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The computation of displacement damage cross sections of silicon, carbon and silicon carbide for high energy applications
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 16501–16508
www.materialstoday.com/proceedings
SCICON 2016
The computation of displacement damage cross sections of silicon, carbon and silicon carbide for high energy applications Uttiyoarnab Sahaa, K. Devana,* a
Reactor Neutronics Division, Reactor Design Group, Indira Gandhi Centre for Atomic Research, Homi Bhabha National Institute, Kalpakkam – 603102, India
Abstract The polyatomic material SiC is being considered as a potential structural constituent for applications in advanced high temperature nuclear reactors. The radiation damage in these materials due to neutron irradiation requires to be assessed accurately to predict their residence time in these reactors. It is generally quantified by a parameter called displacement per atom (dpa). Knowledge of dpa cross section is essential to compute its dpa rate in a known neutron flux field. Nowadays methods like Monte Carlo and Molecular Dynamics are used to compute dpa cross sections. In fast reactors, the dpa model given by Norgett, Robinson and Torrens (NRT-dpa) is used for design studies. In this paper, we discuss the methodology of computing dpa cross sections of SiC by using the NRT model based dpa cross sections of constituent elements (Si and C) from a more recent evaluated nuclear data library called ENDF/B-VII.1. The contributions of all important partial reactions including radiative capture are considered for the calculation of NRT-dpa by developing an indigenous code. Our results are compared with the published values. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advanced Materials (SCICON ‘16). Keywords: dpa cross section; damage efficiency; recoil; PKA; lattice displacement energy; sub-lattice interaction
1. Introduction Silicon carbide has gained notable importance over the past decade for its potential applications in high temperature and radiation environments. Some of the interesting features of this material like chemical inertness, good resistance to corrosion, stability under neutron irradiation and low neutron capture cross section make it
* Corresponding author. Tel.: +91 44 274 80088; fax: +91 44 274 80088. E-mail address: [email protected]; [email protected] 2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advanced Materials (SCICON ‘16).
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suitable for applications in nuclear systems [1]. Also the proposal for SiC composite as structural component in fusion reactor systems can find an economic benefit in its low inherent induced radioactivity to neutron irradiation (less generation of potential waste) and low decay heat production [2, 3]. The low permeability of tritium for SiC is explained in [4]. SiC, a wide band gap (~3 eV) semiconductor with high breakdown field and thermal conductivity can find application in high performance electronics required in high temperature and radiation units [5,6]. The advanced high temperature nuclear reactors and fusion applications demand materials of its type. In spite of having so many positive sides, SiC shows few behavior of concern on the other way too. The formation of hydrogen and helium in fusion environment is seen to be prominent in SiC. Although further investigations of the effects of hydrogen and helium are needed, discussions of different experiments and observations can be found in [7-10]. Long term operation in the radiation environment introduces defects in the atomic scale of the material by displacing the lattice atoms. Ultimately the behavior of these defects in non equilibrium conditions determines the performance of the material. Many experiments and simulation efforts have been put into understanding the production and evolution of defects in SiC [11–20]. The properties of the material after irradiation have been observed carefully by a number of workers [21, 22]. One of the most important parameters to estimate radiation damage due to defect production is the threshold atom displacement energy Ed. It is the minimum energy to displace a lattice atom and create a stable Frenkel pair. The Ed values in SiC have been estimated by experiments and simulations mentioned in the above references. These values have range as broad as 20 - 50 eV for C and 35 - 110 eV for Si. It is mainly due to uncertainty in the sub-lattice interaction considered in the measurement of Ed and basic differences in the experimental methods [11]. Moreover, there are more than 150 known polytypes of SiC which increases the difficulty of analyzing the data from an experiment. The suggested minimum Ed value for C is 21 eV [11] or 20 eV [14] and for Si it is 35 eV [11, 14]. In the displacement damage phenomena, the recoil energy of the primary knock on atom (PKA) is dissipated into electronic excitation and displacements of atoms in the lattice. The PKA generates further atom displacements through atomic collisions with the lattice atoms. This latter part of PKA energy is considered in the damage energy calculation from an incident neutron. This partitioning of the PKA energy is usually done by the Robinson partition function [23] and the number of stable Frenkel pairs generated is calculated from analytical expression of NRT model by Norgett et al. in [24]. It is based on the electronic and nuclear stopping distribution given by Lindhard et