Solid State Communications 149 (2009) 1772–1776
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Structural phase transition and elastic properties of Curium and Uranium monobismuthides under pressure effect D. Rached a,∗ , M. Rabah a , R. Khenata b , B. Abidri a , S. Benalia a a
Laboratoire des Matériaux Magnétiques, Faculté Des Sciences, Université Djillali Liabes de Sidi Bel-Abbès, Sidi Bel-Abbès (22000), Algérie
b
Laboratoire de Physique Quantique et de Modélisation Mathématique de la Matière (LPQ3M), Centre universitaire de Mascara, Mascara (29000), Algérie
article
info
Article history: Received 31 March 2009 Received in revised form 11 July 2009 Accepted 16 July 2009 by C. Lacroix Available online 24 July 2009 PACS: 71.15.Ap 71.15.Nc 74.62.Fj 62.20.Dc
abstract A density functional (DFT) calculations of the structural, elastic and high pressure properties of the cubic XBi (X = U, Cm) compounds, has been reported using the full potential linear muffin-tin orbital (FP-LMTO) method. In this approach the local density approximation (LDA) is used for the exchangecorrelation (XC) potential. Results are given for lattice constant, bulk modulus and its pressure derivatives. The pressure transitions at which these compounds undergo structural phase transition from NaCl-type (B1) to CsCl-type (B2) phase were found to be in good agreement with the available theoretical results. We have determined the elastic constants C11 , C12 ,C44 and their pressure dependence which have not been established experimentally or theoretically. © 2009 Elsevier Ltd. All rights reserved.
Keywords: B. FP-LMTO C. Pressure effects D. Ground state D. Elastic constants
1. Introduction Nowadays, quantum mechanical calculations have reached a sufficient degree of sophistication to reproduce satisfactorily experimental data and (or) to obtain interesting properties in the case in which experimental measurements are absent. Among these quantities, elastic constants represent a good test for estimating the quality of a theoretical approach. As a matter of fact, they require computations, point by point, of the hypersurface of the total energy for appropriate lattice deformations and numerical calculation of the second derivatives of the energy with respect to strain compounds. The structural phase transition at high pressure in rare-earth chalcogenides and pnictides (REX; X = O, S, Se, Te and REY; Y = N, P, As, Sb and Bi) have been investigated in recent years with great interest. The majority of these compounds exhibit no integer valence at high pressure and display numerous allotropic structures and properties which can be in general interpreted in terms of valence fluctuations arising from instability of felectrons of the heavy RE atoms [1–7]. A phenomena of interest
∗
Corresponding author. Fax: +213 48 54 43 44. E-mail addresses:
[email protected] (D. Rached),
[email protected] (R. Khenata). 0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.07.033
in these compounds is the hybridization of the 5f-electrons with conduction electrons which is an important parameter and leads to complex properties such us Kondo-like, magnetic anisotropy, heavy fermion or intermediate valence behavior and is responsible for their unusual properties [1,8,9]. Like the most of the lanthanides, actinides and pnictides, the curium and uranium monobismuthides XBi (X = Cm, U) compounds crystallize in NaCltype structure (B1) at ambient conditions with space group Fm3m (225). In this structure, the X atoms and Bi atoms occupy (0; 0; 0) and (1/2; 1/2; 1/2) positions, respectively. At pressures 12 and 5 GPa, as it has been shown by X-ray diffraction measurements [10], the CmBi and UBi compounds have been found to undergo a firstorder phase transition from the sixfold-coordinated NaCl-type (B1) structure to eightfold-coordinated CsCl-type (B2) structure (S.G.S.: Pm3m-(221)). To our knowledge, there is no theoretical report in the literature on the structural phase transformation except one report by Jha and co-workers [11] who have used the interatomic potential approach which is not a full potential method. Moreover, as we know, theoretical and experimental data on the elastic constants and their pressure dependence are not available. Therefore, we think it is worth to perform calculations for these properties, using the full potential linear muffin-tin orbital (FP-LMTO) method, based on the density functional theory (DFT), in order to provide
D. Rached et al. / Solid State Communications 149 (2009) 1772–1776 Table 1 Parameters used in the calculations: number of plane wave (NPLW), cut-off energy (Ecut in Rydbergs) and the muffin-tin radius (RMT in atomic units).
