Materials Sciences and Applicatio ns, 2011, 2, 1355-1366
doi:10.4236/msa.2011.210184 Published Online October 2011 (
Copyright © 2011 SciRes. MSA
Theoretical Investigations of Ti-Based Binary
Shape Memory Alloys
Rita John1, Hannah Ruben2,3
1Department of Theoretical Physics, University of Madras, Chennai, India; 2Post Graduate Department of Physics, Women’s Chris-
tian College, Chennai, India; 3Mother Teresa Women’s University, Kodaikanal, India.
Received June 15th, 2011; revised July 8th, 2011; accepted August 17th, 2011.
The electronic structure and ground state properties of TiX (X = Fe, Ni, Pd, Pt and Cu) type Shape Memory alloys have
been calculated using the self consistent Tight-Binding Linear Muffin Tin Orbital (TB-LMTO) method. The systematic
total energy studies made on TiX alloys in both B2 and (B19/B19’) structures successfully explain the structural stabil-
ity of these compounds. The equilibrium lattice parameters, bulk moduli (Bo), cohesive energy (Ecoh) and heat of forma-
tion (H) are calculated for these systems and compared with the available experimental and other theoretical results.
The bonding nature of these TiX alloys is analyzed via the density of states (DOS) histogram.
Keywords: Shape Memory Alloys, TB-LMTO, B2-B19 Phases, Structural Parameters
1. Introduction
Martensitic alloys have been a hot topic for several dec-
ades due to shape memory effect and many other peculiar
properties present during their martensitic phase trans-
formation [1,2]. A lot of experimental and theoretical
works have been devoted to study this phenomenon yet
many aspects of the transformation are still elusive. It has
stimulated many investigations in exploring the differ-
ence in electronic structure of different structural phases
involved in the transformation from martensite phase
(B19/B19’) to austenite phase (B2).
To investigate Martensitic phase transformation from
the electronic point of view, Ye et al. [3] have used first
principles total energy calculations to show some inter-
esting correlations between the relative stability of B2
and (B19/B19’) phases with the electronic structure for
TiNi, TiPd and TiPt alloys.
The main objective of the paper is to study the elec-
tronic and structural properties of TiX (X = Fe, Ni, Pd, Pt
and Cu) alloys in both (B2) and (B19/B19’) phases cal-
culated using the self consistent Tight-Binding Linear
Muffin Tin Orbital (TB-LMTO) method. The paper is
divided into six sections. Section 2 gives a brief outline
of the computational details of the Tight-Binding Linear
Muffin Tin Orbital scheme. The results of the total en-
ergy calculations obtained for cubic (B2) and ortho-
rhombic/monoclinic (B19/B19’) phases of TiX alloys are
presented in Section 3. The band structure and density of
states (DOS) for TiX (X = Fe, Ni, Pd, Pt and Cu) alloys
in both phases are reported in Section 4. Section 5 deals
with the theoretically calculated cohesive energy, heat of
formation which are compared with the experimentally
reported available values. The bulk modulus values ob-
tained for the alloys using the universal equation of state
(UEOS) analysis are reported in Section 6. The important
conclusions arrived from the above studies are given in
the last Section.
2. Method of Calculation
The band structure and total energy studies are made
within the atomic-sphere approximation by means of
Tight Binding-Linear Muffin Tin Orbital method (TB-
LMTO) [4], which is the exact transformation of Ander-
sen’s linear muffin-tin orbitals [5] to localize short-
ranged or tight-binding orbitals. The potential is calcu-
lated within the density-functional prescription under
local-density approximation (LDA) using the parame-
terization scheme of von Barth and Hedin [6]. The TB
(screened) representation of the LMTO method makes
the computation fast for the following reasons: 1) the
MTO’s are linear in energy and hence, unlike the aug-
mented plane-wave or Korringa-Kohn-Rostoker methods
we can get the eigen values within single diagonalization;
2) a solution to an eigen value equation of size only 9 × 9
Theoretical Investigations of Ti-Based Binary Shape Memory Alloys
(for s, p, d electron elements) per atom at each point in
reciprocal space is required; 3) The screened structure
constant for each atom needs only up to second-nearest-
neighbor atoms. In the band structure calculations, the
valence electronic configurations for TiFe, TiNi, TiPd,
TiPt and TiCu are as Ti: 3d2 4s2, Ni: 3d8, 4s2, Pd: 4d10, Pt:
5d9 6s1 and Cu: 3d10, 4s1 respectively. They are chosen to
represent the basis set for our calculations. The d elec-
trons are treated as valence electrons unlike earlier pseu-
dopotential based calculations. In our calculations, the s,
p, d partial waves have been used (i.e., maximum angular
momentum lmax = 2). Apart from this, the combined cor-
rection terms are also included, which account for the
non-spherical shape of the atomic cells and the truncation
of higher partial waves (l > 2) inside the sphere so as to
minimize the errors in the LMTO method. To exclude
additional freedom in the choice of computational pa-
rameters the same Weigner-Seitz (WS) radius is chosen
for all atoms and the calculated overlaps between the
various atomic spheres in this WS radius are within the
allowed range of the atomic-sphere approximation. The
atomic position coordinates used for B2 and B19 struc-
tures are (0, 0, 0), (0.5, 0.5, 0.5) and (0, 0, 0) and (0.5,
0.0, 0.5) all alloys respectively.