al. in [25, 26]. The total damage due to atom displacement in a neutron irradiated material is found by integrating the neutron displacement cross sections over the energy spectrum of the neutron irradiation source. For a polyatomic material the displacement function is determined by numerically solving the coupled integro–differential equations given by Parkin and Coulter [27–30]. These equations give the total number of atoms displaced. They do not give the spatial distribution of these atoms. To simulate transport, scattering and slowing down of energetic particles in matter, the Monte Carlo code TRIM (Transport of Ions in Matter) [31–33] is widely used. It can calculate the full cascade of the recoil and total displacements generated by the PKA. Radiation damage in metals is generally quantified by displacement per atom (dpa), meaning average number of times an atom is displaced from its lattice site during the period of irradiation. The dpa due to neutron irradiation is computed from the calculated neutron dpa cross section and the measured neutron flux. For compound materials the dpa cross sections can be obtained from computer code SPECOMP [34]. As in the case of pure elements the primary recoil spectra in a compound material can be estimated from the interaction between neutron and individual atoms. But the generation of secondary atom displacements involves the consideration of the different sub-lattices of the separate elements. Again the lattice displacement energy of the element differs in a compound. Therefore, while considering the secondary interactions among different sub-lattices the corresponding lattice displacement energy needs to be used. The aim of the present work is to compute the dpa cross sections of SiC from Si and C dpa cross sections computed from the neutron interaction data given in the ENDF/B-VII.1 [35] (processed for 0K using PREPRO 2015). Here we use an indigenously developed computer code to compute the dpa cross sections from the NRT model. All the important partial neutron reactions (elastic: MT=2; inelastic: MT=51-91; (n, 2n): MT=16; (n, n): MT=22; (n, np): MT=28; (n, γ): MT=102; (n, p): MT=103; (n, d): MT=104; (n, α): MT=107) along with their anisotropy effects [36] are included in the calculations. Further the effect of using different Ed values for Si and C is seen in the dpa cross section of SiC. Finally, the results obtained by us are compared with the available data.
Uttiyoarnab Saha and K. Devan / Materials Today: Proceedings 5 (2018) 16501–16508
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2. Computation of dpa cross section of SiC The general formalism to compute dpa cross section will be discussed in brief in the Appendix A. The interaction of an energetic neutron with one of the atoms in SiC is probabilistic in nature and depends on the relative interaction cross sections of the nuclei with neutron of a particular energy. For simplicity and understanding purpose, here we assume that a neutron incident on SiC target can hit either a Si atom or a C atom with 50% probabilities. So there are 50% chances for the generation of a Si PKA and C PKA. Although further collisions of the PKA depend on the in– depth picture of the lattice and full collision cascade, let us assume that either of these two PKAs has again 50% chance to hit any one type of the constituent atoms. Thus for a projectile – target combination there is 25% possibility for each of these following pairs: Si – Si, Si – C, C – C, C – Si. It is illustrated in the Fig. 1 given below.
Fig. 1. Possible interaction pairs in the displacement damage event. The chance of each interaction denoted by labels a1, a2, etc. is 50% and that of a combined event a1c1or a2b2, etc. is 25% each.
Hence, we can give an analytical expression for the dpa cross section of SiC (containing two atoms in the molecule) in Eq. (2.1).
σD (SiC) (E) 1 0.25σD (Si / Si) (E) 0.25σD (Si / C) (E) 0.25σD (C / Si) (E) 0.25σD (C / C) (E)
(2.1)
2 A term like σ D ( A / B) ( E ) in this equation means the dpa cross section of type A atom hit by the incident neutron of
energy E, when the secondary collision of the type A atom occurs with an atom of type B. While considering the interactions among sub-lattices to find the damage energy, the values of Z1, Z2, A1 and A2 in Eq. (A.2) are changed accordingly. The total dpa cross section of Si is found from the individual isotopes with their isotopic abundances. Data for natural C is available in ENDF/B-VII.1. Three cases have been carried out to compute the dpa cross section of the compound SiC from Eq. (2.1), which are discussed in the following sub-sections. The reference case is done with the Ed values for Si and C given by Devanathan et al. in [14] and a similar method as given by Heinisch et al. in [37] is followed for the different sub-lattice interactions. In the other two cases dpa cross sections are computed with Ed values different from the reference case in at least one of the sub-lattice interactions. 2.1. Reference case In SiC, Si and C PKA can be formed with a neutron of energy at least 35 eV and 20 eV respectively. The maximum energy that can be transferred from a Si atom to C atom and vice versa is about 84% (4MSiMC/(MSi+MC)2) of the projectile energy. Hence, the effective displacement energy for the projectile/ target pairs will become: Ed(Si/Si) = 35 eV, Ed(Si/C) = 24 eV, Ed(C/C) = 20 eV, Ed(C/Si) = 42 eV. The net numbers of displacements of the species are calculated with the corresponding Ed values.
