First-principles Study of Electronic Structures, Elastic ... · unit cell, and finally, the Gibbs free energy is calculated by using the standard statistical thermodynamic formulas.

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First-principles Study of Electronic Structures, Elastic Properties andThermodynamics of the Binary Intermetallics in MgeZneReeZr Alloy

Gang CHEN a,b,*, Peng ZHANG b

aDepartment of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UKb School of Materials Science and Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, China

Received 25 April 2012; revised 20 August 2013; accepted 27 August 2013

Available online 3 October 2013

Abstract

The electronic structures, elastic properties and thermodynamics of MgZn2, Mg2Y and Mg2La have been determined from the first-principlecalculations. The calculated heats of formation and cohesive energies show that Mg2La has the strongest alloying ability and structural stability.The structural stability mechanism is also explained through the electronic structures of these phases. The ionicity and metallicity of the phasesare estimated. The elastic constants are calculated; the bulk moduli, shear moduli, Young’s moduli, Poisson’s ratio value and elastic anisotropyare derived; and the brittleness, plasticity and anisotropy of these phases are discussed. Gibbs free energy, Debye temperature and heat capacityare calculated and discussed.Copyright � 2013, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Magnesium alloy; Electronic structure; Elastic property; Thermodynamic property; First-principles

1. Introduction

MgeZneReeZr alloy is a commonly used deformationmagnesium alloy in aerospace industry because of its excellentproperties, such as high-temperature strength, excellentcorrosion resistance and anti-oxidation properties [1e6].MgZn2, Mg2Y and Mg2La phases, as strengthening phase, arebelieved to be the key components affecting the properties ofMgeZneReeZr alloy. They belong to the typical AB2 typeLaves phase and play an important role in refining the grainsand improving the mechanical properties and creep resistance[7]. Therefore, it is important to study the electronic structures,

* Corresponding author. School of Materials Science and Engineering,

Harbin Institute of Technology at Weihai, Weihai 264209, China.

E-mail address: [email protected] (G. Chen).

Peer review under responsibility of China Ordnance Society

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2214-9147/$ - see front matter Copyright � 2013, China Ordnance Society. Produ

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elastic properties and thermodynamics of MgZn2, Mg2Y andMg2La.

The first-principles techniques have been widely applied tostudy the physical properties of intermetallic compound. Thereare many reports on the properties of MgZn2, Mg2Y andMg2La. Zhou et al. [8] researched the structure stabilities andelastic properties of MgZn2 phase. Wang et al. [9] studied thestructural stabilities and electronic characteristics of Mg2Laphase. Zhang et al. [10] calculated the enthalpies of formationof Mg2Y and Mg2La. Ganeshan et al. [11] calculated theelastic constants of Mg2La. Wrobel et al. studied the ther-modynamic and mechanical properties of Mg2La [12]. Almostall of the above research focused on the enthalpies of forma-tion, electronic structures and the physical properties such aselastic constants, elastic modulus and Poisson’s ratio. How-ever, minimal focus has been placed on the thermodynamicsproperties of MgZn2, Mg2Y and Mg2La. In general, the ther-modynamic properties of intermetallic compounds areimportant because some thermodynamic properties, such asGibbs free energy, Debye temperature and heat capacity, candetermine the thermodynamic stability of system. The

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132 G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

measurement of thermodynamic property of intermetalliccompounds is often challenging. Recently, the first-principlesbased on the density functional theory investigations of ther-modynamic properties of metals have given some quitesatisfactory results for Gibbs free energy and heat capacity[13e19]. Therefore, it seems to be interesting to investigatethe thermodynamic properties of MgZn2, Mg2Y and Mg2Laphases by using the first-principles.

In this paper, the electronic structures, elastic propertiesand thermodynamics of MgZn2, Mg2Y and Mg2La will beanalyzed with first-principles method. The results are dis-cussed in comparison with the available experimental values.The structures are optimized by full relaxation, and the latticeparameters are obtained. The heat of formation and thecohesive energy are calculated and discussed. The structuralstability mechanism is also explained through the electronicstructures of these phases. The elastic constants are calculated,the bulk modulus, shear modulus, Young’s modulus, Poisson’sratio value and elastic anisotropy are derived, and the brittle-ness and plasticity of these phases are researched by empiricalcriterion. Gibbs free energy, Debye temperature and heat ca-pacity from phonon calculations are discussed. This study willprovide useful data for analysis and design of MgeZneReeZralloy, and also for future measurements of MgZn2, Mg2Y andMg2La.

