Structural, electronic and mechanical properties of alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba)

Author’s Accepted Manuscript Structural, electronic and mechanical properties of alkaline earth metal Oxides MO (M=Be, Mg, CA, Sr, Ba) A. Jemmy Cinthia, G. Sudha Priyanga, R. Rajeswarapalanichamy, K. Iyakutti www.elsevier.com/locate/jpcs

PII: DOI: Reference:

S0022-3697(14)00261-3 http://dx.doi.org/10.1016/j.jpcs.2014.10.021 PCS7415

To appear in: Journal of Physical and Chemistry of Solids Received date: 17 June 2014 Revised date: 20 October 2014 Accepted date: 24 October 2014 Cite this article as: A. Jemmy Cinthia, G. Sudha Priyanga, R. Rajeswarapalanichamy and K. Iyakutti, Structural, electronic and mechanical properties of alkaline earth metal Oxides MO (M=Be, Mg, CA, Sr, Ba), Journal of Physical and Chemistry of Solids, http://dx.doi.org/10.1016/j.jpcs.2014.10.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Structural, Electronic and Mechanical Properties of Alkaline Earth Metal Oxides MO (M=Be, Mg, Ca, Sr, Ba). A.Jemmy Cinthiaa, G. Sudha Priyangaa, R. Rajeswarapalanichamya,*, K. Iyakuttib a b

Department of physics, N.M.S.S.V.N college, Madurai, Tamilnadu-625019, India

Department of physics and Nanotechnology, SRM University, Chennai, Tamilnadu-603203. India.

Abstract: The structural, electronic and mechanical properties of alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) in the cubic (B1, B2 and B3) phases and in the wurtzite (B4) phase are investigated using density functional theory calculations as implemented in VASP code. The lattice constants, cohesive energy, bulk modulus, band structures and the density of states are computed. The calculated lattice parameters are in good agreement with the experimental and the other available theoretical results. Electronic structure reveals that all the five alkaline earth metal oxides exhibit semiconducting behaviour at zero pressure. The estimated band gaps for the stable wurtzite phase of BeO is 7.2 eV and for the stable cubic NaCl phases of MgO, CaO, SrO and BaO are 4.436eV, 4.166 eV, 4.013 eV, and 2.274 eV respectively. A pressure induced structural phase transition occurs from wurtzite (B4) to NaCl (B1) phase in BeO at 112.1 GPa and from NaCl (B1) to CsCl (B2) phase in MgO at 514.9 GPa, in CaO at 61.3 GPa, in SrO at 42 GPa and in BaO at 14.5 GPa. The elastic constants are computed at zero and elevated pressures for the B4 and B1 phases for BeO and for the B1 and B2 phases in the case of the other oxides in order to investigate their mechanical stability, anisotropy and hardness. The sound velocities and the Debye temperatures are calculated for all the oxides using the computed elastic constants.

Keywords: C. Ab-initio calculations; D. Crystal structure; D. Phase transitions; D. Electronic structure; D. Mechanical properties. PACS No.: 31.15.A- , 61.50 Nw, 61.50.Ks, 31.15.ae, 62.20.- x

*Corresponding author E-mail: [email protected] Address: Department of Physics, N.M.S.S.V.N College, Madurai, Tamilnadu-625019, India. Phone : 0452-2459187, Fax : 0452-2458358

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1. Introduction. The alkaline earth oxides MO (M=Be, Mg, Ca, Sr, Ba) have gained the attention of researchers owing to their various unique physical and chemical properties and potential technological applications [1-5]. All these oxides except BeO crystallize in the cubic NaCl (B1) structure at ambient conditions. BeO crystallizes in the wurtzite (B4) structure at ambient condition. Most of these alkaline earth oxides are widely found in the lower mantle of the earth [6, 7], and therefore high pressure study becomes essential to predict their stability and to create relevant geophysical models of the interior of the earth. Amrani et al. [8] have found the transition pressure of BeO from B4 to B1 phase to be 107.4GPa; while Formichev [9] has experimentally determined its band gap as 10.3eV. Duffy et al. [10] have experimentally established that MgO is stable in the B1 phase up to a pressure of 230 GPa; while the theoretical prediction goes up to 1050 GPa [11, 12].The temperature dependence of the elastic properties of MgO has been studied experimentally by Isaak et al.[13] at ambient pressure by the resonance method; whereas the pressure dependence study that has been carried out by Bogardus [14] is limited to a pressure of less than 3 GPa. Karki et al [15] have studied the structural and optical properties of MgO up to a pressure of 150 GPa only by the first principles pseudo-potential method. The electronic band structure of oxides are generally affected by structural changes as confirmed by Errandonea et al.[16] in their study of PbWO4 and by Lacomba Perales et al. [17] in their combined high pressure experimental and theoretical studies of AWO4 (A=Ca, Sr, Ba, Pb). The effect of pressure on the band gap of AWO4 (A=Cd, Mg,Mn, Zn) has been studied experimentally by Ruiz-Feurtes et al.[18]. Jeanloz et al [19] have reported a pressure induced phase transition of CaO from B1 to B2 phase around 60 GPa at room temperature by the diamond-anvil cell experiment and between 63 GPa and 70 GPa at 1350K by the shock wave experiment; whereas the theoretically predicted values are around 55GPa[20,21]. Speziale et al. 2

