Computational Materials Science 74 (2013) 129–137

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

First-principles study of structure and properties of x-Ti2Zr

0927-0256/$ - see front matter � 2013 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.commatsci.2013.03.025

⇑ Corresponding authors at: National Key Laboratory of Science and Technologyon Reliability and Environment Engineering, Beijing Institute of Spacecraft Envi-ronment Engineering, Beijing 100094, China (Z. Gong). Tel.: +86 10 68746609 (Z.Gong), tel.: +86 021 34204573 (D. Wei).

E-mail addresses: gongzz@263.net (Z. Gong), dqwei@sjtu.edu.cn (D.-q. Wei).

Pinliang Zhang a, Fanchen Meng a, Zizheng Gong b,a,⇑, Guangfu Ji c, Shouxin Cui d, Dong-qing Wei d,⇑a Key Laboratory of Advanced Technologies of Materials of Ministry of Education, School of Material Science and Engineering, Southwest Jiaotong University, Chengdu 610031, Chinab National Key Laboratory of Science and Technology on Reliability and Environment Engineering, Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, Chinac Laboratory for Shock Wave & Detonation Physics, Institute of Fluid Physics, CAEP, Mianyang 621900, Chinad College of Life Science and Biotechnology and Research Center Astronautics, Shanghai Jiaotong University, Shanghai 200240, China

a r t i c l e i n f o

Article history:Received 20 November 2012Received in revised form 13 March 2013Accepted 14 March 2013

Keywords:x-Ti2ZrStructureElastic constantsElectronic propertiesThermodynamic properties

a b s t r a c t

The structural, elastic, electronic properties, and Debye temperature of x-Ti2Zr under compression wereinvestigated by the first-principles pseudopotential method based on density functional theory (DFT).The calculated structural parameters at zero pressure are in consistent with experimental values. Theelastic constants and their pressure dependence were obtained using the static finite strain technique.We derived the bulk modulus, Young’s modulus and Poisson’s ratio for x-Ti2Zr. The Debye temperaturewas obtained by the average sound velocity, and compared with other Ti–Zr metals and alloys. The pres-sure dependence of electron distribution, as well as the s ? d electron transfer indicates that there is ax ? b phase transition in the high pressures regime. Finally, the heat capacity at the constant pressureand the linear thermal expansion coefficient as a function of temperature had been obtained.

� 2013 Published by Elsevier B.V.

1. Introduction

Group IV metals and their alloys have attracted tremendous sci-entific and technological interests, due to its high strength-to-weight ratio, high rigidity-to-weight ratio and excellent resistanceto corrosion [1]. Ti–Zr alloys, as one of the group IV alloys, have po-tential uses in aerospace, medical, nuclear industries. Many solidphases and allotropes have been observed in the phase diagrams[2,3]. At ambient condition, it possesses a hexagonal close-packed(hcp) structure (a-phase), and transforms to body-centered cubic(bcc) b-phase under high temperature and a three atoms hexagonalstructure (x-phase) under high pressure [4]. Recently, there aremany investigations related to the a ? x phase transitions of theTi–Zr systems, for example, studies of pure Ti [5–10] and Zr[5,9,11–13], TiZr alloys by Bashkin [14–16] and Aksenenkov [17].For group IV metals, which has a narrow d band in the midst of abroad s–p band [18,19], the electrons transfer from s–p band to dband induced by pressure plays an important role in the phasestability.

The elastic properties are closely related to many fundamentalsolid-state properties, and have a critical impact on many practicalapplications related to the mechanical properties of a solid [20].

Therefore, investigation of their elastic properties is importantfor technological applications. Early investigations on Ti–Zr system(mostly on Ti [21–24], Zr [13,25–28] and TiZr [16,29]) showed thatbrittleness of x-phase was more predominant than that of a- andb-phase [27,29,30], due to the unique structure of x-phase whichcrystallizes in the AlB2 structure, in which Al atom occupied the 1aWychoff site (0,0,0) and two B atoms held the 2d Wychoff site (1/3,2/3,1/2) and (2/3,1/3,1/2) [9]. This structure forms a sequence oftype ABAB. . . describes by (0001) plane, and with graphite-likenets of B-plane which has three nearest neighbors [31,32]. Othercomposites with this structure [33] also show unique physicaland chemical properties such as hardness, high melting point,and chemical inertness, and belong to the most promising engi-neering materials.

