First-principles investigation of the elastic and electronic properties of the binary intermetallics in the Al–La alloy system

Physica B 407 (2012) 4706–4711

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First-principles investigation of the elastic and electronic properties of the binary intermetallics in the Al–La alloy system Zhi-Sheng Nong a,b, Jing-Chuan Zhu a,b,n, Xia-Wei Yang a,b, Yong Cao a,b, Zhong-Hong Lai a, Yong Liu a,b a b

School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China National Key Laboratory for Precision Hot Processing of Metals, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 July 2012 Received in revised form 8 September 2012 Accepted 10 September 2012 Available online 20 September 2012

The structural, elastic and electronic properties of Al2La, AlLa3 and Al3La binary intermetallics in the Al–La alloy system were investigated using the first-principles method. The calculated lattice constants were consistent with the experimental values. Formation enthalpy and cohesive energy showed that the studied Al2La, AlLa3 and Al3La all have a higher structural stability, and the alloying ability of Al2La and Al3La is stronger than that of AlLa3. The single-crystal elastic constants (Cij) as well as polycrystalline elastic parameters (bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio u and anisotropy value A) were calculated by the Voigt–Reuss–Hill (V–R–H) approximations, and the relationship of these elastic parameters between Al2La, AlLa3 and Al3La phases were discussed in detail. The results showed that Al2La and Al3La which are anisotropic materials are absolutely brittle, while the isotropic AlLa3 is slightly ductile. Finally, the electronic density of states (DOS) was also calculated to reveal the underlying mechanism of structural stability. & 2012 Elsevier B.V. All rights reserved.

Keywords: First-principles Elastic constants Electronic structure Al–La alloy system

1. Introduction In recent years, aluminum and its alloys have been receiving a great deal of attention in the fields of aerospace industry, buildings and electronic industry due to their light weight, good corrosion resistance, reasonably high strength and favorable economics [1,2]. However, the application of aluminum alloys in some special industries is still limited because of the restrained wear and mechanical properties. Therefore, much work has been carried out in order to improve the properties of the alloys. As an effective method, addition of rare earth (RE) elements to Al-based alloys has been made to improve the wear resistance, mechanical properties and hot deformation behavior of the materials [3–5]. Furthermore, the intermetallic compounds in Al-based alloys have excellent physical and chemical properties [6–8], which is important for the design of novel Al-based materials and further scientific investigations. Recently, the first-principles method based on the density functional theory was successfully adopted to study groundstate properties such as the structure and phase stabilities, formation of enthalpy, electronic structure and elastic properties for the Al-RE system [9,10]. Tao et al. [11] calculated the total

n Corresponding author at: School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China. Tel.: þ86 451 86413792; fax: þ 86 451 86413922. E-mail address: [email protected] (J.-C. Zhu).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.09.017

energy and elastic properties of C15 Al2RE by using the projector augmented wave method. The results of calculation predicted the thermodynamics properties of Al2RE phases at 300 K successfully. The common intermetallics compounds in the Al–La alloy system such as Al2La, AlLa3 and Al3La phases have been emphasized and studied extensively by many researchers [12–15]. However, the electronic structures and elastic properties of Al2La, AlLa3 and Al3La are not well compared and reported at the same time. Thus, for the further developing the novel Al-based materials, it is necessary to investigate the ground-state properties of the intermetallics in the Al–La alloy system. In this paper, the first-principles approach is used to calculate the structural, elastic, and electronic properties of the binary intermetallics Al2La, AlLa3 and Al3La in the Al–La alloy system. This study will provide useful data for analysis and design of Al-based alloys, and also for future scientific comparison of Al2La, AlLa3 and Al3La phases. 2. Calculation method For theoretical analyses on the structural and electronic properties of Al2La, AlLa3 and Al3La, a Cambridge Serial Total Energy Package (CASTEP) was used for first-principles calculation. This software package employed the first principles plane-wave pseudopotential method based on the density functional theory (DFT) [16,17], in which the exchange and correlation terms were described with the generalized gradient approximation (GGA) of

