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Estimation of point defect formation energies in the l12-type intermetallic compound Ni3Ga
Scripta mater. 42 (2000) 567–572 www.elsevier.com/locate/scriptamat
ESTIMATION OF POINT DEFECT FORMATION ENERGIES IN THE L12-TYPE INTERMETALLIC COMPOUND Ni3Ga O.P. Semenova, W. Yuan, R. Krachler and H. Ipser Institut fu¨r Anorganische Chemie, Universita¨t Wien, Wa¨hringerstrasse 42, A-1090 Wien, Austria (Received July 26, 1999) (Accepted in revised form October 13, 1999) Keywords: Compounds, intermetallic; Theory and modeling, defects; Thermodynamics Introduction There has been considerable interest in recent years in the properties of intermetallic compounds with the cubic L12-structure. One prominent example is the compound Ni3Al which has gained technological importance in the development of the so-called superalloys due to some of its unique properties like high temperature strength and excellent corrosion resistance. Ni3Al has a homogeneity range of up to 4.5 at% (1), and it is well known that the deviation from stoichiometry is caused by anti-structure atoms both on the nickel and the aluminum sublattice whereas the concentration of vacancies is generally very small over the entire composition range at all temperatures (2– 4). This was recently confirmed by application of a statistical-thermodynamic model to experimental aluminum activities available in the literature (5). Obviously, many of the outstanding properties of Ni3Al will be related to the amount of defects present in thermodynamic equilibrium and to the variation of the different defect concentrations with composition and temperature. Much less is known about the properties of the isostructural compound Ni3Ga which may have similar interesting properties even if its (peritectic) melting point is considerably lower. This lack of information is certainly caused by the much higher cost of metallic gallium, nevertheless, Ni3Ga might still be of interest for specific applications. The compound has an even larger homogeneity range (nearly 8 at% (6)) than Ni3Al, however, not much seems to be known about the type of defects that are mainly responsible for the deviation from stoichiometry and that usually constitute also the main type of thermal (or intrinsic) defects. An early estimate of the disorder in this phase was presented by Chang and Hsiao (7) who evaluated experimental thermodynamic activity data by Katayama et al. (8) and by Pratt and co-workers (9 –11) in terms of the model equations derived by Chang and co-workers (12,13). Assuming only substitutional disorder in Ni3Ga and neglecting vacancies entirely, the authors arrived at values for the disorder parameter of ␣ ⫽ 0.009 at 1223 K and ␣ ⫽ 0.0015 at 873 K where ␣ was defined as Ga Ni ␣ ⫽ 共N Ni /N兲 stoich ⫽ 共N Ga /N兲 stoich
(1)
with NGaNi, NNiGa being the number of gallium atoms on nickel sublattice sites and vice versa, and N being the total number of atoms. Unfortunately, the number of experimental data points to be used for the evaluation was extremely limited, and the possible existence of vacancies on the two sublattices was a priori completely neglected. Therefore it was considered worthwhile to determine carefully the composition dependence 1359-6462/00/$–see front matter. © 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(99)00391-7
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POINT-DEFECT FORMATION ENERGIES IN Ni3Ga
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of the gallium activity in the Ni3Ga-phase and to apply a newly developed statistical-thermodynamic model (5); this would not only yield a more accurate disorder parameter (since it would be based on a higher number of experimental data points) but also values for the energies of formation of the various point defects (see below). Statistical-Thermodynamic Model For the derivation of the statistical-thermodynamic model the L12-type crystal lattice is divided into two different sublattices of which one (the faces of the cubic unit cell) is preferentially occupied by nickel atoms, the other one (the corners of the unit cell) by gallium atoms. Four types of point defects are allowed in the lattice, i.e. anti-structure atoms and vacancies on both sublattices; the possibility of interstitial defects is neglected. Since we are interested in the interrelationship between lattice disorder and thermodynamic properties our Ni3Ga-crystal is an open system which is allowed to exchange both energy and matter with its surroundings. Therefore we have to use the grand partition function ⌶ ⌶⫽