2. Computational methods

The total energy, elastic properties and electronic structureare calculated by using Cambridge sequential total energypackage (CASTEP), in which the first-principles plane-wavepseudo-potentials method based on density functional theory(DFT) is adopted [20]. The ultrasoft pseudo potentials [21] areemployed to represent the interactions between ionic core andvalence electrons. For Mg, Zn, Y and La, the valence electronsconsidered are 2p63s2, 3d104s2, 4s24p64d15s2 and5s25p65d16s2, respectively. A special k-point sampling methodwas used for the integration by setting 6 � 6 � 4 for MgZn2,5 � 5 � 4 for Mg2Y and 5 � 5 � 5 for Mg2La, respectively.PerdeweWang (PW91) [22] version of the generalizedgradient approximation (GGA) [23] is used for calculating theexchange-correlation energy. A kinetic energy cut-off value of340 eV [24] is used for plane wave expansion. The total en-ergy changes during the optimization finally converging to lessthan 1 � 10�6 eV and the forces per atom are reduced to0.02 eV/�A. The BroydeneFletchereGoldfarbeShannon(BFGS) algorithm is applied to relax the whole structure toreach the ground state where both cell parameters and frac-tional coordinates of atoms are optimized simultaneously. Allcalculations are performed with the non-spin polarized densityfunctional theory. The DMol program is used for the calcu-lation of thermodynamic properties of phases [25e27]. As foras the calculation of Gibbs free energy is concerned, at first,the simulation of molecular dynamics is done, the motiondirection and strength of each atom and other information areobtained based on the frequency analysis by the use of celloptimization configuration, some energies of vibration,

rotation and translation at different temperatures are given bythe integrated treatment of the movement information of allthe atoms in order to calculate the entropy and enthalpy of theunit cell, and finally, the Gibbs free energy is calculated byusing the standard statistical thermodynamic formulas. In thecalculation of Gibbs free energy, each compound is relaxed for0.01 ps with a time step of 1.0 fs through NVT moleculardynamics. The BLYP exchange-correlation function is adop-ted for GGA correction. All-electron KohneSham wavefunctions are expanded in a double numerical basis withpolarized orbit (DNP).

3. Results and discussion

3.1. Crystal structures and lattice constants

The crystal structures of MgZn2, Mg2Y and Mg2La phasesare shown in Fig. 1. The crystal structural parameters andlattice constants are listed in Tables 1 and 2. The structures areoptimized by full relaxation, and the equilibrium lattice pa-rameters of MgZn2, Mg2Y and Mg2La phases are derived andlisted in Table 2. The calculated results are compared withothers experimental and analytical results [8,10,12,28,29]. Itcan be found that the calculated lattice parameters are in goodagreement with the experimental values and other theoreticalvalues, with the difference between them being less than 0.5%,especially, our calculated lattice parameters are closer to theexperimental values compared with other theoretical values, sothe present calculations are highly reliable.

3.2. Heat of formation and cohesive energy

Negative heat of formation usually means an exothermicprocess, the lower the heat of formation is, the stronger thealloying ability is. To analyze the alloying abilities of MgZn2,Mg2Y and Mg2La phases, their heats of formation (DH ) arecalculated by the following expression [30]

DH ¼ 1

xþ y

�Etot � xEA

solid � yEBsolid

� ð1Þ

where Etot is the total energy of the unit cell, x and y are thenumbers of Mg, Zn, Y and La atoms, respectively, EA

solid andEBsolid are the energies of each Mg, Zn, Y and La atoms in the

solid states. The calculated energies of Mg, Zn, Y and Laatoms for our considered systems are �978.4696,�1711.1724, �1053.1388 and �863.0929 eV/atom,respectively.

The obtained heats of formation of MgZn2, Mg2Y andMg2La calculated by Eq. (1) are listed in Table 2. Thecalculated heats of formation of Mg2Y are in good agreementwith the experimental result and other theoretical value, butthe calculated heats of formation of MgZn2 and Mg2La arequite different from the experimental results. This differencemay be attributed to that our calculation method is differentfrom others and the temperature calculated here is 0 K. Furtheranalysis found that the heats of formation of MgZn2, Mg2Y

Fig. 1. The crystal structures.

133G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

and Mg2La were all negative, confirming that the structures ofthese phases can exist stably. The values of heats of formationdecrease in the sequence of Mg2Y > MgZn2 > Mg2La. It canbe seen that Mg2La phase has the strongest alloying abilityamong them.

Cohesive energy is defined as the energy which releaseswhen the crystal is decomposed into single atoms. The lowerthe cohesive energy is, the higher the structure stability is. Toestimate their phase stabilities, their cohesive energies (Ecoh)are calculated by the following expression

Ecoh ¼ 1

xþ y

�Etot � xEA

atom � yEBatom

� ð2Þ

where Ecoh is the total energy of the unit cell used in thepresent calculation, x and y are the numbers of Mg, Zn, Y andLa atoms in unit cell, respectively, and EA

atom and EBatom are the

energies of isolated Mg, Zn, Y and La atoms in the free state,respectively. The calculated energies of Mg, Zn, Y and Laisolated atoms are �976.0772, �1708.0388, �1047.8932 and�857.8028 eV/atom, respectively.