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Perdew–Burke–Ernzerhof (PBE) parameterized by Perdew [18,19]. Ultrasoft pseudopotentials were used to describe the electron–ion interaction. The cutoff energies were all set at 450 eV for these calculations under a series of tests. Each calculation was considered converged when the maximum force on the atom was below ˚ the maximum displacement between cycles was below 0.01 eV/A, ˚ the maximum stress was below 0.02 GPa and the 5.0  10  4 A, energy change was below 5.0  10  6 eV/atom. For the calculation of density of states (DOS), the k-point meshes for Al2La, AlLa3 and Al3La were set to 4  4  4, 6  6  6 and 4  4  6, respectively, which were determined according to the Monkhorst–Pack scheme [20]. Although both Al2La and AlLa3 have a cubic crystal structure, Al2La with space group Fd3mS (no. 227) contains 16 Al atoms and 8 La atoms, while AlLa3 with space group Pm3m (no. 221) has 1 Al atom and 3 La atoms in the unit cell (as shown in Fig. 1a and b). Al3La has a hexagonal structure with space group P63/mmc (no. 194). There are 8 atoms in the unit cell. 2 La atoms occupy the 8g sites and 6 Al atoms occupy the 4a and 4c sites (as shown in Fig. 1c).

3. Results and discussions 3.1. Crystal structure and stability To obtain the lattice constants and equilibrium unit cell volumes, the structure was optimized by the Broyden–Fletcher– Goldfarb–Shanno (BFGS) method [21] to reach the ground state, in which both cell parameters and fractional coordinates of atoms were optimized simultaneously. The calculated equilibrium structural parameters for the Al2La, AlLa3 and Al3La intermetallics compounds are derived and listed in Table 1. It can be seen that the calculated lattice constants agree well with the experimental data with the error of less than 2%, which shows that the present calculation is highly reliable. From the view of thermodynamics, a lower formation enthalpy Hf defined as the total energy difference between the compounds and its constituents in proportion to the composition means the stronger alloying ability. Similarly, a lower cohesive energy means better structural stability. Cohesive energy Ec is defined as the work to decompose the crystal into single free atoms. In order to analyze the alloying ability and structural stability of Al2La, AlLa3 and Al3La phases, the formation enthalpy Hf and cohesive energy Ec were calculated by the following equation. [22,23]:

Fig. 1. The crystal structures of Al2La (a), AlLa3 (b) and Al3La (c) (the white spheres refer to La atoms and the purple spheres refer to Al atoms). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Hf ¼

1 La ðEtot nEAl solid mEsolid Þ nþm

ð1Þ

Ec ¼

1 La ðEtot nEAl atom mEatom Þ n þm

ð2Þ

Table 1 The calculated and experimental lattice constants, volumes of unit cell V0 as well as the cohesive energy Ec and formation enthalpy Hf for Al2La, AlLa3 and Al3La binary phases. Materials

Al2La AlLa3 Al3La a b c

From Ref. [12]. From Ref. [13]. From Ref. [14].

˚ Lattice constants (A) Cal.

Exp.

a¼ b¼ c¼ 8.120 a¼ b¼ c¼ 5.068 a¼ b¼ 6.635

a¼ b¼ c¼ 8.148a a¼ b¼ c¼ 5.093b a¼ b¼ 6.667

c ¼4.613

c ¼4.619c

V0 (A˚ 3)

Hf (eV/atom)

Ec (eV/atom)