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2.2. Case 1 Consider the lattice displacement energies of Si and C in their element forms as 25 eV and 31 eV respectively. Then we compute the dpa cross section of SiC assuming that a Si atom can be displaced from its lattice site either by a neutron or a C atom or a Si atom with 25 eV energy and that a C atom can be displaced by any one of the species with energy 31 eV. Here Ed(Si/Si) = 25 eV, Ed(Si/C) = 31 eV, Ed(C/C) = 31 eV, Ed(C/Si) = 25 eV. 2.3. Case 2 Consider the values of Ed for Si and C in their compound form (SiC) as 35 eV and 20 eV respectively. Then as in case 1, we assume that any projectile with energy 35 eV can displace a Si atom and that with energy 20 eV can displace a C atom from their lattice sites in the compound. Here Ed(Si/Si) = 35 eV, Ed(Si/C) = 20 eV, Ed(C/C) = 20 eV, Ed(C/Si) = 35 eV. Fig. 2 shows the profile of the computed cross sections in the reference case. The cross sections obtained in all the cases along with the percentage differences of values in cases 1 and 2 with respect to the reference case are shown in Fig. 3.
Fig. 2. The dpa cross sections as obtained in the reference case for the four interaction pairs and for SiC.
In Fig. 2, σD(Si/Si) is obtained by adding the dpa cross section of all Si isotopes with their abundances. While considering one isotope, its basic neutron interaction cross sections are taken. In this case Z1, Z2, A1, A2 correspond to that of the Si isotope and Ed = 35eV. σD(C/C) is also obtained in the same way from the data given for natural C, with Ed = 20eV. For σD(Si/C), the basic data for all silicon isotopes along with corresponding Z1 and A1 are used, but Z2 and A2 used in each case are that of natural C. Then contributions from all isotopes of Si are added with abundances. The threshold energy to form a Si PKA is taken as 35eV and Ed = 24eV for further displacements of C atom by the PKA. The reverse is done for σD(C/Si), with neutron data and Z1, A1 for natural C and Z2, A2 that of natural Si. Threshold to form C PKA is taken as 20eV and Ed = 42eV for further displacements of Si atom by the PKA. Then σD(SiC) is obtained using Eq. (2.1). From the percentage differences in Fig. 3, it is seen that on an average the values in case 2 differs from that in reference case by about 12%. In case 1, results differ from reference case by about 2% at most of the energy points and within 10% at high energies. The elastic scattering contribution to the dpa cross section in SiC effectively starts around 100 eV. The Ed values used in case 1 result into a comparatively higher effective value of this threshold energy in SiC (first plot in Fig. 3). Due to this, a large percentage deviation (maximum value of 95%) is observed around 100 eV for case 1 with respect to the reference case.
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Fig. 3. The dpa cross sections of SiC in three cases and percentage differences of values in cases 1 and 2 from the reference case.
3. Comparison with available data The dpa cross section of SiC has been computed by H.L. Heinisch et al. in [37]. Here the SPECOMP code has been used to arrive at the cross sections. Here the displacement functions for polyatomic materials, νij(E) defining the number of displacements of atom type j started by the PKA of atom type i, is obtained from the solutions of Parkin–Coulter integro–differential equations for νij(E). The lattice displacement energies used for pair-wise interaction between the two atom species in the compound are similar as done in the reference case of the present work. The total dpa cross section of SiC is given in 100 energy group structure. In a more recent work of Chang et al. in [38] dpa cross section of SiC has been computed by using combined NJOY/TRIM model. They have treated the Si in SiC as 28Si and calculated the cross sections at 300K. They have given the dpa cross section of SiC (their own and Heinisch et al.) in 47 energy groups.
Fig. 4. Comparison of dpa cross sections of SiC between our study and that of Heinisch et al. (in linear scale).
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We compare the results obtained in the reference case with these data. Since the present results are not applied to any particular reactor system, a flat neutron spectrum is considered for group averaging. The 100 and 47 group dpa cross sections are shown in Figs. 4 and 5 respectively. The dpa cross sections of SiC obtained purely from the NRT model in the present work are smaller than that obtained in the other two works (‘reference case’ curves in Figs. 4 and 5). However, if we ignore the NRT damage efficiency factor 0.8 in our calculation by dividing reference case values by 0.8, other details remaining same, then we get relatively higher values of cross section (‘(1/0.8) * reference case curves’ in Figs. 4 and 5). As can be seen in Figs. 4 and 5, the damage efficiency factors for the net number of displacements effectively going in their estimations may be greater than 0.8.
Fig. 5. Comparison of dpa cross sections of SiC with the results of Heinisch et al. and Chang et al. (in linear scales).