冘e 冋
Ni
册冒
共NNi兲n 共NGa兲n ⫹Ga ⫺En NL NL
kT
(2)
n
and correpondingly the grand potential ⍀ ⍀ ⫽ ⫺ kT1n ⌶
(3)
In these equations Ni and Ga are the chemical potentials of Ni and Ga, resp.; (NNi)n and (NGa)n are the numbers of Ni- or Ga-atoms in our crystal and En is its energy when it is in the state n; NL is Avogadro’s number. Using the Wagner-Schottky approach (14) for non-interacting (i.e. isolated) point defects we obtain
冉 冊
冉 冊
3 1 1 1 N ! N ! 4 4 䡠 ⫹ ⍀ ⫽ ⫺kT 䡠 ln Ga Ni Ni Ga 共N Ni ⫺ N Ni 兲!N Ga !N VNi! 共N Ga ⫺ N Ga 兲!N Ni !N VGa! Ni Ga ⫹ U共N Ga , N Ni , N VNi, N VGa兲
⫺
Ni Ga Ni Ga N Ni共N Ga N Ga共N Ga , N Ni , N VNi, N VGa) , N Ni , N VNi, N VGa) Ni ⫺ Ga NL NL
(4)
In this equation, Nl denotes the total number of lattice sites and V denotes a vacancy; all subscripts define the species (Ni, Ga, V), whereas the superscripts define the corresponding type of sublattice, e.g., NNiGa is the number of nickel atoms on the gallium sublattice (i.e. anti-structure nickel atoms). As usual in the Wagner-Schottky approach, any changes in the vibrational entropy due to the introduction of defects into the lattice are neglected. All details of the derivation of the model which follows the procedure outlined earlier for B2-phases by Krachler and Ipser (15) can be found in Ref. (5). As parameters we use the energies of formation (at the stoichiometric composition) of the four types of point defects: Ef(NiGa), Ef(GaNi), EfV(Ni), EfV(Ga), which are, in the corresponding sequence, the energies of formation of an anti-structure nickel atom, an anti-structure gallium atom, a vacancy on the nickel-sublattice, a vacancy on the gallium sublattice. A concise definition of these energy parameters can be found, for example, in the paper by Gao et al. (16).
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TABLE 1 Temperature Interval of Measurements and Gallium Activity at 1123 K in the Ni3Ga-Phase xGa
T (°C)
ln aGa (1123 K)
ln (aGa/aGa,0) (1123 K)
0.235 0.240 0.245 0.250 0.255 0.265 0.270 0.275 0.280
800–1000 800–900 800–1000 800–1000 800–1000 800–1000 800–930 800–1000 800–1000
⫺8.56 ⫺8.48 ⫺8.15 ⫺8.12 ⫺7.78 ⫺7.45 ⫺7.34 ⫺7.16 ⫺6.97
⫺0.62 ⫺0.54 ⫺0.21 ⫺0.18 0.16 0.49 0.60 0.78 0.97
As frequently done, the chemical potentials of the components, i, are referred to their values i,0 at the exactly stoichiometric composition Ni3Ga (xGa ⫽ 0.25). Since various thermodynamic measurements yield more or less directly the activity values, the corresponding activity ratio 1n
i ⫺ i,0 ai ⫽ a i,0 RT
(5)
can be calculated and is conveniently plotted as a function of composition. Experimental Gallium Activities Thermodynamic activities of gallium in the Ni3Ga-phase were determined by emf-measurements using yttria stabilized zirconia as a solid electrolyte. All experimental details will be published elsewhere (17). The results of the measurements, i.e. the gallium activities at 1123 K are collected in Table 1 together with the temperature range for which they were measured. It is estimated that the compositions of the samples are accurate within ⫾0.25 at%. Results and Discussion Based on the model outlined above, the theoretical composition dependence of the gallium activity in the Ni3Ga-phase was calculated and compared with the experimental activity values from Table 1. Since, other than in the case of Ni3Al (5), no literature values were available for the energies of formation of the different point defects in Ni3Ga to guide the calculations, various sets of parameters were tried in a first approximation. It turned out immediately that, similar as for Ni3Al, the energies to form vacancies in the Ni3Ga-phase must be considerably larger than those to create anti-structure defects. As can be seen from Fig. 1, good agreement between theoretical curve and experimental gallium activities was finally obtained with the following parameter set: E f(NiGa) ⫽ 0.57 eV, E f(GaNi) ⫽ 0.57 eV, E fV(Ni) ⫽ 1.5 eV, E fV(Ga) ⫽ 2.0 eV To the best of our knowledge, this is the first set of values for the defect formation energies in Ni3Ga. (It should be pointed out that for the situation of very low vacancy concentrations –as in the present case –the shape of the theoretical activity curve becomes sensitive to the sum of the formation energies of
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Figure 1. Experimental gallium activity data in Ni3Ga at 1123 K and theoretical activity curve, calculated with the parameters Ef(NiGa) ⫽ 0.57 eV, Ef(GaNi) ⫽ 0.57 eV, EfV(Ni) ⫽ 1.5 eV, EfV(Ga) ⫽ 2.0 eV.