533.49 130.15 181.29

 0.47  0.24  0.48

 4.56  4.61  4.52

4708

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where Etot is the total energy of the unit cell, n (or m) is the atom La number of Al (or La) in a unit cell, and EAl solid (or Esolid ) is the energy per Al (or La) element in the solid state of the crystal structure. La During calculation for EAl solid and Esolid , Al and La are of fcc and dhcp structure, respectively. EAl and ELa atom atom are the energy per atom of Al and La under isolated state, respectively. They are calculated by methods of a large box within which the corresponding element Al and La are put, respectively. The calculated results of the formation enthalpy Hf and cohesive energy Ec for Al2La, AlLa3 and Al3La phases are listed in Table 1. The negative formation enthalpy and cohesive energy of Al2La, AlLa3 and Al3La indicate that both of them have a strong alloying ability and structural stabilization. From the calculated values, it can be found that the formation enthalpy of AlLa3 is 0.24 eV/atom, higher than that of Al2La and Al3La, which indicates that the alloying ability of Al2La and Al3La is stronger than that of AlLa3, also indicating that the latter is more stable than the former. Furthermore, the similar cohesive energy of Al2La, AlLa3 and Al3La shows that they all have a higher structural stability. 3.2. Elastic properties Elastic constants are the measure of the resistance of a crystal to an externally applied stress. The single-crystal elastic constants can be obtained by first-principles calculations by applying small strains to the equilibrium unit cell and determining the corresponding variations in the total energy [24]. The elastic strain energy is given as follows: U¼

DE V0

¼

1 2

6 X 6 X i

Table 3 The elastic constants Cij (GPa) of Al2La, AlLa3 and Al3La. Materials

Structure

Al2La

Cubic

AlLa3 Al3La

Hexagonal

a

Elastic constants

Cal. Exp.a Cal. Cal.

C11

C12

C13

147.43 148.2 72.88 117.72

30.92 31.9 28.72 28.58

30.22

C33

C44

C66

164.60

36.49 43.6 22.07 63.71

44.57

From Ref. [28].

are given by Eqs. (4) and (5), respectively: C 12 4 0,

C 44 4 0, C 11 C 12 4 0,

C 12 4 0,

C 44 4 0,

C 11 C 12 4 0,

C 11 þ 2C 12 4 0, ðC 11 þC 12 ÞC 33 2C 213 40

ð4Þ ð5Þ

From Table 3 it can be seen that the calculated elastic constants Cij for Al2La and AlLa3 with cubic crystal and Al3La with hexagonal crystal satisfy well the Born stability criteria, suggesting Al2La, AlLa3 and Al3La phases are all mechanically stable structure at zero pressure. From calculated single crystal elastic constants, the polycrystalline structural properties such as bulk modulus (B), shear modulus (G), Young’s modulus (E) and Poisson’s ratio (u) can be derived using the Voigt–Reuss–Hill (V–R–H) approximations. The final values can be taken as Eqs. (6) and (7) [30]: B ¼ ðBV þBR Þ=2; G ¼ ðGV þ GR Þ=2; E ¼ 9BG=ð3B þ GÞ; u ¼ ð3BEÞ=6B;

ð6Þ

where V and R refer to the model of Voigt and Reuss, respectively: C ij ei ej

ð3Þ

j

Here DE is the energy difference, V0 is the volume of the unit cell, Cij are the components of the elasticity tensor, and ei, ej are applied strains. The indispensable number of strains is determined by the crystal symmetry [25]. For cubic and hexagonal crystal, there are three (C11, C12, C13(¼ C12), C33(¼C11), C44, C66(¼C44) ) and five (C11, C12, C13, C33, C44, C66(¼(C11–C12)/2) ) independent elastic constants, and the corresponding strains are shown in Table 2 [26,27]. For each kind of the above different strains of the different crystals, we have calculated the total energies by imposing appropriate strain. The calculation results of elastic constants are listed in Table 3. Although there are no available experimental data for elastic constants of AlLa3 and Al3La phases, the calculation results of elastic constants for Al2La agree well with the experimental values [28]. Therefore, our calculations of elastic constants can be supposed to be reliable. For a stable crystal structure, its elastic constants Cij should satisfy the Born stability criteria [29]. The mechanical stability of the crystal implies that the strain energy must be positive. The stability criteria for cubic and hexagonal crystal at zero pressure