The tetrahedron method for the Brillouin zone (i.e., k spa-
ce) integration has been used with its latest version, which
avoids misweighing and corrects error due to the linear
approximation of the bands inside each tetrahedron [7].
3. Total Energy Calculations
In order to determine the phase stability of both B2 and
(B19/B19’) crystal structures in Figure 1, we have cal-
culated the total energy for each alloy with different lat-
tice constants in the reduced and extended experimental
volumes for all the compounds. The self consistent itera-
tions were carried out with an accuracy of 10–4 Ryd for
eigen values, using 96 k points in the irreducible wedge
of the first Brillouin zone (IBZ) of orthorhombic/mono-
clinic structures and 72 k points in the IBZ of cubic
The total energy curves for TiX (X = Fe, Ni, Pd, Pt and
Cu) alloys in B2 and (B19/B19’) structures for different
reduced and extended volumes are shown in Figure 2. The
equilibrium lattice constants of the above mentioned sys-
tems in both phases are calculated. From Table 1, the
theoretically obtained equilibrium lattice constants are
underestimated compared to the experimental values [8],
(a) (b)
Figure 1. (a) Austenite (B2) lattice structure and (b) mart-
ensite (B19/B19’) lattice structure.
Table 1. Experimental and Theoretical lattice constants (a, b, c) in Å of Ti X (X = Fe, Ni, Pd, Pt, Cu) alloys.
a b c Theoretical Experimental
Alloy Type
Present work a b c a b c
B2 2.9156 2.987[a] 2.976[a]
B19 2.7581 4.4204 3.934
TiNi B2 2.9714 3.023[a] 3.015[a]
B19 2.512 4.028 3.582 2.859[b] 4.582[b] 4.078[b]
B19’ 2.8801 4.5792 4.0321 2.898[c] 4.108[c] 4.646[c]
TiPd B2 3.1263 3.191[a] 3.18[a]
B19 2.7030 4.6010 4.3780 2.79[b] 4.81[b] 4.52[b] 2.81 4.89 4.52
TiP B2 3.1374 3.205[a] 3.192[a]
B19 2.6464 4.5476 4.2843 2.81[b] 4.83[b] 4.55[b] 2.76[a] 4.84 4.59
TiCu B2 3.0275 3.07[a]
B19 2.8966 4.6427 4.1320
[a]Reference [9]; [b]Reference [10]; [c]Reference [11].
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys1357
Figure 2. Total energy as a function of volume for TiX (X = Fe, Ni, Pd, Pt and Cu) alloys Continuous line indicate B2 struc-
ture and line with points indicate B19’/B19 structure.
and this is partly ascribed to the local density approxima-
tion (LDA) used in the calculations. To minimize the
deviations, it is argued that the zero point vibrations have
to be included in the calculation [12].
The total energy curves of TiX (X = Fe, Ni, Pd Pt and
Cu) alloys in B2 and (B19/B19’) phases, as seen in Fig-
ure 2 show that these alloys are stable in B2 structure [13]
though the energy difference between the two is very
small of the order of 0.2502 Ry/F.u.