[22] have studied the high pressure elasticity of CaO by the single-crystal Brillouin scattering up to 25.2 GPa and up to a pressure of 65.2 GPa by the X-ray diffraction method. Tsuchiya et al. [23] have studied its elasticity at ambient conditions. SrO and BaO are far less studied when compared to MgO and CaO both experimentally and theoretically. The experimental band gap of SrO was found to be 5.9 eV by Rao et al [24]; whereas the theoretical values are as low as 3.83 eV[25] and 3.9eV[26]. Amorim et al [27] have studied the energetics of phase transition through DFT calculations and Ghebouli et al. [28] have analysed the structural, elastic and optical properties of CaO, SrO and BaO by the first principles method up to a pressure of 40 GPa only. The structural properties of MgO, CaO and SrO have been theoretically studied by Cortona et al.[29]. In spite of these studies, the high pressure electronic and elastic properties of these alkaline earth oxides in their stable B4 phase for BeO and B1 phase for the other oxides up to their respective transition pressures are not yet investigated theoretically. In this paper, we present the ground state and phase transition studies of the alkaline earth oxides MO (M=Be, Mg, Ca, Sr, Ba), in addition to the high pressure study of the electronic structure and mechanical properties of these oxides in their stable phase below their respective transition pressures, which we assume, would be of great use to the experimentalists involved in the high pressure study of these alkaline earth oxides. The paper is organized as follows: 2nd section describes the computational approach; the 3rd describes the results and discussion, followed by the conclusion in the last section. 2. Computational approach The first principles calculations were all carried out using the Vienna Ab-initio Simulation Package (VASP) [30, 31] within the density functional theory framework, where PBE form of GGA[32-34] are employed to describe the electron exchange and correlation. The 3

ground state geometries are determined using the conjugate – gradient algorithm with a force convergence less than 10-3 eV /A0, with a plane wave cut-off energy of 600 eV and the sampling of the brillouin zone are performed by Monkhorst-Pack scheme [35] with the k-point meshes 6x6x6 for the cubic phases and 8x8x6 for the wurtzite phase for well converged results. The Beryllium 2s2; Magnesium 3s2; Calcium 3s2,3p6,4s2; Strontium 4s2,4p6,5s2; Barium 5s2,5p6,6s2 and oxygen 2s22p4 orbitals are treated as valence electrons. Ground state properties were studied for these alkaline earth oxides MO (M=Be, Mg, Ca, Sr, Ba) in the following four phases: B1(NaCl); B2(CsCl); B3(Zinc Blende,( ZB) and B4 (wurtzite)), wherein the anions (Be, Mg, Ca, Sr, Ba) are positioned at (0,0,0) and the cation oxygen at (1/2,1/2,1/2) for the B1 and B2 phases, while for the B3 phase the anions are at (0,0,0) and the cation is at (1/4,1/4,1/4). As for the B4 phase, there are four atoms per hexagonal unit cell in which the anions are located at (1/3,2/3,0); ( 2/3,1/3,1/2)and the oxygen at (1/3,2/3,u); (2/3,1/3,1/2+u).The unit cell of the proposed phases of MO (M=Be, Mg, Ca, Sr, Ba) are depicted in Fig1. 3. Results and Discussion 3.1 Structural properties The physical properties of any material considered are predominantly determined by its structure and the lattice parameter is the essential quantity related to the structure of the material. The optimized lattice parameters of the alkaline metal oxides (MO), are determined by computing the total energies for various volumes for the four phases considered. The volume corresponding to the minimum energy is the equilibrium volume V0 (Fig 2), from which the optimized lattice constant ao is computed. Obviously, the energy is minimum for the B4 phase for BeO and the energy is minimum in the B1 phase for the other oxides revealing that the B4 phase is the most stable phase for BeO, while the B1 phase is the most stable phase at zero pressure for the other oxides. The data (Energy E and volume V) are then fitted to the universal 4

