The electronic, elastic, and structural properties of Ti

Intermetallics 18 (2010) 998–1006
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Intermetallics
journal homepage: www.elsevier.com/locate/intermet
The electronic, elastic, and structural properties of Ti–Pd intermetallics and
associated hydrides from first principles calculations
Xing-Qiu Chen a, *, C.L. Fu a, James R. Morris a, b
a
b
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6114, USA
Department of Materials Science and Engineering, University of Tennessee, Oak Ridge, TN 37996-2200, USA
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 1 September 2009
Received in revised form
20 January 2010
Accepted 25 January 2010
Available online 4 March 2010
Using an ab initio density functional approach, we report on the ground-state phase stabilities, enthalpies
of formation, electronic, and elastic properties of the Ti–Pd alloy system. The calculated enthalpies of
formation are in excellent agreement with available calorimetric data. We found a linear dependence
between the calculated enthalpies of formation of several intermetallic structures and the
Pd-concentration, indicating that each of these compounds has a very limited composition range. The
elastic constants for many of these Ti–Pd intermetallics were calculated and analyzed. The B2 TiPd phase
is found to be mechanically unstable with respect to the transformation into the monoclinic B190
structure. A series of hydrides, Ti2PdHx (x ¼ 1, 1.5, 2, 3, 4), have been investigated in terms of electronic
structure, enthalpies of hydrogen absorption, and site preference of H atoms. Our results illustrate the
physical mechanism for hydrogen absorption in term of the charge transfer, and explain why TiPd2 does
not form a stable hydride.
Ó 2010 Elsevier Ltd. All rights reserved.
Keywords:
B. Elastic properties
B. Thermodynamic and thermochemical
properties
B. Hydrogen storage
B. Phase transformation
E. Ab initio calculations
1. Introduction
There is a strong interest in Ti–Pd alloys due to their interesting
mechanical properties and technological importance for engineering and medicine, particularly for the shape-memory effect
and for the property of reversible hydrogen storage. Extensive
research on this system has been stimulated by the existence of
a martensitic transformation in alloys near equiatomic stoichiometry [1–6]. For Ti-rich alloys, Ti2Pd forms a hydride (Ti2PdH1.5 and
Ti2PdH2) with good hydrogen storage capability [7]. For Pd-rich
alloys, particular for TiPd2, uncertainties still exist for their stable
crystal phase [8,9]. Many properties (including phase stabilities, as
well as electronic and mechanical properties) of Ti–Pd alloys still
need to be understood at an atomic level. Here, we present the
results of a comprehensive study of ground-state structural, cohesive, electronic, and elastic properties of Ti–Pd intermetallics as
well as associated hydrides using ab initio calculations within the
framework of density functional theory. In Section 2 we briefly
review experimental data of intermetallic compounds, including
structures and calorimetric enthalpies of formation, and also
* Corresponding author. Present address: Shenyang National Laboratory for
Materials Science, Institute of Metal Research, Chinese Academy of Sciences,
Shenyang 110006, China.
E-mail address: [email protected] (X.-Q. Chen).
0966-9795/$ – see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.intermet.2010.01.027
previous studies of phase stability by ab initio methods. In Section 3
we present the details of the computational method used in the
current study. In Section 4, the equilibrium structures, phase
stabilities, and cohesive properties of the intermetallic compounds
are discussed. The electronic structures and elastic constants are
presented in Sections 5 and 6, respectively. In Section 7, the hydride
phases of hydrogen in Ti2Pd are analyzed and discussed in detail.
Finally, the summary and discussion are given in Section 8.
2. Literature review
Based on the currently available experimental data and the
latest version of the Ti–Pd phase diagram [10], there are seven
reported compounds at various alloy compositions. Most of them
are found experimentally to have extremely narrow concentration
ranges, typically 2% or less. The only exception is TiPd, which has
a range of 6% for most temperatures below 1280 C. The phase
diagram [10] is rich in structures; the crystallographic data of their
compounds are listed in Table 1. TiPd3 is known to exist in two
stable forms: the hexagonal TiNi3-type (hP16, D024) structure
[16–18] and (with excess Pd) the cubic AuCu3-type (cP4, L12)
structure [19]. For TiPd2, Krautwasser et al. [8] suggest an orthorhombic ReSi2-type (oI6, Immm) structure at low temperatures and
a tetragonal MoSi2-type structure at high temperatures. Recently,
using perturbed angular correlation, Wodniecki et al. [9] confirmed
X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
999
Table 1
Lattice constants (Å) and internal structural parameters of Ti–Pd intermetallics. The comparisons between calculated and measured (where available) lattice parameters are
also listed. The last column corresponds to the optimized atomic positions obtained in this work.
Phase
Structure
Ti3Pd
A15 (Cr3Si)
Ti2Pd
TiPd
Ref.