The calculated results of the cohesive energies of MgZn2,Mg2Y and Mg2La are shown in Fig. 2. The negative cohesiveenergies of MgZn2, Mg2Y and Mg2La show their energeticstabilization. From the calculated values, it can be furtherfound that the cohesive energy decreases in the sequence ofMgZn2 > Mg2Y > Mg2La. Hence, Mg2La phase has thestrongest structural stability among them.

3.3. Electronic structures

In the present work, the total and partial densities of states(DOS) are calculated to have a further insight into the bondingof MgZn2, Mg2Y and Mg2La phases and then to reveal thestructural stability mechanism of these phases. The total andpartial densities of states of MgZn2, Mg2Y and Mg2La phasesare shown in Fig. 3. It can be seen from Fig. 3 that the mainbonding peaks of all phases basically fall within �10 eV to0 eV, and originate from the contribution of valence electronnumbers of Zn(s), Zn(d ), Mg(s) and Mg( p) orbits for MgZn2(Fig. 3(a)); but for Mg2Y, those are the result of the bondingorbits of Y(s), Y( p), Y(d ), Mg(s) and Mg( p) orbits(Fig. 3(b)); for Mg2La, those mainly originate from thecontribution of valence electron numbers of La(s), La( p),La(d ), Mg(s) and Mg( p) orbits (Fig. 3(c)). From Fig. 3(d), itis found that the number of bonding electrons (per atom) ofMgZn2 is 1.5445 between the Fermi level and �10 eV, whichis smaller than 2.2197 of Mg2Y and 2.3002 of Mg2La. Thesmaller the number of bonding electrons is, the weaker theinteraction of charges is [31]. Hence, Mg2La has the strongeststructural stability among them. The same result can be ob-tained through cohesive energy analysis, confirming theconclusion from the electronic structure point of view.

The population analytical results can provide moreinsightful information on chemical bonding. The calculatedresults are listed in Table 3. It is seen from Fig. 3 that thecharges of Mg are transferred to Zn in MgZn2, and the

Table 1

Crystal structural parameters of MgZn2, Mg2Y and Mg2La phases.

phase Structure type Atom number in cell Group(no.) Atom Atom coordinates

x y z

MgZn2 C14 12 P63/mmc(194) Mg 0.333 0.667 0.0628

Zn(I) 0 0 0

Zn(II) �0.169 �0.339 0.25

Mg2Y C14 12 P63/mmc(194) Mg(I) 0 0 0

Mg(II) 0.841 0.682 0.25

Y 0.333 0.667 0.626

Mg2La C15 24 Fd-3m(227) Mg 0.625 0.625 0.625

La 0 0 0

134 G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

transferred charges (1.05 � 4) are 4.2; for Mg2Y, the charges(0.06 � 4) of Mg(I) transferred to Mg(II) and Y are 0.24; butfor Mg2La, the charges (0.18 � 4) of La transferred to Mg are0.72. The calculated results indicate that the ionicity of thesecompounds gradually increases in the order of Mg2Y <Mg2La < MgZn2.

The metallicity of the compound is estimated by Li [32]

fm ¼ nmne

¼ kBTDf

ne¼ 0:026Df

neð3Þ

where Df is the DOS value at the Fermi level in unit state/eV cell, T is the temperature, nm and ne are the thermallyexcited electrons and the valence electron density of cell,respectively, kB is the Boltzmann constant, ne is calculated byne ¼ N/Vcell, N is the total number of valence electrons, andVcell is the cell volume. From the related parameters andcalculated results listed in Table 4, we can observe that fmincreases in the following sequence: MgZn2 < Mg2La <Mg2Y. Thus, the maximal and minimal “metallicities” corre-spond to Mg2Y and MgZn2, respectively.

The electron density difference, which is defined as theelectron density difference between the isolated atoms andtheir bonding states, reflects directly their bonding nature, asshown in Fig. 3. The contour lines are plotted from�0.02833 e/�A3 to 0.08057 e/�A3 with the interval of0.01729 e/�A3. The red (in web version) lines correspond tohigher density region, and the blue (in web version) linescorrespond to lower density region. From Fig. 4(a), it is foundthat the metallic MgeMg bonds, the covalent ZneZn bondsand the ionic MgeZn bonds exist in MgZn2. In Fig. 4(b), themetallic YeY bonds, the covalent MgeMg bonds and theionic MgeY bonds are found in Mg2Y. Fig. 4(c) also showsthe metallic LaeLa bonds, the covalent MgeMg bonds and

Table 2

Lattice constants and heats of formation of MgZn2, Mg2Y and Mg2La phases.

phase Lattice constant

Present Ref. Ex

a/nm c/nm a/nm c/nm a/

MgZn2 0.5186 0.8624 0.516 [8] 0.856 [8] 0.