3778 3294 3942 4168
MTS (a.u) Actinide atom (U and Cm)
Bi
2.836 3.052 2.919 3.101
2.617 2.817 2.804 2.862
74.16 73.66 73.20 83.22
reference data for the experimentalists and to complete existing theoretical and experimental works on these compounds. The rest of the paper is organized as follows: In Section 2, we briefly describe the computational techniques used in this study. The most relevant results obtained for the structural and elastic properties of CmBi and UBi compounds are presented and discussed in Section 3. Finally, in Section 4, we summarize the main conclusions of our work. 2. Computational details The calculations reported in this work were carried out by means of the full potential linear muffin-tin (FP-LMTO) [12,13] method based on the density functional theory (DFT). In this method the space is divided into an interstitial region (IR) and nonoverlapping (MT) spheres centered at the atomic sites. In the IR region, the basis set consists of plane waves. Inside the MT spheres, the basis sets are described by radial solutions of the one particle Schrödinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. In order to achieve energy eigenvalue convergence, the charge density and potential inside the muffin-tin spheres are represented by spherical harmonics up to lmax = 6. The exchange-correlation potential was treated by the local density approximation (LDA) developed by Perdew and Wang [14]. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10−5 Ry. The integrals over the irreducible Brillouin zone (IBZ) are performed up to 56 k-points for both NaCl and CsCl phases using the tetrahedron method [15]. The values of sphere radii (MTS) and the number of plane waves used in our calculations are listed in Table 1. 3. Results and discussion 3.1. Structural properties In order to calculate the ground state properties of CmBi and UBi compounds, the total energies are calculated in both phases for different volumes around the equilibrium cell volume V0 . The plots of the calculated total energies versus reduced volume for these compounds in both (B1) and (B2) phases are given in Fig. 1 (a)–(b). It is seen from these E–V curves, that the NaCl-(B1) phase is stable than the CsCl-(B2) phase at ambient pressure. This fact is consistent with the experimental and theoretical works. The calculated total energies are fitted to Murnaghan’s equation of state [16] to determine the ground state properties such as the equilibrium lattice constant a0 , the bulk modulus B0 and its pressure derivative B0 0 . The calculated equilibrium parameters (a0 , B0 and B0 0 ) in both structures are given in Table 2, which also contains experimental and theoretical data for comparison. The calculated equilibrium lattice constant values for the two compounds agree well with the available theoretical results [11]. In comparison with the experimental data we find that the lattice parameters are underestimated whereas the bulk moduli are overestimated. This is attributed to our use of the local density
-99277.3 -99277.4
Ecut (Ry) Total
UBi B1 Structure B2 Structure
-99277.5 Energy Total (Ryd)
UBi (B1) UBi(B2) CmBi (B1) CmBi(B2)
NPLW (Total)
-99277.6 -99277.7 -99277.8 -99277.9 -99278.0 -99278.1 200
250
300
350
400
450
Volume (a.u)3
b
-105841.65 CmBi
-105841.70 Energy Total (Ryd)
Material
a
1773
B1 Structure B2 Structure
-105841.75
-105841.80
-105841.85
-105841.90
200
250
300
350
400
450
500
Volume (a.u)3
Fig. 1. Energy versus volume curves of B1 and B2 phases for UBi (a) and CmBi(b).
