Computer simulation of displacement energies for several ceramic materials

Computer simulation of displacement energies for several ceramic materials

Nuclear Instruments and Methods in Physics Research B 141 (1998) 94±98 Computer simulation of displacement energies for several ceramic materials R.E...

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Nuclear Instruments and Methods in Physics Research B 141 (1998) 94±98

Computer simulation of displacement energies for several ceramic materials R.E. Williford *, R. Devanathan, W.J. Weber Paci®c Northwest National Laboratory, Material Sciences Department, M.S. K2-44, P.O.Box 999, Richland, WA 99352, USA

Abstract Displacement energies …Ed † are fundamental parameters controlling the production of radiation damage in materials, and as such, are useful for understanding and modeling the e€ects of radiation on materials. These energies are not easily determined experimentally for many ceramic materials. However, advances in computational methodologies and their application to ceramic materials provide a means to determine these energies in a number of materials of interest. Although computationally intensive molecular dynamics methods can be used to determine Ed for the various cations and anions, energy minimization methods can also provide a more expedient means to obtain reasonable estimates of these energies. In this paper, the energy minimization code General Utility Lattice Program (GULP), which uses a Mott±Littleton approximation to simulate isolated defects in extended solids, is used to calculate displacement energies. The validity of using this code for these computations is established by calculating Ed for several ceramics for which these energies are known. Computational results are in good agreement with the experimental values for alumina, MgO, and ZnO. Results are also presented for two ceramic materials, zircon and spinel, for which there are little or no experimental values yet available. Published by Elsevier Science B.V. PACS: 61.80.-X; 61.80.AZ; 61.82.MS Keywords: Radiation damage; Displacement energies; Ceramics; GULP code

1. Introduction The displacement threshold energy …Ed † is an important parameter that controls the extent of damage production in irradiated materials. Many experimental techniques are available for measuring Ed in ceramics, including optical and thermally stimulated spectroscopies, electron paramagnetic

* Corresponding author. Fax: +1 509 375 2186; e-mail: [email protected]

0168-583X/98/$19.00 Published by Elsevier Science B.V. PII S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 0 6 6 - 4

resonance, positron annihilation, and electron microscopy. However, because of the level of e€ort and expense required to overcome experimental diculties, reliable Ed values are not available for many ceramics. An alternative approach is to quantify the defect structures and energetics using computer modeling techniques. The most modern of these techniques rely heavily on quantum mechanical formulations. Although computing costs are decreasing with constant improvements in computing power, it can still be expensive to develop

R.E. Williford et al. / Nucl. Instr. and Meth. in Phys. Res. B 141 (1998) 94±98

computer codes appropriate for all the cases that need study. Less demanding computational approaches use empirical potentials in molecular dynamics (MD) codes, but can still be computationally non-trivial because of the limited real times accessible to MD. A third computational alternative is to use energy minimization techniques to obtain estimates of displacement energies. This method is mature enough so that several energy minimization codes are available. One of the more recent is the GULP code [1], which incorporates several particularly useful features for ®tting potentials and for calculating displacement and defect migration energetics. GULP is brie¯y described in the following section. 2. Energy minimization method The General Utility Lattice Program (GULP) [1] was authored by Prof. Julian D Gale (Department of Chemistry, Imperial College, London, UK). Originally intended as a code for interatomic potential ®tting, GULP evolved to include energy minimization and many other useful features. Although similar in some ways to other codes e.g., [2±4], GULP is di€erent in that it maximizes use of the crystal symmetry to make structure generation easier and to speed up calculations. GULP uses the Mott±Littleton [5] approximation to calculate defect energies in extended solids. In this well-known method, the crystal surrounding the defect is divided into three spherical regions of progressively larger radii. Region 1 contains the defect per se, interactions are treated explicitely at the atomic level, and ions are relaxed in response to the defect. In Region 2a, the ions are assumed to reside in a harmonic well and respond accordingly to the defect forces. This approximation is only valid for small perturbations and requires that the bulk lattice be relaxed prior to introduction of the defect. Individual ion displacements are still treated in Region 2a, whereas only the implicit polarization of sublattices is considered in Region 2b. The defect energy is calculated as the energy di€erence between the defected …Edef † and perfect …Eperf † lattices, and corrected for the energy of interstitials or vacancies at in®nite sepa-

