High Pressure Research
Vol. 28, No. 4, December 2008, 651–663
High-pressure structural phase transition and elastic properties
of yttrium pnictides
N. Kaurava *, Y.K. Kuoa , G. Joshib , K.K. Choudharyc and Dinesh Varshneyb
of Physics, National Dong Hwa University, Hualien, Taiwan; b School of Physics, Vigyan
Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore, India; c Department of Physics,
Shri Vaishnav Institute of Technology and Science, Baroli, Indore, India
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a Department
(Received 30 April 2008; final version received 17 July 2008 )
Pressure-induced structural aspects of NaCl-type (B1) to CsCl-type (B2) structures in yttrium pnictides
(YX; X = N, P and As) are presented. An effective interionic interaction potential with the long-range
Coulomb and van der Waals (vdW) interaction and the short-range repulsive interaction up to secondneighbor ions within the Hafemeister and Flygare approach with a modified ionic charge is developed.
Particular attention is devoted to evaluate the vdW coefficients following the Slater–Kirkwood variational
method, as both the ions are polarizable. Our results on vast volume discontinuity in the pressure–
volume phase diagram identify the structural phase transition from B1 to B2 structure. The estimated
value of the phase-transition pressure (Pt ) and the magnitude of the discontinuity in volume at the transition pressure are consistent when compared with the reported data. The variations of elastic constants
and their combinations with pressure follow a systematic trend identical to that observed in other compounds of the NaCl-type structure family, and the Born relative stability criteria are valid in yttrium
pnictides.
Keywords: high pressure; structural phase transition; elastic properties
1.
Introduction
The pressure-induced structural phase transition in binary compounds of lanthanides, actinides
and pnictides with a NaCl-type structure has received considerable attention in the recent past
[1,2]. In particular, rare-earth (RE) pnictides have various anomalous physical properties, in terms
of structural, magnetic and phonon properties [3]. These properties are related to their electronic
structure where the 4f band is partially filled and extended valence states coexist in the same energy
range. Pressure is believed to be an attractive thermodynamical variable to reveal the mechanical
properties of most of the solids and alloys. Since cohesion of solids is crucial, the interatomic
distance changes as a consequence of the application of pressure. Recent studies using synchrotron
and power X-ray diffraction demonstrated that under pressure, a majority of binary compounds
with a NaCl-type (B1) structure undergo a phase transition to the CsCl-type (B2) structure with
eightfold co-ordination at high pressure [4–6].
*Corresponding author. Email: [email protected]
ISSN 0895-7959 print/ISSN 1477-2299 online
© 2008 Taylor & Francis
DOI: 10.1080/08957950802348542
http://www.informaworld.com
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N. Kaurav et al.
Considerable theoretical but few experimental studies of yttrium pnictide (YX; X = N, P, As
and Sb) compounds have been published during the last few years [6–19]. Hayashi et al. [6]
have shown that the first-order phase transition from the NaCl-type (B1) to a CsCl-type (B2)
structure began to occur at around 26–28 GPa for YSb. However, no high-pressure experimental
studies of YN, YP and YAs are known at present. Band-structure calculations for these crystals
have been performed by various techniques and, in some cases, the computed properties include
the structural, electronic, phonon dynamics, equations of state and the transition pressures [7–
14]. The structural properties of YN have been determined by Takeuchi and co-workers [9–11],
Stampfl et al. [12] and De La Cruz et al. [13] using the full potential linearized augmented plane
wave (FP-LAPW) method. Amrani and El Haj Hassan [7] have performed ab initio self-consistent
calculations based on the FP-LAPW method with the generalized gradient approximation (GGA)
to investigate the structural and electronic properties of yttrium pnictide (YX; X = N, P, As and
Sb) compounds in B1 and B2 phases.
In addition, the transition pressures at which these compounds undergo structural phase transition from the NaCl to the CsCl phase are 136.39, 55.94, 50.45 and 28.61 GPa for YN, YP, YAs and
YSb, respectively [7]. Recently, Soyalp and Ugur [8] have also studied the structural, electronic
and phonon properties of the YP and YAs compounds in NaCl and CsCl structures using the
density functional theory (DFT) within the GGA. However, they have predicted smaller transition
pressures, 65.55 and 54.55 GPa, for YP and YAs, respectively. Further, the authors argued that the
main source of the difference between the two DFT calculations is due to the different k-points
sampling in the calculations. Moreover, the structural and elastic studies on alkaline earth [19,20],
RE chalcogenides [21,22] and other compounds [23] have further widened the scope of future
theoretical and accurate experimental investigations of crystallographic phase transition from B1
to B2 in RE compounds.
