Materials Transactions, Vol. 44, No. 1 (2003) pp. 204 to 210
#2003 The Japan Institute of Metals
EXPRESS REGULAR ARTICLE
Valence Electron Concentration and Phase Transformations
of Shape Memory Alloys Ni–Mn–Ga–X
Kenichi Yamaguchi1; *, Shoji Ishida1 and Setsuro Asano2
1
2
Department of Physics, Faculty of Science, Kagoshima University, Kagoshima, 890-0065 Japan
The Graduate School of=College of Arts and Sciences, The University of Tokyo, Tokyo, 153-0041 Japan
In the Ni2 MnGa based alloys with additions of transition element Ni–Mn–Ga–X, the martensitic transformation temperature TM was
observed as a function of the valence electron concentration per atom e=a. The TM ðe=aÞ strongly depends on e=a and increases with increasing
e=a. In this paper, to examine the effect of X atom on the phase transformation in Ni–Mn–Ga–X alloys, the electronic structures for six systems
were calculated for four phases, that is, the paramagnetic cubic, the paramagnetic monoclinic, the ferromagnetic cubic and the ferromagnetic
monoclinic phases. Moreover, the total energy differences Eðe=aÞ between two phases among four phases were calculated as a function of e=a.
The variations of TM ðe=aÞ were predicted by the difference Eðe=aÞ between the cubic and monoclinic structures in a ferromagnetic state. It was
found that their correspondence is good for some systems and that the features of TM ðe=aÞ reflect the changes of the density of states of X atoms.
(Received October 9, 2002; Accepted December 13, 2002)
Keywords: shape memory, martensitic transformation temperature, valence electron concentration, electronic structure, total energy, nickel,
manganese, gallium, Curie temperature
1.
Introduction
Many researchers have reported on the crystal structures
and the phase transformations of the Ni–Mn–Ga alloys. The
tetragonal structure was observed in the martensitic phase
around valence electron concentration e=a ¼ 7:50 (stoichiometric Ni2 MnGa). On the other hand, the orthorhombic and
monoclinic structures were observed.1–4) For example, the
orthorhombic structure was observed at the e=a ¼ 7:635 and
the monoclinic structures at e=a ¼ 7:64, 7.67, 7.72 and
7.78.2,3) It was also reported that the tetragonal phase in the
lower e=a can be suppressed by Ni excess.5) Furthermore, it is
also predicted theoretically that the tetragonal and orthorhombic structures may be metastable and the monoclinic
structure is the most stable state among these structures for
Ni2:17 Ni0:83 Ga and Ni2 (Pd0:17 Ni0:83 )Ga.6) Thus, it is possible
for the monoclinic structure to appear in martensitic phase in
wide e=a range.
The martensitic transformation temperature TM and the
Curie temperature Tc for Ni2 MnGa (e=a ¼ 7:50) are 202 and
376 K, respectively.7) The various values of TM were
observed in the wide range from 175 to 626 K in the range
e=a ¼ 7:45{8:10.8–10) For Ni–Mn–Ga alloys, Tc decreases
and TM increases with increasing e=a.5,9) They merge in the
range e=a ¼ 7:635{7:70. It was also shown that TM is lower
than Tc in the lower e=a and higher than Tc in the higher
e=a.5,9) Moreover, Xin et al. reported that the values of TM for
ferromagnetic alloys Ni–Mn–Ga are higher than 300 K and
lower than Tc and that TM is represented by the equation
TM ¼ 702:5ðe=aÞ 5067 K as a function of e=a.11) These
results indicate that the martensitic transformation occurs in a
ferromagnetic phase for the lower e=a and in a paramagnetic
state for the higher e=a.
Moreover, Tsuchiya et al. reported three types of transformations: (I) paramagnetic parent phase , ferromagnetic
parent phase , intermediate phase , ferromagnetic
*Graduate
Student, Kagoshima University.
martensitic phase in the range e=a < 7:62, (II) paramagnetic
parent phase , (ferromagnetic parent phase) , ferromagnetic martensitic phase in the range 7:62 < e=a < 7:65 and
(III) paramagnetic parent phase , paramagnetic martensitic
phase , ferromagnetic martensitic phase in the range
7:65 < e=a.12) The symbol ‘‘,’’ denotes the process of
transformation between two phases.