4. Conclusions The dpa cross section of the polyatomic material SiC is computed from the individual dpa cross sections of its constituent elements. The dpa cross sections of Si and C are computed including all the important partial neutron interaction data along with their anisotropy from ENDF/B-VII.1 library by developing an indigenous code based on the NRT model. Three cases are investigated with various known Ed values for the elements in the elemental and compound forms and the results are compared. The dpa cross section of SiC obtained by using the neutron-nucleus interactions and the subsequent energy partitioning of the PKA by the NRT formalism is compared with the published results of Heinisch et al. and Chang et al. The deviations observed here are similar to that reported by Chang et al. in their work. Considering the differences in the procedures followed, the agreements between these results are quite satisfactory. However, further investigations are necessary to establish a standard procedure for computing dpa cross sections in compound materials. Appendix A. General formalism to compute dpa cross section from NRT model The parameter dpa characterizing neutron irradiation is evaluated from the knowledge of dpa cross section and neutron flux as follows:
Uttiyoarnab Saha and K. Devan / Materials Today: Proceedings 5 (2018) 16501–16508
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E max
DPA t
dE ( E )
total D
(E )
E min
total where D ( E )
i D (E)
is calculated by taking the contribution from all partial neutron – nucleus interactions i.
i
φ is the neutron flux and t is the irradiation time. The total dpa cross section is computed from
total D
(E )
E R max i
i
(E )
dE
R
K i ( E , E R ) [ T ( E R )]
(A.1)
0
The function Ki (E, ER) is the kernel of energy transfer ER through a nuclear reaction of type i with cross section σi(E). ν[T(ER)] is the displacement damage function. Damage energy T(ER) is found from Robinson partition function, Eq. (A.2) and the number of displacement per atom is found from the NRT displacement damage model function ν(T) in Eq. (A.3).
ER 1 FG 0.0793 Z12/3Z 21/2 (A1 A2 )3/2
T (ER ) F
Z
2/3 1
Z 22/3
3/4
A 13/ 2A1/2 2
G 3.4008 1/6 0.40244 3/4 ER 1/2 2/3 30.724 Z1Z 2 Z1 Z 22/3 A1 A2 / A2
(A.2)
Z and A with suffixes 1 and 2 denote the atomic and mass numbers of the recoil and lattice nuclei respectively.
0 ; (T ) 1; 0.8T / 2E ; d
T Ed Ed T 2Ed T 2Ed
References [1] R.H. Jones, L.L. Snead, A. Kohyama, P. Fenici, Fusion Eng. Des. 41 (1998) 15 – 24. [2] H.L. Heinisch, Fusion Mater. DOE/ER-0313/18 (1995) 91 – 96. [3] T. Noda, M. Fujita, H. Araki, A. Kohyama, Fusion Eng. Des. 61-62 (2002) 711 – 716. [4] R.A. Causey, W.R. Wampler, J. Nucl. Mater. 823 (1995) 220 – 222. [5] W. Wesch, Nucl. Instrum. and Meth. B 116 (1996) 305 – 321. [6] W.J. Choyke, G. Pensl, MRS Bull. 22 (1997) 25 – 29. [7] Y. Katoh, H. Kishimoto, A. Kohyama, J. Nucl. Mater. 307 (2002) 1221 – 1226. [8] S. Kondo, T. Hinoki, A. Kohyama, Mater. Trans. 46 (2005) 1923 – 1927. [9] T. Taguchi, N. Igawa, S. Miwa, E. Wakai, S. Jitsukawa, L.L.Snead, A. Hasegawa, J. Nucl. Mater. 335 (2004) 508 – 514. [10] S. Kondo, K.H. Park, Y. Katoh, A. Kohyama, Fusion Sci. Tech. 44 (2003) 181 – 185. [11] R. Devanathan, W.J. Weber, J. Nucl. Mater. 278 (2000) 258 – 265. [12] B. Hudson, B.E. Sheldon, J. Microsc. 97 (1973) 113 – 119. [13] I.A. Honstvet, R.E. Smallman, P.M. Marquis, Philos. Mag. A 41 (1980) 201 – 207. [14] R. Devanathan, W.J. Weber, F. Gao, J. Appl. Phys. 90 (2001) 2303 – 2309. [15] R. Devanathan, T. Diaz de la Rubia, W.J. Weber, J. Nucl. Mater. 253 (1998) 47 – 52. [16] H. Inui, H. Mori, H. Fujita, Philos. Mag. B 61 (1990) 107 – 124. [17] A.L. Barry, B. Lehmann, D. Fritsch, D. Braunig, IEEE Trans. Nucl. Sci. 38 (1991) 1111 – 1115. [18] I.I. Geiczy, A.A. Nestorov, L.S. Smirnov, Radiat. Eff. 9 (1971) 243 – 246.