Mg2Y 0.6064 0.9786 0.6049 [10] 0.9831 [10] 0.

Mg2La 0.8817 0.8776 [10] 0.

0.8800 [12] 0.

the ionic MgeLa bonds. Generally, the charge density distri-butions show that, for the AB2-type phase, there are mostlymetallic bonding between A and A, the covalent bondingbetween B and B and the ionic bonding between A and B,which is a common feature for the electronic structure in AB2

type binary phase [33].

3.4. Elastic properties

The elastic properties of Mg, MgZn2, Mg2Y and Mg2Lawill be discussed in this section. Mg, MgZn2, Mg2Y andMg2La belong to the hexagonal and cubic structures, respec-tively. The hexagonal structure has 5 independent elasticconstants: C11, C12, C13, C33 and C44, and the cubic structurehas 3 elastic constants: C11, C12 and C44. The correspondingmechanical stability conditions are C11 > 0, C11 � C12 > 0,C44 > 0, (C11 þ C12) C33 � 2C13

2 > 0 for the hexagonalstructure, and are C44 > 0, C11 > jC12j, C11 þ 2C12 > 0 for thecubic structure. The calculated elastic constants of Mg,MgZn2, Mg2Y and Mg2La are listed in Table 5. As can be seenin Table 5, these conditions are easily satisfied, and the ob-tained elastic constants are close to the available theoreticaland experimental values. Therefore, the calculated elasticconstants are credible, and the calculated conditions selectedin this paper should be suitable.

Mg2La phase is cubic structure. The bulk modulus B andshear modulus G are calculated as follows [34]

B¼ 1

3ðC11 þ 2C12Þ ð4Þ

G¼ 1

5ð3C44 þC11 �C12Þ ð5Þ

Heat of formation/(kJ mol�1)

p. Present Ref. Exp.

nm c/nm

521 0.854 �14.34 �12.95 [8] �10.90 [28]

6037 0.9752 �11.99 �9.17 [29] �10.37

8809 �16.04 �12.55 [10] �11.88

8806 [12] �11.66 [12] �12.3 [12]

Fig. 2. Cohesive energies (Ecoh) of MgZn2, Mg2Y and Mg2La phases.

135G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

The structures of Mg, MgZn2 and Mg2Y phases are hex-agonal. The bulk modulus B and shear modulus G are calcu-lated as follows [35]

B¼ 2

9

�C11 þC12 þ 2C13 þC33

2

�ð6Þ

G¼ 1

30ð7C11 þ 2C33 þ 12C44 � 5C12 � 2C13Þ ð7Þ

The Poisson’s ratio n and Young’s modulus E of Mg,MgZn2, Mg2Y and Mg2La are deduced according to thefollowing formula

n¼ ðE� 2GÞ=2G ð8Þ

E¼ 9BG=ð3BþGÞ ð9ÞTable 6 lists the calculated elastic moduli Poisson’s ratios

and universal elastic anisotropy indexes. The bulk modulus is

Fig. 3. Densities of states of MgZ

usually assumed to be a measure of resistance to volumechange by applied pressure, so the larger bulk moduli ofMgZn2 and Mg2Y show that they have stronger resistance tovolume change by applied pressure. Besides, the shearmodulus is a measure of resistance to reversible deformationupon shear stress. If the value of shear modulus is larger, thedirectional bonding between atoms is more pronounced. Thecalculated results demonstrate that Mg2Y has the largest shearmodulus, and then followed by Mg2La and MgZn2. Hence, thedirectional bonding in Mg2Y would be much stronger thosethat in Mg2La and MgZn2. Furthermore, Young’s modulusprovides a measure of stiffness of the solid. The larger theYoung’s modulus is, the stiffer the material is. From thecalculated values we found that Young’s modulus of Mg2Y is59.48 GPa larger than those of Mg2La and MgZn2, indicatingthat Mg2Y is much stiffer than Mg2La and MgZn2. As indi-cated above, the elastic moduli of the three phases are largerthan that of pure Mg. Hence, it is obvious that their me-chanical properties are improved after alloying.

The ratio of shear modulus to bulk modulus of phase can beused to predict the brittle and ductile behaviors of materials. Ahigh (low) G/B value is associated with brittleness (ductility).The critical value separating ductility from brittleness is about0.57. In the present work, the values of Mg, MgZn2, Mg2YandMg2La are 0.539, 0.328, 0.569 and 0.613, respectively,implying that Mg2La is brittle, while Mg, MgZn2 and Mg2Yare ductile. On the other hand, the Poisson’s ratio is used toquantify the stability of the crystal against shear, which usu-ally ranges from �1 to 0.5. The bigger the Poisson’s ratio is,the better the plasticity is. Most of the calculated Poisson’sratios are very close to 0.25, which means that most of the

n2, Mg2Y and Mg2La phases.