Previously, paying attention to only two systems in a
ferromagnetic state, the authors calculated total energy
differences E between the cubic and monoclinic structures
as a function of e=a and related the E with the e=a
dependence of TM .13) It was found that Eðe=aÞ changes like
a straight line in the range e=a ¼ 7:50{7:625 for the case
where X atoms occupy Ni sites, while like a parabolic line in
the range e=a ¼ 7:625{7:77 for the case where X atoms
occupy Mn sites. The characteristic behavior of Eðe=aÞ is
similar to the behavior of TM ðe=aÞ of Ni2:16x Cox Mn0:84 Ga,
Ni2:20z Fez Mn0:80 Ga and Ni2:16 Mn0:84y Coy Ga. However, for
Ni2þx Mn1x Ga, the correspondence between Eðe=aÞ and
TM ðe=aÞ is good in the range e=a ¼ 7:50{7:625 but not good
in the range e=a > 7:625.
In this paper, new four systems and a paramagnetic state
will be considered to investigate in more detail the effect of X
atom on the phase transformation in Ni–Mn–Ga–X systems.
2.
Crystal Structure and Method of Calculation
As described in the previous section, it was reported
theoretically that the monoclinic structure is the most stable
among the cubic, tetragonal, orthorhombic and monoclinic
structures for Ni2:17 Ni0:83 Ga and Ni2 (Pd0:17 Ni0:83 )Ga. Then,
we consider the cubic structure and the monoclinic structure
as the parent phase and the martensitic phase, respectively.
The symmetry of the monoclinic structure is lower than that
of the cubic structure. The cubic structure is treated as a
monoclinic structure with an angle of shown in Fig. 1 to
calculate under the same condition. The angle is 71.565
and 98.461 or the cubic structure and the monoclinic
Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X
205
(a)
Monoclinic Structure
Ni
Mn
Ga
y=1 plane
y=3/4 plane
Mn,Ga : y=0 or 1
Ni
: y=1/4
2
3
4
Cubic Structure
1
3
y=1/2 plane
Mn(2)
3
1
y=0 plane
1
y=1/4
Ni(1)
(b)
x
Monoclinic Structure
Mn, Ga : y=1/2
Ni
: y=3/4
2
4
z
Fig. 2 Monoclinic structure. The monoclinic structure has twenty-four
atoms in the unit cell, which corresponds to the observed monoclinic
structure having the shuffling of 6 layers of (2 2 0) planes.
Cubic Structure
2
3
4
Mn(1)
y
4
1
2
Fig. 1 Relation between the cubic and monoclinic structures of Ni–Mn–
Ga–X alloy. The constituent atoms on the y ¼ 0 and y ¼ 1=4 planes are
shown in (a) and ones on y ¼ 1=2 and y ¼ 3=4 planes in (b). The numbers
denote the atomic sites in the monoclinic structure. The cubic structure is
treated as the monoclinic structure with an angle of ¼ 71:565 .
structure of Ni2:17 Ni0:83 Ga.6) When we assume that y-axis is
vertical to this paper, Mn and Ga atoms are located on the
y ¼ 0 (or 1) and 1=2 planes and Ni atoms on the y ¼ 1=4 and
3=4 planes shown in Fig. 1. Each of nickel, manganese and
gallium in Ni2 MnGa has the four different atomic sites in the
unit cell with the P2/m symmetry of the tenth space group.
For example, the sites of Ni and Mn atoms are distinguished
by such as the symbols of Ni(1), Mn(1) and Mn(2). The
Ni(1), Mn(1) and Mn(2) are located at the 2j, 1a and 1f sites.
The monoclinic structure has twenty-four atoms in the unit
cell, which corresponds to the observed monoclinic structure
having the shuffling of 6 layers of (2 2 0) planes.2) This
monoclinic structure is shown in Fig. 2 and the symbols open
circles, solid circle and circle with slants denote Ni, Mn and
Ga, respectively. The Mn(1), Mn(2) and Ni(1) are the sites
where X atoms occupy. Recently, it was confirmed that the
monoclinic structure is equivalent to the tetragonal structure
of the named of 2M.14)
The alloy where a sixth of Mn atoms of Ni2 MnGa were
replaced with Ni atoms was described as Ni2:17 Ni0:83 Ga in the
previous papers.6) In this paper, the alloy is described as
Ni2 (X1=6 Mn5=6 )Ga where the Ni atoms at the Mn(1) sites are
described in parentheses with the Mn atoms. Here, we
consider six systems of Ni–Mn–Ga–X alloys where Ni or Mn
atoms in Ni2 MnGa or Ni2 (Ni1=6 Mn5=6 )Ga are replaced with
other transition element. They are listed in Table 1, where the
name of the systems, the molecular formula and atoms at the
Mn(1), Mn(2) and Ni(1) sites are shown. For example, in
(Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 )Ga, Ni atoms are replaced with X
atoms and Mn(1) atoms with Ni atoms. The s-Nm1-n1 and sNm1-m2 are new notation for sys-N1 and sys-M2 in the
previous paper, respectively.13) When transition elements are
chosen as the X atoms, these alloys are in the range of
e=a ¼ 7:50{7:77. When we cannot choose a real element as
the X atom for the special value of e=a, we adopt an artificial
atom. For example, the artificial atom is described like Z27.5
where the number of 27.5 means the atomic number and the
number of electrons.
