![](https://csdnimg.cn/release/download_crawler_static/12472747/bg7.jpg)
S. C. Wu et al.
potentials together with bare exchange-correlation functionals neglecting
gradient corrections [34] [35] [36]. Since elastic constants are calculated from
the second derivatives of the total energy, an accurate calculation of total energy
is extremely important.
The exchange-correlation functional was taken in the generalized gradient
approximation of Perdev, Burke, and Enzerhoff (GGA-PBE) [37] [38]. The
number of plane waves was restricted by
, and the number of
k
-points was set to 8000
k
-points in the full Brillouin zone. As discussed in
Section 3.1, these criteria ensure the convergence of the calculated elastic
properties for the investigated systems.
The lattice parameters were optimized before calculating the elastic constants.
The results of the structural optimizations are summarized in
Table 2 along with
some previously reported experimental and theoretical values. Here, the
c
/
a
ratio
was obtained by a full optimization of the Heusler compounds in tetragonal
space groups 119 or 139. In other words, to find the energy minimum, not only
the
c
/
a
ratio changed (the elongation of
c
) [32] [39] (see also Figure 2) but also
the volume of the structures was relaxed. The assignment of lattice parameters
should be performed carefully since the optimization was performed in the
tetragonal symmetry. When reducing the cubic
fcc
cell to a tetragonal
fct
cell,
the cubic lattice parameter
becomes
c
, and the tetragonal parameter
becomes
. To better understand the distortion of the cubic
Table 2. Results of the structural optimization. Lattice parameters
a
and
c
are given in Å,
ε
is dimensionless, and total magnetic moments
m
are in
μ
B
. All structures are fully
optimized (
V
,
c
/
a
, magnetic state) for the given symmetries. Column sym gives the
number of the corresponding space group, and cubic and tetragonal structures are
assigned by the Pearson symbols
cF
16 and
tI
4, respectively. Experimental values are given
for comparison. Note that the cubic variant of Rh
2
FeSn is only hypothetical.
Compound sym
a
c
,
c
a
t
ε m
tot
Mn
2
NiGa
tI
4 119 6.91 3.78 0.293 1.0
Ni
2
MnGa
tI
4 139 6.80 3.78 0.272 4.0
Exp.
[40] 6.44 3.90 0.168 4.09
Ni
2
MnGa
cF
4 225 5.81 0 4.1
Exp.
[41] 5.82 0 4.17
Mn
2
FeGa
tI
4 119 7.30 3.68 0.403 0.77
Fe
2
MnGa
tI
4 139 7.31 3.62 0.428 0.13
Rh
2
CrSn
tI
4 139 7.33 4.08 0.270 2.40
Exp.
[27] 7.16 4.09 0.238
Rh
2
FeSn
tI
4 139 7.12 4.13 0.219 3.92
Exp.
[27] 6.91 4.15 0.177 3.70
Rh
2
FeSn
cF
4 225 6.25 0 3.54
Rh
2
CoSn
tI
4 139 7.22 4.05 0.261 2.25
Exp.
[27] 6.90 4.14 0.179 2.44
DOI: 10.4236/jmp.2018.94050 781 Journal of Modern Physics