Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure
The martensitic α → ω transition was investigated in Ti under hydrostatic pressure. The calculations were carried out using the density functional theory (DFT) framework in combination with the Birch-Murnaghan equation of state. The calculated ground-state properties of α and ω phases of Ti, their b...
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irk-123456789-1208382017-06-14T03:04:28Z Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure Jafari, M. Nobakhti, M. Jamnezhad, H. Bayati, K. The martensitic α → ω transition was investigated in Ti under hydrostatic pressure. The calculations were carried out using the density functional theory (DFT) framework in combination with the Birch-Murnaghan equation of state. The calculated ground-state properties of α and ω phases of Ti, their bulk moduli and pressure derivatives are in agreement with previous experimental data. The lattice constants of α and ω-phase at 0 K were modeled as a function of pressure from 0 to 74 GPa and 0 to 119 GPa, respectively. It is shown that the lattice constants vary in a nonlinear manner upon compression. The calculated lattice parameters were used to describe the α → ω transition and show that the phase transition can be obtained at 0 GPa and 0 K. Дослiджувався мартенситовий перехiд α → ω в Ti при гiдростатичному тиску. Обчислення здiйснювалися iз використанням теорiї функцiоналу густини (DFT) в комбiнацiї з рiвнянням стану Берча-Мурнагана. Обчисленi властивостi основного стану α та ω фаз Ti, їх об’ємнi модулi i тисковi похiднi узгоджуються iз попереднiми експериментальними даними. Постiйнi ґратки α i ω-фази при 0 K моделювалися як функцiя тиску, що змiнювався вiд 0 до 74 GPa та вiд 0 до 119 GPa, вiдповiдно. Показано, що ґратковi сталi змiнюються з концентрацiєю нелiнiйно. Обчисленi ґратковi параметри використовувалися для опису переходу α → ω i було показано, що фазовий перехiд може бути отриманий при 0 GPa та 0 K. 2013 Article Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure / M. Jafari, M. Nobakhti, H. Jamnezhad, K. Bayat // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33703:1-6. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 71.15.Mb, 71.20.Be, 05.70.Ce, 64.70.K- DOI:10.5488/CMP.16.33703 arXiv:1310.1233 http://dspace.nbuv.gov.ua/handle/123456789/120838 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The martensitic α → ω transition was investigated in Ti under hydrostatic pressure. The calculations were carried out using the density functional theory (DFT) framework in combination with the Birch-Murnaghan equation of state. The calculated ground-state properties of α and ω phases of Ti, their bulk moduli and pressure derivatives are in agreement with previous experimental data. The lattice constants of α and ω-phase at 0 K were modeled as a function of pressure from 0 to 74 GPa and 0 to 119 GPa, respectively. It is shown that the lattice constants vary in a nonlinear manner upon compression. The calculated lattice parameters were used to describe the α → ω transition and show that the phase transition can be obtained at 0 GPa and 0 K. |
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Jafari, M. Nobakhti, M. Jamnezhad, H. Bayati, K. |
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Jafari, M. Nobakhti, M. Jamnezhad, H. Bayati, K. Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure Condensed Matter Physics |
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Jafari, M. Nobakhti, M. Jamnezhad, H. Bayati, K. |
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Jafari, M. |
| title |
Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure |
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Інститут фізики конденсованих систем НАН України |
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Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure / M. Jafari, M. Nobakhti, H. Jamnezhad, K. Bayat // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33703:1-6. — Бібліогр.: 31 назв. — англ. |
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Condensed Matter Physics |
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2025-07-08T18:42:37Z |
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1837105320422277120 |
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Condensed Matter Physics, 2013, Vol. 16, No 3, 33703: 1–6
DOI: 10.5488/CMP.16.33703
http://www.icmp.lviv.ua/journal
Density functional theory study of the α→ω
martensitic transformation in titanium induced by
hydrostatic pressure
M. Jafari∗, M. Nobakhti, H. Jamnezhad, K. Bayati
Department of Physics, K.N. Toosi University of Technology, Tehran, Iran
Received June 10, 2012, in final form February 7, 2013
The martensitic α→ω transition was investigated in Ti under hydrostatic pressure. The calculations were car-
ried out using the density functional theory (DFT) framework in combination with the Birch-Murnaghan equation
of state. The calculated ground-state properties of α and ω phases of Ti, their bulk moduli and pressure deriva-
tives are in agreement with the previous experimental data. The lattice constants of α and ω-phase at 0 K were
modeled as a function of pressure from 0 to 74 GPa and 0 to 119 GPa, respectively. It is shown that the lat-
tice constants vary in a nonlinear manner upon compression. The calculated lattice parameters were used to
describe the α→ω transition and show that the phase transition can be obtained at 0 GPa and 0 K.