4. Band Structure and Density of State
Studies on TiX (X = Fe, Ni, Pd, Pt and Cu)
The correlation of structural stability with electronic
structure is performed using band structure calculations
[14]. The band structures of TiX (X = Fe, Ni, Pd, Pt and
Cu) alloys have the same generic nature in B2 and B19
structures respectively. They are plotted along high sym-
metry lines; Figures 3(a) and 3( b) show the band struc-
tures of TiX alloys in both phases for several symmetry
directions in k-space. The (B19/B19’) structure of TiX
alloys possesses lower symmetry leading to splitting of
degenerate bands in the interior of brillouin zone as seen
in Figure 3(b). The size of the brillouin zone is twice as
big as B2 structure since it possesses four atoms per unit
cell. Hence the band structure of (B19/B19’) phase has
complicated brillouin zone compared to B2.
The band structure of TiNi in B2 structure is shown in
Figure 3(a). The s-orbitals of Ti and Ni sites around 8
eV are the lowest lying valence bands and do not con-
tribute much in deciding the properties of these alloys.
The top most valence band below EF between 4 eV and
2 eV are due to Ti-d states followed by Ni-d states. Be-
low them, the contribution from Ni-p state predominates.
The band dispersions of (B19/B19’) phase have simi-
lar generic nature as B2 at more bound states as seen in
Figure 3(b), yet there are significant differences that can
be detected at higher energies near the Fermi level EF.
This is reflected in the corresponding density of states
(DOS) curve there by influencing various physical prop-
erties such as susceptibilities and optical conductivities in
both phases [15].
The widths of the valence band for all alloys are tabu-
lated in Table 3. We observe the width of the band to be
narrower for (B19/B19’) phase compared to B2. Hence
the interactions between Ti and X (X = Fe, Ni, Pd, Pt and
Cu) atoms are much stronger in (B19/B19’) phase com-
pared to B2, which is also reflected in their correspond-
ing DOS curves [15].
In both phases TiPt possess wider bandwidth com-
pared to all other TiX (X = Fe, Ni, Pd and Cu) alloys.
The higher localization of t atom towards the bound P
Copyright © 2011 SciRes. MSA
Theoretical Investigations of Ti-Based Binary Shape Memory Alloys
Figure 3. (a) Band structure of TiX (X = Fe, Ni, Pd, Pt and Cu) in B2 phase at V/Vo = 1; (b) Band structure of TiX (X = Fe, Ni,
d, Pt and Cu) in B19 phase at V/Vo = 1.
Copyright © 2011 SciRes. MSA
Theoretical Investigations of Ti-Based Binary Shape Memory Alloys1359
state in comparison to other X (X = Fe, Ni, Pd, and Cu)
atoms results in decrease in overlap of bands between Ti
and Pt. This weakens the interaction between the two,
and consequently the strength of the covalent bonding
between them is decreased.
Density of States
To study the phase stability at microscopic level, the
DOS are calculated for the TiX (X = Fe, Ni, Pd, and Cu)
alloys at their equilibrium volumes and are plotted using
linear tetrahedron method. In present study, the double
peak structure is the most typical two peak structure of
the total DOS curve which is shown in Figures 4(a) and
The DOS curves for B19/B19’ structures are similar in
nature to that of the B2. Due to lower symmetry the DOS
peaks of the B19 and B19’ structures tend to be broader
than B2. There are some changes noticed in the lower
portion of the DOS curves of B19/B19’ structure at a
range of ±1.5 eV around Fermi. From Figures 4(a) and
4(b) we observe the dividing dip of the DOS in B2
structure at about 1.5 eV around Fermi becomes less
conspicuous in B19 structures. In B19’ structure, near the
Fermi level within a range of ±1.5 eV there is an upward
shift of the dip of the DOS curve.
The DOS curve of TiFe alloy in both phases is found
to be in good agreement with the reported results of Y.
Ye et al. The typical feature of the total DOS curve of
TiFe in B2 phase is the presence of pseudo gap. Two
mechanisms were proposed for the formation of pseudo
gap in the binary alloys [16]. One is of ionic origin, and
other is owing to hybridization effects. As the electro
negativity difference between Ti and X is low, the ionic-
ity does not play a major role on bonding behavior of
these compounds. Consequently the pseudo gap present
in TiFe alloys is believed to be due to covalent hybridi-
zation between Ti and Fe atoms. Such a strong hybridi-
zation gives not only an important mixing between the
states of conduction bands but also leads to a separation
of bonding states creating a pseudo gap.