(A.3)
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[19] R.R. Hart, H.L. Dunlap, O.J. Marsh, Radiat. Eff. 9 (1971) 261 – 266. [20] H. Inui, H. Mori, and T. Sakata, Philos. Mag. B 66 (1992) 737 – 748. [21] G. Newsome, L.L. Snead, T. Hinoki, Y. Katoh, D. Peters, J. Nucl. Mater. 371 (2007) 76 – 89. [22] Y. Katoh, L.L. Snead, C.H. Henager Jr., A. Hasegawa, A. Kohyama, B. Riccardi, H. Hegeman, J. Nucl. Mater. 367 – 370 (2007) 659 – 671. [23] M.T. Robinson, I.M. Torrens, Phys. Rev. B 9 (1974) 5008 – 5024. [24] M.J. Norgett, M.T. Robinson, I.M. Torrens, Nucl. Eng. Des. 33 (1975) 50 – 54. [25] J. Lindhard, M. Scharff, H. E. Schiøtt, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (14) (1963) 1 – 42. [26] J. Lindhard, V. Nielsen, M. Scharff, P.V. Thomsen, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (10) (1963) 1 – 42. [27] D.M. Parkin, C.A. Coulter, J. Nucl. Mater. 85 – 86 (1979) 611 – 615. [28] D.M. Parkin, C.A. Coulter, J. Nucl. Mater. 101 (1981) 261 – 276. [29] D.M. Parkin, C.A. Coulter, J. Nucl. Mater. 103 – 104 (1981) 1315 – 1318. [30] D.M. Parkin, C.A. Coulter, J. Nucl. Mater. 117 (1983) 340 – 344. [31] J.P. Biersack, L.G. Haggmark, Nucl. Instrum. Methods 174 (1980) 257 – 269. [32] J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Range of Ions in Solid; Pergamon Press: New York, 1985. [33] J.F. Ziegler, In: Handbook of Ion Implantation Technology; J.F. Ziegler, Ed.; North Holland: Amsterdam, 1992; pp 1 – 68. [34] L.R. Greenwood, J. Nucl. Mater. 216 (1994) 29 – 44. [35] M. Herman, A. Trkov, Ed.; ENDF-6 Formats Manual, available at: https://www.oecd-nea.org/dbdata/data/manual-endf/endf102.pdf [36] Uttiyoarnab Saha, K. Devan, Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 61 (2016) 644 – 645. [37] H.L. Heinisch, L.R. Greenwood, W.J. Weber, R.E. Williford, J. Nucl. Mater. 327 (2004) 175 – 181. [38] Jonghwa Chang, Jin-Young Cho, Choong-Sup Gil, Won-Jae Lee, Nucl. Eng. Tech. 46, 4 (2014) 475 – 480.
ScienceDirect Materials Today: Proceedings 5 (2018) 16501–16508
www.materialstoday.com/proceedings
SCICON 2016
The computation of displacement damage cross sections of silicon, carbon and silicon carbide for high energy applications Uttiyoarnab Sahaa, K. Devana,* a
Reactor Neutronics Division, Reactor Design Group, Indira Gandhi Centre for Atomic Research, Homi Bhabha National Institute, Kalpakkam – 603102, India
Abstract The polyatomic material SiC is being considered as a potential structural constituent for applications in advanced high temperature nuclear reactors. The radiation damage in these materials due to neutron irradiation requires to be assessed accurately to predict their residence time in these reactors. It is generally quantified by a parameter called displacement per atom (dpa). Knowledge of dpa cross section is essential to compute its dpa rate in a known neutron flux field. Nowadays methods like Monte Carlo and Molecular Dynamics are used to compute dpa cross sections. In fast reactors, the dpa model given by Norgett, Robinson and Torrens (NRT-dpa) is used for design studies. In this paper, we discuss the methodology of computing dpa cross sections of SiC by using the NRT model based dpa cross sections of constituent elements (Si and C) from a more recent evaluated nuclear data library called ENDF/B-VII.1. The contributions of all important partial reactions including radiative capture are considered for the calculation of NRT-dpa by developing an indigenous code. Our results are compared with the published values. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advanced Materials (SCICON ‘16). Keywords: dpa cross section; damage efficiency; recoil; PKA; lattice displacement energy; sub-lattice interaction
1. Introduction Silicon carbide has gained notable importance over the past decade for its potential applications in high temperature and radiation environments. Some of the interesting features of this material like chemical inertness, good resistance to corrosion, stability under neutron irradiation and low neutron capture cross section make it
* Corresponding author. Tel.: +91 44 274 80088; fax: +91 44 274 80088. E-mail address: [email protected]; [email protected] 2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advanced Materials (SCICON ‘16).