anti-structure atoms rather than to the individual values; therefore, Ef(NiGa) and Ef(GaNi) were assumed to be be identical.) Fig. 2 shows the variation of the defect concentrations in the Ni3Ga-phase with composition at 1123 K, calculated with the obtained energy parameters. Due to the rather high values for the energies of formation of both types of vacancies, their concentrations are several orders of magnitude smaller (of the order of 10⫺11 to 10⫺7) than those of the anti-structure atoms, and the curves of their composition dependences coincide with the axis of the diagram. From Fig. 2 one can also see that the main type of defects are anti-structure atoms, and that the deviation from stoichiometry is in principle caused entirely by a substitutional (or anti-structure) mechanism. Defining a disorder parameter similar to the one of Chang and co-workers (7,12) as
␣ ⬘ ⫽ 共N NiGa/N1)stoich ⫽ (N GaNi/N1)stoich
(6)
Figure 2. Defect concentrations in Ni3Ga at 1123 K as functions of composition; curve 1— concentration of nickel anti-structure atoms (NNiGa/Nl); curve 2— concentration of gallium anti-structure atoms (NGaNi/Nl); curve 3— concentration of nickel vacancies coinciding with the curve for the gallium vacancies; at the stoichiometric composition a value of ␣⬘ ⫽ 0.0012 is obtained.
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571
TABLE 2 Values of the Disorder Parameter in the Ni3Ga-Phase at Different Temperatures T (K)
␣⬘ (this work)
␣ (Ref. (7))
873 1123 1223 1400
0.00022 0.0012 0.0019 0.0038
0.0015 — 0.009
a value of ␣⬘ ⫽ 0.0012 is obtained for a temperature of 1123 K as can also be seen from Fig. 2. Referring this value to the individual sublattices, one obtains that 0.48% of the gallium sublattice sites are occupied by nickel atoms and 0.16% of the nickel sites by gallium atoms. In order to compare our results with those of Chang and Hsiao (7) the disorder parameter was also calculated for 873 and 1123 K based on the optimized set of defect formation energies (see above). Although our defect concentrations (and with them the disorder parameter ␣⬘) are referred to the total number of lattice sites and not to the total number of atoms as in Refs. (7,12), they can still be compared with each other since, due to the very small vacancy concentration, the difference between the total numbers of atoms and of lattice sites is negligible. As can be seen from Table 2, the values of the disorder parameter obtained in the present study are considerably smaller than those of Chang and Hsiao (7): they differ by nearly one order of magnitude at 873 K and still by a factor of about five at 1223 K. This corresponds to a higher degree of ordering, i.e. to a smaller concentration of anti-structure defects in Ni3Ga than derived in Ref. (7). However, due to the much higher number of experimental data points used in the present evaluation, our defect concentrations, and with them our disorder parameter values, are thought to be more reliable. Furthermore, the value of ␣⬘ ⫽ 0.0038 at 1400 K (see Table 2) can be compared with a value of 0.0075 for Ni3Al at the same temperature (5). These results indicate that the disorder in Ni3Al is by about a factor of two higher, i.e. the concentrations of anti-structure defects in Ni3Al are twice as high as in Ni3Ga at 1400 K. Very recently Fa¨hnle and co-workers (18 –20) presented an extension of the statistical model approach used here including the effects of the vibrational entropy. However, to keep the present model as simple as possible and to avoid additional parameters, these entropy contributions are neglected here. Following similar arguments as given in Ref. (5) for Ni3Al it is thought that the error caused by that should be rather minute. Acknowledgment Financial support of this investigation by the Austrian Science Foundation (Fonds zur Fo¨rderung der wissenschaftlichen Forschung) under project Nos. P10739-CHE and P12962-CHE is gratefully acknowledged. References 1. 2. 3. 4. 5.