The modeling of lattice models in RE compounds is a complicated task and, in many
instances, must be guided by experimental evidence of the low degree of freedom in order to
obtain a correct minimal model which will capture the observed effect and will make useful
predictions. The first-principle DFT, microscopic tight binding models and effective Hamiltonian models have been used successfully to address the electronic, magnetic and structural
properties of binary compounds. On the hand, phenomenological lattice models [24–26] have
proved to be very successful in obtaining qualitative and quantitative understanding with proper
parameterization.
Among the lattice models, which have been invoked so far, to discuss the mechanical properties
of several solids and alloys, charge transfer approach [24], following the Hafemeister and Flygare
[25] type overlap repulsion extended up to second-neighbor ions besides short-range interactions.
We refer to the pioneering work of Tosi [26], who properly incorporated van der Waals (vdW)
interaction along with dipole–dipole (d–d) and dipole–quadrupole (d–q) interactions to reveal
the cohesion in several ionic solids. In trying to understand the structural aspects, we admit
that the vdW attractions are the corner stone of lattice phenomenological models that is ignored
in the first-principle microscopic calculations.
Motivated by the remarks on band structure studies [7–14] and the charge transfer effect
approach [24] for the successful description of the phase-transition and high-pressure behavior
of binary compounds, we have chosen a two-body interaction potential because it includes vdW
attractions, which are not well described by the standard methods currently used in first-principle
microscopic calculations. It is worth noting that the vdW interaction appears to be effective in
revealing the elastic and structural properties of RE compounds.
The article is organized as follows. In Section 2, the phase-transition pressures and elastic
constants are deduced within the framework of the Shell model, which incorporates the long-range
Coulomb and vdW interaction and the short-range overlap repulsive interaction up to secondneighbor ions within the Hafemeister and Flygare approach. Theoretical results are compared
High Pressure Research
653
and discussed in Section 3 with the existing first-principle data, and the concluding remarks are
presented in Section 4.
2. Theory and method of computation
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The understanding of thermodynamical properties for yttrium pnictides needs the formulation of
an effective interionic potential. The idea is as follows: the change in force constants is small,
the short-range interactions are effective up to the second-neighbour ions, and the atoms are held
together with harmonic elastic forces without any internal strain within the crystal. The effective
interionic potential between a pair of ions (i and j th) is expressed as
Zm e 2 ri + rj − rij
U (r) =
+
bβij exp
cij rij−6 +
dij rij−8 .
(1)
+
r
ρ
ij
ij
ij
ij
ij
Here, the long-range Coulomb energy is represented by the first term. The second term corresponds
to the Hafemeister and Flygare [25] type of short-range overlap repulsive energy. vdW multipole
energies are represented by third and fourth terms, respectively. Here, the Pauling coefficients βij
are defined as: βij = [1 + (zi /ni )/(zj /nj )], with zi (zj ) and ni (nj ) as the valency and number
of outermost electrons in the anions (cations), respectively. The symbols cij and dij represent the
dipole–dipole (d–d) and dipole–quadrupole (d–q) vdW coefficients. Zm e is the modified ionic
charge and parametrically includes the effect of Coulomb screening effects. b (hardness) and ρ
(range) are short-range parameters. Thus, the effective interionic potential contains only three
free parameters (Zm , b and ρ), which can be determined from the crystal properties [27].
An isolated phase is stable only when its free energy is minimum for the specified thermodynamic conditions. As the temperature or pressure or any other variable acting on the systems is
altered, the free energy changes smoothly and continuously. A phase transition is said to occur
when the changes in structural details of the phase are caused by such variations of free energy.
The test materials transform from their initial B1 to B2 structure under pressure. The stability of a
particular structure is decided by the minima of the Gibbs free energy, G = U + PV − TS, where
U is the internal energy, which at 0 K corresponds to the cohesive energy. S is the vibrational
entropy at the absolute temperature T , pressure P and volume V .