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suitable for applications in nuclear systems [1]. Also the proposal for SiC composite as structural component in fusion reactor systems can find an economic benefit in its low inherent induced radioactivity to neutron irradiation (less generation of potential waste) and low decay heat production [2, 3]. The low permeability of tritium for SiC is explained in [4]. SiC, a wide band gap (~3 eV) semiconductor with high breakdown field and thermal conductivity can find application in high performance electronics required in high temperature and radiation units [5,6]. The advanced high temperature nuclear reactors and fusion applications demand materials of its type. In spite of having so many positive sides, SiC shows few behavior of concern on the other way too. The formation of hydrogen and helium in fusion environment is seen to be prominent in SiC. Although further investigations of the effects of hydrogen and helium are needed, discussions of different experiments and observations can be found in [7-10]. Long term operation in the radiation environment introduces defects in the atomic scale of the material by displacing the lattice atoms. Ultimately the behavior of these defects in non equilibrium conditions determines the performance of the material. Many experiments and simulation efforts have been put into understanding the production and evolution of defects in SiC [11–20]. The properties of the material after irradiation have been observed carefully by a number of workers [21, 22]. One of the most important parameters to estimate radiation damage due to defect production is the threshold atom displacement energy Ed. It is the minimum energy to displace a lattice atom and create a stable Frenkel pair. The Ed values in SiC have been estimated by experiments and simulations mentioned in the above references. These values have range as broad as 20 - 50 eV for C and 35 - 110 eV for Si. It is mainly due to uncertainty in the sub-lattice interaction considered in the measurement of Ed and basic differences in the experimental methods [11]. Moreover, there are more than 150 known polytypes of SiC which increases the difficulty of analyzing the data from an experiment. The suggested minimum Ed value for C is 21 eV [11] or 20 eV [14] and for Si it is 35 eV [11, 14]. In the displacement damage phenomena, the recoil energy of the primary knock on atom (PKA) is dissipated into electronic excitation and displacements of atoms in the lattice. The PKA generates further atom displacements through atomic collisions with the lattice atoms. This latter part of PKA energy is considered in the damage energy calculation from an incident neutron. This partitioning of the PKA energy is usually done by the Robinson partition function [23] and the number of stable Frenkel pairs generated is calculated from analytical expression of NRT model by Norgett et al. in [24]. It is based on the electronic and nuclear stopping distribution given by Lindhard et al. in [25, 26]. The total damage due to atom displacement in a neutron irradiated material is found by integrating the neutron displacement cross sections over the energy spectrum of the neutron irradiation source. For a polyatomic material the displacement function is determined by numerically solving the coupled integro–differential equations given by Parkin and Coulter [27–30]. These equations give the total number of atoms displaced. They do not give the spatial distribution of these atoms. To simulate transport, scattering and slowing down of energetic particles in matter, the Monte Carlo code TRIM (Transport of Ions in Matter) [31–33] is widely used. It can calculate the full cascade of the recoil and total displacements generated by the PKA. Radiation damage in metals is generally quantified by displacement per atom (dpa), meaning average number of times an atom is displaced from its lattice site during the period of irradiation. The dpa due to neutron irradiation is computed from the calculated neutron dpa cross section and the measured neutron flux. For compound materials the dpa cross sections can be obtained from computer code SPECOMP [34]. As in the case of pure elements the primary recoil spectra in a compound material can be estimated from the interaction between neutron and individual atoms. But the generation of secondary atom displacements involves the consideration of the different sub-lattices of the separate elements. Again the lattice displacement energy of the element differs in a compound. Therefore, while considering the secondary interactions among different sub-lattices the corresponding lattice displacement energy needs to be used. The aim of the present work is to compute the dpa cross sections of SiC from Si and C dpa cross sections computed from the neutron interaction data given in the ENDF/B-VII.1 [35] (processed for 0K using PREPRO 2015). Here we use an indigenously developed computer code to compute the dpa cross sections from the NRT model. All the important partial neutron reactions (elastic: MT=2; inelastic: MT=51-91; (n, 2n): MT=16; (n, n): MT=22; (n, np): MT=28; (n, γ): MT=102; (n, p): MT=103; (n, d): MT=104; (n, α): MT=107) along with their anisotropy effects [36] are included in the calculations. Further the effect of using different Ed values for Si and C is seen in the dpa cross section of SiC. Finally, the results obtained by us are compared with the available data.