Theoretically, the first-principles density functional theory(DFT) is very useful in studying the structure and properties ofTi–Zr metals and alloys. Hao [21,30] and Mei [22] predicted thehigh pressure phase transitions and elastic properties of Ti basedon the Perdew–Burke–Ernzerhof (PBE) generalized-gradientapproximation (GGA) [34], which is consistent with the experi-ments. Wang [25,29] used the frozen-core projector augmentedwave (PAW) method [35] with the VASP code [36] to study thephase transitions, elastic modulus and superconductivity of Zrand TiZr successfully. The bulk modulus and shear modulus arecalculated from the Voigt–Reuss–Hill (VRH) approximations [37].In addition, the plane-wave pseudopotential DFT method has beensuccessfully applied to investigate the thermodynamic propertiesof MgB2 [38], AlB2 [31], ZrB2 [39] and HfB2 [40].

130 P. Zhang et al. / Computational Materials Science 74 (2013) 129–137

To the best of our knowledge, although the x-Ti2Zr alloys havebeen synthesized and the crystal structures have been determined[41–43], the mechanical and thermodynamic properties, especiallyunder high pressure, are rarely reported up to now. In this study,we calculated the structure, elastic constants, electron distributionand thermodynamic properties of x-Ti2Zr in the regime of 0–50 GPa by using the first-principles plane-wave pseudopotentialdensity function theory [34] based on the Cambridge Serial TotalEnergy Package (CASTEP) [44]. In Section 2, we make a brief reviewof the theoretical method. The results and discussions are pre-sented in Section 3. The conclusions are summarized in theSection 4.

2. Theoretical method

2.1. Computational details

In present calculation, the electron exchange and correlationpotential was treated by the generalized gradient approximation(GGA) using the Perdew–Burke–Ernzerhof (PBE) functional [34].Full geometry optimization for each structure was performed bythe Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimizationscheme [45]. The ultrasoft pseudopotentials [46] were used, andthe pseudo-atom calculations were performed for Ti 3s2 3p6 3d24s2 and Zr 4s2 4p6 4d2 5s2. The cut-off energy for plane-wavewas set to 550 eV. The k-point mesh sampled in the Brillouin-zonewere 16 � 16 � 18 (for x-Ti, x-Zr and x-Ti2Zr) according to theMonkhorst–Pack scheme [47]. The self-consistent convergence ofthe total energy was set to 5 � 10�6 eV/atom. Atoms were allowedto relax until Hellman–Feynman forces were less than 0.01 eV/Å,the maximum strain value was 0.0005 Å, and all the stress compo-nents within 0.02 GPa.

2.2. Elastic properties

The single-crystal elastic constants can be obtained by applyingstatic finite strains to the equilibrium unit cell [48]. The elastic en-ergy is given by:

DE ¼ V0

2

X6

i¼1

X6

j¼1

Cijeiej ð1Þ

where V0 is the volume of the undistorted lattice cell, DE representsthe energy different from the strain with vector e, and C is the ma-trix of the elastic constants. The strain tensor e links with the strainvector e by:

e ¼e1 e6=2 e5=2

e6=2 e2 e4=2e5=2 e4=2 e3

0B@

1CA ð2Þ

For hexagonal crystals, there are five independent elastic con-stants: C11, C12, C13, C33 and C44 [48]. The elastic constants can beobtained by calculating the second derivatives of the DE with re-spect to strain:

Cij ¼1

V0

@2DE

@d2

!ð3Þ

We apply the strain e = (d,d,0,0,0,0), (0,0,0,0,0,d), (0,0,d,0,0,0),(0,0,0,d,d,0) and (d,d,d,0,0,0) to the hexagonal crystal, respec-tively, Then C11 + C12, C11 � C12, C33, C44 and C13 are calculated.