The Gibbs free
GB1 (r) = UB1 (r) + 2Pr3 for the NaCl (B1) phase and GB2 (r ) =
√ energy
3
UB2 (r ) + [8/3 3]Pr for the CsCl (B2) phase becomes equal at the phase-transition pressure P
and at zero temperature, i.e. G(= GB2 − GB1 ). Here, UB1 and UB2 represent cohesive energies
for B1 and B2 phases:
2
e 2 Zm
+ 6Vij (r) + 6Vii (r) + 6Vjj (r),
r
e2 Z 2
UB2 (r ) = −1.7627 m + 8Vij (r ) + 3Vii (r ) + 3Vjj (r ).
r
UB1 (r) = −1.7475
(2)
(3)
r and r are nearest-neighbor (nn) separations corresponding to NaCl and CsCl phases, respectively. Here, Vij (r) and Vii (r)[Vjj (r)] are the overlap potentials between the nearest (anion–cation)
and the next nearest neighbors (anion–anion or cation–cation), respectively, and are defined as
ri + rj − rij
Vij (r) = bβij exp
(4)
− cij rij−6 − dij rij−8 , i, j = 1, 2.
ρ
The study of the second-order elastic constants (SOEC) (C11 , C12 and C44 ) at 0 K is quite important
for understanding the nature of the interatomic forces in them. Since these elastic constants are
654
N. Kaurav et al.
functions of the first- and second-order derivatives of the short-range potentials, their calculations
will provide a further check on the accuracy of short-range forces in these materials. Following
[24] and subjecting the dynamical matrix to the long-wavelength limit, we find the expressions
for the SOEC for the B1 phase as:
e2
(A2 + B2 )
2
C11 = 4 −5.112Zm + A1 +
,
(5)
2
4r0
e2
(A2 − 5B2 )
2
,
(6)
C12 = 4 0.226Zm − B1 +
4
4r0
e2
(A2 + 3B2 )
2
C44 = 4 2.556Zm + B1 +
.
(7)
4
4r0
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and for the B2 phase as:
e2
(A1 + 2B1 ) A2
2
0.7010Z
+
+
,
m
6
2
4r04
e2
(A1 − B1 )
2
= 4 −0.5201Zm
+
,
6
4r0
e2
(A1 + 3B1 ) B2
2
+
.
= 4 −0.3505Zm
+
6
2
4r0
C11 =
(8)
C12
(9)
C44
(10)
Here, (A1 , B1 ) and (A2 , B2 ) are the short-range parameters for the nearest and the next nearest
neighbors, respectively. These parameters are defined in terms of overlap potentials Vij (r) and
Vii (r)[Vjj (r)] between the nearest (anion–cation) and the next nearest neighbors (anion–anion or
cation–cation) as
4r03 d 2
Vij (r)
,
(11)
A1 = 2
e
dr 2
r=r0
√
4(r0 2)3 d 2
d2
A2 =
V
(r)
+
V
(r)
,
(12)
ii
jj
√
e2
dr 2
dr 2
r=r0 2
4r 2 d
B1 = 20
,
(13)
Vij (r)
e
dr
r=r0
√
4(r0 2)2 d
d
B2 =
,
(14)
Vii (r) +
Vjj (r)
√
e2
dr
dr
r=r0 2
We shall now compute numerically the high-pressure phase-transition and elastic properties
for B1 and B2 phases in the next section.
3.
Results and discussion
Knowledge of force constants is crucial to the understanding of the stability of the different structures at different volumes. Two different factors determine the response of any crystal structure
to pressure: first, changes in nearest-neighbor distances, which affect the overlapping of adjacent ions and the bandwidth of the conduction band, and second, changes in symmetry, which
affect the hybridization and bond repulsion. The formalism described above is applied to yttrium
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655
compounds belonging to the cubic crystal system. We have undertaken such structural and elastic
properties in an ordered way. For such purposes, we have then three free parameters, namely modified ionic charge, range and hardness parameters (Zm , ρ and b). To estimate the free parameters,
we first deduce the vdW coefficients from the Slater–Kirkwood variational method [28], which
are listed in Table 1. C and D represent the overall vdW coefficients due to vdW interactions as
mentioned in Equation (1), defined as [26]:
1
C = cij S6 (r) + (cii + cjj )S6 (0)
2
(15)
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and
1
(16)
D = dij S8 (r) + (dii + djj )S8 (0).
2
The values of the overall vdW coefficients are obtained using Equations (15) and (16) and weighted
in terms of appropriate lattice sums [S6 (0), S6 (r), S8 (0) and S8 (r)], as argued earlier by Tosi [26].
