Phase stability and electronic structure in ZrAl3 compound

Phase stability and electronic structure in ZrAl3 compound

Journal of Alloys and Compounds 319 (2001) 154–161 L www.elsevier.com / locate / jallcom Phase stability and electronic structure in ZrAl 3 compoun...

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Journal of Alloys and Compounds 319 (2001) 154–161

L

www.elsevier.com / locate / jallcom

Phase stability and electronic structure in ZrAl 3 compound a, b C. Colinet *, A. Pasturel a

´ ` , France Laboratoire de Thermodynamique et Physico-Chimie Metallurgiques , CNRS /INPG /UJF, ENSEEG, B.P. 75, 38402 Saint Martin d’ Heres b ´ ` ` , CNRS, B.P. 166, 38042 Grenoble Cedex 09, France Laboratoire de Physique Numerique des Systemes Complexes, Maison des Magisteres Received 4 December 2000; accepted 14 December 2000

Abstract The relative stabilities of L1 2 , D0 22 , and D0 23 structures in the ZrAl 3 intermetallic compound have been investigated employing the Vienna ab initio simulation package (VASP). The effects due to the tetragonal distortion of the D0 22 and D0 23 structures are important. The effect of the cell-internal displacements of the atoms in the D0 23 structure is studied. The more stable structure is D0 23 followed by D0 22 and by L1 2 . The calculations show that at high pressure L1 2 becomes more stable than D0 22 . The energies of formation of ZrAl 3 in the L1 2 , D0 22 , and D0 23 structures are calculated. The computed electronic densities of states show that each structure has a pseudo gap in the density of states distribution. The preferred crystal structure is the one in which the Fermi level lies in the pseudo gap and for which the density of states at the Fermi level is the lowest. The energetic results are discussed in the framework of the ANNNI model.  2001 Elsevier Science B.V. All rights reserved. Keywords: Transition metal compounds; Intermetallics; Electronic band structure; Enthalpy

1. Introduction Trialuminides of early transition metals MAl 3 where M is a group III, IV, or V transition metal are promising structural materials because of their high melting points, low densities, and oxidation resistances. Most of these compounds crystallize in the D0 22 (or TiAl 3 -type) structure such as TiAl 3 , NbAl 3 , VAl 3 , and TaAl 3 [1,2]. ZrAl 3 and HfAl 3 crystallize in the D0 23 (or ZrAl 3 -type) structure [1,2]. The L1 2 (or AuCu 3 -type) structure is observed for ScAl 3 , YAl 3 at high temperature [1,2], and for metastable TiAl 3 , ZrAl 3 and HfAl 3 [3]. Experiments on many face centered cubic (fcc) metallic alloys reveal that those which develop L1 2 atomic order are significantly more ductile than those that form D0 22 order, presumably due to the lack of a sufficient number of slip systems in the D0 22 structure [4]. Consequently the L1 2 alloys are more likely to be suitable in structural applications because of their mechanical properties [5]. In the present work we will focus our attention on the ZrAl 3 compound in the L1 2 , D0 22 and D0 23 structures shown in Fig. 1. As quoted above ZrAl 3 crystallizes in the

*Corresponding author. E-mail address: [email protected] (C. Colinet).

tetragonal D0 23 structure but it can be produced as a metastable cubic L1 2 structure either by precipitation during rapid solidification of supersaturated Al rich solid solutions [5–9] or by mechanical alloying [3,10,11]. When

Fig. 1. Crystal structures for ZrAl 3 compounds: cubic L1 2 , tetragonal D0 22 at ideal c /a52, tetragonal D0 23 at ideal c /a54. (s) Al atoms; (d) Zr atoms.