Key words: titanium, martenstic transition, hydrostatic pressure, Gibbs free energy, DFT calculation
PACS: 71.15.Mb, 71.20.Be, 05.70.Ce, 64.70.K-
1. Introduction
Most of the elements of the periodic table display a transformation of their crystal structures under
the application of temperature, pressure or both [1]. Group-IV transition metals, such as titanium, zirco-
nium and hafnium, have attracted considerable interest due to their appealing electrical and mechanical
properties. Their smaller density and higher strength make them the preferred metal of choice in many
high-technology industries.
These electrical and mechanical properties may be changed by controlling the crystal structure. Such
phase changes occur in pure titanium in response to the changes in pressure and temperature. The pur-
pose of this work is to study its α → ω martensitic transition under hydrostatic pressure. It should be
noted that it is a scalar hydrostatic pressure rather than uniaxial mechanical stress.
Martensitic transformation is a class of diffusionless, displacive and athermal transformations which
is of great technological interest. The first-order martensitic transformations were considered in the
present study where a unit cell of the parent phase was continuously distorted into a unit cell of the
daughter phase. Certain vectors and planes in the parent and daughter phases were parallel and the
transformation happened extremely quickly, giving the atoms insufficient time to travel very far.
At ambient pressure, theα-phase (hexagonal close-packed structure or hcp) with space group P63/mmc
has two atoms per unit cell at the Wyckoff positions of (1/3, 2/3, 3/4) and (1/3, 2/3, 1/4) [2, 3]. At high pres-
sure, the ω-phase (hexagonal structure) with the space group P6/mmm, has three atoms per unit cell and
is composed of two planes, a hexagonal plane at z = 0 (A-plane) and a honeycomb plane at z = 1/2c (B-
plane). This is the AlB2 type structure. Atoms are at (0, 0, 0) position in the A-plane and at (1/3, 2/3, 1/2),
and (2/3, 1/3, 1/2) positions in the B-plane.
The α-phase is known as the stable phase of Ti at room temperature and atmospheric pressure. By in-
creasing the pressure, the α→ω transition was observed. At high temperatures and zero pressure, the ω
phase transforms to β-phase (body-centered cubic structure or bcc) below the melting point (1933 K) [4].
∗E-mail: Jafari@kntu.ac.ir
©M. Jafari, M. Nobakhti, H. Jamnezhad, K. Bayati, 2013 33703-1
http://dx.doi.org/10.5488/CMP.16.33703
http://www.icmp.lviv.ua/journal
M. Jafari et al.
Except in Nguyan-Manh et al. [3], theoretical calculations have shown that the ω phase is more stable
than the α phase at 0 K [5–9]. Xia et al. showed that the ω phase remains stable up to 116 GPa [10]. Aka-
hama et al. [11] reported a new η phase with Fmmm space group in the range of 51÷80 GPa with the b/a
ratio ranging from 0.704 to 1.007 and the c/a ratio ranging from 0.704 to 0.789. Vohra et al. [12] investi-
gated Ti-structures up to 146 GPa and obtainedω→ γ transformation at 116± 4 GPa. The γ-phase belongs
to the orthorhombic lattices within the Cmcm space group. Akahama et al. [11] also obtained the ω→ γ
phase transition, but they also reported for this transformation to occur at 124÷130 GPa. Furthermore,
they reported the transition from the γ-phase to a new δ-phase at 140÷145 GPa [11].