BV ¼ 1=9ðC 11 þC 22 þ C 33 Þ þ 2=9ðC 12 þ C 13 þ C 23 Þ; 1 BR ¼ ðS11 þ S22 þ S33 Þ þ2ðS12 þ S13 þ S23 Þ GV ¼ 1=15ðC 11 þ C 22 þC 33 C 12 C 13 C 23 Þ þ1=5ðC 44 þ C 55 þC 66 Þ; 15 ð7Þ GR ¼ 4ðS11 þS22 þ S33 Þ4ðS12 þ S13 þ S23 Þ þ3ðS44 þ S55 þ S66 Þ where Cij and Sij are the elastic constants and elastic compliances, respectively. Sij is the inverse matrix of Cij. By applying the above V–R–H approximations, the calculated results for Al2La, AlLa3 and Al3La phases with mechanical stability are listed in Table 4. The bulk modulus B is a measure of resistance to volume change by applied pressure, and shear modulus G is a measure of resistance to reversible deformations upon shear stress [31]. From Table 4, it can be seen that the Al2La phase displays larger bulk modulus than AlLa3 and Al3La, suggesting that Al2La have stronger resistance to volume change by applied pressure. Regarding shear modulus, Al3La phase has the largest value, followed by Al2La and AlLa3 phases, which indicates that the directional bonding in Al3La is much stronger than that in Al2La and AlLa3 phases. In addition, the larger Young’s modulus E corresponds to the stiffer material. The calculated results from

Table 2 The strains used to calculate the elastic constants of Al2La, AlLa3 with cubic structure and Al3La with hexagonal structure. Structure

Strain types

Parameters (unlisted eij ¼ 0)

DE/V0 to 0 (d2)

Cubic

e(1) e(2) e(3)

e11 ¼ e22 ¼ d, e33 ¼  2d e11 ¼ e22 ¼ e33 ¼ d e12 ¼ e21 ¼ d/2

3(C11  C12) d2 3/2(C11 þ 2C12) d2 (C44d2)/2

Hexagonal

(d, (0, (0, (0, (d,

e11 ¼ e22 ¼ d e12 ¼ d/2 e33 ¼ d e13 ¼ e23 ¼ d/2 e11 ¼ e22 ¼ e33 ¼ d

C11 þC12 1/4(C11  C12) 1/2C33 C44 C11 þC12 þ 2C13 þ C33/2

d, 0, 0, 0, 0) 0, 0, 0, 0, d) 0, d, 0, 0, 0) 0, 0, d, d, 0) d, d, 0, 0, 0)

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Table 4 The elastic modulus (GPa) and Poisson’s ratio for Al2La, AlLa3 and Al3La using the Voigt, Reuss and Hill’s approximations. Materials

BV

BR

BH

GV

GR

GH

E

u

B/G

Al2La AlLa3 Al3La

69.42 42.77 64.23

69.40 42.77 62.76

69.41 42.77 63.49

45.40 22.94 55.13

43.33 22.94 53.43

44.36 22.94 54.28

109.69 57.25 126.73

0.24 (0.22a) 0.28 0.17

1.57 1.86 1.17

a

From Ref. [28].

Table 4 indicate that Young’s modulus of Al3La is 126.73 GPa, larger than that of Al2La and AlLa3, thus, Al3La is much stiffer than the others. Poisson’s ratio u which usually ranges from 1 to 0.5 is used to quantify the stability of the crystal against shear. The bigger Poisson’s ratio corresponds to better plasticity. From the calculated values we find that comparing to Al2La and AlLa3, Al3La has a smaller value of Poisson’s ratio, suggesting Al3La has a poorer plasticity than Al2La and AlLa3. Moreover, the ratio of bulk to shear modulus B/G has been proposed by Pugh [31] to predict brittle or ductile behaviors of materials. If B/G41.75, ductile behavior can be predicted for this material, otherwise the material shows a brittle behavior. In our present work, the values of Al2La, AlLa3 and Al3La are 1.57, 1.86 and 1.17, respectively, suggesting that Al2La and Al3La are absolutely brittle and AlLa3 is slightly ductile. The results are in accordance with the conclusions of Poisson’s ratio discussed above. 3.3. Elastic anisotropy The elastic anisotropy of crystals correlated with the possibility of inducing microcracks in materials has an important application in engineering science [32]. The method of measuring the elastic anisotropy is the percentage of anisotropy in compression and shear [33]. The expressions are taken as Eq. (8). AB ¼