0925-8388 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )00879-9

C. Colinet, A. Pasturel / Journal of Alloys and Compounds 319 (2001) 154 – 161

metastable L1 2 –ZrAl 3 is heated, it transforms to the stable D0 23 structure. Several theoretical studies of the structures and phase stabilities of ZrAl 3 have been performed. Carlsson and Meschter [12] used the augmented spherical wave (ASW) method [13,14] to obtain the total energies of the L1 2 , D0 22 , and D0 23 structures. For these last two structures the calculations were performed for an ideal ration c /a and for a non-ideal ratio c /a which was the experimental value for the D0 23 structure and a typical c /a ratio for the D0 22 structure. Carlsson and Meschter [10] found that the D0 22 structure at c /a ratio of 2.23 is more stable than the D0 23 one at the experimental c /a ratio, and that the D0 22 structure is more stable than the L1 2 one. Xu and Freeman [15–17] performed calculations using the linear muffin tin orbitals (LMTO) method [18] associated with the atomic sphere approximation (ASA). The calculated total energy results show that D0 22 structure at c /a ratio of 2.23 is more stable than L1 2 and that D0 23 at c /a ratio of 4.32 is more stable than the D0 22 structure. As well Nicholson et al. [19] used the LMTO [18] method in the ASA approximation to determine the relative stability of the three structures; the values of c /a in the tetragonal structures were assumed. The ground state of ZrAl 3 was incorrectly given to be D0 22 . Using the muffin tin approximation Nicholson et al. [19] found the correct ground state and D0 22 to be less stable than L1 2 . Amador et al. [20] performed calculations using the full potential linear muffin tin orbitals (FPLMTO) method [21–23]. For the D0 22 and D0 23 structures the c /a ratio was optimized. Moreover, in the case of the D0 23 structure an optimization was performed additionally with respect to cell-internal displacements; the D0 23 structure was found to be the ground state after the last stage of relaxation, and the L1 2 structure to be more stable than the distorted D0 22 structure. More recently Alatalo et al. [24] used two methods to obtain the total energies of compounds in the Zr–Al system: the full potential linearized augmented Slater-type orbital (LASTO) method [25] and a pseudo potential method [26] (called by the authors PWPP). In both calculations the D0 23 structure was found to be the ground state for the compound ZrAl 3 . In the LASTO method, L1 2 is more stable than the distorted D0 22 structure whereas in the PWPP method it is the opposite. In the purpose of a clarification and a better undestanding of the relative stability of the L1 2 , D0 22 , and D0 23 structures for the compound ZrAl 3 , we have performed calculations of total energies and electronic structures, and studied the effect of the tetragonal distortion of the D0 22 and D0 23 structures and of internal cell displacements of the atoms in the D0 23 structure.

2. Ab initio calculations The calculations presented here have been done using the Vienna ab initio simulation package (VASP) which has

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been described elsewhere [27,28]. The establishment of the VASP code is based on the density-functional theory (DFT) within the local density approximation (for more details concerning these theories and a review of the various methods used in ab initio calculations see Hafner [29]). VASP is a plane-wave code based on ultrasoft pseudopotentials, and uses iterative strategies based on residual minimization and preconditioned conjugate-gradient techniques for the diagonalization of the Kohn–Sham Hamiltonian. The method [30] used to build the pseudopotentials is derived from the Vanderbilt’s [31] recipe for ultrasoft potentials. The pseudopotentials allow the use of a modest cutoff for the construction of the plane-wave basis for the transition metals. The calculations of the Hellmann–Feynman forces acting on the atoms and of the stresses on the unit cell are possible in VASP. Hence the total energy may be optimized with respect to the volume and shape of the unit cell, and to the positions of the atoms within the cell with no other restrictions than those imposed by space-group symmetry. For the Brillouin-zone integration the Methfessel–Paxton [32] technique with a modest smearing of the one-electron levels (0.1 eV) is used. For the total energy calculations of the ZrAl 3 compound in the L1 2 (four atoms in the primitive cell), D0 22 (eight atoms in the primitive cell), and D0 23 (16 atoms in the primitive cell) structures a 83838, 83836, and 83834 k-point mesh, respectively, was sufficient. A finer mesh (10310310, 1031036, and 1031036, respectively, for the L1 2 , D0 22 , and D0 23 structures) was used for computing the electronic densities of states. All calculations were performed with the generalized gradient approximation (GGA) proposed by Perdew and Wang [33]. Throughout this study, Zr 4p states are treated as valence electrons [34].