In the experimental study with energy-dispersive X-ray diffraction (with a diamond anvil cell), the
stability of different phases of Ti [10, 12–14], transition between α and ω phases [10, 12–15], the effects
of impurity [12], uniaxial stress and heating [14] on pressure-induced transition have been investigated.
Moreover, the observed results of the X-ray diffraction show that the stability range of α phase varies
between the room temperature to around 923 K [16].
All theoretical works on transformation have been based on finding the minimum energy or Gibbs
free energy. To obtain the Gibbs free energy for each structure, researchers have used different meth-
ods such as the linear Muffin-Tin orbital and the local-density approximation (LDA) methods within the
framework of the DFT [5], the full-potential linearized-augmented-plane wave (FP-LAPW) method and
LDA [3, 13], the FP-LAPW and generalized-gradient approximation (GGA) [7], first-principles energy cal-
culations [17], the tight-binding total energy (TBE) method with first-principles calculations [18], the epi-
taxial bain path (EBP) combined with FP-LAPW and GGA [9] and the modified embedded-atom potential
method and DFT [19].
2. Methodology
This study is solely concerned with the α and ω structures of Ti. The Gibbs free energy per atom
obtained as a function of the volume in the two phases was fitted to the Birch-Murnaghan’s equation of
state (EOS). At zero temperature, the calculations of total energy were carried out using the FP-LAPW
method supplemented by local orbitals (FP-LAPW+lo) method based on the DFT [20]. The exchange and
correlation functionals were given by the GGA of Perdew-Wang [21, 22]. The calculations were carried out
using the WIEN2K package [20]. It is well known that the application of the FLAPW method to metallic-
like compounds always underestimates the inter-band energy distances [23] and very often a scissors
operator is used to shift the conduction bands in order to achieve an agreement with the experimental
value.
Since the bulk Ti exhibits paramagnetic properties, the spin effects were neglected. The value of
RMTKmax = 8.5 obtained after optimization was selected for both α and ω, where RMT is the average
radius of the muffin-tin spheres and Kmax is the maximum value of the wave vector K .
The optimized RMT for the highest pressure was calculated and it was used for the remaining pres-
sures in each phase. Although this was not necessary for low pressures (because the muffin-tin spheres
did not intersect at these pressures), it facilitated the subsequent comparison of the results from various
pressures. The radii of the spheres used in this study were 2.208 Å for theα phase and 2.300 Å and 2.060 Å
for the ω phase.
A desirable charge convergence was determined as 1× 10−4
e and the largest vector in the charge-
density Fourier expansion was fixed at Gmax = 12 bohr−1.
Following the k-point optimization at P = 0, 731 points were found for the hcp and 2460 for the
simple hexagonal structures, which corresponded to 64 and 171 k-point mesh in the irreducible wedge
of the Brillouin zone, respectively.
In the first step, k-mesh was optimized at the equilibrium volume. For each volume smaller than V0,
the same k-point as that generated for the zero-pressure case was used. The hexagonal structure could
be described by two parameters a and c (which included the axial ratio c/a). The following procedure
outlines the current method used to optimize the c/a ratio at each phase and volume. In the first step, by
using the WIEN2K code, the total energy was calculated for several values of c/a and keeping the volume
(V ) fixed. Then, the obtained results were fitted to a polynomial form in order to find the equilibrium
value of the c/a ratio. The minimum energy related to the minimum value of c/a in the polynomial
33703-2
Density functional theory study of the α→ω martensitic transformation in titanium
form was compared with the energy given by WIEN2K. If the energy related to the minimum c/a in the
polynomial form was the minimum energy provided by WIEN2K, it was accepted; otherwise, this proce-
dure was repeated with other values of c/a until the minimum energy was found. This procedure was
repeated for various volumes. To investigate the thermodynamic properties of the considered structures,
the following form representing the third-order Birch-Murnaghan EOS was used:
E (V ) = E0 +
9B0V0
16
[
(
V0
V
) 2
3
−1
]3
+
[
(
V0
V
) 2
3
−1
]2 [
6−4
(
V0
V
) 2
3
]
, (1)
where E0 and V0 are the energy and volume at equilibrium, respectively, B0 is the zero-pressure bulk
modulus and B
′
0
is the pressure derivative of B0. For all cases, B0 and B
′
0
can be determined by fitting the
E (V ) points from DFT to the Birch-Murnaghan’s EOS. Moreover, it was found that for each volume, the
pressure depends on the third-order Birch-Murnaghan’s EOS as follows:
P (V ) =
3B0
2
[
(
V0
V
) 7
3
−
(
V0
V
) 5
3
]{
1+
3
4
(B0 −4)
[
(
V0
V
) 2
3
−1
]}
. (2)
The theoretical pressure dependences of the lattice constants and the axial ratios c/a can be calculated.