Birch-Murnaghnan equation of state to determine the bulk modulus B0 and its first derivative B0' at zero pressure. The calculated values of the lattice parameters a0, bulk modulus B0 and its first derivative B0' for all the four phases of (MO) are listed in Table 1, together with the available experimental and other theoretical works [9,11,12,20,36-48].The computed equilibrium lattice parameters of the alkaline earth oxides are in agreement with the experimental values, and are found to increase with the size of the metal atoms; while their corresponding bulk modulus values decrease with the increase in size of the metal atoms. The bulk modulus values are highest for BeO, for all the four phases, indicating that it is the hardest amongst the oxides considered; while they are the lowest for BaO, revealing its highest compressibility at zero pressure. The cohesive energy of a solid is the energy required to break the atoms of the solid into isolated atoms. MO Ecoh

M [ Eatom

O MO Eatom Etotal ]

(1)

MO M where Etotal is the total energy of the compound at the equilibrium lattice constant and Eatom

O and Eatom are the atomic energies of the pure constituent atoms. The cohesive energies for

alkaline earth metals (M) and their oxides (MO) are shown in Fig.3 .The cohesive energies of these oxides are observed to be higher than those of the host elements and BeO owing to its highest cohesive energy is the most stable compound among the considered oxides. A first order structural phase transition occurs from B4 to B1 in BeO and from B1 to B2 in the other oxides at elevated pressures, due to the displacement of the atoms in them. That is, the lattice parameters of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) decrease on compression, resulting in a phase transformation. To determine the transition pressures of these oxides, enthalpy (H) is calculated for various pressures using the expression 5

H = E + PV

(2)

where E is the total energy for the pressure P and volume V. The total energies are computed for various pressures for the B1 and B4 phases for BeO and for the B1 and B2 phases for the other oxide compounds, to determine the enthalpy values. The intersection in the enthalpy versus pressure plots is the transition pressures (PT) for these oxides (Fig.4); the transition pressure values together with the available experimental and other theoretical works [9, 11, 12, 20, 42, 49-54] are tabulated in Table 2 for comparison. Amrani et al. [9] have predicted the transition pressure of BeO from B4 to B1 phase to be 107.4 GPa and our computed values of this transition pressure is 112.1 GPa. Experimentally magnesium oxide is found to be stable in the B1 phase up to a high pressure of 200 GPa as predicted by Vassiliou et al. [49] and our results are very close to that predicted by Mehl et al.[12]. The transition from B1 to B2 phase was observed at 63 GPa in CaO by Mammone et al. [50]; while the theoretical predicted values are around 55GPa[20,21] and our calculated value 61.27GPa, is much closer to the experimental value. Sato et al. [51] have found a transition from B1 to B2 phase at a pressure of 36 GPa in SrO by the x-ray diffraction method, while our computed value is 42 GPa. However, for BaO, the calculated transition pressure coincides with the one observed by Liu [53]. The transition pressures from B1 to B2 in MO (M=Mg, Ca, Sr, Ba) decrease with increase in the size of the metal atoms, i.e.; PT (MgO) >PT (CaO) > PT (SrO) > PT (BaO), resembling the trend shown for the hardness of these oxide materials.

3.2. Electronic properties The electronic band structure, an important property of any material provides information about electronic and optical properties. The band structure changes if the composition or atomic arrangements in a material are changed. The first Brillouin zone figures of the NaCl (B1) and 6