Wyckoff Site: (x,y,z)
[16]
Ti(6c): (0.25,0.5,0)
Pd(2a): (0,0,0)
[11]
Ti(4e): (0,0,0.333)
Pd(2a): (0,0,0)
3.118
3.090
Calc
Expt
Calc
3.168
3.180
2.855
Calc
Expt
Calc
Calc
4.573
4.550
2.799
g ¼ 93.21
2.796
2.780
4.570
4.899
4.860
5.903
Clac
Expt
14.364
14.330
4.622
4.610
4.717
4.640
[8]
Calc
Expt
3.296
3.263
11.512
11.436
[16]
3.271
3.240
3.456
3.097
8.571
8.480
8.729
[8]
oI6
Calc
Expt
Calc
oI6
Expt
3.410
3.070
8.560
[10]
D024 (TiNi3)
Calc
Expt
Expt
Expt
Calc
Expt
5.555
5.489
5.489
5.489
3.926
3.820
9.063
8.964
8.614
8.964
[16]
[17]
[18]
B2 (CsCl)
0
B19 (NiTi)
TiPd2
c
Calc
Expt
C11b (Si2Mo)
B19 (AuCd)
Ti3Pd5
b
5.044
5.055
L10 (AuCu)
Ti2Pd3
a
Calc
Expt
oC20
tI8
C11b (Si2Mo)
9.895
10.054
[12]
3.907
[14]
Ti(1a): (0,0,0)
Pd(1b): (0.5,0.5,0.5)
Ti(1a): (0,0,0)
Pd(1d): (0.5,0.5,0.5)
Ti(2e): (0.25,0,0.703)
Pd(2f): (0.25, 0, 0.703)
Ti(2e): (0.446,0.195,0.25)
Pd(2e): (0.011,0.683,0.25)
Ti(8g): (0.11, 0.23, 0.25)
Pd(8g): (0.19, 0.72, 0.25)
Pd(4c): (0, 0.71, 0.25)
Ti(1b): (0,0,0)
Ti(2h): (0.5,0.5,0.136)
Pd(1a): (0,0,0.5)
Pd(2g): (0,0,0.242)
Pd(2h): (0.5,0.5,0.380)
Ti(4e): (0,0,0.331)
Pd(2a): (0,0,0)
Ti(2a): (0,0,0)
Pd(4g): (0,0,0.333)
TiPd3
L12 (AuCu3)
an orthorhombic distortion of the MoSi2-type structure in the
temperature range from 24 to 1023 K, supporting the early studies
of Krautwasser et al. [8]. TiPd undergoes a martensitic transformation at about 800 K, but the transition temperature can be
reduced to 410 K with 8% substitution of Cr for Pd [5,6]. The hightemperature phase of TiPd has a simple cubic CsCl-type (cP2, B2)
structure [14,15], while at ambient temperature the martensite
phase has been reported as the orthorhombic AuCd-type (oP4, B19)
structure [13,14]. For Ti2Pd, the MoSi2-type (tI6, C11b) structure has
been reported by various experiments [10,12,16,20]. For Ti3Pd,
a cubic Cr3Si-type (cP8, A15) structure has been reported in vapor
deposited thin film [11] at a temperature range between 450 and
550 C. Earlier, the compound Ti3.2Pd0.8 shows the same Cr3Si-type
structure [16] where the 2a site is found to share a disordered
mixture of 0.2Ti and 0.8Pd. The enthalpies of formation of TiPd
[19,21–24], TiPd2 [19], Ti3Pd5 [19] and TiPd3 [19,23,24] have been
measured by mass-spectrometric technique [21], solute-solvent
dropping calorimetry [22], or high-temperature mixing calorimetry [19,23,24].
Previous ab initio studies on Pd–Ti system focused mainly on
shape-memory alloy TiPd. These studies include the elastic
constants of the B2 phase [25], the structural properties of the B2,
B19, and B190 phases [26], the optical properties of the B2 phase
[27], and the structural energies and phonon dispersion of the B2
phase [28]. More recently, Curtarolo et al. calculated structural
stabilities and enthalpies of formation for five intermetallics (Ti3Pd,
Ti2Pd, TiPd, TiPd2, TiPd3) using an ultrasoft pseudopotential within
the local density approximation (LDA) [30]. They found that the
[19]
Ti(2a): (0,0,0)
Ti(2b): (0.333, 0.667, 0.25)
Pd(6g): (0.5,0,0)
Pd(6h): (0.166, 0.332, 0.25)
Ti(2a): (0,0,0)
Pd(3c): (0, 0.5, 0.5)
energies of the L10 and B19 phases of TiPd are essentially degenerate, which is inconsistent with previous calculations [26–28].
3. Computational details
The present results were obtained using the Vienna ab initio
simulation package (VASP) [31–33] with the projector augmented
wave potential (PAW) [34,35] for the ion–electron interaction. An
energy cutoff of 400 eV was chosen for the plane-wave expansion
of eigenfunctions. Ti and Pd semi-core s and p states were treated as
valence states. For the exchange correlation functional, the generalized gradient approximation (GGA) of Perdew and Wang [36] was
applied. Brillouin-zone integrations were performed for suitably
large sets of k points according to Monkhorst and Pack (e.g.,
orthorhombic – 11 13 9; hexagonal – 11 11 9; cubic –
13 13 13) [37]. Optimization of structural parameters (atomic
positions and lattice parameters) was achieved by the minimization of forces and stress tensors.
Elastic constants were calculated for cubic, hexagonal, tetragonal, and orthorhombic lattice structures. Since these lattices have
three, five, six, and nine independent elastic constants, respectively,
it is necessary to calculate three, five, six, and nine energy-versusstrain curves to obtained all the elastic constants [39–41]. For all
structures, we carried out calculations for strains in the range of
0.02 to 0.02 with the step of 0.005 for each distortion. The
distortion energies were then fitted to third-order polynomials to
calculate their elastic constants. This approach relies on an accurate
total energy determination from first principles because the
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X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
distortion energy involved is very small. This requires the use of
a fine k mesh to guarantee a high degree of precision. During these
calculations, all internal atomic parameters are allowed to relax.
A number of competing structures were considered for five of
these compounds. The ground-state equilibrium properties were
obtained from the total energy minimization of different phases.
Enthalpies of formation were calculated [38] through the energy
difference
(oI6), I2Cd (hP3, C6), Pt2Si (tI6, L02b)). In agreement with experiment
[12,16,20], the tetragonal MoSi2-type structure has the lowest
enthalpy of formation (45.0 kJ (mol of atoms)1) among these
structures (Table 2) whereas the energies of the I2Cd, Al2Cu, MoPt2,
and Pt2Si-type structures are much less stable by 14.0, 16.0, 29.9,
and 45.9 kJ (mol of atoms)1 with respect to the energy of the
MoSi2-type structure, The Pt2Si-type structure is unstable, having
a positive formation energy. The calculated lattice parameters of
the MoSi2-type structure are in excellent agreement with experiment (Table 1), with a deviation within 1.5%. So far, the enthalpy of
formation of Ti2Pd has not been experimentally measured.