2.3. Thermodynamic properties

For a system with a given averaged atomic volume V and tem-perature T, the Helmholtz free energy can be approximated as[49,50]:

FðV ; TÞ ¼ E0ðVÞ þ FphðV ; TÞ þ FelðV ; TÞ ð4Þ

where E0(V) is the static zero temperature energy, Fel(V,T) repre-sents the electronic contribution, Fph(V,T) is the vibrational contri-butions to the free energy of the lattice ions. In thequasiharmonic approximation, the Fph(V,T) is given by:

FphðV ; TÞ ¼ kBTX

q

Xj

ln 2 sinh�hxjðq;VÞ

2kBT

� �� �ð5Þ

where ⁄ and kB are Planck’s constant and Boltzmann’s constant,respectively. xj(q,V) represents the frequency of the jth phononmode at wave vector q in the Brillouin zone. The charge density istemperature dependent through occupation numbers according tothe Fermi–Dirac distribution, the electronic entropy can be writtenas [49]:

SelðV ; TÞ ¼ �kB

Znðe;VÞ½f ln f þ ð1� f Þ ln ð1� f Þ�de ð6Þ

where n(e,V) is the electronic density of states with f being the Fer-mi distribution, e is mean vibrational energy.

The energy Eel can be given by:

EelðV ; TÞ ¼Z

nðe;VÞf ede�Z eF

nðe;VÞede ð7Þ

where eF is the Fermi energy.The thermal properties of the system has been deduced from

the Helmholtz free energy F(V,T) [51]. One can easily obtain thelinear thermal expansion coefficient aL, the heat capacity at con-stant volume CV, and the heat capacity at constant pressure CP:

aL ¼ ð1=3VÞð@V=@TÞP ð8Þ

CV ¼ �Tð@2F=@T2ÞV ð9Þ

CP ¼ CV þ 9a2L BT VT ð10Þ

where BT is the isothermal bulk modulus.

3. Results and discussion

3.1. Structural properties

The group IV metals and their alloys in the x-phase have hex-agonal AlB2 structure and belong to space group P6/mmm (No.191). Usually there is no preferential occupancy of A-site and B-site(equivalent positions are A = (0,0,0) and B = (2/3,1/3,1/2), (1/3,2/3,1/2)) by any one of the constituents [9]. However, it has beendemonstrated that for x-phase Ti2ZrO [52] and Ti2HfO [53], theatoms with smaller size are preferential occupancy of B-site. Thestructure of x-Ti2Zr were constructed (Fig. 1), with Zr atom at 1a(0,0,0), and two Ti atoms which had smaller size at 2d (1/3,2/3,1/2) and (2/3,1/3,1/2).

In order to determine the theoretical equilibrium geometry, aseries of different lattice parameters were used to calculate the to-tal energy E and the corresponding volume V, and then fitted theenergy–volume (E–V) data with the third-order Birch–Murnaghanequation of state (EOS) [54]. The equilibrium structural parame-ters, bulk modulus B0 and its pressure derivative B00 for x-Ti2Zr,x-Ti and x-Zr are listed in Table 1, together with experimentaland other theoretical values for comparisons. From Table 1, it isclear that our present values are consistent well with experimental

Fig. 1. The structure model of x-Ti2Zr (Zr atoms are black, and white for Ti).

Table 1The equilibrium lattice parameters, B0 and B00 of x-Ti2Zr, x-Ti, x-Zr, and x-TiZr.

Phase a (Å) c (Å) c/a B0 (GPa) B00

x-Ti2Zr This worka 4.721 3.011 0.638 104.4 3.50Exptb 4.795 3.020 0.630

x-Zr This worka 5.041 3.146 0.624 95.2 3.00Wangc 5.036 3.152 0.626 96.9 3.39Haod 5.056 3.150 0.623 101.1 3.27Expt 5.039e 3.150e 0.625e 104.0f 2.80f

104.0g 2.05g

x-Ti This worka 4.581 2.832 0.618 111.8 3.50Haoh 4.578 2.829 0.618 112.6 3.60Meii 4.575 2.828 0.618 118.0 3.40Exptj 4.598 2.823j 0.614j 138.0k

4.588l 2.837l 0.618l 115.0m 3.35m

x-TiZr Wangn 4.825 3.011 0.624 100.5 3.55Trubitsino 4.561 2.814 0.617Exptp 4.840 2.991 0.618 146.0 1.70

a Present work.b Ref. [43].c Ref. [25].d Ref. [26].e Ref. [56].f Ref. [13,27].g Ref. [28].h Ref. [21,30].i Ref. [22].j Ref. [1].k Ref. [23].l Ref. [57].

m Ref. [24].n Ref. [29].o Ref. [55].p Ref. [16].

Fig. 2. The normalized lattice parameters a/a0, c/c0 and primitive cell volume V/V0

as a function of pressure (P) for x-Ti2Zr.