The lattice parameters of two phases are shown in figure 1. For both phases a nonlinearity was ob-
served with an increasing pressure. The c/a ratios in both phases increased upon compression, but in a
different manner (see figure 2). In the α phase, there was a positive concavity and it was negative in the
ω phase.
The Gibbs free energy was derived using the equation, G = H −T S, where H stands for the enthalpy
of a system. At T = 0 K, the enthalpy is equal to the Gibbs free energy and thus G = H = E +PV . The
transition pressure was obtained by comparing the enthalpies of the two phases.
3. Results and discussion
The electronic densities of states (DOS) for the α and ω phases are shown in figures 3 and 4. As can
be seen, the Fermi energy level (Ef) is at the minimum of the DOS curves in both phases. Furthermore,
the minimum around Ef in the ω phase is narrower than that in the α phase. These features are in good
agreement with the previous works [24]. The total energy of −1707.6354 Ryd/atom was calculated for
α and −1707.63600 Ryd/atom for ω phases at T = 0 K and P = 0 GPa. In our previous work [25], the
electrical properties were determined from the location of the Fermi level and the DOS per atom at N
(Ef). Furthermore, in both phases, the d -states dominate at N (Ef) [26].
Figure 1. Theoretical unit cell parameters vs. pres-
sure for the α andω-Ti (lines are for the eye’s guide).
Figure 2. Theoretical c/a ratio vs. pressure for the α
and ω-Ti (line is for the eye’s guide).
33703-3
M. Jafari et al.
Figure 3. (Color online) Total and partial DOS of the
α-Ti.
Figure 4. (Color online) Total and partial DOS of the
ω-Ti.
Table 1. The optimized lattice constants, B0, B
′
0
at T = 0 K and P = 0 GPa for the α and ω-Ti.
α phase ω phase
This work Experiment This work Experiment
B0 101±6 GPa 117±9 GPa [28], 137±3 GPa 138±10 GPa [28]
117.4 GPa [29]
B
′
0
3.59±0.02 3.9±0.4 [28], 2.94±0.04 3.8±0.5 [28]
3.444 [29]
a 2.948017 Å 2.956±0.002 Å [29] 4.591137 Å 4.598 ± 0.009 [12]
c 4.650202 Å 4.734±0.025 Å [29] 2.834109 Å —
c/a 1.57740 1.601 [29] 0.61730 0.614 [12]
Figure 5. Gibbs free energies as a function of pres-
sure for α and ω phases of Ti.
This indicates that the chemical bonding of
the hexagonal structure is quite different from
that expected for transition metals. The hexago-
nal structure shows a large degree of covalency
in the B site with sd
2 hybridization. These prop-
erties cause more hardiness and low conductivity
due to some electrons localized in the hybrid or-
bital (Doherty and Gibbons model). It should be
noted that the anharmonic electron-phonon inter-
action also plays an important role in the pressure
induced phase transition [27].
The optimized lattice constants and the values
of B0 and B
′
0
are given in table 1. The optimized
theoretical lattice parameters at zero pressure in
the α-phase were within 95% of the experimental
data. The pressure range for the α-phase and the
ω-phase were 0÷74 GPa and 0÷119 GPa, respec-
tively. The results are presented in figures 1 and 2.