I. Ansara, N. Dupin, H. L. Lukas, and B. Sundman, J. Alloys Comp. 247, 20 (1997). K. Badura-Gergen and H.-E. Schaefer, Phys. Rev. B. 56, 3032 (1997). S. M. Foiles and M. S. Daw, J. Mater. Res. 2, 5 (1987). C. L. Fu and G. S. Painter, Acta Mater. 45, 481 (1997). R. Krachler, O. P. Semenova, and H. Ipser, Phys. Stat. Sol. accepted for publication.
572
6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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S. Y. Lee and P. Nash, Phase Diagrams of Binary Nickel Alloys, p. 133, ASM International, Materials Park, OH (1991). Y. A. Chang and Y. Hsiao, Scripta Metall. 10, 201 (1976). I. Katayama, Sh. Igi, and Z. Kozuka, Trans. JIM. 15, 447 (1974). J. N. Pratt, J. M. Bird, and S. Martosudirjo, Final Technical Report, United States Army, Contract DAJA37-72-C-3034, March 1973, Department of Physical Metallurgy and Science of Materials, University of Birmingham, England; this report was later on published as Refs. (10 and 11). S. Martosudirjo and J. N. Pratt, Thermochim. Acta. 17, 183 (1976). J. N. Pratt and J. M. Bird, J. Phase Equil. 14, 465 (1993). I. Gyuk, W. W. Liang, and Y. A. Chang, J. Less-Common Metals. 38, 249 (1974). Y. A. Chang, in Treatise on Materials Science and Technolgy, ed. H. Herman, vol. 4, p. 173, Academic Press, New York (1974). C. Wagner and W. Schottky, Z. Phys. Chem. B11, 163 (1931). R. Krachler and H. Ipser, Intermetallics. 7, 141 (1999). F. Gao, D. J. Bacon, and G. J. Ackland, Phil. Mag. A. 67, 275 (1993). W. Yuan, O. Diwald, A. Mikula, and H. Ipser, Z. Metallkd. submitted for publication. J. Mayer and M. Fa¨hnle, Acta Mater. 45, 2207 (1997). B. Meyer and M. Fa¨hnle, Phys. Rev. B. 59, 6072 (1999). B. Meyer and M. Fa¨hnle, Phys. Rev. B. 60, 717 (1999).