DH ¼ UDFT ðTi1x Pdx Þ ð1 xÞUDFT ðTiÞ xUDFT ðPdÞ
4.3. TiPd
4. Structural and phase stability
(1)
between the DFT total energy UDFT of the compounds and those of
the constituent elements, i.e., hcp Ti and fcc Pd. Table 2 presents the
enthalpies of formation of all the structures studied here. The
calculated lattice parameters together with the relaxed atomic
positions for the most energetically stable structure of each of these
Ti–Pd compounds are summarized in Table 1.
4.1. Ti3Pd
The stoichiometric Ti3Pd compound has the cubic ordered Cr3Sitype (cP8, A15) structure. Relative structural stabilities of Ti3Pd
were studied by considering four competing structures: Cr3Si (cP8,
A15), Ti3Cu (tP4, L60), AuCu3 (cP4, L12), TiNi3 (hP16, D024). In
agreement with experiment, the Cr3Si-type structure is found to be
the most energetically favorable of all the structures considered
here. A global optimization approach also has predicted this same
structure to be the lowest in energy [42]. The energies of the AuCu3,
Ti3Cu, TiNi3-type structures are 12.8, 12.7, and 13.7 kJ (mol of
atoms)1 higher than that of the Cr3Si-type structure, respectively.
The ground-state enthalpies of formation are listed in Table 2, but
no experimental value was reported for comparison. The calculated
lattice constant of the Cr3Si-type structure is a ¼ 5.044 Å which is in
good agreement with that of the experimental value of 5.055 Å [16].
However, it is larger by 4% than that of the thin film sample [11].
The relatively large deviation may be attributed to the fact that the
experimental sample was synthesized as thin film by vapor phase
deposition [11] at 450 C, whereas our calculation corresponds to
the perfect bulk material at zero temperature.
4.2. Ti2Pd
We investigated the relative structural stabilities of Ti2Pd among
five different structures (MoSi2 (tI6, C11b), Al2Cu (tI12, C16), MoPt2
Table 2
Enthalpies of formation from ab initio calculations, DHDFT, for the Ti–Pd intermetallics compared with the available experimental values DHexp and the estimated
values DHMie derived from Miedema’s model [42]. All energies are given in kJ (mol of
atoms)1.
Phase
Structure
DHDFT
Ti3Pd
Ti2Pd
TiPd
A15 (Cr3Si)
C11b (Si2Mo)
B190 (NiTi)
36.2
45.0
53.3
Ti2Pd3
Ti3Pd5
TiPd2
TiPd3
oC20
tI8
C11b (Si2Mo)
D024 (TiNi3)
57.9
59.1
60.8
62.9
DHexp
DHMie
53.3
51.6
53.1
53.0
80.0
1.8 [24]
6.7 [22]
3.0 [19]
15.0 [23]
10.0 [21]
58.7
58.1
65.0
50.7
1.2
1.3
0.9
1.7
[19]
[19]
[24]
[21]
60.3
78.7
99.1
95.1
92.1
85.5
67.2
Experimentally, the high-temperature phase of TiPd is a cubic
CsCl-type (cP2, B2) structure; the low-temperature martensitic
phase is an orthorhombic AuCd-type (oP4, B19) structure
[14,43,44]. We calculated the total energies and structural properties of several other competing phases [CrB (oC8, B33), AuCu (tP2,
L10), NiTi (mP4, B190 )]. Although the calculated equilibrium lattice
parameters are in good agreement with experiment [14] (Table 1)
and with previous ab initio calculations [26–28], the ground-state
structure found in our calculation actually is not the experimentally reported AuCd-type (B19) structure. More specifically, the
formation energies of the AuCd-, AuCu-, CsCl-, and CrB-type
structures are less stable by 1.0, 1.4, 8.0, and 13.3 kJ (mol of
atoms)1 than that of the NiTi-type structure. In agreement with
a previous ab initio study of Huang et al. [28], the most energetically
stable structure is found to be the monoclinic NiTi-type (B190 )
structure. The calculated lattice parameters for the NiTi-type
structure agree well with previous ab initio calculations [28,29].
The enthalpy of formation of TiPd has been measured by direct
reaction synthesis [19,21–24] (Table 2). Calorimetric measurements
gave very similar results, with a spread of 2 kJ (mol of atoms)1
[19,22–24]. Among these results, the recent data of 53.3 1.8 kJ
(mol of atoms)1 [24] agrees very well with our ab initio value of
53.3 kJ (mol of atoms)1 for the B190 structure. However, Choudary et al. [21] obtained a much more negative value of 80.0 kJ
(mol of atoms)1 using the mass-spectrometric technique.
4.4. Ti2Pd3 and Ti3Pd5
The Ti2Pd3 and Ti3Pd5 compounds have the orthorhombic (oS20)
[8] and tetragonal (tP8) [10,16] structures, respectively. As shown in
Table 1, the calculated structural parameters are in very good
agreement with available experimental data [8,16]. Using calorimetry, Selhaoui et al. [19] measured the enthalpy of formation of Ti3Pd5
and yielded a value of 58.7 1.2 kJ (mol of atoms)1, in good
agreement with our calculated value of 59.1 kJ (mol of atoms)1. The
enthalpy of formation of Ti2Pd3 was calculated to be 57.9 kJ (mol of
atoms)1; but no experimental value is available for comparison.