Least-squares curve fitting was used to determine a model for these data. It was found that a linear
model was not capable of accurately capturing the parameter variations, and instead a nonlinear model
was evaluated. Nonlinear behavior was visible in all the parameters (a, c, and c/a).
33703-4
Density functional theory study of the α→ω martensitic transformation in titanium
Moreover, the thermodynamic parameters (E ,V ) calculated using the DFT theory were fitted to the
third-order Birch-Murnaghan’s EOS and then the Gibbs free energy was obtained with these three pa-
rameters (E ,V and P ).The transition pressures were found by comparing the enthalpy values of the two
phases. Furthermore, our results show that these two Gibbs free energy curves (figure 5) intersect at
0 GPa and 0 K.
The extrapolation of experimental data [30] predicted that a martensitic α → ω transition occurs
near −1 GPa at 0 K. Our results are within the hysteresis interval (−5 to +8 GPa) and remain in excellent
agreement with the other experimental and theoretical works [9, 18, 31]. Theoretical investigations [9]
were carried out using first-principles total-energy calculations and the EBP for finding the equilibrium
structures directly as a function of the pressure from the minima of the particular Gibbs free energies.
These studies reported that the α-phase was stable up to 50 GPa and it transformed to the ω structure at
50 GPa. In another theoretical work [18], theα→ω transition in Ti was studied using the TBEmethod and
it was reported that theα→ω transition in Ti occurred at 6 GPa. First-principles total-energy calculations
with GGA approximation were also used to investigate the α→ω transition in Ti [31]. They compared the
enthalpy values of different structures and obtained −3 GPa for the α to ω transition. These results are
summarized in table 2.
Table 2. Calculated pressure-induced α→ω transition.
Theoretical work Pressure
[9] 50 GPa
[18] 6 GPa
[31] −3 GPa
This work 0 GPa
Based on the comparison between this paper’s calculated pressure-induced transition and the extrap-
olated data from experiments, it can be suggested that this method can also be used for explaining the
structural sequence in Ti under pressure for other phases.
4. Conclusion
This study investigated themartensitic α toω transition in Ti under hydrostatic pressure. The calcula-
tions were carried out using FP-LAPW+lo method. Accordingly, the lattice constants, c/a ratios, electronic
structures, the bulk moduli and pressure derivatives of the two Ti phases were calculated. Thereby, it
was shown that most of the structural parameters vary in a nonlinear manner. The calculated parame-
ters were used for describing the α→ω martensitic transition under hydrostatic pressure and showing
that this transition was obtained at 0 GPa and at zero temperature. This value was extremely close to the
extrapolated experimental data.
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Дослiдження методом функцiоналу густини мартенситового
переходу α в ω в титанi пiд дiєю гiдростатичного тиску
M. Джафарi, M. Нобахтi, H. Джамнезхад, K. Баятi
Факультет фiзики, Технологiчний унiверситет iм. К.Н. Тусi, Тегеран, Iран
Дослiджувався мартенситовий перехiд α→ ω в Ti при гiдростатичному тиску. Обчислення здiйснювали-
ся iз використанням теорiї функцiоналу густини (DFT) в комбiнацiї з рiвнянням стану Берча-Мурнагана.
Обчисленi властивостi основного стану α та ω фаз Ti, їх об’ємнi модулi i тисковi похiднi узгоджуються iз
попереднiми експериментальними даними. Постiйнi ґратки α i ω-фази при 0 K моделювалися як функцiя
тиску, що змiнювався вiд 0 до 74 GPa та вiд 0 до 119 GPa, вiдповiдно. Показано, що ґратковi сталi змiню-
ються з концентрацiєю нелiнiйно. Обчисленi ґратковi параметри використовувалися для опису переходу
α→ω i було показано, що фазовий перехiд може бути отриманий при 0 GPa та 0 K.
Ключовi слова: титан, мартенситовий перехiд, гiдростатичний тиск, вiльна енергiя Гiббса, DFT
обчислення
33703-6
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Introduction
Methodology
Results and discussion
Conclusion
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