approximation (LDA) which is known to slightly underestimate the lattice constant and overestimate the bulk modulus values compared to the measured ones. Under compression, the calculation shows that CmBi and UBi will undergo a structural phase transition from (B1) to (B2) structures. The structural phase transition is determined by calculating the Gibbs free energy (G) [19] for the two phases, which is given by G = E0 + PV − TS. Since, the theoretical calculations are performed at T = 0 K, Gibbs free energy becomes equal to the enthalpy, H = E0 + PV . For a given pressure, a stable structure is one for which the enthalpy has its lowest value. The enthalpy versus pressure curves for both structures for CmBi and UBi are displayed in Fig. 2(a)–(b). The phase transition pressure values Pt are found to be 4.93 GPa and 5.32 GPa for UBi and CmBi, respectively. These pressures are accompanied by a volume reduction of 0.124 for UBi and 0.246 for CmBi. Similar to the experimental observation, our calculations and those of Jha et al. [11] predict large volume change in these compounds. This is due to the delocalisation of 5f-electrons of the heavy Cm and U atoms. Let us observe that for UBi compound the transition pressure Pt agrees well with the theoretical and experimental results [10,11]. However, for CmBi, our calculated transition pressure Pt value is twice smaller than the available experimental and theoretical data. It is interesting to note that the calculated transition pressure decreases with the size of the cation atoms. The difference in the total energy per volume unit between the two phases (B1 and B2) is about 0.0819 Ry for UBi and 0.0098 Ry for CmBi. 3.2. Elastic properties To further confirm the mechanical stability under pressure effect, we calculated the elastic constants of UBi and CmBi in the B1 phase. The elastic moduli require knowledge of the derivative
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D. Rached et al. / Solid State Communications 149 (2009) 1772–1776
Table 2 Calculated equilibrium lattice constant a0 , bulk modulus B0 and its pressure derivative B0 , phase transition pressure Pt , transition volumes VtB1 and VtB2 , for UBi and CmBi compounds, compared to the available experimental and theoretical data. UBi
CmBi
Present
Expt.
NaCl-B1 a0 (Å) B0 (GPa) B00
6.097 108.50 3.34
6.362a 93a , 89a −7a , −3a
CsCl-B2 a0 (Å) B0 (GPa) B00 Pt (GPa) Vt(B1)/V0 Vt(B2)/V0 ∆V %
3.682 86.59 3.88 4.93 0.916 0.91 0.124
a b c d
Theoretical calculations
3.673d 93d 5.0d
4.0d
0.94d 0.12d
0.96d 0.16d
Present
Expt.
6.057 69.24 4.05
6.328a , 6.333b , 6.23c 54a , 52a 7 a , 9a
3.644 72.63 4.55 5.32 0.98 0.84 0.246
Theoretical calculations
3.65d 54d 12.0d
12.0d
0.88d 0.16d
0.70d 0.13d
Ref. [10]. Ref. [17]. Ref. [18]. Ref. [11].
a
The present values of elastic constants in the NaCl-(B1) phase for UBi and CmBi are given in Table 3. From Table 3, we can remark that the elastic constants increase in magnitude as a function of the cation chemical identity, i.e. in going from CmBi to UBi. For these compounds, with the increase of the atomic number the metallic characteristic increases from U–Bi bond to Cm–Bi bond, accompanied by a decrease of the magnitude of elastic constants. Furthermore, these compounds exhibit small values of C12 . This is a consequence of the hybridization of 5f and conduction electrons. This hybridization is responsible for the decrease in the cation–cation (Cm–Cm; U–U) distance thereby to small value of elastic constant C12 . A given crystal structure cannot exist in a stable or metastable phase unless its elastic constants obey certain relationship. The requirement of mechanical stability in a cubic structure leads to the following restrictions on the elastic constants, C11 − C12 > 0; C44 > 0; C11 + 2C12 > 0. These criteria are satisfied, indicating that these compounds are stable against elastic deformations. The shear modulus G, Young’s modulus E and Poisson’s ratio σ and, which are the important elastic properties for applications, are also calculated in terms of the computed data (see Table 3) using the following relations:
2 UBi B1 Structure B2 Structure
ΔH (eV)
1
0 4.93 GPa
-1
-2 0
2
4
6
8
10
12
14
16
18
20
Pressure (GPa)
b
2 CmBi B1 Structure B2 Structure
ΔH (eV)
1
0
G = (C11 − C12 + 3C44 )/5
5.32 GPa
-1
-2 0
2
4
6
8
10
12
14
16
18
20
Pressure (GPa)
Fig. 2. Variation of enthalpies as a function of hydrostatic pressure for Ubi(a) and CmBi (b) in both structures, B1 and B2. The arrow marks the calculated transition pressure Pt .
of the energy as a function of the lattice strain. In the case of cubic system, it is possible to choose this strain in such a way that the volume of the unit cell is preserved. Thus for the calculation of elastic moduli C11 and C12 we have used the Mehl method [20], (discussed in detail in Refs. [6,21–23]). In order to avoid the sensitivity of the shear modulus C44 value to the relaxation of the positions of anionic atoms in the [001] directions [24–26], we have calculated it from the obtained C11 and C12 elastic constants using the Harisson formula [27,28]. C44 =
(3C11 + 2C12 )(C11 − C12 ) . 7C11 + 2C12
(1)
(2)
E = 9BG/(3B + G)
(3)
ν = (3B − E )/(6B).