In the present work we observe that in B2 structure the
shape of the DOS curve of TiNi is similar to TiFe. For
both alloys the EF falls on a dip as shown in Figure 4(a).
It is well known fact that if the Fermi level EF falls on the
dip, the corresponding structure may be regarded as a
save energy system as compared with one whose Fermi
level EF does not fall on a dip. Therefore TiFe and TiNi
alloys are more stable than other transition metals Pd, Pt
and noble metal Cu. On going from TiPd, TiPt and TiCu
the B2 structure becomes relatively less stable, as the
Fermi level EF shifts from the dip towards higher peaks
of the DOS [17]. This is accounted by the increase in
valence electrons which will tend to shift the Fermi level
EF from the dip towards the anti-bonding states. From
Figure 4(b) similar dip is observed for TiPd, wherein the
Fermi level EF falls on the dip. Hence TiNi and TiFe in
B2 structure and TiFe and TiPd in B19 structure have
low N (EF) values at Fermi and are considered to be most
stable structures. However the stability of TiPd in B19
contradicts the inference for the total energy calculations
shown in Figure 2. As the energy of TiPd at Fermi is
lower for B2 (0.0523 eV) when compared to B19
(0.0271 eV) structure, TiPd can be considered to be sta-
ble in B2 structure. This is in line with the results re-
ported by Ravindran et al. [17] in the case of Ni3Al and
As the electrons around EF play an important role in
deciding the electronic, structural and mechanical prop-
erties of the alloys, we carry out the investigations on the
electronic structure histogram using the projected DOS
of TiX alloys. General nature of the DOS histograms of
all TiX (X = Fe, Ni, Pd, Pt and Cu) alloys are observed to
be similar of which the DOS histogram of TiNi alloy is
shown in Figure 4(c). consists of three parts 1) the peak
present in the lower energy part of the DOS curve is
mainly due to the localized or tightly bound s-electrons
of Ni and Ti atoms; 2) the bonding sates of Ti-d and Ni-p
orbitals are near (to the left of) the Fermi level EF; 3) the
DOS curve due to anti-bonding states. It is found that
Ti-s state and Ni-s-state electrons in TiX alloy are local-
ized and its effect in bonding is very small. Thus the
electrons from Ti-d and Ni-d, Ni-p states predominately
contribute to the density of states at the Fermi level EF.
In order to explore the role of X (X = Fe, Ni, Pd, Pt
and Cu) atoms in the Ti-based alloys we compare the
d-partial DOS curves of these alloys as shown in Figures
4(d) and 4(e). We observe that the d-states of Ti atom at
Fermi shows a dominance compared to X (X = Ni, Pd, Pt,
Cu) atoms. The XPS studies by Shabaloskaya et al. [18]
show that as elemental X atoms combine with Ti atom to
form TiX (X = Ni, Pd, Pt, Cu) compound, it results in
enhancement of localization of d-electron of X atoms
towards bound state. Hence the intensity of the d-band of
X atom in TiX (X = Ni, Pd, Pt, Cu) alloys considerably
decreases at Fermi and that of Ti atom increases. How-
ever in case of TiFe alloy, we observe the d-states of Fe
atom to dominate that of Ti at Fermi. This is because as
elemental Fe forms TiFe compound on combination with
Ti atom, the position of d-band peaks of Fe atom does
not change [19], (i.e.) (the Fe atom does not undergo
much localization as other X (X = Ni, Pd, Pt, Cu) atoms).
Thus the d-band of Fe atom shows much dominance at
Fermi when compared to that of d-band of Ti.
This is also well supported by the DOS values of TiX
(X = Ni, Pd, Pt, Cu) alloys at EF which are tabulated in
Table 2. From Table 2 in both B2 and B19 structure, we
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys1361
Figure 4. (a) Total DOS structure of TiX (X = Fe, Ni, Pd, Pt and Cu) in B2 Phase; (b) total DOS structure of TiX (X = Fe, Ni,
Pd, Pt and Cu) in B19 Phase; (c) (a’) s-Partial DOS (b’) p-Partial DOS (c’) d-Partial DOS of TiNi in B2 phase. Line with
points indicate Ti atom and continuous line indicate Ni atom; (d) d-Partial DOS of TiX (X = Fe, Ni, Pd, Pt and Cu) alloys in
B2 phase. Line with points indicate Ti atom and continuous line indicates X atoms; (e) d-Partial DOS of TiX (X = Fe, Ni, Pd,
Pt and Cu) alloys in B19 phase. Line with points indicate Ti atom and continuous line indicates X atoms.