4.5. TiPd2
As already mentioned in Section 2, experimental investigations
show that TiPd2 has two possible structures: the tetragonal MoSi2type (tI6, C11b) and the closely related orthorhombic ReSi2-type
(oI6, Immm) structures. Note that the ReSi2-type structure is
equivalent to the MoPt2-type structure, which has been found to be
a stable phase for many other Pd alloys, such as TaPd2. In addition,
we consider other three competing structures (hexagonal CrSi2
(hP9, C40); orthorhombic ZrSi2 (oC12, C49); tetragonal Pt2Si (tI6,
L02b)). Our calculations show that the most stable structure is the
orthorhombic MoPt2-type (i.e., ReSi2-type) structure, which is
slightly lower in energy by about 0.8 kJ (mol of atoms)1 than the
X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
1001
tetragonal MoSi2-type structure. In fact, our calculations of the
elastic constants, discussed in Section 6. A below, show that at T¼0,
TiPd2 in the MoSi2 structure is mechanically unstable to an orthorhombic distortion, as indicated by c11–c12 < 0. The CrSi2- and ZrSi2type structures are energetically less stable by 2–3 kJ (mol of
atoms)1 compared with the MoPt2-type structure. The least stable
phase among these five structures is the Pt2Si-type structure. Table 1
presents comparisons between calculations and measurements for
lattice parameters and atomic coordinates. For the MoSi2-type
structure we find the calculated lattice parameters agree to within
1% of experimental measurements [8]. However, for the orthorhombic MoPt2-type structure, the c lattice parameter is overestimated by 2% with respect to the experimental value [10]. The
measured enthalpy of formation from calorimetry for TiPd2 is
58.1 1.3 kJ (mol of atoms)1 (Ref. [19]), which agrees well with
our calculated result of 61.6 kJ (mol of atoms)1 (Table 2) [45].
4.6. TiPd3
Among the three structures (TiNi3 (hP16, D024), AuCu3 (cP4, L12),
TiAl3 (tI8, D022)) that we have considered, the hexagonal TiNi3-type
is found to have the lowest energy. This result is consistent with the
experimental findings of Raub et al. [16], Harris et al. [17], and
Schulz et al. [18]. The energy of the cubic AuCu3-type structure is
only slightly higher by 0.6 kJ (mol of atoms)1 than that of the
TiNi3-type structure, suggesting that the cubic AuCu3-type structure could exist as a metastable phase. In fact, the stoichiometric
TiPd3 compound with the cubic AuCu3-type structure has been
synthesized by Selhaoui et al. [19]. Also, according to the Ti–Pd
phase diagrams [10,20], the precipitation of TiNi3-type TiPd3 from
the supersaturated solid solutions of Pd is preceded by a metastable
phase called g with the AuCu3 structure. The energy of the TiAl3type structure is about 5.2 kJ (mole of atoms)1 higher than that of
the TiNi3-type structure. Our calculated lattice parameters (Table 1)
for the TiNi3 and AuCu3 structures are in good agreement with the
experimental results [16–19]. Enthalpies of formation for TiPd3
have been determined by calorimetry [24] and mass-spectrometric
[21] methods. Our calculated result (62.9 kJ (mol of atoms)1)
agrees very well with the measured values at 298 K (65.0 0.9 kJ
(mol of atoms)1) by high-temperature mixing calorimetry [24].
The value obtained from the mass-spectrometric method [21] is
smaller by 10 kJ/(mol of atoms) compared with our result.
0
0.2
0.4
0.6
0.8
1.0
Ti xPd 1-x
Fig. 1. Trends in the enthalpies of formation (kJ (mol of atoms)1) as a function of Pdconcentration in Ti–Pd alloys. The solid circles and open triangles are the results for
stable and metastable alloy phases (see text for details) calculated from first principles.
The experimental calorimetry data are shown as solid diamonds and the results
derived from Miedema’s model [46] are shown as open circles.
not fall on this common straight line, but with a deviation
Eh ¼ 4.8 kJ (mol of atoms)1 above the straight lines (Fig. 1). This
straight line implies that the concentrations ranges for different
compounds are quite narrow and that the phase diagram consists
mainly of two phase regions, in agreement with experiments
[10,20,49]. Also, supporting this argument is the fact that the phase
diagrams of the isoelectronic systems Zr–Pd and Hf–Pd [52,53] are
very similar to Ti–Pd; however, both systems have fewer experimentally observed compounds than the Ti–Pd system.
5. Electronic structures
The total and projected local DOS’s for several of the Ti–Pd
intermetallic compounds are shown in Figs. 2 and 3. There are two
common features in the DOS profiles for these compounds: (i) Ti
a
c
b
d
4.7. Trend of enthalpies of formation for Ti1xPdx
In Fig. 1 we plot the calculated formation enthalpies (DHDFT) as
a function of Pd content. The trend in the calculated DHDFT as
a function of alloy composition is consistent with that of a previous
calculation [30]; however, DHDFT of Ti2Pd3 and Ti3Pd5 were not
included in Ref. [30] and the calculated DHDFT in Ref. [30] are slightly
less negative compared to our results. The B19 and L10 structures of
TiPd were found to be degenerate in energy in Ref. [30], whereas our
calculations clearly indicate that the enthalpy of formation of B19 is
slightly lower by 0.4 kJ (mol of atoms)1 than that of L10 TiPd. Also
shown in Fig. 1, the DH values derived from the Miedema’s semiempirical model [46] deviate considerably from those of ab initio
and experimental results [22,24], except for the case of TiPd3. The
core of Miedema’s theory is too simplified and the shortcomings
have been seen in many cases [38,47,48].
Fig. 1 shows that the convex hull of ground-states is asymmetric
and skewed toward the Pd side, with the most negative DHDFT at
TiPd3. A remarkable feature in Fig. 1 is that the enthalpies of
formation of Ti2Pd, TiPd, Ti2Pd3, Ti3Pd5, TiPd2 and TiPd3 at their
respective ground-state phases all nearly fall on a common straight
line. The only exception is Ti3Pd whose enthalpy of formation does
Fig. 2. Total and d-like partial density of states for the Ti2Pd and TiPd2 compounds:
Ti2Pd with the MoSi2-type (panel (a)) and MoPt2-type (panel (b)) structures; TiPd2
with the MoSi2-type (panel (c)) and MoPt2-type (panel (d)) structures. The energy
scale is adjusted so that the Fermi energy is zero. The 2a, 4e, and 4g sites denote the
Wyckoff atomic sites in these structures.