Table 1 Six systems classified by the site of X atom (Mn or Ni site) in the shape memory alloys Ni–Mn–Ga–X. The symbols, the
molecular formula used in this paper are listed. The atoms at Mn(1), Mn(2) and Ni(1) sites are also shown and the other atoms occupy the
regular sites.
Symbol
of system
s-m1
Constituent atom
Molecular formula
Replaced
Mn(1)
Mn(2)
Ni(1)
Ni2 (X1=6 Mn5=6 )Ga
X
Mn
Ni
Ni2 (Mn5=6 X1=6 )Ga
Mn
X
Ni
s-Nm1-m2
s-Cm2-m1
Ni2 (Ni1=6 Mn4=6 X1=6 )Ga
Ni2 (X1=6 Mn4=6 Co1=6 )Ga
Mn
X
X
Co
Ni
Ni
s-Nm1-n1
(Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 )Ga
Ni
Mn
X
s-Nm12-n1
(Ni5=6 X1=6 )2 (Ni1=6 Mn4=6 Ni1=6 )Ga
Ni
Ni
X
s-m2
site
Mn
Ni
K. Yamaguchi, S. Ishida and S. Asano
Results and Discussion
3.1 Total energy and valence electron concentration
In a previous paper, total energy differences E between
cubic and monoclinic structures were calculated for s-Nm1n1 (old notation: sys-N1) and s-Nm1-m2 (old notation: sysM2).13) Here, E were newly calculated for four systems
listed in Table 1 except for above two systems. To calculate
the e=a dependence of E, transition elements were chosen
as X atoms such as Mn, Fe, Z26.5, Co, Z27.5 and Ni for s-m1
where the value of e=a changes from 7.50 to 7.625. In this
study, a paramagnetic state is newly considered. Therefore,
band calculations were performed for four phases; paramagnetic cubic (PC), ferromagnetic cubic (FC), paramagnetic
monoclinic (PM) and ferromagnetic monoclinic (FM) phases. The obtained total energies of six systems are the lowest
for FM phase among four phases.
At first, we consider the transformation in the ferromagnetic state, that is, the transformation between FC and FM
phases. The total energy differences E between FC and FM
phases is described as EFC-FM . The curves of EFC-FM ðe=aÞ
are shown in Fig. 3(a) for four systems where X atoms
occupy Mn sites and in Fig. 3(b) for two systems where X
atoms occupy Ni sites. The cases of X = Ni in s-m1 (s-m2)
and s-Nm1-n1 are equivalent to the case X = Mn in s-Nm1m2 which correspond to e=a ¼ 7:625. Also, the case of X =
Co in s-m1 (s-m2) and the case of X = Ni in s-Cm2-m1 are
equivalent to those of X = Mn in s-Cm2-m1 and X = Co in sNm1-m2. The curves for s-m1, s-m2 and s-Cm2-m1 are
similar to that of s-Nm1-m2 shown in the previous paper13)
and the curve of s-m2 overlaps with that of s-m1 each other.
Their shapes are like a parabola with a top at Co. On the other
hand, the curves of s-Nm12-n1 is similar to that of s-Nm1-n1
in Fig. 3(b) and the EFC-FM ðe=aÞ increases linearly with
increasing e=a.
Thus, the change of EFC-FM ðe=aÞ depends on the site of X
atom (Mn or Ni site) and the value is not unique for e=a.
Now, we will discuss the relation between EFC-FM ðe=aÞ and
the martensitic transformation temperature TM .