3. Results

3.1. Phase stability From L1 2 to D0 22 , and to D0 23 symmetry elements are progressively lost so that relaxation degrees of freedom increase correspondingly. In the L1 2 structure, energy may be optimized with respect to the lattice parameter a only. In the D0 22 structure, energy optimization may be performed with respect to lattice parameter a and with respect to c /a ratio (tetragonal distortion of the lattice). Finally, in the D0 23 structure, energy optimization may be performed with respect to lattice parameter a, with respect to c /a ratio (tetragonal distortion of the lattice), and additionally with respect to cell-internal displacements of Zr and Al atoms which are in the mixed Al–Zr planes (see Fig. 1) since the z coordinates of these atomic positions are not fixed by symmetry (Wyckoff positions 4e for D0 23 space group I4 /mmm). The total energy calculations versus lattice parameter, a,

C. Colinet, A. Pasturel / Journal of Alloys and Compounds 319 (2001) 154 – 161

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distorted D0 22 structure is more stable that the L1 2 one. The crossing of the energy–volume curves of these two structures at 17.3310 23 nm shows that the L1 2 structure is energetically favorable by increasing the pressure at roughly 2 GPa (this value has been estimated from the slope of the common tangent to the two energy–volume curves).

3.2. Lattice parameters and bulk modulus

Fig. 2. Total energies per mol of atoms versus the volume per mol of atoms for L1 2 (d), distorted D0 22 (m), and fully relaxed D0 23 (j) structures.

have been performed for the structures L1 2 , D0 22 at ideal c /a (in the following we will denote this case ideal D0 22 ), D0 22 at optimized c /a (distorted D0 22 ), D0 23 at ideal c /a (ideal D0 23 ),D0 23 at optimized c /a (distorted D0 23 ),and D0 23 by optimizing both c /a and the internal cell displacements of the atoms (fully relaxed D0 23 ). In Fig. 2, the calculated total energy as function of the atomic volume is shown for L1 2 , distorted D0 22 , and fully relaxed D0 23 structures. In order to keep things clear, the curves corresponding to the other cases have not been reported. From Birch–Murnaghan [35,36] least-square fits of the total energy versus volume, we obtain the equilibrium volume, the corresponding energy, and the bulk modulus. From Fig. 2, it can be observed that the fully relaxed D0 23 structure is the more stable one. At zero pressure the

The lattice parameters obtained at the minimum of the energy–volume curves are reported in Table 1, where they may be compared with the values found in the literature. These values have been obtained either experimentally or by ab initio calculations. For the L1 2 structure, our value of the parameter a is slightly higher than the other values but is in very good agreement with the experimental value of Desch et al. [11]. For the D0 22 structure, there is a good agreement with Xu and Freeman [18] for the parameter a, our value of c /a is higher than the value chosen by these authors to perform their calculations. For the D0 23 structure, our values of the a parameter and c /a ratio are in excellent agreement with the experimental values of Schuster and Nowotny [38] and Tsunekawa and Fine [39]. In the optimization of the D0 23 structure, the cellinternal displacements of the Zr and Al atoms situated on the mixed Al–Zr planes have been obtained. The actual cell-internal displacements are reported in Fig. 3. The indicated directions are in agreement with those found by Amador et al. [20]. In absolute value, the cell-internal displacement of the Al atoms is dAl 53310 24 nm and that 22 of the Zr atoms is d Zr 51.09.10 nm. The ratio d Zr / dAl 5 36 may be compared with the one calculated (9) and the one measured (68) by Amador et al. [20]. The agreement with the experimental value is relatively good. The calculated values of the bulk modulus are 99 GPa

Table 1 Experimental and calculated lattice parameters of the ZrAl 3 compound in the L1 2 , D0 22 (distorted) and D0 23 (fully relaxed) structures a (nm) L1 2

D0 22 D0 23

c (nm)

c /a

0.411 0.407 0.408 0.405 0.40731 0.4093 0.4077 0.396 0.395 0.402 0.391 0.397 0.40012 0.40080

0.904 0.882 1.736 1.704 1.713 1.72444 1.72824

2.283 2.23 4.318 4.358 4.315 4.310 4.312

0.40074 0.4015 0.4014 0.4009

1.7286 1.7316 1.7315 1.7280

4.314 4.313 4.314 4.310

Method

Reference

Calculated VASP Calculated LMTO Experimental Experimental Experimental Experimental Experimental Calculated VASP Calculated LMTO Calculated VASP Calculated FPLMTO Calculated LMTO Experimental 12K Experimental room temperature Experimental Experimental Experimental Experimental

Present work Xu and Freeman [17] Nes [8] Ryum [6] Ohashi and Ichikawa [7] Desch et al. [10] Srinivasan et al. [3] Present work Xu and Freeman [17] Present work Amador et al. [20] Xu and Freeman [17] Amador et al. [20] Amador et al. [20] Kematick and Frantzen [37] Schuster and Nowotny [38] Tsunekawa and Fine [39] Srinivasan et al. [3]

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Fig. 4. Formation energies referred to Zr hcp and Al fcc of ZrAl 3 in the three structures at the different stages of relaxation. (d) L1 2 , (m) D0 22 , and (j) D0 23 .