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X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
a
increased Pd–Pd interactions in TiPd2. In addition, for Ti2Pd the
Fermi level falls about 3 eV above the filled Pd-4d band, whereas in
TiPd2 the Fermi level falls at less than 1 eV above the top of the
filled Pd-4d-band. A similar trend was observed in the DOS profiles
of Ti3Pd and TiPd3 (c.f., Fig. 3). Although these two compounds have
different structures, the trend in the change of the DOS profile from
Ti3Pd (CrSi3, A15) to TiPd3 (TiNi3, D024) is very similar to that from
Ti2Pd to TiPd2. As shown in Fig. 3, Ti3Pd has a considerably high
density of states (1.20 states eV1 atoms1) at the Fermi level,
whereas the Fermi level of TiPd3 lies in the deep valley of the DOS
profile. This may explain the fact that TiPd3 has the highest
enthalpy of formation among all these Ti–Pd compounds considered here
Using N(EF), we can derive the linear term in the specific heat
coefficient at low temperature, g, which is summarized in Table 3.
Again, except for the case of TiPd3, the value of g is shown to be the
smallest for the most stable structure. No experimental value is
available for comparison.
b
Fig. 3. Total and d-like partial density of states for the Ti3Pd (the CrSi3-type structure)
and TiPd3 (the TiNi3-type structure) compounds. The energy scale is adjusted so that
the Fermi energy is zero. The 2a, 2b, 6c, 6h and 6g sites denote the Wyckoff atomic sites
in the CrSi3- and TiNi3-type structures.
and Pd have similar DOS profiles in the whole energy region,
indicating the presence of hybridization between Ti and Pd delectrons; (ii) The Pd-4d bands are nearly fully occupied and the
states near the Fermi level are mainly the Ti 3d states.
Table 3 compiles the density of states at the Fermi level, N(EF),
for all seven compounds, which is often used as a good indicator for
the structural stability. Interestingly, there seems to be a correlation
between N(EF) and structural stability for most of these Ti–Pd
compounds. For each of these compounds considered here, except
for the case of TiPd3, the lowest energy structure is also found to
have the lowest N(EF) value. For instance, for Ti2Pd, the Fermi level
of the most stable MoSi2-type structure is located in a deep valley
(Fig. 2(a)), whereas in the least stable MoPt2-type structure the
Fermi level lies at a very sharp peak as shown in Fig. 2(b). For TiPd2,
the DOS profiles and their N(EF) values of the MoSi2-type and
MoPt2-type structures are very similar (c.f. Fig. 2(c) and (d)), which
explains their tiny energy difference and suggests the possibility of
the co-existence of both structures. It is interesting to compare the
electronic structures of the Ti2Pd and TiPd2 in the MoSi2-type
structure [8,12]. The bandwidth (about 4.9 eV) of the filled Pd-4d
bands in TiPd2 (Fig. 2(c)) is much wider than that of Ti2Pd (2.3 eV).
The increased Pd d-band width from Ti2Pd to TiPd2 is due to the
Table 1
Table 3Total density of states N(EF) (in states per eV and per atom) and specific heat
coefficient g ¼ ð1=3Þp2 NðEF Þk2B (in mJ mol1 K2) at T ¼ 0 K.
Phase
Structural types
Ti3Pd
N(EF); g
Ti2Pd
N(EF); g
TiPd
N(EF); g
Ti2Pd3
N(EF); g
Ti3Pd5
N(EF); g
TiPd2
N(EF); g
TiPd3
N(EF); g
CrSi3
1.20; 2.82
MoSi2
0.69; 1.63
TiNi
1.18; 2.78
Ti2Pd3
0.56; 1.32
Ti3Pd5
0.67; 1.58
MoPt2
0.53; 1.24
TiNi3
0.43; 1.01
TiNi3
2.67; 6.29
MoPt2
1.86; 4.38
AuCd
1.43; 3.37
MoSi2
0.81; 1.91
AuCu3
0.35; 0.82
CsCl
1.71; 4.02
6. Elastic properties
6.1. Elastic constants
6.1.1. Ti3Pd, Ti3Pd5, and TiPd3
In Table 4 we present the calculated elastic constants of the
cubic Ti3Pd (CrSi3), tetragonal Ti3Pd5, and hexagonal TiPd3 (TiNi3).
Their elastic constants clearly obey the mechanical stability
conditions for cubic, hexagonal, and tetragonal crystals [54]. The
precision of our calculation for the bulk modulus can be estimated
by comparing the value directly obtained from the elastic
constants, namely B0 ¼ 1/3 (c11 þ 2c12) for cubic, B0 ¼ 2/
9(c11 þ c12 þ 2c13 þ c33/2) for hexagonal, and B0 ¼ 2/
9(2c11 þ c33 þ 2c12 þ 4c13) for tetragonal lattices, with the value
derived from the volume derivative of the total energy (c.f. Table 4).
In all cases, the difference is 2% or smaller. For the compound TiPd3,
our calculated results are in reasonable agreement with those
obtained from interatomic potentials [55].
6.1.2. Ti2Pd and TiPd2
The calculated elastic constants of Ti2Pd and TiPd2 with the
tetragonal MoSi2-type structure are compiled in Table 5. For
tetragonal crystals, the mechanical stability conditions are
c11 c12 > 0, c11 þ c33 2c13 > 0, c11 > 0, c33 > 0, c44 > 0, c66 > 0, and
2c11 þ c33 þ 2c12 þ 4c13 > 0 [40,54,56]. The elastic constants of Ti2Pd
satisfy all of the above conditions, whereas those of TiPd2 do not
satisfy the condition c11 c12 > 0. This means that the MoSi2-type
TiPd2 compound would distort spontaneously to the orthorhombic
lattice to lower the energy [45], confirming recent experimental
investigations [9]. The elastic constants of the orthorhombic TiPd2
with the MoPt2-type structure were calculated and compiled in
Table 5; these nine elastic constants clearly obey the stability
conditions [56]: c11 þ c22 2c12 > 0, c22 þ c33 2c23 > 0, c11 > 0,
c22 > 0, c33 > 0, c44 > 0, c55 > 0, c66 > 0, and
c11 þ c22 þ c33 þ 2c12 þ2c13 þ 2c23 > 0. The bulk moduli calculated
Table 4
Elastic constants (in GPa) of the Ti3Pd, Ti3Pd5, and TiPd3. B0 is the bulk modulus
calculated directly from the elastic constants and from the fit of the energy-versusvolume curve to the Murnaghan equation of state (in parentheses).