observe the d-DOS value of Ti at EF in general increases
from TiFe to TiCu along which the stability of phases
decreases. The increase in degeneracy of the d-states of
Ti at Fermi level EF decreases the phase stability as sug-
gested by Wang et al. [20]. Except for Fe atom, the d-
DOS contribution of X atomto DOS at Fermi level EF, is
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys
Table 2. The DOS value at the EF in TiX (X = Fe, Ni, Pd, Pt and Cu) alloys.
PHASE B2 B19 B2 B19 B19’ B2 B19 B2 B19 B2 B19
Nd(Ef) of Ti 0.1841 0.8607 1.353 1.9886 0.7651 2.40002.1904 1.92302.1428 3.5596 2.9885
Nd(Ef) of X 0.2957 1.8116 0.882 0.7272 0.6262 0.82850.3714 0.75000.4000 0.8217 0.2952
Np(Ef)of Ti 0.0377 0.1435 0.117 0.1181 0.0198 0.1950 0.1795 0.1034 0.1568 0.3397 0.1551
Np(Ef) of X 0.0442 0.1982 0.205 0.1948 0.0982 0.19710.2431 0.25710.2857 0.3586 0.2665
Ns(Ef) of Ti 0.0032 0.0781 0.029 0.0925 0.0188 0.05580.1121 0.04650.1037 0.0174 0.0958
Ns(Ef) of X 0.0045 0.1040 0.047 0.1318 0.0256 0.09410.0636 0.06550.0409 0.0174 0.0443
NT(Ef) of Ti 0.2161 1.1177 1.470 1.9318 0.7810 2.63002.4761 2.07692.3714 3.9669 3.3383
NT(Ef) of X 0.3135 2.0951 1.117 1.0455 0.7555 1.22800.6984 1.13630.7619 1.1467 0.6021
NT(Ef) of TiX 0.4427 3.4932 2.588 3.4285 3.9149 3.86363.6000 3.20003.5600 5.0767 4.3034
relatively small when compared to Ti. We observe Ti-d
state is virtually twice as high as the corresponding state
of X atom, resulting in weakening of dTi-d X directional
bond between them. The main part of the bonding state
of d-electrons of X site gradually move towards the bot-
tom of the valence band while the anti-bonding state of
d-electrons at Ti site becomes gradually strong, thereby
weakening the dTi-dX directional bond between them.
Thus as the atomic number of X (X = Fe, Ni, Pd, Pt,
Cu) atom increases, its d-state in both B2 and B19 phases
become more localized, the maximum of d-bands shifts
towards the bottom of the valence band, and the X con-
tribution to the density of states (DOS) at the Fermi level
N (EF), degrades (Table 2). While the d electron contri-
bution of X to the DOS at EF decreases, the d electron
contribution of Ti increases to such an extent that in TiPd
and TiCu, the X contribution to N (EF) is almost negligi-
ble. Hence the d-DOS localization accompanied by a
spatial localization of d electrons of X (X= Fe, Ni, Pd, Pt,
Cu) atom results in weakening of the d-d covalent bonds
between the alloy components thus, destabilizing the
phase.Similarly from Table 2 we observe the p contribu-
tion of X atom to DOS at EF are relatively smaller than d
contribution of X. Hence the p-d hybridisation will be
less pronounced for all these alloys.
In order to compare the nature of hybridisation in each
alloy we have plotted graphs between the differnce in d
DOS of Ti and X with d electrons of X atoms, and dif-
ference in d DOS of Ti and p of X atom with d electron
of X atoms as shown in Figures 5 and 6 In On compar-
ing Figures 5 and 6 we observe that both hybridisations
follow the similar trend. The d-d and p-d hybridisations
are well pronounced for TiFe in B2 and B19 structure.
In case of TiNi d-d hybridization in B19’ structure
dominates. For TiPd and TiPt p-d hybridization in B19
Figure 5. d-d hybridation of TiX (X = Fe, Ni, Pd, Pt and Cu)
in B2 and B19/B19’ phase.