BV BR ; BV þBR

AG ¼

GV GR GV þ GR

ð8Þ

The value of 100% means that the crystal is maximum anisotropic, while 0 represents the isotropy of material. The calculated results of percentage of anisotropy in compression and shear for Al2La, AlLa3 and Al3La listed in Table 5 show that Al2La and Al3La phases display small anisotropy in shear and compression, while AlLa3 phase is isotropy in shear and compression. Furthermore, for anisotropic Al2La and Al3La phases, AB shows that Al2La has better isotropy in compression, while AG indicates that Al3La is more isotopic in shear. It is worth to mention that for cubic crystal, another way of characterizing the anisotropy value A is given as following [34]: A¼

2C 44 C 11 C 12

ð9Þ

If A¼1, a completely perfect isotropic material can be predicted, while a value smaller or larger than unity indicates the degree of elastic anisotropy. The calculated anisotropy results of Al2La and AlLa3 with cubic crystal are listed in Table 5. It can be seen that Al2La and AlLa3 have an anisotropic factor of A¼0.64 and 0.99, respectively, indicating that AlLa3 phase can be seen as an isotropic material while Al2La is an anisotropic material. Overall, AlLa3 is an isotropic material while Al2La and Al3La all exhibit small anisotropy. 3.4. Thermodynamics properties As a fundamental parameter, the Debye temperature yD of a crystal correlated with specific heat, elastic constants and melting

Table 5 The anisotropic factors, the percentage of anisotropy in the compression AB and shear AG (%) and the Debye temperature (K) of Al2La, AlLa3 and Al3La. Materials

Structure

A

AB (%)

AG (%)

yD (K)

Al2La AlLa3 Al3La

Cubic

0.63 0.99 –

0.014 0 1.16

2.33 0 1.57

333.4 (374a) 402.9 450.5

a

Hexagonal

From Ref. [28].

temperature is an important material parameter to describe phenomena of solid state physics. It depends basically on the elastic constants of the crystal, and can be given as following [35]:    h 3n NA r0 1=3 YD ¼ um ð10Þ k 4p M where h is the Planck constant, k is the Boltzmann constant, NA is the Avogadro number, n is the number of atoms in the molecule, M is the molecular weight, and r0 is the density. The average sound velocity um can be defined as follows [35]: " !#1=3 1 2 1 um ¼ þ 3 u3s u3l  us ¼

G

r0



1=2 ;

ul ¼

Bþ

 1=2 4G =r0 3

ð11Þ

where ul is the longitudinal sound velocity, and us is the shear sound velocity. In Table 5, the calculated Debye temperature for Al2La, AlLa3 and Al3La are 333.4, 402.9 and 450.5 K, respectively. Although there are no available experimental data for the Debye temperature of AlLa3 and Al3La, the calculated Debye temperature of Al2La agrees with the reported 374 K [28]. Nevertheless, the calculated data would be helpful for the further investigation. 3.5. Electronic structure The density of states (DOS) is an important theoretical quantity to reveal the nature of chemical bonding interactions within compounds, and to understand their structural stability mechanisms. The total and partial DOS at equilibrium lattice constants including Al2La, AlLa3 and Al3La phases were calculated, as shown in Fig. 2, where the energy zero is set as the Fermi energy (EF). It is found that the calculated total DOS exhibits obvious metallic behavior because the values of total DOS of these phases at the Fermi level are all higher than 0, and the total DOS of these phases is mainly dominated by Al-s, Al-p, La-s, La-p, and La-d states. All of these phases display a strong hybridization nearby the Fermi level, being characteristic of Al-p and La-d states. From Fig. 2, it is found that the main bonding peaks between 10 and  3 eV mainly originate from the contribution of valence electron numbers of Al-s and Al-p orbits with a mall number of La-s, La-p and La-d states, while the p states of Al hybridize strongly with the La-d states in the range between  3 eV and 5 eV. The characteristic of hybridization around the Fermi level indicates the presence of the directional covalent bonding. It is also generally considered that the