Fig. 3. Variation of the Ti–Ti and Ti–Zr bond length and ratio with pressure.

P. Zhang et al. / Computational Materials Science 74 (2013) 129–137 131

and other theoretical values for x-Ti and x-Zr. However, there ex-ists little difference between calculated and experimental for x-Ti2Zr. This discrepancy ascribed to the imperfect crystals, becausethe metastable x-phase was obtained at ambient pressure by cool-ing of the b-phase under high pressures with subsequent unload-ing at room temperature [29,42,55]. Moreover, the temperatureeffects maybe a critical factor.

Fig. 2 shows the normalized lattice parameters a/a0, c/c0 andprimitive cell volume V/V0 as a function of pressure in the rangeof 0–50 GPa. The subscript ‘‘0’’ means equilibrium lattice parame-ter and volume at zero pressure. It is found that the equilibrium ra-tios a/a0 and c/c0 decrease with the pressure, the compressionratios are very close, indicating a quite rare nearly uniform com-pression in the basal plane. The changes of Ti–Ti and Ti–Zr bonddistances with the applied pressures are plotted in Fig. 3. The bondlength ratios dTi–Ti/d(Ti–Ti)0 are equal to dTi–Zr/d(Ti–Zr)0, showing thatthe compression of the Ti–Ti bond which determines the latticeconstant a is virtually the same as that of Ti–Zr which determinesthe lattice constant c.

3.2. Elastic properties

Elastic constants determine the mechanics properties ofmaterials, such as Young’s modulus, bulk modulus and shear mod-ulus. For hexagonal crystals, the mechanical stability conditions ofcrystals can be written as [58,59]: C44 > 0, C11>|C12|,(C11 + C12)C33 > 2C13

2. The calculated theoretical Cij values have tosatisfy these conditions for any pressure to ensure the mechanicalstability of x-Ti2Zr under pressure of 0–50 GPa.

The independent components as a function of pressure are illus-trated in Fig. 4 (and are listed in Table 2). It is obvious that the C11,C33, C12, C13 increase monotonically with pressure, while C44 de-crease with increasing pressure. The C11 and C33 are more relativelyimportant because they are related to the atomic bonding charac-teristics and deformation behavior of transition metal. From Fig. 4,it is observed that C33 > C11, which is due to the fact that the atomicbonding of neighboring atoms along the {0001} planes is strongerthan that along the {10 �10} plane.

Since the bulk modulus (B) is related to the cohesive energy(bonding energy) of atoms in crystals, it can be used to describethe average atomic bond strength [60]. Furthermore, the shearmodulus and Young’s modulus are also important quantities deter-mining the strength of the materials [61].

The isotropic polycrystalline bulk modulus and shear moduluscan be obtained from the VRH method [62]. For hexagonal crystals,the bulk modulus and shear modulus is shown to be [63]:

Fig. 4. The calculated independent elastic constants Cij as a function of pressure (P)for x-Ti2Zr.

132 P. Zhang et al. / Computational Materials Science 74 (2013) 129–137

BVoigt ¼ ð1=9Þ½2ðC11 þ C12Þ þ 4C13 þ C33� ð11Þ

BReuss ¼ C2=ðC11 þ C12 þ 2C33 � 4C13Þ ð12Þ

GVoigt ¼ ð1=30ÞðC11 þ C12 þ 2C33 � 4C13 þ 12C44 þ 12c66Þ ð13Þ

GReuss ¼ ð5=2Þ½C2C44C66�=½3BVoigtC44C66 þ C2ðC44 þ C66Þ� ð14Þ

where

C66 ¼ ðC11 � C12Þ=2 ð15Þ

C2 ¼ ðC11 þ C12ÞC33 � 2C213 ð16Þ

Table 2The elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s mo

Phase P C11 C12 C13 C

x-Ti2Zr This worka 0 173.4 81.6 60.3 15 189.5 96.1 73.2 2

10 206.3 117.4 87.4 215 218.7 136.8 100.5 220 229.7 152.9 112.4 225 244.5 179.3 126.8 230 257.4 193.8 138.0 335 270.8 218.8 151.0 340 275.0 236.0 160.0 345 284.1 255.2 170.1 350 296.3 277.0 181.1 3

x-Zr This worka 0 164.5 71.6 46.2 2Wangb 0 161.7 72.6 53.5 1Expt 0

x-Ti This worka 0 196.2 82.2 51.7 2Haoe 0 192.3 77.7 54.6 2Expt 0

x-TiZr Wangh 0 170.9 76.2 51.9 2Expti 0

a Prensent work.b Ref. [25].c Ref. [13,27].d Ref. [28].e Ref. [21,30].f Ref. [23].g Ref. [24].h Ref. [29].i Ref. [16].