Uttiyoarnab Saha and K. Devan / Materials Today: Proceedings 5 (2018) 16501–16508
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2. Computation of dpa cross section of SiC The general formalism to compute dpa cross section will be discussed in brief in the Appendix A. The interaction of an energetic neutron with one of the atoms in SiC is probabilistic in nature and depends on the relative interaction cross sections of the nuclei with neutron of a particular energy. For simplicity and understanding purpose, here we assume that a neutron incident on SiC target can hit either a Si atom or a C atom with 50% probabilities. So there are 50% chances for the generation of a Si PKA and C PKA. Although further collisions of the PKA depend on the in– depth picture of the lattice and full collision cascade, let us assume that either of these two PKAs has again 50% chance to hit any one type of the constituent atoms. Thus for a projectile – target combination there is 25% possibility for each of these following pairs: Si – Si, Si – C, C – C, C – Si. It is illustrated in the Fig. 1 given below.
Fig. 1. Possible interaction pairs in the displacement damage event. The chance of each interaction denoted by labels a1, a2, etc. is 50% and that of a combined event a1c1or a2b2, etc. is 25% each.
Hence, we can give an analytical expression for the dpa cross section of SiC (containing two atoms in the molecule) in Eq. (2.1).
σD (SiC) (E) 1 0.25σD (Si / Si) (E) 0.25σD (Si / C) (E) 0.25σD (C / Si) (E) 0.25σD (C / C) (E)
(2.1)
2 A term like σ D ( A / B) ( E ) in this equation means the dpa cross section of type A atom hit by the incident neutron of
energy E, when the secondary collision of the type A atom occurs with an atom of type B. While considering the interactions among sub-lattices to find the damage energy, the values of Z1, Z2, A1 and A2 in Eq. (A.2) are changed accordingly. The total dpa cross section of Si is found from the individual isotopes with their isotopic abundances. Data for natural C is available in ENDF/B-VII.1. Three cases have been carried out to compute the dpa cross section of the compound SiC from Eq. (2.1), which are discussed in the following sub-sections. The reference case is done with the Ed values for Si and C given by Devanathan et al. in [14] and a similar method as given by Heinisch et al. in [37] is followed for the different sub-lattice interactions. In the other two cases dpa cross sections are computed with Ed values different from the reference case in at least one of the sub-lattice interactions. 2.1. Reference case In SiC, Si and C PKA can be formed with a neutron of energy at least 35 eV and 20 eV respectively. The maximum energy that can be transferred from a Si atom to C atom and vice versa is about 84% (4MSiMC/(MSi+MC)2) of the projectile energy. Hence, the effective displacement energy for the projectile/ target pairs will become: Ed(Si/Si) = 35 eV, Ed(Si/C) = 24 eV, Ed(C/C) = 20 eV, Ed(C/Si) = 42 eV. The net numbers of displacements of the species are calculated with the corresponding Ed values.
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2.2. Case 1 Consider the lattice displacement energies of Si and C in their element forms as 25 eV and 31 eV respectively. Then we compute the dpa cross section of SiC assuming that a Si atom can be displaced from its lattice site either by a neutron or a C atom or a Si atom with 25 eV energy and that a C atom can be displaced by any one of the species with energy 31 eV. Here Ed(Si/Si) = 25 eV, Ed(Si/C) = 31 eV, Ed(C/C) = 31 eV, Ed(C/Si) = 25 eV. 2.3. Case 2 Consider the values of Ed for Si and C in their compound form (SiC) as 35 eV and 20 eV respectively. Then as in case 1, we assume that any projectile with energy 35 eV can displace a Si atom and that with energy 20 eV can displace a C atom from their lattice sites in the compound. Here Ed(Si/Si) = 35 eV, Ed(Si/C) = 20 eV, Ed(C/C) = 20 eV, Ed(C/Si) = 35 eV. Fig. 2 shows the profile of the computed cross sections in the reference case. The cross sections obtained in all the cases along with the percentage differences of values in cases 1 and 2 with respect to the reference case are shown in Fig. 3.
Fig. 2. The dpa cross sections as obtained in the reference case for the four interaction pairs and for SiC.
In Fig. 2, σD(Si/Si) is obtained by adding the dpa cross section of all Si isotopes with their abundances. While considering one isotope, its basic neutron interaction cross sections are taken. In this case Z1, Z2, A1, A2 correspond to that of the Si isotope and Ed = 35eV. σD(C/C) is also obtained in the same way from the data given for natural C, with Ed = 20eV. For σD(Si/C), the basic data for all silicon isotopes along with corresponding Z1 and A1 are used, but Z2 and A2 used in each case are that of natural C. Then contributions from all isotopes of Si are added with abundances. The threshold energy to form a Si PKA is taken as 35eV and Ed = 24eV for further displacements of C atom by the PKA. The reverse is done for σD(C/Si), with neutron data and Z1, A1 for natural C and Z2, A2 that of natural Si. Threshold to form C PKA is taken as 20eV and Ed = 42eV for further displacements of Si atom by the PKA. Then σD(SiC) is obtained using Eq. (2.1). From the percentage differences in Fig. 3, it is seen that on an average the values in case 2 differs from that in reference case by about 12%. In case 1, results differ from reference case by about 2% at most of the energy points and within 10% at high energies. The elastic scattering contribution to the dpa cross section in SiC effectively starts around 100 eV. The Ed values used in case 1 result into a comparatively higher effective value of this threshold energy in SiC (first plot in Fig. 3). Due to this, a large percentage deviation (maximum value of 95%) is observed around 100 eV for case 1 with respect to the reference case.