We consider the yttrium compound to be partially ionic. It is perhaps worth remarking that we
have deduced the values of free parameters, modified ionic charge (Zm ), range (ρ) and hardness
(b) from the knowledge of equilibrium distance and the bulk modulus following the equilibrium
conditions [29]. The input data along with their relevant references and the model parameters for
yttrium compounds are given in Table 2. We have found a negative gradient in the modified ionic
charge, which is a direct consequence of the decreasing trend of bulk modulus. We appeal that
the consistency of the results can be tested as a measure of the validity of these assumptions.
In an attempt to reveal the structural phase transition of the test materials, we minimize Gibbs
free energies GB1 (r) and GB2 (r ) for the equilibrium interatomic spacing (r) and (r ). Figure 1
shows the Gibbs free energy difference G[= GB2 (r) − GB1 (r )] as functions of pressure (P )
using the interionic potential discussed above for yttrium compounds. Let us summarize the results
of the plot. The pressure corresponding to G approaching to zero is the phase-transition pressure
(Pt ) (indicated by arrows in the figure).At zero pressure, the B1 crystal phase is thermodynamically
and mechanically stable, whereas B2 is not. As pressure increases, beyond the phase-transition
pressure (Pt ), the B2 system becomes mechanically and thermodynamically stable.
Eventually, at a pressure higher than the theoretical thermodynamic transition pressure, the
B1 crystal becomes thermodynamically unstable, whereas the B2 phase remains stable up to the
greatest pressure studied. In yttrium compounds, a crystallographic transition from B1 to B2
occurs. The phase-transition pressure (Pt ) thus obtained are listed in Table 3 and compared with
other available theoretical results [7,8,10]. We note that the transition pressure increases with the
Table 1.
vdW coefficients of yttrium compounds (cij in units of 10−60 erg cm6 and dij in units of 10−76 erg cm8 ).
Compound
YN
YP
YAs
cii
cij
cjj
C
dii
dij
djj
D
263.6
263.6
263.6
99.0
107.0
142.3
40.9
47.6
79.9
927.9
992.2
1249.0
273.9
273.9
273.9
83.2
91.5
126.3
22.8
28.1
55.8
629.7
683.2
908.1
Note: C and D are the overall van der Waals coefficients.
Table 2.
Input crystal data and model parameters for yttrium compounds.
Input parameters
Model parameters
Compound
ri (Å)
rj (Å)
a0 (Å)
BT (GPa)
2
Zm
YN
YP
YAs
1.00
1.00
1.00
0.89
0.92
0.93
4.915 [7]
5.683 [7]
5.835 [7]
154.4 [7]
86.28 [7]
76.20 [7]
3.11
2.88
2.87
b(10−12 erg)
ρ(10−1 Å)
10.49
19.77
21.89
2.89
2.59
3.07
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656
N. Kaurav et al.
Figure 1. Variation of the Gibbs free energy difference with pressure.
Table 3. Calculated lattice constant a, Gibbs free energy, transition pressures and transition volume of YX (X = N, P
and As) compounds.
Gibb’s free
energy (kJ/mol)
Lattice constant (Å)
Compounds
YN (B1 → B2)
Present
Others
YP (B1 → B2)
Present
Others
YAs (B1 → B2)
Present
Others
a (B1)
a (B2)
GB1 (r)
GB2 (r )
5.20
4.915 [7]
4.93 [10]
4.85 [12]
2.75
3.002 [7]
3.01 [10]
−2613
−2460
5.86
5.683 [7]
5.667 [8]
3.08
3.473 [7]
3.471 [8]
−2158
6.04
5.835 [7]
5.815 [8]
3.02
3.582 [7]
3.577 [8]
−2084
Transition
pressure (GPa)
Trasition volumes (Å3 )
V (B1)
V (B2)
135
136.39 [7]
138 [10]
23.08
20.07 [7]
20.0 [10]
21.56
18.15 [7]
18.3 [10]
−2041
55
55.95 [7]
65.55 [8]
35.969
33.09 [7]
33.45
30.42 [7]
−1970
50
50.45 [7]
54.55 [8]
39.37
35.64 [7]
36.02
33.07 [7]
decrease in the lattice constant for yttrium compounds. The consistency between band structure
calculation data and lattice model calculation is attributed to the formulated effective interionic
potential which considers the various interactions as well as the use of free parameters from the
reported data.