Table 3

Mulliken electron populations of MgZn2, Mg2Y and Mg2La phases.

phase Species s/e p/e d/e Total/E Charge/E

MgZn2 Mg 0.51 6.44 0.00 6.95 1.05

Zn(I) 0.73 1.80 9.95 12.47 �0.47

Zn(II) 0.79 1.80 9.95 12.54 �0.54

Mg2Y Mg(I) 0.82 7.12 0.00 7.94 0.06

Mg(II) 0.83 7.19 0.00 8.02 �0.02

Y 2.57 6.70 1.74 11.01 �0.01

Mg2La Mg 0.87 7.22 0.00 8.09 �0.09

La 2.59 6.04 2.19 10.82 0.18

Fig. 4. The contour plots of density difference of electronic charges on (001)

plane.

136 G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

materials are with predominantly central interatomic forces[36]. MgZn2 has the biggest Poisson’s ratio among the threematerials considered in this paper.

All single crystals in practice are anisotropic, so anappropriate parameter is needed to characterize the extent ofanisotropy. Recently, Ranganathan and Ostoja-Starzewski [37]summarized the existing anisotropy theories, and concludedthat most of them are lack of universality because of their non-uniqueness and ignoring a large part of the elastic stiffnesstensor. Then they developed a new universal anisotropy index,AU, which can be calculated by the following equation

AU ¼ 5GV

GRþBV

BR� 6 ð10Þ

where GV, BV, GR and BR are the shear moduli and bulkmoduli estimated using Voigt and Reuss methods, respectively.The anisotropy indexes of Mg, MgZn2, Mg2Y and Mg2La arelisted in Table 6, from which, we can conclude that the bulkand shear moduli values calculated by Voigt method are closeto those calculated by Reuss method, and the difference be-tween shear moduli calculated with Voigt and Reuss methodshas a significant effect on AU. The anisotropy decreases in thefollowing sequence: Mg >MgZn2 >Mg2La >Mg2Y. Besidespure Mg, MgZn2 behaves more anisotropy than Mg2Y andMg2La.

3.5. Thermodynamic stability

Table 5

The calculated and experimental elastic constants of MgZn2, Mg2Y and

Mg2La phases.

phase Source Elastic constant/GPa

C11 C12 C13 C33 C44

Here, the thermodynamic property is used to describe thestructural stabilities of these compounds with the elevatedtemperature, especially their Gibbs free energy (G). The Gibbsfree energy G of the calculated Mg, MgZn2, Mg2Y and Mg2Laphases as a function of temperature from 297 to 573 K isdepicted in Fig. 5. It can be seen from Fig. 5 that the values ofthe Gibbs free energy at the same temperature graduallydecrease in the following sequence: Mg > Mg2Y >MgZn2 > Mg2La. The smaller the Gibbs free energy is, the

Table 4

Df, N, Vcell and fm of MgZn2, Mg2Y and Mg2La phases.

Phase Df/(state eV�1 cell�1) N/e Vcell/nm3 fm

MgZn2 4.2988 152.4361 200.522 0.1471

Mg2Y 14.5952 131.8208 311.665 0.8972

Mg2La 6.0671 78.2333 171.383 0.3456

better the thermal stability of compound is [34]. Hence, thecalculated results of the Gibbs free energy show that thethermal stabilities of these compounds gradually increase inthe following sequence: Mg <Mg2Y <MgZn2 <Mg2La. Thethermal stabilities of MgZn2, Mg2Y and Mg2La phases are

Mg Present 62.74 23.16 15.09 78.41 11.57

Calc. [36] 52.27 37.19 20.69 70.46 23.78

Exp. [33] 59.74 26.14 21.67 70.60 18.42

MgZn2 Present 85.84 84.17 19.86 133.35 19.81

Calc. [8] 91.25 85.27 23.38 198.31 24.88

Mg2Y Present 76.83 25.41 21.11 83.96 17.78

Mg2La Present 58.43 27.48 28.28

Calc. [11] 58.4 24.9 21.8

Calc. [12] 58.0 25.0 22.2

Table 6

The calculated and experimental elastic moduli, Poisson’s ratios and universal elastic anisotropy indexes of MgZn2, Mg2Y and Mg2La phases.

phase Source Modulus/GPa G/B n GV/GR BV/BR AU

B G E

Mg Present 34.51 18.62 47.34 0.539 0.271 1.143 1.002 0.717

Calc. [36] 36.71 17.83 46.05

Exp. [33] 37.05

MgZn2 Present 61.42 20.17 54.54 0.328 0.3521 1.007 1.385 0.420

Calc. [8] 70.71 16.124 45.57 0.228 0.3926

Mg2Y Present 41.43 23.59 59.48 0.569 0.2607 1.001 1.049 0.054

Mg2La Present 37.81 23.16 57.69 0.613 0.2455 1.000 1.089 0.089

Calc. [11] 36.0 19.7 50.1 0.547 e

Calc. [12] 36.0 19.8 49.9 0.550

Fig. 5. The Gibbs free energies of Mg, MgZn2, Mg2Y and Mg2La phases as

functions of temperature.