ESTIMATION OF POINT DEFECT FORMATION ENERGIES IN THE L12-TYPE INTERMETALLIC COMPOUND Ni3Ga O.P. Semenova, W. Yuan, R. Krachler and H. Ipser Institut fu¨r Anorganische Chemie, Universita¨t Wien, Wa¨hringerstrasse 42, A-1090 Wien, Austria (Received July 26, 1999) (Accepted in revised form October 13, 1999) Keywords: Compounds, intermetallic; Theory and modeling, defects; Thermodynamics Introduction There has been considerable interest in recent years in the properties of intermetallic compounds with the cubic L12-structure. One prominent example is the compound Ni3Al which has gained technological importance in the development of the so-called superalloys due to some of its unique properties like high temperature strength and excellent corrosion resistance. Ni3Al has a homogeneity range of up to 4.5 at% (1), and it is well known that the deviation from stoichiometry is caused by anti-structure atoms both on the nickel and the aluminum sublattice whereas the concentration of vacancies is generally very small over the entire composition range at all temperatures (2– 4). This was recently confirmed by application of a statistical-thermodynamic model to experimental aluminum activities available in the literature (5). Obviously, many of the outstanding properties of Ni3Al will be related to the amount of defects present in thermodynamic equilibrium and to the variation of the different defect concentrations with composition and temperature. Much less is known about the properties of the isostructural compound Ni3Ga which may have similar interesting properties even if its (peritectic) melting point is considerably lower. This lack of information is certainly caused by the much higher cost of metallic gallium, nevertheless, Ni3Ga might still be of interest for specific applications. The compound has an even larger homogeneity range (nearly 8 at% (6)) than Ni3Al, however, not much seems to be known about the type of defects that are mainly responsible for the deviation from stoichiometry and that usually constitute also the main type of thermal (or intrinsic) defects. An early estimate of the disorder in this phase was presented by Chang and Hsiao (7) who evaluated experimental thermodynamic activity data by Katayama et al. (8) and by Pratt and co-workers (9 –11) in terms of the model equations derived by Chang and co-workers (12,13). Assuming only substitutional disorder in Ni3Ga and neglecting vacancies entirely, the authors arrived at values for the disorder parameter of ␣ ⫽ 0.009 at 1223 K and ␣ ⫽ 0.0015 at 873 K where ␣ was defined as Ga Ni ␣ ⫽ 共N Ni /N兲 stoich ⫽ 共N Ga /N兲 stoich
(1)
with NGaNi, NNiGa being the number of gallium atoms on nickel sublattice sites and vice versa, and N being the total number of atoms. Unfortunately, the number of experimental data points to be used for the evaluation was extremely limited, and the possible existence of vacancies on the two sublattices was a priori completely neglected. Therefore it was considered worthwhile to determine carefully the composition dependence 1359-6462/00/$–see front matter. © 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(99)00391-7
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Vol. 42, No. 6
of the gallium activity in the Ni3Ga-phase and to apply a newly developed statistical-thermodynamic model (5); this would not only yield a more accurate disorder parameter (since it would be based on a higher number of experimental data points) but also values for the energies of formation of the various point defects (see below). Statistical-Thermodynamic Model For the derivation of the statistical-thermodynamic model the L12-type crystal lattice is divided into two different sublattices of which one (the faces of the cubic unit cell) is preferentially occupied by nickel atoms, the other one (the corners of the unit cell) by gallium atoms. Four types of point defects are allowed in the lattice, i.e. anti-structure atoms and vacancies on both sublattices; the possibility of interstitial defects is neglected. Since we are interested in the interrelationship between lattice disorder and thermodynamic properties our Ni3Ga-crystal is an open system which is allowed to exchange both energy and matter with its surroundings. Therefore we have to use the grand partition function ⌶ ⌶⫽
冘e 冋
Ni
册冒
共NNi兲n 共NGa兲n ⫹Ga ⫺En NL NL
kT
(2)
n
and correpondingly the grand potential ⍀ ⍀ ⫽ ⫺ kT1n ⌶
(3)
In these equations Ni and Ga are the chemical potentials of Ni and Ga, resp.; (NNi)n and (NGa)n are the numbers of Ni- or Ga-atoms in our crystal and En is its energy when it is in the state n; NL is Avogadro’s number. Using the Wagner-Schottky approach (14) for non-interacting (i.e. isolated) point defects we obtain
冉 冊
冉 冊
3 1 1 1 N ! N ! 4 4 䡠 ⫹ ⍀ ⫽ ⫺kT 䡠 ln Ga Ni Ni Ga 共N Ni ⫺ N Ni 兲!N Ga !N VNi! 共N Ga ⫺ N Ga 兲!N Ni !N VGa! Ni Ga ⫹ U共N Ga , N Ni , N VNi, N VGa兲
⫺
Ni Ga Ni Ga N Ni共N Ga N Ga共N Ga , N Ni , N VNi, N VGa) , N Ni , N VNi, N VGa) Ni ⫺ Ga NL NL
(4)
In this equation, Nl denotes the total number of lattice sites and V denotes a vacancy; all subscripts define the species (Ni, Ga, V), whereas the superscripts define the corresponding type of sublattice, e.g., NNiGa is the number of nickel atoms on the gallium sublattice (i.e. anti-structure nickel atoms). As usual in the Wagner-Schottky approach, any changes in the vibrational entropy due to the introduction of defects into the lattice are neglected. All details of the derivation of the model which follows the procedure outlined earlier for B2-phases by Krachler and Ipser (15) can be found in Ref. (5). As parameters we use the energies of formation (at the stoichiometric composition) of the four types of point defects: Ef(NiGa), Ef(GaNi), EfV(Ni), EfV(Ga), which are, in the corresponding sequence, the energies of formation of an anti-structure nickel atom, an anti-structure gallium atom, a vacancy on the nickel-sublattice, a vacancy on the gallium sublattice. A concise definition of these energy parameters can be found, for example, in the paper by Gao et al. (16).