Chernenko et al.17) have measured the temperature
dependence of the transformation stress to be
d=dT ¼ 13 MPa/K for the alloy Ni–23.5Mn–23.9Ga. Tsuchiya et al.12) have studied the e=a dependence of TM and Tc
for Ni–Mn–Ga alloys and estimated the transformation
entropy S to be 48:2 1018 aJ/molK, using the value
d=dT ¼ 13 MPa/K. When the e=a changes from 7.50 to
7.625, corresponding change in TM was observed to be about
100 K. In the same interval, the increase of EFC-FM ðe=aÞ is
5:71 1021 aJ/mol. The value is converted to the increase in
TM to be 118 K, using S ¼ 48:2 1018 aJ/molK. Thus, the
correspondence between the variation of the EFC-FM ðe=aÞ
and that of TM is fairly well.
The six curves of EFC-FM ðe=aÞ shown in Fig. 3 are again
shown by the solid and broken lines in Fig. 4. The theoretical
values EFC-FM ðe=aÞ near the experimental values are plotted
-1
(a)
E , E / aJ unit-cell
3.
0.80
Mn
0.72
s-m1
Fe
0.68
Mn
Co
X at Mn site
Co
s-Cm2-m1
Fe
0.76
0.64
Ni
Fe
Ni
Co
s-Nm1-m2
Mn
s-m2 7.625
Ni
X at Mn(1) or Mn(2)
E =E cub.-E mono.
0.60
7.49
7.54
7.59 7.64 7.69 7.74 7.79
Valence Electron Concentration, e/a
7.84
0.80
X at Ni site
(b)
-1
Band calculations were carried out self-consistently by the
LMTO-ASA method.15) The exchange correlation potential
was treated within the framework of the local-spin-density
(LSD) approximation.16)
E, E / aJ unit-cell
206
0.76
Ni
Z27.5
0.72
Co
s-Nm12-n1
Z27.5 Ni
0.68
Co
s-Nm1-n1
0.64
X at Ni(1)
7.625
E =E cub.-E mono.
0.60
7.49
7.54
7.59
7.64
7.69
7.74
7.79
7.84
Valence Electron Concentration, e/a
Fig. 3 Valence electron concentration (e=a) dependence of difference
(E) of total energies between the cubic and monoclinic structures in a
ferromagnetic state. In (a), the solid curves with solid circles, open
diamonds and crosses distinguish s-m1, s-m2 and s-Cm2-m1, respectively.
A broken line with open circles is for s-Nm1-m2. In (b), a straight line with
solid circles and a broken line with open circles distinguish s-Nm1-n1 and
s-Nm12-n1.
by solid circles. The experimental values of TM for
Ni2:16x Cox Mn0:84 Ga,
Ni2:20z Fez Mn0:80 Ga
and
Ni2:16 Mn0:84y Coy Ga observed by Khovailo are plotted by
open triangle, open square and open circle in the Fig. 4(a),
respectively.18) The values of TM ðE) refer to the left (right)
axis. The values of EFC-FM ðe=aÞ is plotted so that the values
EFC-FM ðe=aÞ of the case X = Ni in the s-Nm1-n1 and X =
Mn in the s-Nm1-m2 are superposed on the values of TM at
e=a ¼ 7:625. The values of TM are distributed near the
EFC-FM ðe=aÞ line for s-Nm1-n1 in the range
e=a ¼ 7:54{7:625, while along the EFC-FM ðe=aÞ curve for
s-Nm1-m2 in the range e=a ¼ 7:625{7:71, as described in the
previous paper.13) Thus, the values of TM for
Ni2:16x Cox Mn0:84 Ga and Ni2:20z Fez Mn0:80 Ga correspond
to those of EFC-FM ðe=aÞ for s-Nm1-n1 and the values of TM
for Ni2:16 Mn0:84y Coy Ga correspond to those EFC-FM ðe=aÞ
for s-Nm1-m2.
In the Fig. 4(b), the experimental values for
Ni2þx Mn1x Ga are plotted by crosses for TM and by open
diamonds for the Curie temperature Tc .12) The TM increases
along the EFC-FM ðe=aÞ line for s-Nm1-n1 with increasing
e=a, while the Tc decreases in the range e=a ¼ 7:50{7:65.