Z.-S. Nong et al. / Physica B 407 (2012) 4706–4711

Al2La

DOS (states/eV)

8 6 4 2 0 4 3 2 1 4 0 3 2 1 0 -10

Al (s) Al (p)

La (s) La (p) La (d) -8

-6

-4

-2 0 2 Energy (eV)

DOS (states/eV)

DOS (states/eV)

4710

10 8 6 4 2 5 0 4 3 2 1 4 0 3 2 1 0 -10

4

6

8

8 6 4 2 2.0 0 1.5 1.0 0.5 6 0.0

AlLa3

Al (s) Al (p)

La (s) La (p) La (d)

4 2

10

0 -10

-8

-6

-4

-2 0 2 Energy (eV)

4

6

8

10

Al3La

Al (s) Al (p)

La (s) La (p) La (d)

-8

-6

-4

-2 0 2 Energy (eV)

4

6

8

10

Fig. 2. Total and partial density of states nearby the Fermi level for Al2La (a), AlLa3 (b) and Al3La (c). The energy 0 is the Fermi energy (EF).

formation of covalent bonding would enhance the strength of material in comparison with the pure metallic bonding [36]. Thus, these compounds all have a pronounced stability.

4. Conclusion In this paper, the structural, elastic and electronic properties of Al2La, AlLa3 and Al3La phases in the Al–La alloy system were investigated by first-principles calculations based on the DFT. The calculated equilibrium lattice constants of both phases were in accordance with experimental values. The calculation results of formation enthalpy and cohesive energy showed that Al2La, AlLa3 and Al3La all have a higher structural stability, and the alloying ability of Al2La and Al3La is stronger than that of AlLa3. The independent elastic constants were calculated, showing that Al2La, AlLa3 and Al3La phases are all mechanically stable at zero pressure. The polycrystalline elastic parameters (B, G, E and u) were obtained by the V–R–H method. The calculated results showed that Al2La and Al3La are absolutely brittle and AlLa3 is slightly ductile. In addition, AlLa3 phase is an isotropic material while Al2La and Al3La are anisotropic materials by discussing elastic anisotropies including percentage of bulk and shear anisotropies and the anisotropy factor A for cubic crystal. According to the calculated electronic structure of Al2La, AlLa3 and Al3La phases, the stability of these phases may be attributed to the hybridization nearby the Fermi level caused between Al-p and La-d states. Overall, this study will provide useful data for further design of Al-based alloys, and also for future scientific comparison of Al2La, AlLa3 and Al3La phases.

Acknowledgments This work was supported by 2012 Opening Funding of National Key Laboratory on Advanced Composites in Special Environment. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