The equations follow the Reuss and Voight definitions, whilethe Hill values are defined as the arithmetic average of the Reussand Voight values [37] (the minimum value for Reuss results andthe maximum value for Voigt results):

B ¼ BHill ¼ ðBVoigt þ BReussÞ=2 ð17Þ

G ¼ GHill ¼ ðGVoigt þ GReussÞ=2 ð18Þ

The Young modulus is defined as the ratio between stress andstrain, and is correlated to the stiffness of the solid. The Poisson ra-tio provides a lot of information about the characteristics of thebonding forces, and can be obtained from the following formulas:

E ¼ 9BG=ð3Bþ GÞ ð19Þ

m ¼ ð3B� 2GÞ=½2ð3Bþ GÞ� ð20Þ

The elastic constants of x-Ti2Zr, x-Ti, x-Zr and x-TiZr at zerotemperature and zero pressure are listed in Table 2. As it can beseen from Table 2, the experimental values for x-Ti, x-Zr and x-TiZr are larger than the calculated results. This may be due toimperfect crystals in the samples of the experiments. So far wehave not found any experimental results for x-Ti2Zr for compari-son. It is found that the elastic constants of x-Ti2Zr and x-TiZr be-tween corresponding of their archetype metals (x-Ti and x-Zr),and tend to the components which has a higher content. Therefore,we can regulate the mechanical behavior of Ti–Zr alloys throughadjusting the content of components. The evolutions of the bulkmodulus and shear modulus as a function of pressure are shownin Fig. 5. It is worth noting that the bulk modulus increase withincreasing pressure, while the shear modulus instead of increaseslightly first and decrease afterwards. The results are different fromx-Ti [30], x-Zr [25] or x-TiZr [29], their B and G both increasewith the increasing pressure.

The G and B relate to the resistance to plastic deformation andfracture, respectively. Pugh [64] proposed an empirically relation-ship to predict the ductile and brittle behavior of solids bases on

dulus E (GPa) and Poisson’s ratio m at various pressures (GPa).

33 C44 B G E m

96.2 50.2 105.3 51.5 132.7 0.29021.3 52.3 120.5 53.4 139.7 0.30743.7 53.5 137.8 53.5 142.1 0.32863.9 53.6 153.0 52.3 140.8 0.34782.4 53.0 166.3 51.1 139.1 0.36199.6 51.5 183.8 47.7 131.8 0.38115.7 49.2 196.7 46.9 130.3 0.39030.7 46.3 212.6 42.6 119.9 0.40644.3 42.7 222.9 37.4 106.2 0.42156.8 38.5 235.1 32.6 93.4 0.43468.1 33.7 248.7 27.5 79.5 0.447

20.7 33.6 97.3 46.0 119.2 0.29695.6 33.7 97.5 43.6 113.7 0.306

104.0c, 104.0d 45.1c 118.3c 0.311c

46.5 54.5 112.1 62.0 157.0 0.26748.7 54.3 112.8 62.1 157.5 0.267

138f, 115.0g

09.2 39.6 100.5 48.7 125.3 0.291146

Fig. 5. The relation of the calculated bulk modulus (B) and shear modules (G) withpressure (P) for x-Ti2Zr.

P. Zhang et al. / Computational Materials Science 74 (2013) 129–137 133

elastic constants, by the ratio R = G/B. If R > 0.57, the solid behavesin a brittle manner, otherwise in a ductile manner. The results inthis study show that the G/B ratio is 0.489 for x-Ti2Zr, indictingthe intrinsic ductility at ambient condition. The trends of G/B forx-Ti2Zr may be discussed as displaying in Fig. 6, which illustratesthe calculated values for x-Ti, x-Zr and the values for x-TiZr from[29]. It can be seen that, with the increasing of pressure, the G/Bvalues for all are decreasing, indicating that all of them lead to lessbrittleness. On the other hand, Frantsevich [61] introduced anothermethod, Poisson’s ratio m, to predict the brittle and ductile behav-iors of solids. According to Frantsevich’s critical, if m < 1/3, the solidbehaves in a brittle manner. However, there are deviation betweenthe results of Pugh and Frantsevich. According to criterion pro-posed by Frantsevich, the x-Ti2Zr behaves in a brittle manner be-low 10 GPa. However, the ductile properties predicted by twomethods have the same tendency of evolution under increasingpressure. In Fig. 6, it is clear that the value of G/B for x-Ti(0.551) is higher than others at ambient pressure, then x-Ti2-