And the TM and Tc are nearly equal in the range
e=a ¼ 7:65{7:71. The values of TM ðe=aÞ distribute along
7.71
s-Nm1-m2
0.780
-1
600
500
0.732
400
300
s-Nm1-n1
0.684
200
100
0.639
7.48 7.53 7.58 7.63 7.68 7.73 7.78
800
7.625
(b)
700
7.71
0.780
600
-1
700
7.625
207
0.732
500
400
s-Nm12-n1
300
0.684
E , E / aJ unit-cell
(a)
Transformation Temperatures, TM, Tc /K
800
E , E / aJ unit-cell
Transformation Temperature, TM /K
Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X
s-Nm1-n1
200
100
0.639
7.48 7.53 7.58 7.63 7.68 7.73 7.78
Valence Electron Concentration, e/a
Valence Electron Concentration, e/a
Fig. 4 Comparison between phase transformation temperatures and total energy differences. The values of martensitic transformation
temperature TM refer to the left axes and those of total energy difference E to the right axes. The solid and broken curves are the curves
of E shown in Fig. 3. The solid curves with solid circles show the curves of EFC-FM ðe=aÞ which are comparable with the experimental
values. In (a), the open squares, open triangles and the open circles indicate the values of TM for Ni2:16x Cox Mn0:84 Ga,
Ni2:20z Fez Mn0:80 Ga and Ni2:16 Mn0:84y Coy Ga, respectively.18) In (b), crosses and diamonds indicate the values of TM and Tc for
Ni2þx Mn1x Ga, respectively.12)
(a)
X at Mn site
E=E-E FM
E, E / aJ unit-cell
-1
1.7
1.4
E PC-FM
1.1
7.625
0.8
E FC-FM
0.5
0.2
E PM-FM
-0.1
7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85
Valence Electron Concentration, e/a
2.0
(b)
X at Ni site
1.7
E=E-E FM
-1
3.2 Intermediate state
In the previous section, we considered above the transformation in the ferromagnetic state and also we will consider
the paramagnetic state in followings. The differences
(E ¼ E EFM ) of total energies between the FM phase
with the lowest total energy and the other phase are plotted as
a function of e=a in Figs. 5(a) and (b). The differences E are
shown in Fig. 5(a) for the case where Mn atoms are replaced
with X atoms and in Fig. 5(b) for the case that Ni atoms are
replaced with X atoms.
In Fig. 5(a), the three curves with solid symbols in the
range e=a ¼ 7:50{7:625 and with open symbols in the range
e=a ¼ 7:625{7:77 are drawn for s-m1 and s-Nm1-m2,
respectively. The symbols ‘‘triangle’’, ‘‘circle’’ and ‘‘square’’
correspond
to
EPC-FM ðe=aÞ,
EFC-FM ðe=aÞ
and
EPM-FM ðe=aÞ, respectively. In Fig. 5(b), the differences
E are drawn by the three straight lines for s-Nm1-n1 and sNm12-n1 as in Fig. 5(a).
Here, we refer to the experimental results that the
martensitic transition occurs in the ferromagnetic state for
the lower e=a. In the range e=a ¼ 7:50{7:70, the
EFC-FM ðe=aÞ varies like a parabolic or straight line as
described above, while the EPC-FM ðe=aÞ and EPM-FM ðe=aÞ
decrease with increasing e=a. The increase of EFC-FM ðe=aÞ
corresponds to the increase of TM and the decrease of
EPC-FM ðe=aÞ and EPM-FM ðe=aÞ corresponds to the decrease
of Tc .
Our results show that the total energy becomes lower in
order of PC, FC, PM and FM phases and suggest the
possibility of four kinds of transitions as follows:
2.0
E, E / aJ unit-cell
Eðe=aÞ line of s-Nm1-n1 in the range e=a ¼ 7:50{7:65. In
the range e=a > 7:625, the values of TM do not distribute near
the broken curve for s-Nm1-m2 but the curve for s-Nm12-n1.
Thus, it was found that the Eðe=aÞ for s-Nm12-n1
corresponds to the TM ðe=aÞ of Ni2þx Mn1x Ga in the range
e=a ¼ 7:65{7:71.
1.4
E PC-FM
1.1
7.625
0.8
0.5
E FC-FM
E PM-FM
0.2
-0.1
7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85
Valence Electron Concentration, e/a
Fig. 5 The e=a dependence of the total energy difference E for four
phases; PC, FC, PM and FM phases. The differences (E ¼ E EFM )
between the FM phase with the lowest total energy and the other phase are
shown for the case (a) where Mn atoms are replaced with X atoms for s-m1
and s-Nm1-m2 and the case (b) where Ni atoms are replaced with X atoms
for s-Nm1-n1 and s-Nm12-n1.