Fig. 3. Unit cell of the D0 23 structure. (s) Al atoms; (d) Zr atoms. The arrows show the actual directions of the displacements of the Al and Zr atoms as calculated in the present work.

for the three structures. To our best knowledge there are no experimental data available for the bulk modulus of ZrAl 3 . Our calculated values are in good agreement with the values calculated by Xu and Freeman [18] (1.0, 1.1, and 1.1 Mbar, respectively, for the L1 2 , D0 22 , and D0 23 structures).

3.3. Total energies and enthalpies of formation The total energies at minimum of each of the energy– volume curves are reported in Table 2. The enthalpies of formation of the ZrAl 3 compound in the various structures are referred to fcc Zr and fcc Al in column 5 and to Zr hcp and Al fcc in column 6.

The values of the energies of formation are shown in Fig. 4 for the three stages of relaxation: ideal c /a, distortion, and fully relaxed. In the ideal case, L1 2 (circles, dot–dash lines) has the lowest energy, followed by D0 23 (squares, unbroken lines), then by D0 22 (triangles, dashed lines). At the distortion stage, the D0 23 becomes the more stable structure followed by D0 22 then by L1 2 . After the complete relaxation, the relative situation remains unchanged, the D0 23 structure is still more stabilized. In the last two stages, our results are in contradiction with those obtained by Amador et al. [20]. Indeed these authors found L1 2 structure more stable than D0 22 . Using the LASTO method, Atalato et al. [24] found also the L1 2 structure more stable than D0 22 . However Carlsson and Meschter [12], Xu and Freeman [18] and Atalato et al. [24] using the PWPP method have derived the same relative stability of L1 2 and D0 22 as those obtained by us. The values of the energies of formation of ZrAl 3 in the different structures are reported in Table 3 where they may be compared with other calculated values and with experimental values of the enthalpies of formation (the values calculated by Carlsson and Meschter [12] and by Amador et al. [20] do not appear in this table because only figures present the results in the respective publications.

Table 2 Total energies of Al fcc, Zr fcc and hcp, ZrAl 3 in the structures L1 2 , D0 22 and D0 23 calculated using VASP. D f E are the energies of formation referred to Zr fcc and Al fcc a Elements

Et (eV/ atom)

ZrAl 3 compound

Et (eV/ atom)

DfE (kJ / mol atoms)

D f E9 (kJ / mol atoms)

Al fcc Zr fcc Zr hcp

23.691 28.360 28.398

L1 2 Ideal D0 22 Distorted D0 22 Ideal D0 23 Distorted D0 23 Fully relaxed D0 23

25.330 25.241 25.335 25.304 25.344 25.361

245.5 236.9 246.0 243.0 246.9 248.5

244.6 236.0 245.1 242.1 246.0 247.6

a

D f E9 are the energies of formation referred to Zr hcp and Al fcc.

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Table 3 Values of the enthalpies of formation referred to Zr hcp and Al fcc of ZrAl 3 in the L1 2 , D0 22 , and D0 23 structures obtained in the present work: comparison with literature values

Calculated values

VASP, present work LMTO-ASA [18] LASTO [24] PWPP [24]

Experimental values

Vap. press [37] Calorimetry [40] Calorimetry [41]

Only differences of total energies are given by Nicholson et al. [19]. The enthalpy of formation of the ZrAl 3 in the D0 23 structure has been derived by Kematick and Franzen [37] from vapor pressure measurements. The enthalpy of formation has also been obtained by direct reaction calorimetry by Meschel and Kleppa [39] and by calorimetry by Argent and Perry [41]. An excellent agreement of our calculated value with the value proposed by Meschel and Kleppa [40] is found. The values obtained by pseudopotential methods (the one obtained by Atalato et al. [24] and our value) are in very good agreement. By differential scanning calorimetry, Desch et al. [10] determined an enthalpy of the transition L1 2 →D0 22 of 22.22 kJ / mol of atoms; our value of the enthalpy of transition is in good agreement.