Let us now estimate the values of relative volumes associated with various compressions
following the Murnaghan equation of state [29]:
−1/B V
B
= 1+
P
,
V0
B0
(17)
V0 being the cell volume at ambient conditions. The estimated value of pressure-dependent radius
for both structures and the curve of volume collapse with pressure to depict the phase diagram
are illustrated in Figure 2a–c for YN, YP and YAs, respectively. It is noticed from the plot that
657
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High Pressure Research
Figure 2.
Equation of state of YX (X = N, P and As).
our approach has predicted correctly the relative stability of competitive crystal structures, as the
values of G are positive. The magnitude of the discontinuity in volume at the transition pressure
is obtained from the phase diagram and tabulated in Table 3, which is in good agreement with
those revealed from other theoretical [7,8,10] works for all the compounds.
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658
N. Kaurav et al.
The elastic properties define the properties of material that undergoes stress, deforms and then
recovers and returns to its original shape after stress ceases. These properties play an important
part in providing valuable information about the binding characteristic between adjacent atomic
planes, anisotropic character of binding and structural stability. Hence, to study the stability of
these compounds in NaCl (B1) and CsCl (B2) structures, we have calculated the elastic constants at normal and under hydrostatic pressure by using developed effective interionic potential.
Figures 3–5 shows the variation of SOEC with pressure in B1 and B2 phases for YN, YP and
YAs, respectively. We note that a weak pressure and a negative pressure coefficient in C44 were
observed in the B1 phase. On the other hand, the reverse is true for the B2 phase. Further, in
B1 and B2 phases, C11 and C12 increase linearly with pressure. At phase-transition pressures, all
the compounds have shown a discontinuity in SOEC, which is in accordance with the first-order
character of the phase transition. The above feature is quite similar to the earlier reported pressure
dependence of elastic stiffness, possessing the NaCl structure with B1 to B2 structural phase
transition [19,20].
It is useful to mention that the Born criterion for a lattice to be in a mechanically stable state is
that the elastic energy density must be a positive-definite quadratic function of strain. This requires
that the principal minors (alternatively the Eigen values) of the elastic constant matrix should all
be positive. Further, the stability of a cubic crystal is expressed in terms of elastic constants as
follows [29]:
(C11 + 2C12 )
> 0,
3
> 0,
BT =
(18)
C44
(19)
CS =
C11 − C12
> 0.
2
(20)
Estimated values of bulk modulus (BT ), shear moduli (C44 ) and tetragonal moduli (CS ) well
satisfied the above elastic stability criteria for yttrium compounds. The calculated values of SOEC,
bulk modulus (BT ), tetragonal moduli (CS ) and the pressure derivatives of SOEC (dBT /dP ,
dC44 /dP and dCS /dP ) are given in Table 4 and are also compared with other available theoretical
Figure 3. Variation of SOEC with pressure for theYN compound. Broad boundary is a guide to the eye at phase transition
(B1 to B2).
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659
Figure 4. Variation of SOEC with pressure for theYP compound. Broad boundary is a guide to the eye at phase transition
(B1 to B2).
Figure 5. Variation of SOEC with pressure for the YAs compound. Broad boundary is a guide to the eye at phase
transition (B1 to B2).
works [7,8,10,12]. Furthermore, Vukcevich [30] proposed a high-pressure stability criterion for
ionic crystals, combining mechanical stability with minimum energy conditions. In accordance,
the stable phase of the crystal is the one in which the shear elastic constant C44 is non-zero (for
mechanical stability) and which has the lowest potential energy among the mechanically stable
lattices.
On the other hand, C44 is a very small quantity, and the calculated value of ((4r0 /e2 )C44 −
2
0.556Zm
) is found to be a negative quantity so that (A2 − B2 ) is negative. This suggests that these
terms belong to an attractive interaction and possibly arise due to the vdW energy. The vdW energy
converges quickly, but the overlap repulsion converges much more quickly. This means that the
second-neighbor forces are entirely due to the vdW interaction and the first-neighbor forces are the
660
N. Kaurav et al.
Table 4. Second-order elastic constants (C11 , C12 and C44 ), bulk modulus (BT ), tetragonal moduli (CS ) (all
are in GPa) and pressure derivatives of SOECs (dBT /dP, dC 44 /dP and dC S /dP) for yttrium compounds in
B1 and B2 phases.