208
K. Yamaguchi, S. Ishida and S. Asano
(Trans.1) PC ! FC ! FM,
(Trans.2) PC ! PM ! FM,
(Trans.3) PC ! FM and
(Trans.4) PC ! FC ! PM ! FM.
Thus, there is the possibility that FC and PM phases become
the intermediate phase between PC and FM phases. Now, we
compare our results with those of Tsuchiya et al.12) As
described in the introduction, they reported three types of
transitions in three regions: (I) 7:5 > e=a, (II) 7:62 < e=a <
7:65 and (III) 7:65 < e=a. We guess that Trans.1 correspond
to the transition in the range e=a < 7:65 where their observed
intermediate state may be a ferromagnetic phase with a
structure different from the monoclinic structure, Trans.2
does to the transition in the range e=a > 7:65 and Trans.3
does to the transition in the range 7:62 < e=a < 7:65. The
magnetic transition is not natural in the Trans.4 among our
four types of transitions. Therefore, Trans.4 may be not
observed. We have to consider entropy in order to discuss
transitions accurately.
3.3 Density of states
In the previous section, it was found that curves of the
differences Eðe=aÞ are similar for the four systems of s-m1,
s-m2, s-Nm1-m2 and s-Cm2-m1 where the X atoms occupy
the Mn sites. The similarities are also seen in the curves of sNm1-n1 and s-Nm12-n1 where the X atoms occupy the Ni
sites. It is natural to pay attention to X atoms in considering
energy differences due to differences of e=a and systems,
because the difference of e=a and systems is due to the X
atoms. It is expected that the change in the total density of
state (DOS) due to the difference of e=a and system mainly
comes from the change in the local DOS of X atom (X-DOS).
The change in the DOS affects the change of total energy
differences E between cubic and monoclinic structures.
Then, we pay attention to the relation between the Eðe=aÞ
and the X-DOS for case that the X atom occupies the Mn
sites. The X-DOS curves for s-m1 are shown in Fig. 6 where
the curves of the cubic structure are shown for the majority
and minority spins in Figs. 6(a) and (b) and those of the
monoclinic structure in Figs. 6(c) and (d). The variation in the
X-DOS for X = Mn, Fe, Co and Ni atoms is quite similar to
that of s-Nm1-m2 (old notation: sys-M2).13) The vertical line
denotes the Fermi energy. Since the X atom in the cubic
structure is surrounded by eight Ni atoms, the X-DOS has the
characteristics of the bcc structure, that is, the X-DOS is
composed of two large peaks. The large valley between the
two peaks disappears in the monoclinic structure and the
occupied states generally move to the states with the lower
energy. Therefore, we can guess that the band energy for the
monoclinic structure is lower than that of the cubic structure.
In the minority spin states, the X-DOS curve shifts from the
higher energy region to the lower energy region beyond the
Fermi energy, when the X atom changes from Mn to Ni in the
order of Mn, Fe, Co and Ni. On the other hand, in the majority
spin state, the two large peaks under the Fermi energy shift to
the higher energy region with increasing e=a except for the
50
States, n / aJ
States, n / aJ
Cub.
20
10
Mn
Fe
Co
Ni
30
Mono.
20
10
40
50
-1
Mn
Fe
Co
Ni
(b) Ni2(X1/6Mn5/6)Ga
X at Mn(1)
atom spin
-1
50
30
State, n / aJ
20
10
0
-0.8
40
Mn
Fe
Co
Ni
(d) Ni2(X1/6Mn5/6)Ga
X at Mn(1)
30
Mono.
-1
Cub.
-1
atom spin
40
0
0
State, n / aJ
(c ) Ni2(X1/6Mn5/6)Ga
X at Mn(1)
-1
30
atom spin
40
Mn
Fe
Co
Ni
-1
atom spin
-1
(a) Ni2(X1/6Mn5/6)Ga
X at Mn(1)
-1
50
20
10
0
-0.6
-0.4
-0.2
0
0.2
-1
Energy, E / aJ unit-cell
0.4
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
-1
Energy, E / aJ unit-cell
Fig. 6 Local DOS of X atoms in s-m1. Four curves distinguish the cases of X = Mn, Fe, Co and Ni in Ni2 (X1=6 Mn5=6 )Ga, respectively.