Phase
c11
c12
Ti3Pd (cub. cP8 Cr3Si)
Ti3Pd5 (tetr. tI8)
TiPd3 (hex. hP16 TiNi3)
TiPd3 (calc. [55])
213.4
199.9
306.4
331.1
105.1 –
179.6 125.8
124.6 112.7
110.3 65.5
c13
c33
c44
–
251.5
317.8
334.8
35.4 –
141.2 (139.7)
57.3 90.6 168.2 (166.4)
71.0 –
181.1 (178.5)
66.4
c66
B0
X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
1003
Table 5
Elastic constants (in GPa) of the Ti2Pd and TiPd2. See Table 4 for notation.
Phase
c11
c12
c13
c22
c23
c33
c44
c55
c66
B0
Ti2Pd (tetr. tI6 MoSi2)
TiPd2 (tetr. tI6 MoSi2) [45]
TiPd2 (ortho. oI16 MoPt2) [45]
227.8
128.8
214.7
106.1
195.1
121.7
122.3
121.4
167.7
–
–
201.7
–
–
72.7
177.6
221.4
229.1
85.1
72.0
62.1
–
66.3
99.7
95.5
145.2 (145.2)
150.5 (152.3)
152.1 (153.3)
from theoretical elastic constants agree well with those derived
from the energy-versus-volume curves (Table 5).
6.1.3. TiPd
In Table 6 the calculated and available experimental elastic
constants of TiPd are compiled. In the case of B2 TiPd, our calculated
results are quite different from those of Bihlmayer et al. [25]
calculated from the full-potential linearized augmented planewave calculations. Our result for c44 (46.6 GPa) is higher by
46 GPa than that (0.6 GPa) obtained by Bihlmayer et al. [25]. Our
result for the shear modulus c0 (18.6 GPa) is also different from
that of their value (25.8 GPa) [25]. A negative c0 value means that
the B2 structure can transform into the B19 orthorhombic structure. Thus, we further calculated the elastic constants of B19 TiPd
and listed their values in Table 6. We found that, however, the B19
TiPd is also mechanically unstable because of a negative
c44 ¼ 34.4 GPa. A c44-type deformation in the B19 orthorhombic
lattice together with the internal atomic displacements along the
[010] and [001] directions results in the structural transformation
to the B190 monoclinic lattice. Indeed, from our total energy
calculations, we found that the monoclinic B190 structure has
a lower energy than both the B2 and B19 structures (see Section
4.3). This result is also in agreement with that obtained by Huang
et al. [28]. They found that the experimentally reported orthorhombic B19 phase will undergo a low-temperature phase transformation to a monoclinic B190 ground state via a soft phonon mode
coupled to strain [28]. Recently, using inelastic neutron scattering,
Shapiro et al. [6] measured the elastic constants of B2 Ti50Pd42Cr8
single crystal. The measured values are c11 ¼ 188, c12 ¼ 167,
c44 ¼ 38 GPa, which are compared with our calculated results for
the B2 TiPd in Table 7.
6.2. Polycrystalline elastic moduli
From the calculated elastic constants, we derived the polycrystalline averages using the Voigt–Reuss–Hill method [57]. The
calculated Young’s modulus E, shear modulus G, Poisson’s ratio y,
and Debye temperature Q are presented in Table 7. The polycrystalline averages were obtained for random orientations of the
crystallites. Here, for the sake of comparison with experiments, we
also present in Table 8 the results for elemental hcp Ti and fcc Pd.
For hcp Ti and fcc Pd, our calculated values of Debye temperatures,
Young’s moduli, and shear modulus are in good agreement with the
experimental values [58] (Table 7) with deviations within 5%.
Table 6
Elastic constants (in GPa) of the TiPd compound. See Table 4 for notation.
Phase
c11
c12
c44
c0
B0
Ref.
TiPd (CsCl-type)
133.3
169.7
45.3
18.2
157.6
(156.2)
143.7
This work
109.3
160.9
0.6
25.8
Ti50Pd42Cr8
188.0
167.0
38.0
10.6
TiPd (AuCd-type)
c11
245.9
c12
87.2
c13
140.3
c22
274.1
c33
242.8
c44
34.4
c55
31.8
c66
54.4
Calc [25]
Exp [5,6]
c23
71.8
B0
151.3
(150.5)
This work
81.4
7. Hydrogen in Ti2Pd and TiPd2 alloys
Early nuclear magnetic resonance (NMR) measurements [7]
indicate that the compound Ti2Pd with the MoSi2-type structure
forms the Ti2PdH1.47 and Ti2PdH1.96 hydrides [51,52]. Many of its
isoelectronic compounds (e.g., Zr2Pd) [7,50,51,52,59] also exhibits
good hydrogen storage capacity. It has been identified by NMR that,
in Ti2PdH1.96, the hydrogen atom occupies a single type of tetrahedral interstitial site, as also found in the isoelectronic compound
Zr2PdH1.84 [7]. The powder X-ray diffraction revealed that
Ti2PdH1.96 has a slightly distorted MoSi2-type structure [7]; but the
site of H occupancy was not identified.
In order to systematically investigate the trend of the H
absorption energy in Ti2Pd, we have considered a series of hydrides,
Ti2PdH, Ti2PdH1.5, Ti2PdH3, and Ti2PdH4. The positions of H atoms in
the unit cell as well as the cell shapes of these hydrides were
determined through the total energy calculations. The corresponding lattice parameters of Ti2PdHx (x ¼ 1–4) and the site
locations of absorbed hydrogen atoms are summarized in Table 8.
We found that, in terms of the H absorption energy, the most stable
hydride form is Ti2PdH2. Thus, our discussion in the following will
focus on Ti2PdH2.