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ration from the lattice: Edefect ˆ Edef ÿ Eperf ‡ Einf : Details of the computational method in GULP are given elsewhere [1]. Charged defects polarize other ions in the lattice, so ionic polarizability is incorporated into the potentials using the shell model of Dick and Overhauser [6]. This is a simple harmonic spring model in the present work, although GULP can accommodate more detail if needed. In several cases below, the core-shell charge split departs from formal charges in order to approximate the e€ects of partial covalency e.g., [7]. In the present work, GULP's 'translate' option was used to simulate atomic displacements, where the displacement energy …Ed † was de®ned as the minimum amount of kinetic energy required to permanently displace an atom from its lattice position [8]. This feature moves a selected atom along a vector in a speci®ed number of steps, and relaxes the lattice at each step. Because the displacement process is inherently very dynamic, it was assumed that the displacement energy was accurately represented by the unrelaxed energy of the lattice with a selected atom in the displaced position, i.e., Ed ˆ Edef (unrelaxed) - Eperf (relaxed). This seems reasonable since the time for a displaced atom to travel one lattice unit is several orders of magnitude smaller than the time for phonons to relax the local lattice. The determination of whether the defect was stable or not was made by relaxing the lattice. If the displaced atom moved back into its original position, the defect was unstable. The defect was stable if a new equilibrium position was assumed, resulting in a change in the overall lattice energy. 3. Validation of method To verify the validity of using GULP for computing Ed , a series of cases were run for ceramics for which these energies are relatively well known. Each is described below. 3.1. Mgo Potentials for MgO were taken from an example case included in the GULP software package

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[1]. The oxygen was modeled as a core-shell system, and Mgcore ±Oshell and Oshell ±Oshell interactions were modeled with two similar Buckingham potentials, the main di€erence being the addition of a dispersion term for the latter. Table 1 compares the displacement energies calculated in the present work to the recommended values of Zinkle and Kinoshita [8] based on the available experimental data. The average Ed values calculated for O and Mg are 47.5 eV and 41.5 eV, respectively. This is in reasonable agreement with the corresponding recommended values of 50 eV and 55 eV. The `ion trajectories' noted in Table 1 (and in the following tables) were selected using computerized graphical aids, and represent directions where the lattice is most open and thus more likely to accommodate a displaced ion. 3.2. Alumina …a-Al2 O3 † Potentials for alumina originated from Gale et al. [9], and were obtained from Gay and Rohl's [10] study of crystal surfaces, which are also known as `the largest defects'. The inherent treatment of reduced coordination numbers in their surface relaxation model was considered an advantage for the present study of point defect modeling. The potentials designated `qm' (quantum mechanically derived) in [10] were used here because they

were a demonstration of GULP without using the shell model. Results (Table 1) for aluminum cation displacement …Ed …Al† ˆ 39:5eV average) are higher than the recommended value of 20 eV [8], but results for the oxygen anion …Ed …O† ˆ 43:3 average) are in reasonable agreement with the recommended average data of 50 eV [8]. 3.3. ZnO Potentials for ZnO were obtained from Binks et al. [11], where they were used for studies of sodium ion migration. Results for displacement energies are shown in Table 1. Although the average of the computed results for Ed …Zn† are about 40% lower than the recommended 50 eV [8], the computed Ed …O† values for oxygen appear to bracket the data range, and the computed average displacement energy of 52.3 eV is in good agreement with the recommended data value of 55 eV [8]. 4. Results for other ceramics Since the above results gave reasonably good agreement with the data in the literature [8], GULP was considered to be an appropriate code for estimating atomic displacement energies in other ceramic structures. It was then applied to

Table 1 Comparison of GULP displacement energies with measured and recommended values Material Calculated value (eV)/ion trajectory MgO

Al2 O3

ZnO

Ed (Mg) ˆ 46.8/[1 2 0] Ed (Mg) ˆ 46.8/[1 3 0] 1] Ed (Mg) ˆ 32.8/[1 2  Ed (Mg) ˆ 45.1/[3 3 1] Ed (O) ˆ 47.5/[1 2 1] Ed (Al) ˆ 27.7/[1 0  1 0] 1 1 0] Ed (Al) ˆ 51.4/[2  1 0] Ed (O) ˆ 54.3/[1 0  1 2 0] Ed (O) ˆ 32.4/[3  4 4 1] Ed (Zn) ˆ 29.3/[8  1 1] Ed (Zn) ˆ 30.6/[1 0  4 1] Ed (Zn) ˆ 30.8/[4 0  4 1] Ed (O) ˆ 38.1/[4 0  1 0] Ed (O) ˆ 75.5/[1 0  4 4 1] Ed (O) ˆ 43.5/[8 