137G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

better than that of pure Mg and do not change with theelevated temperature, that is to say, the addition of Zn, Y, Laaddition to the magnesium alloy can improve the heat resis-tance by forming MgZn2, Mg2Y and Mg2La phases.

Once the elastic constants and electronic structures of thecompound are known, one can calculate the Debye tempera-ture and heat capacity at the low-temperature. Debye tem-perature can be used to predict the thermodynamics ofmaterial from the elastic properties; it can be also used todistinguish between high- and low-temperature regions for asolid. For T > QD, all modes have energy of kBT, and forT < QD, one expects the high-frequency modes to be frozen[38]. QD can be estimated from the average sound velocity bythe following equations [39]

QD ¼ h

kB

�3n

4p

�NAr

M

��1=3nm ð11Þ

nm ¼�1

3

�2

n3sþ 1

n3l

���1=3

ð12Þ

Table 7

Theoretically calculated thermal properties of MgZn2, Mg2Y and Mg2La phases.

phase r/(g cm�3) nl/(m s�1) ns/(m s�1)

MgZn2 5.1265 4146.5155 1981.6325

Mg2Y 2.9307 4986.5884 2837.1141

Mg2La 3.6337 4347.5943 2524.6195

nl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Bþ 4

3G

�1

r

sð13Þ

ns ¼ffiffiffiffiffiffiffiffiffiG=r

pð14Þ

where QD is the Debye temperature, h and kB are the Planckand Boltzmann constants, respectively, n is the total number ofatoms per formula, NA is the Avogadro number, M is themolecular weight per formula, vm is the average sound ve-locity, vl is the longitudinal sound velocity, and vs the shearsound velocity. The calculated results of QD, vl and vs arelisted in Table 7. The largest QD is 317.02 K for Mg2Y. It isknown that QD can be used to characterize the strength ofcovalent bonds in solids. From Table 7, we can conclude thatthe covalent bond in Mg2Y is stronger than those in otherphases. Besides, the sequence of QD for these phases isMg2Y > Mg2La > MgZn2, which is consistent with theobserved trend of Young’s modulus and shear modulus, asshown in Table 6. From the values of QD, Young’s modulusand shear modulus, we may conclude that the mechanicalstability of Mg2Y is the best among three phases.

Besides, since MgZn2, Mg2Y and Mg2La phases studied inthe present paper have the metallic feature at the Fermi level,the calculations of the electronic structures and elastic con-stants enable us to estimate the heat capacity (cp) at the low-temperature, which is given as

cpðTÞ ¼ gTþ bT3 ð15Þ

g¼ 1

3p2k2BDf ð16Þ

b¼ 12p4Rn

5Q3D

ð17Þ

nm/(m s�1) QD/K�1 g (10�3) b (10�5)

2228.5777 259.46 10.1431 33.3674

3153.7097 317.02 34.4376 18.2925

2801.3751 272.81 14.3154 28.7046

Fig. 6. Heat capacities of MgZn2, Mg2Y and Mg2La phases in the range of

0e25 K.

138 G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

where g and b are the coefficients of electronic and lattice heatcapacities, respectively; n is the total number of atoms performula unit; R is the molar gas constant. Heat capacity is aninvaluable tool to explore the fundamental properties of ma-terials. Note that QD, as a rule, only describes the temperaturedependence of cp for T < QD/10 [40]. Fig. 6 shows cp vs T inthe range of 0e25 K for three phases. As can be seen fromFig. 6, the main contribution to cp is the excitation of electronsat 13 K, the values of cp change in the sequence ofMg2Y > Mg2La > MgZn2, and the change of cp is consistentwith the change of g, indicating the heat capacities aredetermined by the electrons at first; besides, in Table 7, thesmallest value of g ¼ 10.1431 � 10�3 J/(K2 mol) is attributedto MgZn2, and because it has the smallest DOS value at theFermi level (Df), MgZn2 has the weakest metallic natureamong three phases. In the range of 13e25 K, the contributionfrom phonon excitation must be taken into account. As aresult, the growth trend of cp is MgZn2 > Mg2La > Mg2Y,which is the same as the sequence of b. Thus, it can concludethat the heat capacity is determined by the electron excitationat very low temperatures (near 0 K), and the contribution fromphonon excitation is significant at high temperature.