In our calculation, we consider absorbed H atoms at the octahedral and/or tetrahedral interstitial sites in the MoSi2-type
structure. Among all the configurations considered for Ti2PdH2, the
most stable configuration is found to be the one with all of the
absorbed H atoms at the octahedral 4e site (Fig. 5(b)). Note that,
however, this configuration is just marginally more stable (by only
2 meV/H atom) than the configuration with all of the absorbed H
atoms at the tetrahedral 4d sites (Fig. 5(a)). Due to the extremely
small energy difference between H occupying these octahedral and
tetrahedral sites, it is likely that, in TiPd2H2, the absorbed H atoms
have a nearly equal probability to be in either one of these two
configurations.
The energy of hydrogen absorption can be obtained from
Ef ðTi2 PdH2 Þ ¼ EðTi2 PdH2 Þ EðTi2 PdÞ EðH2 Þ;
(2)
where E(Ti2PdHx), E(Ti2PdHx), and E(H2) are the calculated total
energies. We obtained an exothermic energy (103.3 kJ/mol H2) for
the lowest energy configuration of Ti2PdH2. Of course, this value
Table 7
The calculated polycrystalline elastic moduli at the respective predicted equilibrium
volumes. The symbols E, G, y, vm, QD are Young’s modulus, shear modulus (all in
units of GPa), Poisson’s ratio, average sound velocity (in m s1), and Debye
temperature (in K), respectively.
Ti (hcp)
Exp [59]
Ti3Pd (CrSi3)
Ti2Pd (MoSi2)
TiPd (TiNi)
Ti3Pd5
TiPd2 (MoPt2)
TiPd3 (TiNi3)
Pd (fcc)
Expt [59]
E
G
y
vm
QD
116.3
116
114.6
164.0
113.5
120.0
172.0
209.9
131.9
121
43.9
44
41.9
62.4
41.3
43.5
65.4
80.3
47.9
44
0.33
3445.3
0.36
0.32
0.37
0.38
0.32
0.31
0.38
2870.6
3345.8
2529.3
2484.9
2962.5
3158.2
2309.3
401.2
420.0
268.9
395.0
301.3
295.9
354.0
239.4
276.3
274.0
1004
X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
Table 8
Calculated lattice constants (a, b, c in Å), hydrogen atomic positions and the
hydrogen absorption enthalpies (Ef in kJ/mol H2) without the inclusion of zero-point
energies of Ti2PdHx (x ¼ 1–4). In Ti2PdH, Ti2PdH1.5, Ti2PdH3, since the symmetric
sites are not fully occupied by hydrogen atoms, each hydrogen position has been
specified. However, if H fully occupies the symmetric interstitial sites (i.e., 4d or 4e),
only its representative is listed.
Compound
Atom
Wyckoff
x
y
z
Ti2PdH
a ¼ 2.899
c ¼ 11.894
Ef ¼ 93.8
Pd
Ti
H
H
2d
4e
0
0
0.5
0.5
0
0
0.5
0.5
0
0.3406
0.3192
0.6808
Ti2PdH1.5
a ¼ 2.876
b ¼ 2.877
c ¼ 12.189
Ef ¼ 102.3
Pd
Ti
H
H
H
2d
4e
0
0
0.5
0.5
0.5
0
0
0.5
0.5
0.5
0
0.3391
0.0
0.3241
0.6758
Ti2PdH2
a ¼ 2.853
b ¼ 2.847
c ¼ 12.508
Ef ¼ 103.3
Pd
Ti
H
2d
4e
4e
0
0
0
0
0
0
0
0.3406
0.8139
Ti2PdH3
a ¼ 2.909
c ¼ 12.934
Ef ¼ 81.0
Pd
Ti
H
H
H
2d
4e
4d
0
0
0
0.5
0.5
0
0
0.5
0.5
0.5
0
0.3391
0.2401
0.3631
0.6368
a ¼ 2.936
c ¼ 13.249
Ef ¼ 58.2
Pd
Ti
H
H
2d
4e
4d
4e
0
0
0
0
0
0
0.5
0
0
0.3458
0.25
0.8685
Ti2PdH4
does not consider the contribution from the hydrogen zero-point
motion, which could be significant. Thus, we have also calculated
the phonon spectra of Ti2PdH2 and Ti2Pd, and H2. Their zero-point
energies are 27.1, 9.7 and 25.7 kJ (mol of f.u.)1, respectively. Our
calculated zero-point energy for the hydrogen gas (H2) agrees well
with that obtained by van Setten et al. [63]. With the zero-point
energies included, the energy of hydrogen absorption becomes
95.1 kJ/mol H2 for Ti2PdH2.
For the purpose of comparison with the case of Ti2PdH2, it would
be interesting to investigate if the hydride form of TiPd2H2 can be
stabilized in MoSi2-type TiPd2. Again, among all configurations that
we have considered, the lowest energy one is the configuration
with the absorbed hydrogen atoms occupying the octahedral 4e
site. However, the calculated energy of hydrogen absorption is
strongly endothermic (123.1 kJ/mol H2). Therefore, TiPd2H2 is
unstable.
This raises the question as to why Ti2Pd exhibits such good
hydrogen absorption properties in contrast with TiPd2, although
a
Fig. 5. Optimized atomic configurations of Ti2PdH2 by ab initio calculation. Atomic
configurations (a) and (b) depict H atoms occupying symmetric sites 4d and 4e in the
MoSi2-type structure (i.e., I4/mmm space group), respectively. Note that both configurations obey the so-called minimum H–H separation 2-Å rule [60–62].
the same MoSi2-type structure was adopted in the calculations. In
Fig. 4 we show the DOS profiles of Ti2PdH2 and TiPd2H2. In both
cases, the bonding states between hydrogen and the host are found
in energies ranging from 10 to 6 eV below the Fermi level.
However, there are some differences in the H-induced features
between Ti2PdH2 and TiPd2H2. These differences will be discussed
in conjunction with the calculated electronic localization function.