wurtzite (B4) structures along the relevant symmetry points in which the band structures are computed is shown in the Fig.5 (a) and Fig. 5(b) respectively. The electronic band structures of MO (M=Be, Mg, Ca, Sr, Ba) are computed for their stable phases at zero and elevated pressures below their respective transition pressures [Fig. 6 (a-b)].The horizontal dotted line represents the Fermi energy level. The occurrence of band gaps indicates that these oxides are not metallic below their transition pressures. In the band structure of BeO, the two lowest valence bands around -15 eV are due to the s state of oxygen; and the cluster of bands just below the Fermi level are mainly due to the 2p state of oxygen with relatively small contribution from the orbitals associated with Beryllium. The energy band gap of BeO is found to be 7.2 eV which is very close to the value reported by Amrani et al. [9].The band structures of MgO and CaO and that of SrO and BaO seem to be similar. The deep low-lying band around -13eV is due to the 2s states of the oxygen atom in MgO and CaO. The energy bands just below the Fermi level are due to the 2p states of oxygen in MgO and with additional 3p states of Ca in the case of CaO with very little contribution from the 2s and 2p states of metal atoms. The computed valence band width and the energy band gap values of all these oxides, along with the available experimental and other theoretical values [9, 11, 24-26, 55-62] are shown in Table 3. The upper valence band widths are found to be 5eV for MgO and 3.92 eV for CaO. These values are in agreement with earlier calculations [11, 49] as listed in Table 3.The energy band gap is 4.436eV for MgO, which much lesser than the experimental value but very close to the theoretical value reported by Chang et al. [11] and it is 4.166eV for CaO. The band structure of SrO and BaO are similar with a deep low lying band around -30 eV for SrO and -25 eV for CaO, due to the s-state of the metal anions .Further above this band, just below and above -10 eV in SrO and BaO respectively, the bands are due to the hybridization of the 2s state of oxygen and the p-states of the metals. The bands just below the Fermi level are 7

less dispersed and are mainly due to the 2p states of oxygen with additional contribution from the metal p states. The valence band width computed for SrO is 2.55eV, which is close to the value reported by Taurian et al. [25]; and it is 2.42eV for BaO. The band gaps are 4.013eV and 2.274eV for SrO and BaO respectively. We do find that the valence band width and the energy band gaps narrows down as one moves from BeO to BaO, a trend which is followed in other theoretical works also. The existence of band gaps up to their corresponding phase transition pressures, reveal that these metal oxides do not crossover to metallic nature in their stable phases (B4 for BeO and B1 for the other oxides) below their respective transition pressures. Thus pressure affects the band gap of these oxides which is similar to the experimental works on oxides [16-18]. The variation of the band gaps with pressure for the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba), in the B4 phase for BeO in the B1 phase for the others are shown in Fig. 7. All the calculated band gap values obey the analytical expression; Eg(P) = Eg(0) + aP + bP2

(3)

where Eg(P) is the band gap value at a pressure P in GPa, Eg(0) is the band gap value at zero pressure and a & b are the first and second order pressure derivatives. The calculated values of the coefficients „a‟ and „b‟ are presented in Table 4.The positive values of the coefficient „a‟ for all the alkaline earth oxides considered indicate that the energy band gap increases with increasing pressure, resulting in the delocalization of the p and d states of the metal and the p states of oxygen. This delocalization causes the valence band maximum to shift to a lower energy level and the conduction band minimum to rise up to a higher energy level, thereby increasing the band gap. The value of „b‟ which is either zero or very low, predict that the energy gap exhibits a strong linearity as a function of pressure for all the oxides.

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The total and partial density of states (DOS) of the metal oxides in their stable phases at zero and an elevated pressure is shown in Fig.8, Fig.9 and Fig.10.The existence of band gaps as suggested by their band plots are confirmed by their respective DOS plots. In the DOS plot of BeO, the strong peak around -15 eV is predominantly due to the oxygen 2s state. The peaks just below the Fermi level, are mainly due to the 2p state of oxygen with little contributions from the Be s and p states. The DOS plots of MgO and CaO and that of SrO and BaO are quite similar to each other. The peaks near -13 eV is due to the oxygen 2s states in the case of MgO and CaO; whereas it is more complex in SrO and BaO due to the hybridization of the 2s state of oxygen with the p and d states of the metal anions. The highest peak just below the Fermi level is mainly due to the oxygen 2p state with little contribution from the p and d states of the metals in all the cases. All the qualitative features of the DOS are very similar to that obtained by Pandey et al. [59] and Taurian et al. [25]. The total valence charge density along the (111) plane is studied to visualize the bond nature of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba), in their stable phase at zero pressure. The plots (Fig. 11) of the electron density distribution in space surrounding the metal and oxygen ions are in the same scale for easy comparison of the nature of the bonds between the atoms. The charge density plots (Fig. 11), show a spherical distribution around the interior of the metal core ions. The electron charge density is more in the interstitial space between Be ion and oxygen ion, featuring the covalent nature of the bond, enabling the valence electrons to wander away from the host atoms. In BaO, the charge density in the interstitial region is very low due to the localization of the electrons, caused by the complete transfer of the valence electrons to the oxygen atom, featuring the ionic nature of the bond between Ba ion and oxygen ion. Our results about the nature of the bonds between the atoms in these oxides are in accord with the discussion in the Ref. [61]. The electron density in the interstitial space between the metal and oxygen ions 9