3.4. Electronic structures The total densities of states have been computed at each stage of the relaxation process. They are displayed in Fig. 5. They look very similar to those presented by Carlsson and Meschter [12] and by Xu and Freeman [18]. The densities of states are characteristic of a strong hybridization between d states of zirconium and p states of aluminum. The salient feature of the plots is the presence of a pseudo-gap in the densities of states with the Fermi energy residing close to the minimum. The left part of the density of states corresponds to the bonding states whereas the right part corresponds to non-bonding states. The antibonding states stand well above the Fermi level and are not presented in the plots. In the L1 2 case, the Fermi level is on the right of the minimum of the density of states. In the D0 22 case, the Fermi level is on the left of the minimum of the density of states. The distortion of the lattice does not change the shape of the density of states, this leads only to a reduction of the density of states at the Fermi level. In the D0 23 case, the Fermi level stands in the pseudo-gap. The distortion of the lattice then the displacement of the Al and Zr atoms in the mixed Al and Zr planes lead to a reduction of the density of states at the Fermi level. The values of the densities of states at the Fermi level at the different stages

D f H (L1 2 ) (kJ / mol atoms)

D f H (D0 22 ) (kJ / mol atoms)

D f H (D0 23 ) (kJ / mol atoms)

244.6 241.9 243.4 245.3

245.1 242.9 242.4 246.3

247.6 246.5 245.3 248.2 240.8 248.4 244.4

of the relaxation process are shown in Fig. 6. It is interesting to compare Fig. 4 and Fig. 6. At the last two stages of the relaxation process the lowest energy of formation corresponds to the lowest density of states at the Fermi level. However it is not the case in the ideal stage where L1 2 is found the more stable, whereas the density of states at the Fermi level is not the lowest one.

4. Discussion and conclusion

4.1. ANNNI model Viewed along the cube axis, the series of structures L1 2 , D0 22 , and D0 23 may be described as an alternate staking of pure layers containing only majority atoms (Al) and mixed layers containing an equal number of minority (Zr) and majority atoms (see Fig. 7). The mixed layers form a centered square lattice, in which the minority atoms occupy the center and the majority atoms the corners or vice-versa. As a result the minority atoms are never nearest neighbors, and it is only their relative positions in subsequent mixed layers which distinguish the structures. In this stacking sequence, two relative positions between subsequent mixed layers are possible. Either the translation [0 0 1] connects minority atoms in subsequent mixed layers or it connects minority atoms to majority ones. The former case is the stacking in the L1 2 structure while the latter is the stacking in the D0 22 structure. By definition the antiphase boundaries (APBs) are the plane boundaries between domains in which the minority atoms occupy different basis positions in the unit cell. For instance the (0 0 1) antiphase boundary in the L1 2 structure is the plane boundary between two domains ordered according to the L1 2 structure but connected by stacking according to the D0 22 structure across the plane of the antiphase boundary. In the same way the (0 0 1) APB in the D0 22 structure may be described as the plane boundary between two regions of stacking according to D0 22 , connected by stacking according to L1 2 . The superstructures like D0 22 and D0 23 may be also viewed as a periodic arrangement of antiphase boundaries and this

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Fig. 5. Density of states n(E) for ZrAl 3 in the structures (a) L1 2 , (b) ideal D0 22 , (c) distorted D0 22 , (d) ideal D0 23 , (e) distorted D0 23 , and (f) fully relaxed D0 23 .

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C. Colinet, A. Pasturel / Journal of Alloys and Compounds 319 (2001) 154 – 161

planes separated by distance c / 4 along the z axis, and J2 an effective interaction coupling sites on the mixed planes separated by distance c / 2 along the z axis. Negative J1 effective interactions favor L1 2 structure, whereas positive J1 favor D0 22 structure. Negative J2 effective interactions favor both L1 2 and D0 22 structures, whereas positive J2 favor D0 23 structure. The corresponding Hamiltonian contains intra- and interplane contributions, and may be written as [43,44]:

O 1O F J O s (r)s

1 H 5 2 ] J0 s (r) sn (r9) 2 n,r,r 9 n r

Fig. 6. Density of states at the Fermi level for ZrAl3 in the three structures at the different stages of relaxation. (s) L1 2 , (m): D0 22 , and (j) D0 23 .

period is commonly given by the separation M of the antiphase boundaries measured in terms of the (0 0 1) lattice parameter of the underlying L1 2 structure [42]. The L1 2 , D0 22 and D0 23 structures correspond to M5`, 1, and 2, respectively. Such an arrangement of identical lattice planes perpendicular to the z axis may be treated in the axial-nextnearest-neighbor Ising (ANNNI) model [43]. Let us quote J0 the effective interactions within the planes normal to z, J1 an effective interaction coupling sites on the mixed

Fig. 7. Schematic representation of the L1 2 , D0 22 , and D0 23 structures. (s) Al atoms; (d) Zr atoms.