The electronic localization function (ELF) [64] is based on
a topological analysis related to the Pauli Exclusion Principle. An
ELF equal to 1 corresponds to perfect electron localization. Here, we
consider only the case that H atoms are absorbed at the octahedral
site. The contours of ELF domains for Ti2PdH2 and Ti2Pd on the (110)
and (220) planes are shown in Fig. 6. For the H-free Ti2Pd phase,
there are four localized ELF domains in the (220) plane (Fig. 6(a))
with the maximum value of ELF ¼ 0.8. These domains are localized
at the ideal tetrahedral interstitial sites formed by the Ti atom, as
visualized by the ELF isosurface in Fig. 6(b). However, upon
hydrogen absorption at the octahedral sites, the maximum ELF
domains are seen at the H absorption site (i.e., 4e sites) with a value
of ELF ¼ 1.0 (c.f., Fig. 6(c)). Furthermore, the ELF domains at the
b
Fig. 4. Total and site-projected density of states (DOS) for Ti2PdH2 (a) and TiPd2H2 (b).
The energy scale is adjusted so that the Fermi energy is zero. The bonding states
between hydrogen and the host occur at energies of 6 to 10 eV below the Fermi
energy.
Fig. 6. Contour and isosurface plots of the calculated electronic localization function
(ELF) domains in Ti2Pd and Ti2PdH2. For Ti2Pd: (a) the contours on the (220) plane, and
(b) the isosurface for ELF ¼ 0.5. For Ti2PdH2: (c) the contours on the (110) plane, (e) the
contours on the (220) plane, and (d) the isosurface for ELF ¼ 0.5.
X.-Q. Chen et al. / Intermetallics 18 (2010) 998–1006
Fig. 7. Contour and isosurface plots of the calculated electronic localization function
(ELF) in TiPd2 and TiPd2H2. For TiPd2: (a) the isosurface for ELF ¼ 0.3. For TiPd2H2: (b)
the isosurface for ELF ¼ 0.5, and (c) the contours on the (220) plane. We note that there
is no localized ELF domain in the interstitial region for TiPd2.
tetrahedral interstitial sites are significantly reduced to 0.45 in
Ti2PdH2 (Fig. 6(e)), resulting in smaller ELF isosurface domains as
shown in Fig. 6(d). This reflects the change of electronic distribution
upon H absorption. Using Bader’s analysis [65,66], we confirmed
that the charges are transferred from Ti to both H and Pd atoms in
Ti2PdH2. Among the 1.21e charge transferred from each Ti to H and
Pd, H gains an excess charge of 0.78e. In fact, the charge transfer
from Ti to H is a key factor for the formation of hydrides. By
comparing the DOS of Ti2Pd and Ti2PdH2 (c.f., Figs. 2(a) and 6(c)), it
can be shown that the charge transfer leads to a change of d-band
occupancy (i.e., a slight downward shift of the Fermi level due to the
Ti to H charge transfer).
By contrast, in terms of ELF, there is no noticeable localized
domain at the tetrahedral interstitial sites of the H-free MoSi2-type
TiPd2, as illustrated in Fig. 7. Also, in terms of electronic structure,
the Pd-4d bands are fully occupied. Thus, in order to form TiPd2H2,
charge transfer from Pd to H is energetically unfavorable, which is
evident by the calculated positive energy of hydrogen absorption
for H in MoSi2-type TiPd2. Indeed, Bader’s analysis [65,66] also
shows that there is no charge transferred from Pd to H in the
hypothetical TiPd2H2 compound. For the same reasons, we do not
expect that TiPd2 can absorb any hydrogen also in its ground-state
MoPt2-type (or ReSi2-type) structure, since the MoPt2-type structure is related to the MoSi2-type structure by a small c11 c12 type
distortion, which does not significantly alter the electronic structure or the local chemistry of H occupying sites.
As the concentration of absorbed hydrogen increases, the
hydrogen atoms start to occupy the 4d tetrahedral sites. In the case
Ti2PdH3, we find that 4 H atoms prefer occupying the 4d tetrahedral
sites and the other two H atoms sit in the 4e octahedral sites. In the
case of Ti2PdH4, we find that both the 4d and 4e sites are fully
occupied by eight H atoms. However, Ti2PdH4 is the least stable
hydride with an energy of hydrogen absorption (Ef ¼ 58.2 kJ/mol
H2) among all the Ti2PdHx hydrides considered here.
8. Summary
Using an ab initio density functional approach, we have studied
the ground-state phase stabilities, enthalpies of formation, electronic, and elastic properties of the Ti–Pd alloy system. We confirm
that Ti3Pd, Ti2Pd, and TiPd3 have the ground-state cubic CrSi3-type,
tetragonal MoSi2-type, and hexagonal TiNi3-type structures,
respectively, which are in agreement with experiments. For TiPd,
we found that the monoclinic B190 structure is the most stable
1005
structure. For TiPd2, the orthorhombic MoPt2-type (or ReSi2-type)
is the ground state, slightly lower in energy than the tetragonal
MoSi2-type structure observed above 1280 C. Among these
compounds, TiPd3 has the most negative enthalpy of formation of
about 63 kJ (mol of atoms)1.
Consider all of the ground-state structures, we find a linear
dependence of the enthalpies of formation on Pd-concentrations,
consistent with the formation of line compounds for most of
these phases. Interestingly, at a fixed alloy stoichiometry, the most
stable structure seems to correlate with the lowest density of states
at Fermi level in most of these Ti–Pd compounds. Elastic constants
for six of the Ti–Pd intermetallics were calculated. The stability of
these compounds was analyzed in terms of the elastic stability
constraints.
Ti2Pd is known to form a very stable hydride. In this study, we
calculated the enthalpies of hydrogen absorption as a function of
hydrogen concentration in the MoSi2-type structure. Our results
illustrate the physical mechanism for hydrogen absorption in term
of the charge transfer from Ti to H atoms. Similar calculations on
TiPd2 show that there is no corresponding charge transfer,
explaining why TiPd2 does not form a stable hydride.
Acknowledgments
Research at Oak Ridge National Laboratory was sponsored by
the Division of Materials Sciences and Engineering, U.S. Department of Energy under contract with UT-Battelle, LLC.
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