Figure 6. p-d hybridation of TiX (X = Fe, Ni, Pd, Pt and Cu)
in B2 and B19/B19’ phase.
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys1363
structure is much pronounced than d-d hybridization.
And both the hybridizations are very less pronounced in
5. Cohesive Energy and Heat of Formation
The cohesive energy of a material is a fundamental prop-
erty which has long been the subject of theoretical and
computational approaches. The chemical bonding is a
mixture between covalent, ionic, and metallic bonding
and therefore the cohesive energy cannot be determined
reliably from simple models. Thus, first principles calcu-
lations based on density functional theory (DFT) have
become a useful tool to determine the cohesive energy of
the solids. In this connection, the cohesive energy of TiX
(X = Fe, Ni, Pd, Pt and Cu) alloy is calculated by using
the expression
cohatom atomtotal
TiX refers to the total energy of the TiX alloy at
equilibrium lattice constants and and are
the atomic energies of the pure Ti and X(X = Fe, Ni, Pd,
Pt and Cu) constituents calculated semi-relativistically.
To determine the heat of formation, we have first calcu-
lated the total energy of Ti element and X (X = Fe, Ni,
Pd, Pt and Cu) corresponding to their respective equilib-
rium lattice parameters. The free energy of formation or
the heat of formation (H) can be obtained from the fol-
lowing relation:
Ti B
 
where refers to the total energy of TiX (X = Fe,
Ni, Pd, Pt and Cu) alloy at equilibrium lattice constants
and and is the total energy of the pure
elemental constituents.
Ti B
The calculated values of the cohesive energies and
heat of formation of all systems are given in Table 3.
The systematic errors in total energy due to the ASA are
cancelled due to the nature of the formula of differences
in total energy leading to a reasonably accurate formation
of energy.
The cohesive energies of the alloys are slightly lower
in (B19/B19’) phase compared to B2 phase except for
TiNi. This indicates that atoms in B2 phase are strongly
bound with better mechanical strength than (B19/B19’)
phase. In the case of TiNi the cohesive energy is much
higher in B19’ phase compared to B2 phase indicating
that bonding effect is much stronger in B19’ phase com-
pared to B2 phase. This is due to strong dTi-dX directional
bonding between Ti and Ni atom which can also be seen
from Table 2. It is observed from Figure 7 that the co-
hesive energies of 5d transition series is higher compared
to 3d series which is due to higher localization of Pd and
Pt atom compared to 3d elements such as Fe, Ni and Cu.
This confirms the experimental studies, that 5d elements
of larger cohesive energies have higher melting point
From Table 3, the study on heat of formation show
that the ordering energy values except for TiFe are much
higher in (B19/B19’) phase compared to B2 phase which
is seen in Figure 8. This is a positive indication of strong
directional bonding leading to brittleness in (B19/B19’)
phase. The present theoretical values of heat of formation
Table 3. The theoretically calculated cohesive energy (Ecoh in eV/F.u), heat of formation (ΔH in eV), bulk modulus (Bo in
Mbar) for TiX (X = Fe, Ni, Pd, Pt and Cu) alloys.
Alloy Type Ecoh (eV/F.u) −ΔH (eV) Bo (Mbar) e/a N (EF) (eV) Width of the valence band (eV)
TiFe B2 1.9955 0.3761 3.3084 6 0.4427 7.9577
B19 1.9953 0.3755 1.6083 3.4931 5.1720
TiNi B2 2.4187 0.640 0.66[a] 2.2984, 1.56[a] 7 2.5881 8.1818
B19 2.4186 0.750 1.5891 3.4285 6.2063
B19’ 2.8262 0.9520 1.5603 3.9149 5.0117
TiPd B2 6.0124 1.033 0.92[a] 2.8446 7 3.8636 7.8409
B19 6.0122 2.582 3.0112 3.6000 5.777
TiPt B2 19.6423 1.630 1.49[a] 2.1717 7 3.2000 9.2045
B19 19.6420 1.879 3.4499 3.5600 7.1379
TiCu B2 2.6187 2.2448 0.8851 7.5 5.0784 8.2688
B19 2.6185 2.2444 0.9847 4.3036 5.8078
eference [3].