4. Conclusions

The electronic structures, elastic properties and thermody-namics of MgZn2, Mg2Y and Mg2La were determined fromfirst-principles calculation. The calculated heats of formationand cohesive energies show that Mg2La has the strongestalloying ability and structural stability. The calculations ofbonding electron numbers show that Mg2La has the strongeststructural stability. The calculated elastic constants show thatthree phases are mechanically stable. The calculated bulkmoduli B, shear moduli G, Young’s moduli E, Poisson ratio n

and elastic anisotropy AU show that Mg2La is brittle, whileMgZn2 and Mg2Y are ductile; the stiffness of Mg2Y is thehighest; MgZn2 has the best plasticity and anisotropy amongthree phases. The calculations of thermodynamic propertiesshow that the Gibbs free energies of MgZn2, Mg2Yand Mg2Ladecrease, while the heat capacity Cv increases with the

elevated temperature. The calculated results of the Gibbs freeenergy show that the thermal stability of these compoundsgradually increases in the order of Mg2Y < MgZn2 < Mg2La.The calculated Debye temperature, Young’s modulus andshear modulus show that the mechanical stability of Mg2Y isthe best among three phases.

Acknowledgments

We are grateful for the support of the National NaturalScience Foundation of China (NSFC) for support under GrantNo. 51005217. Dr. Chen is grateful for the support from ChinaPostdoctoral Science Foundation Grant No. 20100480677.

References

[1] Li YG, Liu SR, Liu HL. Mechanical properties and microstructure of

MB26 alloy under different condition of hot working process. Chin J

Rare Met 2004;28:942e6 [in Chinese].

[2] Chen Q, Shu DY, Zhao ZD, Zhao ZX, Wang YB, Yuan BG. Micro-

structure development and tensile mechanical properties of

MgeZneREeZr magnesium alloy. Mater Des 2012;40:488e96.

[3] Chen Q, Yuan BG, Zhao GZ, Shu DY, Hu CK, Zhao ZD, et al. Micro-

structural evolution during reheating and tensile mechanical properties of

thixoforged AZ91D-RE magnesium alloy prepared by squeeze casting-

solid extrusion. Mater Sci Eng A 2012;537:25e38.

[4] Chen Q, Zhao ZD, Zhao ZX, Hu CK, Shu DY. Microstructure develop-

ment and thixoextrusion of magnesium alloy prepared by repetitive up-

setting-extrusion. J Alloy Compd 2011;509:7303e15.

[5] Chen Q, Lin J, Shu DY, Hu CK, Zhao ZD, Kang F, et al. Microstructure

development, mechanical properties and formability of MgeZneYeZrmagnesium alloy. Mater Sci Eng A 2012;554:129e41.

[6] Bettles CJ, Gibson MA. Microstructural design for enhanced elevated

temperature properties in sand-castable magnesium alloys. Adv Eng

Mater 2003;12:859e65.[7] Mordike BL. Creep-resistant magnesium alloys. Mater Sci Eng A

2002;324:103e12.

[8] Zhou DW, Xu SH, Zhang FQ. First-principles calculations of structural

stabilities and elastic properties of AB(2) type intermetallics in ZA62

magnesium alloy. Acta Metall Sin 2010;46:97e103.

[9] Wang YF, Zhang WB, Wang ZZ. First-principles study of structural

stabilities and electronic characteristics of MgeLa intermetallic com-

pounds. Comput Mater Sci 2007;41:78e85.

[10] Zhang H, Shang SL, Saal JE. Enthalpies of formation of magnesium

compounds from first-principles calculations. Intermetallics

2009;17:878e85.[11] Ganeshan S, Shang SL, Zhang H. Elastic constants of binary Mg

compounds from first-principles calculations. Intermetallics

2009;17:313e8.[12] Wrobel J, Hector LG, Wolf W. Thermodynamic and mechanical prop-

erties of lanthanumemagnesium phases from density functional theory. J

Alloy Compd 2012;512:296e310.

[13] Shang SL, Wang Y, Kim DE. First-principles thermodynamics from

phonon and Debye model: application to Ni and Ni3Al. Comput Mater

Sci 2010;47:1040e8.

[14] Zhao KM, Jiang G, Wang LL. Electronic and thermodynamic properties

of B2-FeSi from first principles. Physica B Condens Matter

2011;406:363e7.