(4)
The value of the Poisson ratio (ν ) for covalent materials is small (ν = 0.1), whereas for ionic materials a typical value of ν is 0.25 [29]. In our case the value of ν vary from 0.336 (UBi) to 0.245 (CmBi), i.e. a higher ionic contribution in intra-atomic bonding for these compounds should be assumed. A simple relationship, which empirically links the plastic properties of materials with their elastic moduli, was proposed by Pugh [30]. The shear modulus G represents the resistance to plastic deformation, while B represents their resistance to fracture [31]. A high B/G ratio is associated with ductility, whereas a low value corresponds to the brittle nature. The critical value which separates ductile and brittle materials is around 1.75; i.e., if B/G > 1.75, the material behaves in a ductile manner; otherwise the material behaves in a brittle manner. Frantsevich and co-workers [32] have suggested B/G = 2.67 as the critical value separating brittle and ductile behaviors. We have found that B/G ratios are 2.72, 1.63 for UBi and CmBi, respectively, classifying UBi as ductile and CmBi as brittle.
D. Rached et al. / Solid State Communications 149 (2009) 1772–1776
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Table 3 Calculated elastic constants Cij (in GPa), shear modulus G (in GPa), Young’s modulus E (in GPa), Poisson’s ratios ν , anisotropy factor A for UBi and CmBi in NaCl-type structure. C11
C12
C44
Cs
G
E
ν
A
217.13 166.80
54.19 20.47
98.92 75.35
81.46 73.16
39.76 42.47
106.31 105.78
0.336 0.245
1.214 1.060
Table 4 Calculated longitudinal, transverse and average sound velocity (vl , vt and vm, in m/s) and Debye temperature (θD , in K) for CmBi and UBi in NaCl-type structure.
vl
vm
θD
3047.359 3529.369
6141.717 6076.134
3420.099 3916.170
166.0446 191.3712
To quantify the elastic anisotropy of these compounds we have calculated the anisotropy factor A using the following expression: A = 2C44 /(C11 − C12 ).
θD =
3n
Na ρ
4π
k
1/3
M
vm
1
2
3
vt3
+
1
vl3
3B + 4G 3ρ
−1/3 (7)
1/2 (8)
and
vt =
1/2 G
ρ
.
C11 B C12 C44 CS
150
B C44 CS C12
50 0
2
4
6
8
10
12
Pressure (GPa)
b
250 C11
200 CmBi C11 B C12 C44 CS
150
B C44 CS C12
100
50 0
2
4
6
8
10
12
Pressure (GPa)
(6)
where vl and vt , are the longitudinal and transverse elastic wave velocities, respectively, which can be obtained using the shear modulus G and the bulk modulus B from Navier’s equation as follows [35]:
vl =
UBi
200
Fig. 3. Calculated pressure dependence of elastic constants (C11 , C12 and C44 ), shear modulus (Cs ) and bulk modulus (B) for UBi (a) and CmBi (b) compounds in B1 phase.