M.Z. Quadir, M. Ferry, O.A. Buhamad, P.R. Munroe, Acta Mater. 57 (2009) 29. S. Zhang, W. Hu, R. Berghammer, G. Gottstein, Acta Mater. 58 (2010) 6695. W.X. Shi, B. Gao, G.F. Tu, S.W. Li, J. Alloys Compd. 508 (2010) 480. K.E. Knipling, D.C. Dunand, D.N. Seidman, Z. Met. Kd. 97 (2006) 246. G. Meng, B. Li, H. Li, H. Huang, Z. Nie, Mater. Sci. Eng. A 516 (2009) 131. D.E. MacLaughlin, M.S. Rose, B.L. Young, Physica B 326 (2003) 387. V.E. Sidorov, O.A. Gornov, V.A. Bykov, L.D. Son, V.G. Shevchenko, V.I. Kononenko, K.Y. Shunyaev, J. Non-Cryst. Solids 353 (2007) 3094. T. Ebiharaa, T. Takahashi, K. Tezukaa, M. Hedob, Y. Uwatokob, S. Uji, Physica B 359 (2005) 272. J. Zhou, B. Sa, Z. Sun, Intermetallics 18 (2010) 2394. V. Srivastava, S.P. Sanyal, M. Rajagopalan, Physica B 403 (2008) 3615. X. Tao, Y. Ouyang, H. Liu, F. Zeng, Y. Feng, Y. Du, Z. Jin, Comput. Mater. Sci. 44 (2008) 392. A.B. Rao, P. Kistaiah, R.N. Rajasekhar, K. Satyanarayana, J. Mater. Sci. Lett. 1 (1982) 432. A. Iandelli, Symposium 1 (1960) 376. I.I. Zalutskii, P.I. Kripyakevich, Kristallografiya 12 (3) (1967) 394. P. Villars, Pearson’s Handbook of Crystallographic Data, ASM International, Materials Park, OH, 1997. P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 384. W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. M. Marlo, V. Milman, Phys. Rev. B 62 (2000) 2899–2907. J.A. White, D.M. Bird, Phys. Rev. B 50 (1994) 4954. H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. C.G. Broyden, J.E. Dennis, J.J. Moref, J. Appl. Math. 12 (1973) 223. M.I. Medvedeva, Y.N. Gornostyrev, D.L. Novikov, O.N. Mryasov, A.J. Freeman, Acta Mater. 46 (1998) 3433. B.R. Sahu, Mater. Sci. Eng. B 49 (1997) 74.

Z.-S. Nong et al. / Physica B 407 (2012) 4706–4711

[24] M.M. Wu, L. Wen, B.Y. Tang, L.M. Peng, W.J. Ding, J. Alloys Compd. 506 (2010) 412. [25] M. Mattesini, R. Ahuja, B. Johansson, Phys. Rev. B 68 (2003) 184108. [26] O. Beckstein, J.E. Klepeis, G.L.W. Hart, O. Pankratov, Phys. Rev. B 63 (2001) 134112. [27] X.H. Deng, B.B. Fan, W. Lu, Solid State Commun. 149 (2009) 441. [28] R.J. SchiltzJr, J.F. Smith, J. Appl. Phys. 45 (1974) 4681. [29] M. Born, K. Huang, Dynamical Theory of Crystal Lattices [M], Oxford University Press, Oxford, 1954, pp. 10–12. [30] H.Z. Yao, L.Z. Ouyang, W.Y. Ching, J. Am. Ceram. Soc. 90 (2007) 3194–3204.

4711

[31] S.F. Pugh, Philos. Mag. Ser. 7 (45) (1954) 823. [32] V. Tvergaard, J.W. Hutchinson, J. Am. Ceram. Soc. 71 (1988) 157. [33] D.H. Chung, W.R. Buessem, in: F.W. Vahldiek, S.A. Mersol (Eds.), Anisotropy in Single Crystal Refractory Compound, vol. 2, Plenum, New York, 1968, p. 217. [34] W.Y. Yu, N. Wang, X.B. Xiao, B.Y. Tang, L.M. Peng, W.J. Ding, Solid State Sci. 11 (2009) 1400. [35] D. Music, A. Houben, R. Dronskowski, J.M. Schneider, Phys. Rev. B 75 (2007) 174102. [36] C. Zhang, Z. Zhang, S. Wang, H. Li, J. Dong, N. Xing, Y. Guo, W. Li, J. Alloys Compd. 448 (2008) 53.