Zr > x-TiZr > x-Zr. The brittleness for x-Ti is more predominantthan that of others, and x-Zr is rather ductile. The Ti–Zr alloysshow moderate manner, which control by the content of compo-nents. In general, the brittle manner of a, b and x-phase for groupIV metals and their alloys is x > a > b.

Fig. 6. The relation of the calculated G/B with pressure (P) for x-Ti2Zr, x-Ti, x-Zr,and x-TiZr.

The stability of the bcc crystals with respect to the shear defor-mations along (110) [1 �10] and (001) [100] directions [65],respectively, and shear deformations along (10 �10) [11 �20] direc-tion of the hcp crystals [66], while there are no slip bands [9] com-bine with graphite-like net structure lead to brittle manner for x-phase. The graphite-like net structure which is shown in Fig. 7 isconfirmed by difference charge density contour along the (11 �20)plane for x-Ti2Zr. There are many charge changes between Tiatoms, indicating a strong directional bonding exists betweenthem. Therefore, the Ti combine with the three nearest neighborsform a graphite-like net structure.

According to the definition of bulk modulus, the anisotropies ofthe bulk modulus along the a-axis (Ba) and the c-axis (Bc) can beexpressed as follows [67–70]:

Ba ¼ aðdP=daÞ ¼ K=ð2þ aÞ ð21Þ

Bc ¼ cðdP=dcÞ ¼ Ba=a ð22Þ

K ¼ 2ðC11 þ C12Þ þ 4C13aþ C33a2 ð23Þ

a ¼ ðC11 þ C12 � 2C13Þ=ðC33 � C13Þ ð24Þ

The calculated Ba and Bc at 0–50 GPa are shown in Fig. 8. Thevalues of Ba and Bc increase with the increasing pressure. The valueof Ba/Bc at zero pressure is 0.989, indicating mechanical behavior ofx-Ti2Zr at zero pressure is anisotropy, and the mechanical behav-ior keep anisotropy (Ba < Bc) with increasing pressure. When thepressure are more than 31.1 GPa, the bulk modulus along the a-axis and the c-axis turn to Ba > Bc, meaning anisotropy. There isan interesting phenomenon occurs around 31.1 GPa, which we findthat the curves for Ba and Bc cross, and the mechanical behavior ofx-Ti2Zr is isotropy, while the x-Ti performs distinct anisotropy,and will gradually weaken with the applied pressure increasing[30].

Now, we investigate anisotropies of the compressional waveand two shear waves. The acoustic velocities are obtained fromelastic constants by solving Christoffel equation [71] for x-Ti2Zr.For hexagonal crystals, the anisotropy of compressional wave (P),the wave polarized perpendicular to the basal plane (S1) and thepolarized one in the basal plane (S2) are calculated:

DP ¼ C33=C11 ð25Þ

DS1 ¼ ðC11 þ C33 � 2C13Þ=ð4C44Þ ð26Þ

DS2 ¼ 2C44=ðC11 � C12Þ ð27Þ

Fig. 7. The total charge density contour (unit in e/Å3) for x-Ti2Zr.

Fig. 8. The calculated bulk modulus along the a-axes Ba and a-axes Bc as a functionof pressure (P) for x-Ti2Zr.

134 P. Zhang et al. / Computational Materials Science 74 (2013) 129–137

The three calculated pressure dependencies of the anisotropiesof elastic waves are plotted in Fig. 9. These results can be under-stood by Born and Huang proposed the central nearest-neighborforces mode (CNNF) [72] for the hexagonal crystals, the elastic con-stants ratio C33:C11:C12:C13:C44 is 32:29:11:8:8. According to CNNFmode, DP = 1.1, DS1 = 1.4 and DS2 = 0.9. In this work, DP, DS1 andDS2 at zero pressure are 1.13, 1.24 and 1.09, respectively. Theyare close to CNNF mode, and indicating that the anisotropy is weakat zero pressure. However, we also note that the values of DP, DS1

and DS2 increase with increasing pressure, in which DS1 and DS2

exhibit abrupt changes but DP is relatively moderate. This indicatesthat the strong anisotropy induced by high pressures.