where vm is the average sound velocity, h is Planck’s constant, k is Boltzmann’s constant, Na Avogadro’s number, n is the number of atoms per formula unit, M is the molecular weight, ρ = M /V is the density. The average sound velocity in the polycrystalline materials is given by [34]:
vm =
C11
100
(5)
The calculated values of the anisotropic factor A for UBi and CmBi are also listed in Table 3. For an isotropic crystal A is equal to 1, while any value smaller or larger than 1 indicates anisotropy. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy possessed by the crystal. From Table 3, we can observe that the UBi compound characterized by a profound anisotropy compared to CmBi. Having calculated Young’s modulus E, bulk modulus B, and shear modulus G, one can calculate the Debye temperature (θD ), which is an important fundamental parameter closely related to many physical properties such as specific heat, elastic constants, thermal coefficient and melting temperature. At low temperature the vibrational excitations arise solely from acoustic vibrations, and the Debye temperature (θD ) calculated from the elastic constant should be the same as that determined from specific heat measurements. One of the standard methods to calculate (θD ) is from the elastic constant data, since the Debye temperature may be estimated from the average sound velocity, vm by the following equation [33]. h
300
250
vt
Elastic Moduli (GPa)
UBi CmBi
a
Elastic Moduli (GPa)
UBi CmBi
(9)
The calculated sound velocities and Debye temperature as well as the density for UBi and CmBi are given in Table 4. In view of Table 4, we can remark that the Debye temperature increases in magnitude as a function of the cation chemical identity as one moves from U to Cm. In this formulation, the Debye temperature is
directly related to elastic constants via average wave velocity and the decreasing elastic constants from UBi to CmBi causes a decrease in Debye temperature from UBi to CmBi. To our knowledge, there are no experimental and theoretical works exploring the Debye temperature for both compounds. Future experimental work will testify our calculated results. In the following, we study the pressure dependence of the elastic properties. In Fig. 3(a)–(b), we present the variation of the elastic moduli C11 , C12 , C44 , Cs and the bulk modulus B of UBi and CmBi with respect to the variation of pressure. Following Fig. 3, we can notice a linear dependence in all curves in the considered range of pressure. Moreover, it is clearly seen that the elastic constants C11 , C12 , C44 and bulk modulus increase when the pressure is enhanced, confirming the idea of Polian et al. [36] and HerreraCabrera et al. [37] that there is no evidence of a soft acoustic mode being responsible for the phase transition in the considered range of pressure. In contrast to our calculated results, the shear mode modulus C44 , which represents the extrema of the transverse moduli in cubic crystals, show a linear decrease with the increase of the pressure away from zero at the phase transition pressure in thorium, cerium and uranium chalcogenides and pnictides [4–7]. It is impossible for us to find any reason for this discrepancy. In Table 5, we list our results for the pressure derivatives ∂ C11 /∂ P, ∂ C11 /∂ P, ∂ C11 /∂ P, ∂ B/∂ P and ∂ Cs /∂ P. The elastic constant C11 is more sensitive to the change of pressure compared to the other elastic moduli. To the best of our knowledge no experimental
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D. Rached et al. / Solid State Communications 149 (2009) 1772–1776
Table 5 Calculated pressure derivatives of the elastic moduli for UBi and CmBi compounds in NaCl-type structure. Compound
∂ C11 /∂ P
∂ C12 /∂ P
∂ C44 /∂ P
∂ Cs /∂ P
∂ B/∂ P
UBi CmBi
6.16 5.20
1.93 3.48
2.94 2.27
2.11 2.005
3.34 4.05
data or theoretical calculation for elastic constants and their pressure derivatives for these compounds have been reported yet. We believe that, our results can serve as a prediction for future investigations. 4. Conclusions Employing the FP-LMTO method within the local density approximation (LDA) in DFT approach we studied the structural phase transformation and the elastic properties of CmBi and UBi compounds under pressure effect. Our calculations show that the structural parameters at zero-pressure are in reasonable agreement with the available experimental data. Our calculated pressure transition for UBi is in excellent agreement with the available experimental and theoretical data. However, for CmBi the pressure is twice smaller than the experimental and theoretical analogue results. Numerical first-principles calculations of the elastic constants were used to compute C11 , C12 and C44 elastic constants. We found a linear pressure dependence of the bulk modulus and the elastic. We have calculated the shear modulus G, Young’s modulus E and Poisson’s ratio ν , for ideal polycrystalline UBi and CmBi compounds. The sound velocity and the Debye temperature are also investigated for the two compounds. References [1] [2] [3] [4] [5]
P.K. Jha, S.P. Sanyal, Phys. Status Solidi B 200 (1997) 13. J.M. Leger, Physica B 190 (1993) 84. M. Aynyas, N. Kaurav, S.P. Sanyal, J. Phys. Chem. Solids 63 (2002) 821. M. Aynyas, S.P. Sanyal, P.K. Jha, Phys. Status Solidi B 229 (2002) 1459. V. Srivastava, A.K. Bandyopadhyay, P.K. Jha, S.P. Sanyal, J. Phys. Chem. Sol 64 (2003) 907.