1

n

n

n11

(r) 1 J2

Os (r)s n

n

n12

(r)

G

(1)

where n labels the planes and r(r9) the sites in the planes. sn 5 1 1 or 21 depending whether or not site n is occupied by Al or Zr. In this model the energies of formation of L1 2 , D0 22 , and D0 23 structures have been given by Amador et al. [20]: Df E (L1 2 ) 5 E0 1 4 J1 1 4 J2 Df E (D0 22 ) 5 E0 2 4 J1 1 4 J2 Df E (D0 23 ) 5 E0 2 4 J2

(2)

From the formation energies referred to Zr fcc and Al fcc (Table 2), system (2) can be inverted to yield E0 and the effective interactions J1 and J2 at each stage of the relaxation process. Results are shown in Fig. 8. The three heavy lines intersecting at the origin delimit the domains

Fig. 8. Ground state phase diagram for the ANNNI model in the space of effective interactions J1 and J2 (kJ / mol of atoms) The three sets of values correspond to the three stages of relaxation process. (d) ideal structures, (m) distorted structures, and (j) fully relaxed structures.

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of stability of the L1 2 , D0 22 , and D0 23 structures. The circle, triangle, and square designate the effective interaction for ideal, c /a distortion, and cell-internal displacements, respectively. The c /a distortion leads to a change of the sign of J1 , whereas the cell-internal displacements of the atoms lead to an increase of the J2 value. These results may be compared with those obtained by Amador et al. [20]. The main difference concern the sign of J1 due to the fact that these authors found D0 22 structure less stable than L1 2 structure.

4.2. Conclusions In the present work we have calculated the total energies of the compound ZrAl 3 in the L1 2 , D0 22 , and D0 23 structures using VASP. D0 22 and D0 23 structures are stabilized with respect to L1 2 structure by tetragonal distortion and cell-internal displacements of the atoms in the case of D0 23 . The relative stability of the three structures is: Et (D0 23 ),Et (D0 22 ),Et (L1 2 ) Guo et al. [9] performed experiments where Al 1002x Zr x (x55, 10, 15 at%) alloys were rapidly solidified into ribbon form by a single roller melt spinning method. They found that: (i) the rapidly solidified alloys are composed of a-Al and L1 2 –ZrAl 3 phases, (ii) when annealing these alloys in a temperature range of 573–773 K, a new phase forms, the size of the phase is less than 100 nm, (iii) X-ray diffraction data show that the new phase has a tetragonal structure with parameters a50.392 nm and c50.906 nm, (iv) when the annealing temperature is higher than 773 K, the tetragonal phase disappears and changes into the equilibrium D0 23 –ZrAl 3 phase. Our theoretical calculations, which show that the D0 22 structure is slightly more stable than the L1 2 structure, authorize to suspect that the tetragonal phase could be D0 22 ; the lattice parameters obtained for this tetragonal phase are very similar to those obtained in the present calculations (see Table 1). Such an interpretation would be not possible if D0 22 was found to be less stable than L1 2 . The densities of electronic states, at each stage of the relaxation process, have been computed. We have shown that the densities of state at the Fermi level and the enthalpies of formation of the ZrAl 3 compound are related for the fully relaxed structures. The energetic results have been discussed in the framework of the ANNNI model. Further theoretical investigations of structural stability in MAl 3 compound are already in progress.

References [1] P. Villars, L.D. Calvert, in: Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, American Society for Metals, Materials Park, OH, 1985.