The DOS for the cubic structure are shown in (a) and (b) and for the monoclinic structure in (c) and (d). The upper ((a) and (c)) and the
lower ((b) and (d)) are of the majority and the minority spin. The vertical line shows the Fermi energy.
atom spin
(a) X=Co
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
FC
FM
20
10
30
-1
(b) X=Z27.5
atom spin
-1
30
FC
FM
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
20
(b) X=Z27.5
20
FC
FM
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
States, n / aJ
-1
-1
States, n / aJ atom spin
20
FC
FM
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
0
10
10
0
0
30
-1
30
atom spin
20
FC
FM
(c) X=Ni
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
(c) X=Ni
20
FC
FM
(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga
States, n / aJ
-1
-1
-1
(a) X=Co
-1
States, n / aJ
10
0
States, n / aJ atom spin
209
30
-1
30
-1
States, n / aJ atom spin
-1
Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X
10
0
-0.8
-0.6
-0.4
-0.2
0
0.2
10
0
-0.8
0.4
-1
Energy, E / aJ unit-cell
Fig. 7 Local DOS of X atoms in s-Nm1-n1. The DOS of X = Co, Z27.5
and Ni in (Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 ) Ga are shown in (a), (b) and (c),
respectively. The DOS curves for the majority spin state in FC and FM
phases are drawn by the solid and dotted lines, respectively. The vertical
line shows the Fermi energy.
case X = Mn. We can roughly guess from these changes that
the difference of band energy between cubic and monoclinic
structures becomes larger with increasing atomic number of
X atom. Therefore, the E increases with increasing atomic
number.
Next, we turn our attention to the case that the X atoms
occupy the Ni sites. The X-DOS curves for s-Nm1-n1 are
shown in Figs. 7 and 8. The curves of Co, Z27.5 and Ni in FC
and FM phases are compared for the majority and minority
spins in Figs. 7 and 8, respectively. The change of the XDOS due to the difference of X atom is small in the majority
spin for both of the FC and FM phases. On the other hand, the
difference of the X-DOS between the FC and FM phases
becomes larger in the minority spin, when X atom changes
from X = Co to Ni. The changes bring the linear increase in
EFC-FM ðe=aÞ.
-0.6
-0.4
-0.2
0
Energy, E / aJ unit-cell
0.2
0.4
-1
Fig. 8 Local DOS of X atoms in s-Nm1-n1. The DOS of X = Co, Z27.5
and Ni in (Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 ) Ga are shown in (a), (b) and (c),
respectively. The DOS curves for the minority spin state in FC and FM
phases are drawn by the solid and dotted lines, respectively. The vertical
line shows the Fermi energy.
4.
Conclusion
To investigate in more detail the effect of X atom on the
phase transformation in Ni–Mn–Ga–X systems, the electronic structures were calculated for six Ni2 MnGa based
systems listed in Table 1. The total energies were also
calculated for four phases, which are PC, FC, PM and FM
phases. Since the total energies become lower in order of the
PM, FC, PM and FM phases, there is possibility that the FC
and PM phases become an intermediate phase between PC
and FM phases.
For the six systems treated in this paper, the total energy
differences EFC-FM ðe=aÞ between FC and FM phases
calculated by changing the X atom in Ni–Mn–Ga–X alloys
from Mn to Ni among transition elements. The EFC-FM ðe=aÞ
have a similar e=a dependence if the X atom occupies the
same atomic site (Ni or Mn site).
It was shown that the increase of the martensitic
transformation temperature TM due to the increase of e=a
210
K. Yamaguchi, S. Ishida and S. Asano
from 7.50 to 7.625 is comparable to that of TM which is
converted from the increase of EFC-FM ðe=aÞ. Therefore, the
e=a dependence of TM ðe=aÞ corresponds to that of
EFC-FM ðe=aÞ for s-Nm1-n1 in the range e=a < 7:625 and
those for s-Nm1-m2 and s-Nm12-n1 in the range
e=a > 7:625. Thus, the TM ðe=aÞ may be predicted from the
Eðe=aÞ which is not unique against the value of e=a. The
variation of Eðe=aÞ due to the difference of the X atoms
mainly comes from the variation of the X-DOS in Ni–Mn–
Ga–X alloys.
Acknowledgments
The authors wish to thank Professor Koichi Tsuchiya of
Toyohashi University of Technology for giving information
and significant discussions. This work was supported by a
Grant-in-Aid (13640638) for scientific Research from the
Ministry of Education, Science and Culture of Japan.
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