3.3. Debye temperature

Debye temperature (H) is one of the important physical param-eters in solid physics. It not only relates to distortion degree ofcrystals lattice, but also represents the adhesion strength betweenthe atoms. Therefore, many physical properties of solids relateswith Debye temperature, such as heat capacity, elastic constants,thermal expansion coefficient and melting temperature. The Debyetemperature can be obtained from elastic constants as follow [73]:

Fig. 9. Anisotropies compressional wave (DP), shear waves (DS1) and (DS2) as afunction of pressure (P) for x-Ti2Zr.

H ¼ hkB

3n4pV

� �1=3

vm ð28Þ

where h is Plank constant, kB is Boltzmann constant, V is the vol-ume, n is the number of atom in the box. The average wave velocity(vm) [73] can be obtained by shear (vs) and compressional wavevelocity (vp) of the polycrystalline solid, obtained from Navier’sequation [74]:

vp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBHill þ 4=3GHillÞ=q

pð29Þ

vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGHill=q

pð30Þ

vm ¼ ½ð2=v3s þ 1=v3

pÞ=3��1=3 ð31Þ

The values of Debye temperature at zero pressure are listed inTable 3, together with other theoretical and available experimentalvalues. The calculated H of x-Zr is in good agreement with exper-imental values. However, there are no experimental values forother materials. The Debye temperature of the four materials asa function of pressure is shown in Fig. 10. For x-Ti2Zr, x-Ti andx-Zr, the Debye temperature increases with increasing pressure,and then decreases at a critical value of pressure. However, forx-TiZr, the Debye temperature increases with pressure under 0–40 GPa. The abrupt changes of Debye temperature maybe corre-lates with structural transformations of these materials. The x-Ti2-

Zr occurs x ? b phase translation under high pressures (25.4–29.1 GPa), and the b-phase has lower Debye temperature [75].

3.4. Electronic properties

Ti and Zr have a narrow d-band in the midst of abroad sp-band.The d-band occupancy is crucial for electronic and structural prop-erties. Total electronic density of state (DOS) and partial density ofstate (PDOS) for x-Ti2Zr at equilibrium geometries under 0 and50 GPa are calculated and presented in Fig. 11. The PDOS inFig. 11 shows that the total DOS near Fermi level is mainly contrib-uted by the contributions of Ti-d and Zr-d states. From the peaksnear the Fermi level, it is clear that the electrons exchange occursbetween Ti(or Zr)-s, Ti(or Zr)-p and Ti(or Zr)-d.

Pressure induced s–d electron transfer plays an important rolein the structural stability [19], and this transfer can be analyzedfrom Mulliken population analysis [76]. We plot DNe (the differentof the number of band occupancy between high pressures and zeropressure) in Fig. 12. The numbers of s electrons (Ns), p electrons(Np), and d electrons (Nd) for x-Ti2Zr at zero pressure are 2.36,6.93, and 2.75 for Ti atoms, for Zr atom are 2.34, 6.78, and 2.78,respectively. The number of d-band electrons for Ti (NTi-d) and Zr(NZr-d) atoms increases with increasing pressure, while the number

Table 3The shear wave velocity (vs), compressional wave velocity (vp), average wave velocity(vm) and Debye temperature at zero pressure and zero temperature of x-Ti2Zr, x-Ti,x-Zr, and x-TiZr.

Phase vp (km/s) vs (km/s) vm (km/s) H (K)

x-Ti2Zr This worka 5.705 3.104 3.462 383.8x-Ti This worka 6.483 3.659 4.038 466.1

Haob 434.9x-Zr This worka 4.917 2.648 2.956 309.1

Wangc 4.868 2.575 2.878 300.93Exptd 4.996 2.616 2.926 06.1

x-TiZr Wange 5.399 2.928 3.267 356.0

a Present work.b Ref. [30].c Ref. [25].d Ref. [27].e Ref. [29].

Fig. 13. The heat capacity at the constant pressure CP of the x-Ti2Zr as a function ofthe temperature at 0, 20 and 40 GPa.