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[2] T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak (Eds.), Binary Alloy Phase Diagrams, 2nd Edition, ASM International, Materials Park, OH, 1990. [3] S. Srinivasan, P.B. Desch, R.B. Schwarz, Scripta Metall. Mater. 25 (1991) 2513. [4] A.T. Paxton, in: D.G. Pettifor, A.H. Cottrell (Eds.), Electron Theory in Alloy Design, The Institute of Materials, 1992, p. 158. [5] R.A. Varin, M.B. Winnicka, Mater. Sci. Eng. A 137 (1991) 93. [6] N. Ryum, Acta Metall. 17 (1969) 269. [7] T. Ohashi, R. Ichikawa, Metall. Trans. 3 (1972) 2300. [8] E. Nes, Acta Metall. 20 (1972) 499. [9] J.Q. Guo, K. Ohtera, K. Kita, T. Shibata, A. Inoue, T. Masumoto, Mater. Sci. Eng. A 181–182 (1994) 1397. [10] P.B. Desch, R.B. Schwartz, P. Nash, J. Less-Common Met. 168 (1991) 69. [11] P.B. Desch, R.B. Schwartz, P. Nash, Scripta Mater. 34 (1996) 37. [12] A.E. Carlsson, P.J. Meschter, J. Mater. Res. 4 (1989) 1060. ¨ [13] A.R. Williams, J.R. Kubler, C.D. Gelatt, Phys. Rev. B 19 (1979) 6094. ¨ [14] M. Methfessel, J.R. Kubler, J. Phys. F 12 (1982) 141. [15] J.-H. Xu, A.J. Freeman, Phys. Rev. B 40 (1989) 11927. [16] J.-H. Xu, A.J. Freeman, Phys. Rev. B 41 (1990) 12553. [17] J.-H. Xu, A.J. Freeman, J. Mater. Res. 6 (1991) 1188. [18] O.K. Anderson, Phys. Rev. B 12 (1975) 3060. [19] D.M. Nicholson, J.H. Schneibel, W.A. Shelton, P. Sterne, W.M. Temmerman, Mater. Res. Soc. Symp. Proc. 186 (1991) 229. [20] C. Amador, J.J. Hoyt, B.C. Chakoumakos, D. de Fontaine, Phys. Rev. Lett. 74 (1995) 4995. [21] M. Methfessel, Phys. Rev. B 38 (1988) 1537. [22] M. Methfessel, C.O. Rodriguez, O.K. Anderson, Phys. Rev. B 40 (1989) 2009. [23] M. Methfessel, M. van Schilfgaarde, Phys. Rev. B 48 (1993) 4937. [24] M. Alatalo, M. Weinert, R.E. Watson, Phys. Rev. B 57 (1998) R2009. [25] G.W. Fernando, J.W. Davenport, R.E. Watson, M. Weinert, Phys. Rev. B 40 (1989) 2757. [26] N. Chetty, M. Weinert, T.S. Rahman, J.W. Davenport, Phys. Rev. B 52 (1995) 6313. ¨ [27] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. ¨ [28] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [29] J. Hafner, Acta Mater. 48 (2000) 71. [30] G. Kresse, J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245. [31] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [32] M. Methfessel, A.T. Paxton, Phys. Rev. B 40 (1989) 3616. [33] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [34] G. Jomard, T. Petit, A. Pasturel, L. Magaud, G. Kresse, J. Hafner, Phys. Rev. B 59 (1999) 4044. [35] F. Birch, J. Geophys. Res. 57 (1952) 227. [36] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [37] R.J. Kematick, H.F. Franzen, J. Solid State Chem. 54 (1984) 226. [38] J.C. Schuster, H. Nowotny, Z. Metallkde. 71 (1980) 341. [39] S. Tsunekawa, M.E. Fine, Scripta Metall. 16 (1982) 391. [40] S.V. Meschel, O.J. Kleppa, J. Alloys Comp. 191 (1993) 111. [41] B.B. Argent, M.J. Perry, quoted by C.B. Alcock, K.T. Jacobs, S. Zador, in: O. Kubaschewski (Ed.), Zirconium: Physico-Chemical Properties of its Compounds, Atomic Energy Review No. 6, IAEA, Vienna, 1976, p. 47. [42] M.E. Fisher, W. Selke, Phys. Rev. Lett. 44 (1980) 1502. [43] D. de Fontaine, J. Kulik, Acta Metall. 33 (1985) 145. [44] F. Ducastelle, in: F.R. de Boer, D.G. Pettifor (Eds.), Order and Phase Stability in Alloys, Cohesion and Structure, Vol. 3, North Holland, Amsterdam, 1991.