Calculated average (eV)

Measured values (eV)

Recommended value (eV) [8]

41.5

46±64

55

47.5 39.5

44±65 18±32

50 20

43.3

41±79

50

30.2

40±70

50

52.3

47±55

55

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calculate displacement energies for spinel, which is an important material for fusion applications [12,13] and zircon, an important material for nuclear waste isolation [14,15]. Almost no experimental data exist for these materials. Results were as follows. 4.1. Spinel …MgAl2 O4 † Potentials for spinel were obtained from the multi-ceramic potential ®tting e€ort by Bush et al. [16], and were also used in GULP by Chen et al. [17], to study various structures of spinel. After reproducing Chen et al's results for the vacancy energy of each ion (Mg,Al,O), GULP was used to compute the displacement energies shown in Table 2. Results for Ed …O† depend on direction (22± 76 eV), and the average is slightly lower than the only known experimental value by Summers et al. [18], as reported in [8]. Displacement energies for the cations, Ed …Al† ˆ 51 eV and Ed …Mg† ˆ 60 eV, are new predictions, and appear reasonable when compared with cation displacement energies Table 2 GULP displacement energies for spinel (MgAl2 O4 ) Calculated value (eV)/ion Average of calcu- Recommended trajectory lation (eV) value (eV) Ed (Mg) ˆ 58.7/[1 0 0] Ed (Mg) ˆ 61.4/[1 0 0] Ed (Al) ˆ 51.0/[1 2 1] Ed (O) ˆ 22.1/[1 1 2] Ed (O) ˆ 40.3/[1 1 2] Ed (O) ˆ 76.3/[1 0 2] a

a

60

30 [19]

51 46.2

30 [19] a 60 [8,19]

a

[8], experimental; [19], assumed.

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for other ceramics (see Table 1), although these numbers are higher than the currently assumed values of 30 eV for each cation [19,20]. 4.2. Zircon …ZrSiO4 † Two sets of potentials were employed for zircon. The ®rst set was obtained from Chandley et al. [21], where they were used to study the site preferences for dopants with the CASCADE code [4]. The O±O short range parameters were derived by Hartree±Fock methods [22], and a shell model was used. The Si±O potentials [23] included a three-body term for the O±Si±O. The Zr±O potential was specially derived [24] to treat seven-coordinated Zr in the monoclinic low-temperature phase, and was used here for the tetragonal phase to test this low-coordination feature for radiation defects. Results shown in Table 3 appear reasonable, but full equilibrium was rarely achieved with these potentials in GULP (denoted by asterisks). For comparison, a second set of potentials for zircon were obtained from Gay and Rohl [10], who used Buckingham potentials for the Zr±O and O±O interactions, a Morse potential for the Si±O bonds, and a three-body term for the O±Si± O interactions. Note that although the latter two were considered intramolecular in [10], we found it necessary to delete this condition for our simulations because displaced atoms moved outside the in¯uence of their original molecules, causing abnormal terminations of GULP. However, agreement with elastic constants was good, and structural parameters were within about 3% of data. The oxygen polarization was again treated with a shell model. Results in Table 3 agree with the

Table 3 Calculated displacement energies for zircon using di€erent potentials Value (eV)/ion trajectory using Chandley et al. [21] potentials Ed (Zr) ˆ 89.2/[1 0 0]



Ed (Si) ˆ 18.2/[1 0 0]  Ed (O) ˆ 42.5/[2 5 2]  

Metastable

Value (eV) / ion trajectory using Gay and Rohl [10] potentials

Average of calculations (eV)

Ed (Zr) ˆ 103.2/[0 1 0] Ed (Zr) ˆ 78.8/[1 2 0] Ed (Si) ˆ 22.6/[1 0 0] Ed (O) ˆ 69.4/[2 5 2] 1] Ed (O) ˆ 47.3/[0 2 

