determining a simple unique formula for a specific material. In fact, the inelastic energy loss has little influence on the atomic displacements when the kinetic energy is comparable with or even smaller than E d . Therefore, the PKA energy and damage energy are two extreme energies for computing the atomic displacements. New efficient energy should be determined for more accurate calculations. Correcting the damage energy in the displacement calculation 2 Robinson and Oen [ 32 32 ] recognized that the inelastic energy loss for atoms with kinetic energy smaller than 2.5 E d does not influence the number of atomic displacements. Thus, the inelastic energy loss when an atom slows down from 2.5 E d to 0 should be added to the damage energy for computing the atomic displacements [ 32 32 ]. Based on this reasoning, they obtained the effective energy as [ 32 32 ] <math display="block" id="M22"><mrow><msub><mi>E</mi><mrow><mtext>D</mtext><mo>,</mo><mtext>eff</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2.5</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><mn>2.5</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow></mrow></mfrac><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> It is noteworthy that the demarcation of 2.5 E d in the NRT formula is used only to ensure the continuity of the displacement function. According to the reasoning of Kinchin and Pease [ 12 12 ], 2 E d is a physically crucial limit. In addition, an atom does not slow down with continuous energy loss. A collision may decrease the energy of an atom from E 1 > 2 E d to E 2 < 2 E d . Assuming the equiprobable energy distribution (i.e., hard-sphere collision [ 11 11 ]) for an atom slowed down to E < 2 E d for the first time, one can introduce a correction factor by <math display="block" id="M23"><mrow><mi>η</mi><mo>=</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></munderover><mfrac><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mtext>d</mtext><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo>/</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mrow><mn>2</mn><msub><mi>E</mi><mtext>d</mtext></msub></mrow></munderover><mtext>d</mtext><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo>.</mo></mrow></math> Because the partition function can be considered as quasi-constant for the PKA energy from 0 to 2 E d , <math id="M24"><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>≈</mo><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub><mo>×</mo><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow></mrow></math> when 0≤ E PKA ≤2 E d . Therefore, the correction factor can be approximated using <math display="block" id="M25"><mrow><mi>η</mi><mo>≈</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mtext>d</mtext></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></math> Using Lindhard’s analytical partition function for monatomic materials, η can be simply calculated as <math display="block" id="M26"><mrow><mi>η</mi><mo>≈</mo><mn>1</mn><mo>+</mo><mn>0.2161</mn><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>18</mn></mrow></msup><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msubsup><mi>E</mi><mtext>d</mtext><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msubsup><mo>.</mo></mrow></math> For atoms from Li to U with <math id="M27"><mrow><msub><mi>E</mi><mtext>d</mtext></msub></mrow></math> of several tens of eV, η ≈1.2. This value is in good agreement with the experimental values of Fe and Ni summarized in the Nuclear Energy Agency (NEA) report [ 33 33 ], the experimental results of Cu obtained by Averback et al. [ 34 34 ], and many MD simulations for energy around 2 E d . It is noticeable that the correction factor proposed by Robinson and Oen [ 32 32 ] is numerically close to the present one because the partition function varies insignificantly between E d and 2.5 E d . The difference is only a factor of 1.16 in the second term of η . Therefore, η ≈1.2 for both corrections. This value leads to ν NRT ( E D ) ≈ E D /2 E d for E D > 2 E d . One obtains exactly the same formula as the KP formula by replacing of the PKA energy with the damage energy. However, it is noticeable that κ≈0.86 or 0.8 in the formula ν NRT ( E D ) ≈ κE D /2 E d is determined by fitting the BCA calculation results [ 15 15 ]. Therefore, if the effective energy E D,eff = ηE D rather than E D is used, the fitted constant (or widely recognized as the correction to the hard-sphere collision cross section) becomes κ ’= κ / η ≈0.7. Effective energy for SRIM simulations 2 Because Lindhard’s partition function is slightly different from that computed by SRIM, the present correction factor for damage energy is calculated with E D ( E d ) from SRIM FC and denoted by η SRIM . The effective energy is computed as follows: <math display="block" id="M28"><mrow><msub><mi>E</mi><mrow><mtext>D</mtext><mo>,</mo><mtext>eff</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>η</mi><mrow><mtext>SRIM</mtext></mrow></msub><msub><mi>E</mi><mtext>D</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>PKA</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> The results shown in Fig. 5 Fig. 5 are rescaled by η SRIM and are illustrated in Fig. 7 Fig. 7 . Excluding W, the six monatomic materials considered in the present work (from C to Cu) almost follow the same law, which uses E D,eff / E d as a unique variable. Therefore, this effective energy provides the possibility to correct the current formulae and describe the results of SRIM FC. Moreover, the use of η satisfies ν ( E PKA = E d )=1, which is consistent with the definition of E d . The fitted line in Fig. 7 Fig. 7 indicates that ν SRIM ≈0.7 ( E D,eff / E d -1)+1. Assuming its applicability up to high energies, once <math id="M29"><mrow><mi>v</mi><mo>≫</mo><mtext>0</mtext><mtext>.3</mtext></mrow></math> , ν SRIM ≈0.7 E D,eff / E d =0.7 η SRIM / E D / E d . For most cases where η