Fig. 10. Debye temperature of x-Ti2Zr, x-Ti, x-Zr, and x-TiZr as a function ofpressure (P).

Fig. 12. The number of electrons on s, p, and d band for x-Ti2Zr as a function ofpressure, relative to their values at zero pressure.

P. Zhang et al. / Computational Materials Science 74 (2013) 129–137 135

of sp-band electrons (Nsp) for both Ti and Zr atoms decrease. It sug-gests that electrons transfer from sp-band to d-band, and then in-crease the number of d-band occupancy. Many theory predicted[16], if the number of d occupancy Nd > 2.2, the bcc structural b-phase are stable in many group IV metal and their alloys [19].

It has been demonstrated that the x ? b transition occur underhigh pressure [29]. For Zr, under pressure of 30–35 GPa [28,77,78],the x ? b phase transition has been observed. For Ti, althoughthere are controversies for the pressure induced phase transitions,it finally remains the more stable b-phase (87 GPa [79]). Bashkin[15,16] employed the energy-dispersive X-ray diffraction to inves-tigate the structural transitions of the TiZr alloy, and found thex ? b transition pressure was 43–57 GPa. Recently, Wang et al.[29] found that the x ? b transition pressure occurs at 43 GPaby first-principles calculations. Therefore, the s ? d electron trans-fer suggests that x-Ti2Zr also eventual transforms to the b-phaseunder high pressures, which is consistent with our early research[75].

3.5. Thermodynamic properties

Thermodynamics properties play an important role in materialsscience and engineering. The heat capacity at constant pressure CP

of the x-Ti2Zr as a function of the temperature at 0, 20 and 40 GPa

Fig. 11. Total and partial electronic density of s

is shown in Fig. 13. It can be seen that CP increases rapidly whenthe temperature increase from 0 to 300 K, and below 300 K the dif-ferences in various pressure are very small, while at high temper-

tates for x-TiZr: (a) 0 GPa and (b) 50 GPa.

Fig. 14. The linear thermal expansion coefficient aL as a function of temperature forx-Ti2Zr.

136 P. Zhang et al. / Computational Materials Science 74 (2013) 129–137

ature, the CP increases monotonously with the temperature, in theaddition, the CP decreases with P at a given temperature.

The variations of linear thermal expansion coefficient aL withtemperature are depicted in Fig. 14. As shown in Fig. 14, the aL in-crease with T at lower temperature (<400 K), and then the incre-ment becomes moderate and approaches to a linear increaseunder further increasing temperature. In addition, the greater ef-fect of temperature on aL at lower pressure. Moreover, the aL de-creases with P at a given temperature.

4. Conclusions

In conclusion, we employed the first-principles pseudopotentialplane-wave method based on DFT, to study the structure, elasticconstants, electron transfers and thermodynamic properties ofx-Ti2Zr under 0–50 GPa, and compared with other group IV metalsand alloys. The calculated values are in good agreement with theexperimental results. We obtained the elastic constants and theDebey temperature of x-Ti2Zr under 0–50 GPa. At zero pressure,the modulus is determined by content of components. The Debyetemperature increases with increasing pressure, and turns to de-crease when pressure get high enough. The values for x-Ti2Zr,x-Ti, x-Zr and x-TiZr display irregularity with increasing pres-sure. The values of G/B for x-Ti2Zr indicate that increasing pressurelead to less brittleness as the same as others. Moreover, stronganisotropy induces by high pressure. In addition, the Mulliken pop-ulation analysis shows that pressure induced s–d electron transferof x-Ti2Zr, and may perform x ? b transition under high pres-sures as the same as other group IV metals and their alloys. Finally,the heat capacity at the constant pressure and the linear thermalexpansion coefficient were obtained, it is clear that both themare sensitive to temperature at lower temperature, and decreaseswith pressure at a given temperature.

Acknowledgements

This research is supported by the National Basic Research Pro-gram of China under Grant No. 2010CB731600, and the SpecializedResearch Project for the protection against Space Debris of Chinaunder Grant Nos. kjsp06209 and kjsp06210, the State Key Labora-tory of Explosion Science and Technology, Beijing Institute of Tech-nology, under the contract No. KFJJ12-2Y and the National NaturalScience Foundation of China under Grant No. 11174201.

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