Ed (Zr) ˆ 90.4 Ed (Si) ˆ 20.4 Ed (O) ˆ 53.1

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above values for Ed …Zr† and Ed …Si†, but are slightly higher for Ed …O†. The overall average Ed …O† ˆ 53:1 eV is reasonable compared to other ceramics, whereas Ed …Si† ˆ 20:4 eV is among the lower values measured in covalently bonded SiC, where Ed …Si† ˆ 18±53 eV [8]. Although somewhat repeatable with GULP calculations, the overall average Ed …Zr† ˆ 90:4 eV is higher than cation displacement energies in other ceramics. This large displacement energy may be due to the relatively large ionic radius of Zr and the strength of the Zr±O bonds. This large value of Ed for Zr in zircon is consistent with recent experimental results that suggest Ed …Zr† in zircon is greater than 41 eV [25]. 5. Conclusions For ceramics with known data bases, results showed that GULP can predict displacement energies with reasonable accuracy. The Ed (minimum calculated) underpredicts the Ed (minimum measured) by about 14% on the average, while the Ed (mean calculated) underpredicts the Ed recommended values [8] by about 2% on the average. Although standard deviations were 36% error and 45% error, respectively, this could be caused by using potentials from a wide variety of sources, and by scatter in the data. The Ed predictions for spinel and zircon also seem reasonable. The overall conclusion is that GULP can be used to estimate displacement energies for a wide range of ceramics that are of interest in radiation damage studies. Acknowledgements The authors gratefully acknowledge Prof. Julian D. Gale of the Chemistry Department, Imperial College, London, UK, for providing a copy of GULP, and for many instructive discussions on how to run the code. This work was supported by the Environmental Management Science Program, US Department of Energy under contract DE AC06-76 RLO 1830.

References [1] J.D. Gale, GULP (General Utility Lattice Program), Royal Institution of Great Britain, London, 1993. [2] S.C. Parker, UKAEA Report AERE-TP968, 1983. [3] C.R.A. Catlow, A.N. Cormack, F. Theobald, Acta. Cryst. B 40 (1984) 195. [4] M. Leslie, Daresbury Laboratory Report DL/SCI/TM 31T, 1982. [5] N.F. Mott, M.J. Littleton, Trans. Faraday Soc. 34 (1938) 485. [6] B.G. Dick, A.W. Overhauser, Phys. Rev. 112 (1958) 90. [7] P. Zapol, R. Pandey, M. Ohmer, J. Gale, J. Appl. Phys. 79 (1996) 671. [8] S.J. Zinkle, C. Kinoshita, J. Nucl. Mater. 251 (1997) 200. [9] J.D. Gale, C.R.A. Catlow, W.C. Mackrodt, Modelling Simul. Mater. Sci. Eng. 1 (1992) 73. [10] D.H. Gay, A.L. Rohl, J. Chem. Soc. Faraday Trans. 91 (1995) 925. [11] D.J. Binks, R.W. Grimes, D.L. Morgenstern, British Ceram. Proc. 5 (1994) 159. [12] S.J. Zinkle, J. Nucl. Mater. 219 (1995) 113. [13] R. Devanathan, N. Yu, K.E. Sickafus, M. Nastasi, J. Nucl. Mater. 232 (1996) 59. [14] R.C. Ewing, W. Lutze, W.J. Weber, J. Mater. Res. 10 (1995) 243. [15] W.J. Weber, R.C. Ewing, W. Lutze, Scienti®c basis for nuclear waste management, XIX, in: W.N. Murphy, D.A. Knecht (Eds.), Mater. Res. Soc. Symp. Proc., vol. 412, Pittsburgh, PA, 1996, p. 25. [16] T.S. Bush, J.D. Gale, C.R.A. Catlow, P.D. Battle, J. Mater. Chem. 4 (1994) 831. [17] S.P. Chen, M. Yan, J.D. Gale, R.W. Grimes, R. Devanathan, K.E. Sickafus, N. Yu, M. Nastasi, Phil. Mag. Lett. 73 (1996) 51. [18] G.P. Summers, G.S. White, K.H. Lee, J.H. Crawford, Phys. Rev. B 21 (1980) 2578. [19] P. Pells, J. Nucl. Mater. 155 157 (1988) 67. [20] C. Parker, J. Nucl. Mater. 133 134 (1985) 741. [21] P. Chandley, R.J.H. Clark, R.J. Angel, G.D. Price, J. Chem. Soc. Dalton Trans. (1992) 1579. [22] C.R.A. Catlow, Defects in Solids: Modern Techniques, Plenum Press, New York, 1986, pp. 264±302. [23] M.J. Sanders, M. Leslie, C.R.A. Catlow, J. Chem. Soc. Chem. Commun. (1985) 1271. [24] I.A. Dwivedi, A.N. Cormack, J. Solid State Chem. 7 (1989) 218. [25] R. Devanathan, W.J. Weber, L. Boatner, in: E. Ma, P. Bellon, R. Trivedi (Eds.), Phase Transformations and Systems Driven Far from Equilibrium, Materials Research Society, Warrendale, PA, 1998, in press.