Compounds
YN Present
B1 phase
B2 phase
Theory
B1 phase
C11
C12
C44
BT
CS
dBT /dP
dC 44 /dP
dC S /dP
133.0
122.0
45.9
14.8
79.2
8.80
74.9
50.6
43.6
53.7
6.169
5.47
1.33
0.78
4.53
3.84
5.9
5.5
0.758
1.02
4.85
3.51
5.91
5.49
0.83
0.99
4.76
3.54
154.4 [7]
157 [10]
163 [12]
149.1 [7]
136 [10]
B2 phase
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YP Present
B1 phase
B2 phase
Theory
B1 phase
116.0
79.9
31.0
12.8
B2 phase
59.5
35.2
86.28 [7]
86.6 [8]
86.92 [7]
86.25 [8]
B2 phase
YAs Present
B1 phase
B2 phase
Theory
B1 phase
42.7
7.8
3.06 [7]
3.5 [10]
4.77 [12]
4.135 [7]
4.11 [10]
96.0
68.5
27.0
10.0
38.0
6.2
76.20 [7]
76.80 [8]
75.43 [7]
75.20 [8]
42.4
33.5
3.805 [7]
3.63 [8]
3.986 [7]
3.90 [8]
50.0
29.5
34.6
29
3.821 [7]
3.90 [8]
3.807 [7]
3.90 [8]
results of the overlap repulsion and the vdW attraction between the nearest neighbors. However,
at high pressure, the short-range forces for these compounds increase significantly, which, in turn,
is responsible for the change in the coordination number and the phase transformation. Other than
deriving the equation of states correctly from a model approach and then analyzing the variation
of short-range forces, at present we have no direct means to understand the interatomic forces at
high pressure.
Apart from the phase transition and pressure dependence of SOEC, we also estimate the Debye
temperature (θD ) from the present approach. We define
3.15 h 3 r 3/2 C11 − C12 1/2 C11 + C12 + 2C44 1/2 1/2
3
C44 ,
(21)
θD =
8π kB
M
2
2
where M is the acoustic mass of the compound. Figure 6a–c shows the variation of the Debye
temperature in B1 and B2 phases for yttrium compounds. It is noticed from the figure that the
θD increase with the increase in pressure for yttrium pnictides in both the phases. This result
motivates us for the definition of an ‘average’ elastic constant as
8π 2/3 kB 2 M
(22)
θD2 ,
C=
3.15
h
r
which in turn is calculated from the Debye temperature, allowing us to correlate the Cauchy
discrepancy in elastic constant as
C12 − C44
,
(23)
C∗ =
C12 + C44
at zero pressure. Figure 7 shows the variation of ‘average’ elastic constant (C) with the Cauchy
discrepancy (C ∗ ) in B1 and B2 phases for yttrium compounds. We have found a negative Cauchy
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661
Figure 6. Debye temperature (θD ) as a function of pressure. Broad boundary is a guide to the eye at phase transition
(B1 to B2).
Figure 7. Shows ‘average’ elastic constant (C) as a function of the Cauchy discrepancy (C ∗ ) in B1 and B2 phases for
YX (X = N, P and As) compounds.
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N. Kaurav et al.
discrepancy in the B1 phase, whereas it is positive in the B2 phase. It is worth mentioning that
strontium chalcogenides (SrS, SrSe and SrTe) [20] and magnesium chalcogenides (MgS, MgSe
and MgTe) [19] show a positive and negative Cauchy deviation C ∗ , respectively.
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4.
Conclusion
An effective interionic interaction potential is formulated to analyze the structural as well as elastic
properties of yttrium compounds. The obtained values of free parameters allow us to predict the
phase-transition pressure and the associated volume collapse. We have found that the vast volume
discontinuity in the pressure–volume phase diagram identifies the structural phase transition from
the NaCl (B1) to the CsCl (B2) structure. From our calculated results, it can be emphasized that
the present approach reproduced the structural properties at high pressure consistently in terms
2
of the screening of the effective Coulomb potential through the modified ionic charge (Zm
).
An immediate consequence of our lattice model calculations is the validity of the Born criterion.
The SOEC C11 and C12 increase with increase in pressure up to the phase-transition pressure
that supports the high-pressure structural stability of yttrium compounds. Further, C44 decreases
linearly with the increase in pressure and does not tend to zero at the phase-transition pressures
in the B1 phase, which is in accordance with the first-order character of the transition. It has been
found that the present simple model, when compared with complicated band structure calculations,
may account for a considerable part of the available results for the high-pressure studies on yttrium
pnictides.
Acknowledgements
Financial support from National Science Council of Taiwan (NSC-96-2112-M-259-003) and Madhya Pradesh Council of
Science and Technology, Bhopal, India, is gratefully acknowledged.
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