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Theoretical Investigations of Ti-Based Binary Shape Memory Alloys
Figure 7. Variation of cohesive energy as function of “d”
electrons of X in TiX (X = Fe, Ni, Pd, Pt and Cu) alloy.
of TiX alloys tend to increase systematically as we go
from 3d to 4d to 5d metals [22]. From Figure 8 we ob-
serve the heat of formation increases from TiFe to TiNi
and then to TiPd. The TiPd alloy possesses highest heat
of formation energy in B19 phase.
6. Equation of State
The total energy of TiX (X = Fe, Ni, Pd, Pt and Cu) al-
loys has been calculated for different reduced and ex-
tended volumes and fitted with the sixth order polyno-
mial. From the first derivative of the polynomial the P-V
data of TiX (X = Fe, Ni, Pd, Pt and Cu) in their stable
structures are generated. Vinet et al. have proposed a
universal equation of state which is valid for all the
classes of solids under compression. The UEOS is ex-
pressed as
 
P3B12exp 1
 
where x denotes (V/Vo)1/3 and Bo refers to the bulk
modulus. If one defines H(x) = x2 P(x)/[3(1 x)], then the
ln[H(x)] versus 1 x curve should be linear and obey the
 
In HInB1
 (4)
The bulk modulus of TiX alloys in B2 and (B19/B19’)
structures are given in Table 3 and shown in Figure 9.
From the bulk modulus curve, for B19 the Bo value is
found to be maximum for 5d TiPt followed by 4d TiPd
and 3d TiNi. Generally, the compounds with high melt-
ing temperature Tm are expected to have high Bo value.
The melting temperatures of TiPt, TiPd and TiNi are
1830 K, 1673 K and 1583 K, respectively. Thus, a sys-
tematic trend between Bo and Tm is observed in B19
Figure 8. Variation of heat of formation as function of “d”
electrons of X in TiX (X = Fe, Ni, Pd, Pt and Cu) alloys.
Figure 9. Variation of Bulk modulus as function of “d” elec-
trons in TiX (X = Fe, Ni, Pd, Pt and Cu) alloy.
phase. In case of B2 phase, we infer bulk modulus values
to decrease gradually as a function of d electrons except
for TiPd. The Bo value is higher for TiPd than for TiPt.
The Tm is defined by both bulk modulus and the shear
modulus. So, the observation of no systematic trend be-
tween Bo and Tm in B2 phase indicates that the shear
contribution varies significantly among these compounds
in this phase. The bulk modulus of TiNi in B2 phase
computed from our total energy study is 2.2984 Mbar
which is higher than the values 1.56 Mbar reported by Y.
Ye et al. It has been experimentally and theoretically
observed that the ternary alloying of V with TiNi [23,24]
will enhance the hardness of the alloy. The present work
show that the hardness of TiPd is larger than that of TiNi
Copyright © 2011 SciRes. MSA
Theoretical Investigations of Ti-Based Binary Shape Memory Alloys1365
hence alloying of TiNi with Pd substitution will improve
the hardness of the material [19,25].
7. Conclusions
We have performed first principles local density func-
tional electronic structure calculation for the five TiX
alloys (X = Fe, Ni, Pd, Pt and Cu) using the TB-LMTO
method. From our theoretical total-energy studies on TiX
alloys we have arrived at the following conclusions:
The calculated lattice constants are found to be in
good agreement with experimental results.
All alloys exist in B2 structure at ground state. The
electronic properties are rather similar for these alloys
as they have same number of valence electrons. If we
look at fine details, the properties of TiFe and TiNi
are similar and that of TiPd and TiPt are closer to
each other than to TiNi.
From the DOS histogram, we observe the lower en-
ergy part of the DOS curve to be dominated by the X
metal d states and the higher energy part dominated
by the Ti d states. The DOS at EF is mainly contrib-
uted by the Ti d states except for TiFe alloy wherein d
states of Fe show more dominance. The DOS local-
ization accompanied by the spatial localization of d
electrons of X (X = Fe, Ni, Pd, Pt, Cu) atoms results
in weakening of the d-d covalent bonds between the
alloy components.
The d-d hybridization is much pronounced for TiNi in
B19’ phase and p-d hybridization for TiPd and TiPt in
B19 phase.
The calculated heat of formation is higher for B19/
B19’ structure showing a positive indication of strong
directional bonding in (B19/B19’) phase.
The bulk modulus value is found to be higher for the
5d series followed by 3d and then by 4d series of
transition elements for B2 phase.
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