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Fig. 3. The dpa cross sections of SiC in three cases and percentage differences of values in cases 1 and 2 from the reference case.
3. Comparison with available data The dpa cross section of SiC has been computed by H.L. Heinisch et al. in [37]. Here the SPECOMP code has been used to arrive at the cross sections. Here the displacement functions for polyatomic materials, νij(E) defining the number of displacements of atom type j started by the PKA of atom type i, is obtained from the solutions of Parkin–Coulter integro–differential equations for νij(E). The lattice displacement energies used for pair-wise interaction between the two atom species in the compound are similar as done in the reference case of the present work. The total dpa cross section of SiC is given in 100 energy group structure. In a more recent work of Chang et al. in [38] dpa cross section of SiC has been computed by using combined NJOY/TRIM model. They have treated the Si in SiC as 28Si and calculated the cross sections at 300K. They have given the dpa cross section of SiC (their own and Heinisch et al.) in 47 energy groups.
Fig. 4. Comparison of dpa cross sections of SiC between our study and that of Heinisch et al. (in linear scale).
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We compare the results obtained in the reference case with these data. Since the present results are not applied to any particular reactor system, a flat neutron spectrum is considered for group averaging. The 100 and 47 group dpa cross sections are shown in Figs. 4 and 5 respectively. The dpa cross sections of SiC obtained purely from the NRT model in the present work are smaller than that obtained in the other two works (‘reference case’ curves in Figs. 4 and 5). However, if we ignore the NRT damage efficiency factor 0.8 in our calculation by dividing reference case values by 0.8, other details remaining same, then we get relatively higher values of cross section (‘(1/0.8) * reference case curves’ in Figs. 4 and 5). As can be seen in Figs. 4 and 5, the damage efficiency factors for the net number of displacements effectively going in their estimations may be greater than 0.8.
Fig. 5. Comparison of dpa cross sections of SiC with the results of Heinisch et al. and Chang et al. (in linear scales).
4. Conclusions The dpa cross section of the polyatomic material SiC is computed from the individual dpa cross sections of its constituent elements. The dpa cross sections of Si and C are computed including all the important partial neutron interaction data along with their anisotropy from ENDF/B-VII.1 library by developing an indigenous code based on the NRT model. Three cases are investigated with various known Ed values for the elements in the elemental and compound forms and the results are compared. The dpa cross section of SiC obtained by using the neutron-nucleus interactions and the subsequent energy partitioning of the PKA by the NRT formalism is compared with the published results of Heinisch et al. and Chang et al. The deviations observed here are similar to that reported by Chang et al. in their work. Considering the differences in the procedures followed, the agreements between these results are quite satisfactory. However, further investigations are necessary to establish a standard procedure for computing dpa cross sections in compound materials. Appendix A. General formalism to compute dpa cross section from NRT model The parameter dpa characterizing neutron irradiation is evaluated from the knowledge of dpa cross section and neutron flux as follows:
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E max
DPA t
dE ( E )
total D
(E )
E min
total where D ( E )
i D (E)
is calculated by taking the contribution from all partial neutron – nucleus interactions i.
i
φ is the neutron flux and t is the irradiation time. The total dpa cross section is computed from
total D
(E )
E R max i
i
(E )
dE
R
K i ( E , E R ) [ T ( E R )]
(A.1)
0
The function Ki (E, ER) is the kernel of energy transfer ER through a nuclear reaction of type i with cross section σi(E). ν[T(ER)] is the displacement damage function. Damage energy T(ER) is found from Robinson partition function, Eq. (A.2) and the number of displacement per atom is found from the NRT displacement damage model function ν(T) in Eq. (A.3).
ER 1 FG 0.0793 Z12/3Z 21/2 (A1 A2 )3/2
T (ER ) F
Z
2/3 1
Z 22/3
3/4
A 13/ 2A1/2 2
G 3.4008 1/6 0.40244 3/4 ER 1/2 2/3 30.724 Z1Z 2 Z1 Z 22/3 A1 A2 / A2
(A.2)
Z and A with suffixes 1 and 2 denote the atomic and mass numbers of the recoil and lattice nuclei respectively.
0 ; (T ) 1; 0.8T / 2E ; d
T Ed Ed T 2Ed T 2Ed
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(A.3)
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