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569
TABLE 1 Temperature Interval of Measurements and Gallium Activity at 1123 K in the Ni3Ga-Phase xGa
T (°C)
ln aGa (1123 K)
ln (aGa/aGa,0) (1123 K)
0.235 0.240 0.245 0.250 0.255 0.265 0.270 0.275 0.280
800–1000 800–900 800–1000 800–1000 800–1000 800–1000 800–930 800–1000 800–1000
⫺8.56 ⫺8.48 ⫺8.15 ⫺8.12 ⫺7.78 ⫺7.45 ⫺7.34 ⫺7.16 ⫺6.97
⫺0.62 ⫺0.54 ⫺0.21 ⫺0.18 0.16 0.49 0.60 0.78 0.97
As frequently done, the chemical potentials of the components, i, are referred to their values i,0 at the exactly stoichiometric composition Ni3Ga (xGa ⫽ 0.25). Since various thermodynamic measurements yield more or less directly the activity values, the corresponding activity ratio 1n
i ⫺ i,0 ai ⫽ a i,0 RT
(5)
can be calculated and is conveniently plotted as a function of composition. Experimental Gallium Activities Thermodynamic activities of gallium in the Ni3Ga-phase were determined by emf-measurements using yttria stabilized zirconia as a solid electrolyte. All experimental details will be published elsewhere (17). The results of the measurements, i.e. the gallium activities at 1123 K are collected in Table 1 together with the temperature range for which they were measured. It is estimated that the compositions of the samples are accurate within ⫾0.25 at%. Results and Discussion Based on the model outlined above, the theoretical composition dependence of the gallium activity in the Ni3Ga-phase was calculated and compared with the experimental activity values from Table 1. Since, other than in the case of Ni3Al (5), no literature values were available for the energies of formation of the different point defects in Ni3Ga to guide the calculations, various sets of parameters were tried in a first approximation. It turned out immediately that, similar as for Ni3Al, the energies to form vacancies in the Ni3Ga-phase must be considerably larger than those to create anti-structure defects. As can be seen from Fig. 1, good agreement between theoretical curve and experimental gallium activities was finally obtained with the following parameter set: E f(NiGa) ⫽ 0.57 eV, E f(GaNi) ⫽ 0.57 eV, E fV(Ni) ⫽ 1.5 eV, E fV(Ga) ⫽ 2.0 eV To the best of our knowledge, this is the first set of values for the defect formation energies in Ni3Ga. (It should be pointed out that for the situation of very low vacancy concentrations –as in the present case –the shape of the theoretical activity curve becomes sensitive to the sum of the formation energies of
570
POINT-DEFECT FORMATION ENERGIES IN Ni3Ga
Vol. 42, No. 6
Figure 1. Experimental gallium activity data in Ni3Ga at 1123 K and theoretical activity curve, calculated with the parameters Ef(NiGa) ⫽ 0.57 eV, Ef(GaNi) ⫽ 0.57 eV, EfV(Ni) ⫽ 1.5 eV, EfV(Ga) ⫽ 2.0 eV.