[15] Pang MJ, Zhan YZ, Wang HZ. Structural, electronic, elastic and ther-

modynamic properties of AlSi2RE (RE ¼ La, Ce, Pr and Nd) from first-

principle calculations. Comput Mater Sci 2011;50:3303e10.[16] Xiao B, Feng J, Zhou CT, Xing JD, Xie XJ, Cheng YH, et al. The

elasticity, bond hardness and thermodynamic properties of X2B (X¼Cr,

Mn, Fe, Co, Ni, Mo, W) investigated by DFT theory. Physica B Condens

Matter 2010;405:1274e8.

139G. Chen, P. Zhang / Defence Technology 9 (2013) 131e139

[17] Hector LG, Herbst JF, Wolf W. Ab initio thermodynamic and elastic

properties of alkaline-earth metals and their hydrides. Phys Rev B

2007;76:014121. 1e18.

[18] Herbst JF, Hector LG. Structural discrimination via density functional

theory and lattice dynamics: monoclinic Mg2NiH4. Phys Rev B 2009;79.

155113-1,7.

[19] Herbst JF, Hector LG. Energetics of the Li amide/Li imide hydrogen

storage reaction. Phys Rev B 2005;72. 125120-1,8.

[20] Shi DM, Wen B, Melnik R. First-principles studies of AleNi interme-

tallic compounds. J Solid State Chem 2009;182:2664e9.

[21] FaganSB,Mota R, Baierle RJ. Stability investigation and thermal behavior

of a hypothetical silicon nanotube. J Mol Struct 2001;539:101e6.

[22] Pack JD, Monkhors HJ. Special points for Brillouin zone integrations’ e

a reply. Phys Rev B 1977;16:1748e9.

[23] Perdew JP, Burke K, Ernzerh M. Generalized gradient approximation

made simple. Phys Rev Lett 1996;77:3865e8.[24] Nyken J, Garcia FJ,Mosel BD. Structural relationships, phase stability and

bonding of compounds PdSnn (n¼ 2, 3, 4). Solid State Sci 2004;6:147e55.

[25] Hector LG, Herbst JF. Electronic structure calculations for LaNi5 and

LaNi5H7: energetics and elastic properties. J Alloy Compd

2003;353:74e85.

[26] Shang SL, Hector LG. First-principles study of elastic and phonon

properties of the heavy fermion compound CeMg. J Phys Condens

Matter 2009;21(24):246001e12.

[27] Hector LG, Herbst JF. Density functional theory for hydrogen storage

materials: successes and opportunities. J Phys Condens Matter

2008;20(6):64229e39.[28] Li CH, Hoe JL, Wu P. Empirical correlation between melting tempera-

ture and cohesive energy of binary Laves phases. J Phys Chem Solids

2003;64:201e12.

[29] Miedema AR, Dechatel PF, Deboer FR. Cohesion in alloys-fundamentals

of a semi-empirical model. Physica B þ C 1980;100:1e28.

[30] Chen L, Peng P, Li GF. First-principle calculation of point defective

structures of B2-RuAl intermetallic compound. Rare Metal Mater Eng

2006;35:1065e70.

[31] Zhou DW, Liu JS, Xu SH. Thermal stability and elastic properties of

Mg3Sb2 and Mg3Bi2 phases from first-principles calculations. Physica B

Conden Matter 2010;405:2863e8.[32] Li YF, Gao YM, Xiao B. Theoretical study on the stability, elasticity,

hardness and electronic structures of WeC binary compounds. J Alloy

Compd 2010;502:28e37.

[33] Yu WY, Wang N, Xiao XB, Tang BY, Peng LM, Ding WJ. First-principles

investigation of the binary AB2 type Laves phase in MgeAleCa alloy:

electronic structure and elastic properties. Solid State Sci 2009;11:1400e7.

[34] Anderson OL. A simplified method for calculating the Debye tempera-

ture from elastic constants. J Phys Chem Solids 1963;24:909e17.[35] Kanoun MB, Goumri-Said S, Reshak AH. Theoretical study of me-

chanical, electronic, chemical bonding and optical properties of Ti2SnC,

Zr2SnC, Hf2SnC and Nb2SnC. Comput Mater Sci 2009;47:491e500.[36] Mattesini M, Ahuja R, Johansson B. Cubic Hf3N4 and Zr3N4: a class of

hard materials. Phys Rev B 2003;68. 184108.

[37] Ranganathan SI, Ostoja-Starzewski M. Universal elastic anisotropy

index. Phys Rev Lett 2008;101(5). 055504.

[38] Mayer B, Anton H, Bolt E. Ab-initio calculation of the elastic constants

and thermal expansion coefficients of Laves phases. Intermetallics

2003;11:23e32.

[39] Deligoz E, Ciftci YO, Jochym PT. The first principles study on PtC

compound. Mater Chem Phys 2008;111:29e33.

[40] Drulis MK, Czopnik A, Drulis H. On the heat capacity of Ti3GeC2. Mater

Sci Eng B 2005;119:159e63.