[6] A. Bouhemadou, R. Khenata, M. Sahnoun, H. Baltache, M. Kharoubi, Physica B 363 (2005) 255. [7] D. Varshney, N. Kaurav, R. Kinge, R.K. Singh, J. Phys.: Condens. Matter 19 (2007) 236204. [8] E.M. Collins, N. Kioussis, S.P. Lim, B.R. Cooper, J. Appl. Phys. 85 (1999) 6226. [9] W. Reim, J. Schoenes, O. Vogt, J. Appl. Phys. 55 (1984) 1853. [10] M. Gensini, R.G. Haire, U. Benedict, F. Hulliger, Physica B 190 (1993) 75. [11] P.K. Jha, S.P. Sanyal, J. Phys. Chem. Solids 64 (2003) 1237. [12] S.Y. Savrasov, Phys. Rev. B 54 (1996) 16470. [13] S. Savrasov, D. Savrasov, Phys. Rev. B 46 (1992) 12181. [14] J.P. Perdew, Y. Wang, Phys. Rev. B 46 (1992) 12947. [15] P. Blöchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [16] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1947) 244; J.R. Macdonald, D.R. Powell, J. Res. Natl. Bur. Stand. A 75 (1971) 441. [17] J.K. Gibson, R.G. Haire, J. Less-Common Met. 132 (1987) 149. [18] J.P. Charvillat, U. Benedict, D. Damien, W. Müller, Radiochem. Radioanal. Lett. 20 (1975) 371. [19] W. Wettling, J. Windscheif, Solid State Commun. 50 (1984) 33. [20] M.J. Mehl, Phys. Rev. B 47 (1993) 2493. [21] R. Sahara, T. Shishido, A. Nomura, K. Kudou, S. Okada, V. Kumar, K. Nakajima, Y. Kawazoe, Comput. Mater. Sci. 36 (2006) 12. [22] D. Rached, M. Rabah, N. Benkhettou, B. Soudini, H. Abid, Phys. Status Solidi B 241 (2004) 2529. [23] D. Rached, M. Rabah, R. Khenata, A. Bouhemadou, N. Benkhettou, M. Dine el Hannani, Phys. Status Solidi B 244 (2007) 1988. [24] H. Meradji, S. Drablia, S. Ghemid, H. Belkhir, B. Bouhafs, A. Tadjer, Phys. Status Solidi B 241 (2004) 2881. [25] V. Kanchana, G. Vaitheeswaran, A. Svane, A. Delin, J. Phys.: Condens. Matter 18 (2006) 9615. [26] R. Dovesi, C. Roetti, C. Freyria-Fara, M. Prencipe, Chem. Phys. 156 (1991) 11. [27] W.A. Harrison, Electronic Structure and Properties of Solids, Freeman, San Francisco, CA, 1980, Reprinted by Dover, New York, 1989. [28] A.E. Merad, H. Aourag, B. Khelifa, C. Mathieu, G. Merad, Superlatt. Microstruct. 30 (2001) 241. [29] J. Haines, J.M. Leger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1. [30] S.F. Pugh, Phil. Mag. 45 (1954) 823. [31] G. Vaitheeswaran, V. Kanchana, R.S. Kumar, A.L. Cornelius, M.F. Nicol, A. Svane, A. Delin, B. Johansson, Phys. Rev. B 76 (2007) 014107. [32] I.N. Frantsevich, F.F. Voronov, S.K. Bokuta, in: I.N. Frantsevich (Ed.), Elastic Constant and Elastic moduli of Metals and Insulator Handbook, Naukova Dumka, Kiev, 1983, pp. 60–180. [33] B. Mayer, H. Anton, E. Bott, M. Methfessel, J. Sticht, J. Harris, P.C. Schmidt, Intermetallics 11 (2003) 23. [34] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909. [35] E. Schreibe, O.L. Anderson, N. Soga, Elastic Constants and their Measurements, McGraw-Hill, New York, 1973. [36] A. Polian, M. Grimsditch, Phys. Rev. B 60 (1999) 1468. [37] M.J. Herrera-Cabrera, P. Rodrıguez-Hernàndez, A. Munõz, Phys. Status Solidi B 223 (2001) 411.