anti-structure atoms rather than to the individual values; therefore, Ef(NiGa) and Ef(GaNi) were assumed to be be identical.) Fig. 2 shows the variation of the defect concentrations in the Ni3Ga-phase with composition at 1123 K, calculated with the obtained energy parameters. Due to the rather high values for the energies of formation of both types of vacancies, their concentrations are several orders of magnitude smaller (of the order of 10⫺11 to 10⫺7) than those of the anti-structure atoms, and the curves of their composition dependences coincide with the axis of the diagram. From Fig. 2 one can also see that the main type of defects are anti-structure atoms, and that the deviation from stoichiometry is in principle caused entirely by a substitutional (or anti-structure) mechanism. Defining a disorder parameter similar to the one of Chang and co-workers (7,12) as
␣ ⬘ ⫽ 共N NiGa/N1)stoich ⫽ (N GaNi/N1)stoich
(6)
Figure 2. Defect concentrations in Ni3Ga at 1123 K as functions of composition; curve 1— concentration of nickel anti-structure atoms (NNiGa/Nl); curve 2— concentration of gallium anti-structure atoms (NGaNi/Nl); curve 3— concentration of nickel vacancies coinciding with the curve for the gallium vacancies; at the stoichiometric composition a value of ␣⬘ ⫽ 0.0012 is obtained.
Vol. 42, No. 6
POINT-DEFECT FORMATION ENERGIES IN Ni3Ga
571
TABLE 2 Values of the Disorder Parameter in the Ni3Ga-Phase at Different Temperatures T (K)
␣⬘ (this work)
␣ (Ref. (7))
873 1123 1223 1400
0.00022 0.0012 0.0019 0.0038
0.0015 — 0.009
a value of ␣⬘ ⫽ 0.0012 is obtained for a temperature of 1123 K as can also be seen from Fig. 2. Referring this value to the individual sublattices, one obtains that 0.48% of the gallium sublattice sites are occupied by nickel atoms and 0.16% of the nickel sites by gallium atoms. In order to compare our results with those of Chang and Hsiao (7) the disorder parameter was also calculated for 873 and 1123 K based on the optimized set of defect formation energies (see above). Although our defect concentrations (and with them the disorder parameter ␣⬘) are referred to the total number of lattice sites and not to the total number of atoms as in Refs. (7,12), they can still be compared with each other since, due to the very small vacancy concentration, the difference between the total numbers of atoms and of lattice sites is negligible. As can be seen from Table 2, the values of the disorder parameter obtained in the present study are considerably smaller than those of Chang and Hsiao (7): they differ by nearly one order of magnitude at 873 K and still by a factor of about five at 1223 K. This corresponds to a higher degree of ordering, i.e. to a smaller concentration of anti-structure defects in Ni3Ga than derived in Ref. (7). However, due to the much higher number of experimental data points used in the present evaluation, our defect concentrations, and with them our disorder parameter values, are thought to be more reliable. Furthermore, the value of ␣⬘ ⫽ 0.0038 at 1400 K (see Table 2) can be compared with a value of 0.0075 for Ni3Al at the same temperature (5). These results indicate that the disorder in Ni3Al is by about a factor of two higher, i.e. the concentrations of anti-structure defects in Ni3Al are twice as high as in Ni3Ga at 1400 K. Very recently Fa¨hnle and co-workers (18 –20) presented an extension of the statistical model approach used here including the effects of the vibrational entropy. However, to keep the present model as simple as possible and to avoid additional parameters, these entropy contributions are neglected here. Following similar arguments as given in Ref. (5) for Ni3Al it is thought that the error caused by that should be rather minute. Acknowledgment Financial support of this investigation by the Austrian Science Foundation (Fonds zur Fo¨rderung der wissenschaftlichen Forschung) under project Nos. P10739-CHE and P12962-CHE is gratefully acknowledged. References 1. 2. 3. 4. 5.
I. Ansara, N. Dupin, H. L. Lukas, and B. Sundman, J. Alloys Comp. 247, 20 (1997). K. Badura-Gergen and H.-E. Schaefer, Phys. Rev. B. 56, 3032 (1997). S. M. Foiles and M. S. Daw, J. Mater. Res. 2, 5 (1987). C. L. Fu and G. S. Painter, Acta Mater. 45, 481 (1997). R. Krachler, O. P. Semenova, and H. Ipser, Phys. Stat. Sol. accepted for publication.
572
6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
POINT-DEFECT FORMATION ENERGIES IN Ni3Ga
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