SRIM ≈1.1 or 1.2, ν SRIM ≈0.8 E D / E d =2ν NRT . This relationship explains the typical discrepancy of a factor of about 2 between the SRIM FC and NRT formula. However, it should be noted that the number of atomic displacements in W still differs from the others. Moreover, the use of η SRIM cannot make the two curves of Si in Fig. 6 Fig. 6 coincide. In fact, it is important to indicate that the damage energy from the SRIM FC depends on the value of E d (e.g., the example on Si shown in Fig. 8 Fig. 8 ), whereas Lindhard’s damage energy is independent of E d [ 16 16 ]. Using the data for E d =36 eV as a reference, we rescale the effective energy for E d =24 eV by a factor of 1.1 to get the same damage energy function. The rescaled data are plotted in Figure 9 Figure 9 together with the data versus the damage energy and effective energy. Rescaling the effective energy to eliminate the bias induced by E d results in similar atomic displacements for Si with E d =24 eV and E d =36 eV. From the case of Si shown above, one can find that none of E D,eff / E d , E D / E d , and E PKA / E d can be the unique variable for describing the number of atomic displacements from the SRIM FC. Therefore, the number of atomic displacements versus E PKA / E d , E D / E d , or E PKA / E d for two arbitrary monatomic materials are not necessarily the same. The only general conclusion is that linear fitting can be used to describe the number of atomic displacements versus E D,eff , E D , and E PKA for energy comparable with E d . One can also adopt a specific value of E d for W to reduce the difference with the other materials, as shown in Fig. 7 Fig. 7 . The difference decreases but still exists using of E d = 55 eV [ 35 35 , 36 36 ] rather than 90 eV. The difference can be further decreased by decreasing E d ; however, using an unphysical value for E d is unnecessary. Further comments on the damage energy 2 Figure 8 Figure 8 shows that the damage energy from SRIM FC depends on the value of E d , whereas the Lindhard’s damage energy is independent of E d . However, these are not physically incompatible. In fact, the original equation governing the damage energy (or atomic vacancies) includes E d as a basic parameter [ 16 16 ]. The currently used Lindhard’s damage energy is TDE-independent because it is obtained according to the numerical solutions after removing E d in the original equation (i.e., their approximation (B): E d is negligeable when compared with kinetic energies of atoms [ 16 16 ]). Table 3 Table 3 summarizes the damage energies and numbers of atomic displacements for Fe PKA in Fe from SRIM FC using two different values for E d : 40 eV and 20 eV. The results show that the damage energy depends on E d for PKA energies up to 100 keV. For a given PKA energy, the damage energy is larger for a larger E d . This is a consequence of knocked-on atoms having a smaller kinetic energy for larger E d . A lower kinetic energy results in lower inelastic energy losses in subsequent collisions. Therefore, once E d is changed, the corresponding number of atomic displacements cannot be directly predicted as the inverse proportion to E d . Taking the Fe PKA in Fe shown in Table 3 Table 3 as an example, the number of atomic displacements is reduced by a factor greater than 2 (the last column in Table 3 Table 3 ) if E d is doubled (20 eV → 40 eV), whereas the NRT model predicts a reduction of a factor of 2 (or smaller than 2 if the slight increase in damage energy is considered). This confirms the conclusion given in Sect. 4.2: E D / E d or E D,eff / E d cannot be the unique variable for computing the number of atomic displacements. A variation of E d by a factor of x does not imply a variation of a factor of 1/ x for the number of atomic displacements. 1 1 PKA energy (keV) 3 1 Damage energy (keV) 3 1 Number of atomic displacements 1 1   1 1 (a) 1 1 (b) 1 1 (a)/(b) 1 1 (a) 1 1 (b) 1 1 (a)/(b)×2 1 1 0.1 1 1 0.0862 1 1 0.0819 1 1 1.053 1 1 1.95 1 1 4.09 1 1 0.952 1 1 1 1 1 0.8408 1 1 0.7927 1 1 1.061 1 1 16.2 1 1 37.1 1 1 0.875 1 1 10 1 1 7.941 1 1 7.466 1 1 1.064 1 1 149 1 1 343 1 1 0.871 1 1 100 1 1 70.65 1 1 66.35 1 1 1.065 1 1 1319 1 1 3040 1 1 0.867 Conclusion 1 In SRIM FC, the number of atomic displacements v > 1 and cannot be uniquely described by E D / E d or E PKA / E d for PKA energy from E d to a few times of E d . Because a part of the inelastic energy loss (when the kinetic energy is smaller than ~ 2 E d ) does not influence the number of atomic displacements, an effective energy E D,eff = ηE D is proposed. An approximate value of η ≈1.2 is obtained for both the present proposal and that of Robinson and Oen [ 32 32 ]. This value is consistent with the experimental data for Fe, Ni, and Cu for damage energies of about 2 E d . Using η SRIM = E d / E D ( E d ), six of the seven monatomic materials considered in the present work have the same number of atomic displacements as a function of E D,eff / E d . However, further investigation shows that the damage energy depends on E d , whereas the currently used analytical damage energy is independent of E d . For a given PKA energy, the damage energy is larger for a larger E d . Consequently, none of E D,eff / E d , E D / E d , and E PKA / E d can be the unique variable for describing the number of atomic displacements. 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Matter. 26 ( 37 ), 375701 ( 2014 ). doi: 10.1088/0953-8984/26/37/375701 http://doi.org/10.1088/0953-8984/26/37/375701 Elastic trapping of dislocation loops in cascades in ion-irradiated tungsten foils 10.1007/s41365-021-00971-2 This work was supported by the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (No. 2021qntd12) . 2021-08-28 2021-10-04 2021-10-11 2021-11-16 2021-11-01 2021-12-06 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00971-2 http://dx.doi.org/10.1007/s41365-021-00971-2 . 2021 Nuclear Science and Techniques 11 32