decreases as one move from BeO to BaO, confirming the increase in their ionic bonding nature. The value of the bulk modulus of BeO is largest among the alkaline earth metal oxides, predicting its covalent nature of bonding and strong incompressibility. To conclude, one can say that the ionic bonding is the strongest in BaO. 3.3. Elastic Properties The elastic constants are essential for applications related to mechanical properties of solids. The high pressure study of these constants is important to predict and understand the material response, its strength and its mechanical stability under compression. In this work the elastic constants at zero and high pressures are calculated for the B4 and B1 phases for BeO and for the B1 and B2 phases for the other oxides, in order to investigate their mechanical stability and other related properties. For cubic crystals, there are only three independent elastic constants C11, C12 and C44; whereas there are five independent elastic constants C11, C12, C13, C33 and C44 for hexagonal crystals. The computed values of the elastic constants, for the alkaline earth oxides MO(M=Be, Mg, Ca, Sr, Ba) in the B1, B2, B3 and B4 phases at zero pressure are listed in Table 5, along with the available experimental and other theoretical works. It is remarkable to note that the values of the elastic constants for the stable phase of MgO, CaO, SrO and BaO are in agreement with the experimental and other theoretical values [13, 21, 37, 48, 62-66]. There were no experimental or theoretical values for the elastic constants to compare with our data values for the other phases. The bulk modulus (B) and shear modulus (G) for both cubic and hexagonal crystals are determined using the Voigt-Reuss- Hill (VRH) averaging scheme [67-69].

B

(C11 2C12 ) 3

G

3C44 C11 C12 5

(4)

(5)

10

B0

2 [C11 C12 2C13 (1/ 2)C33 ] 9

(6)

G

2(C11 C33 ) 15

(C12 2C13 ) 15

3[2C44 (1/ 2)(C11 C12 )] 15

(7) The bulk modulus values obtained by the EOS fit (Table 1) are relatively very close to the values calculated using the elastic constants (Table 5). Mechanical stability criteria for cubic crystal [70] at ambient conditions are C44

C12

0, C33

0, C11

C12 , C44

0, C11

0

C12 , C11 2C12

, C11 C12 C33

0 and

2C 132 for hexagonal crystal. The negative

values of C12 and C44 indicate the instability of BeO, MgO, CaO and SrO in their B2 phase at normal pressure. The elastic constants are computed at high pressures for the B4 and B1 phases for BeO and for the B1 and B2 phases for the other oxides. The variation of the elastic constants, the bulk modulus, Young‟s modulus and shear modulus with pressure are shown in Fig.12; the sudden change in their values is attributed to their structural transition that occurs at their respective transition pressures. In all the cases, C11, the bulk modulus Young‟s modulus and shear modulus increase linearly with pressure; indicating that the B1 structure is increasingly stable against spinodal and tetragonal shear deformation. Also, we find that C44 softens with pressure in the B1 phase and increases slowly with pressure in the B2 phase. The variation of C11, C12 and C44 with pressure for MgO, CaO, SrO and BaO are in agreement with the results obtained by Singh et al. [71]. The mechanical stability is closely related to the structural stability of a compound. The elastic stability criteria for the cubic crystals under pressure are

K

1 (C11 C12 3

P) 0

(8) 11

G'

1 (C11 C12 2 P) 0 2

(9)

G

4(C44 P) 0

(10) which is related to the bulk, tetragonal and shear moduli respectively. The stability criteria for BeO are obeyed for pressures upto 112.1 GPa in the B4 phase and upto a pressure of 200 GPa in the B1 phase. The stability criteria for the other alkaline earth oxides MO (M= Mg, Ca, Sr, Ba), for various pressures are shown in Fig. 13. From the plots we note that both CaO and BaO are elastically stable below their respective transition pressures in the B1 phase and up to a pressure of 100 GPa in the B2 phase, beyond their transition pressures, obeying all the stability criteria. Howbeit, MgO and SrO do not obey the stability criteria, for pressures less than their respective transition pressures in the B1 phase. The shear instability sets in at 229 GPa for MgO, which is close its experimental transition pressure; while it is 38.3GPa for SrO, close to the value 39.2GPa predicted by Ghebouli et al.[ 28]. The Young‟s modulus (E) and the Poisson‟s ratio are calculated using the relations [72]:

E

9 BG (3B G )

(11) and C12 C11 C12

(12)

12

The computed Young‟s modulus values for the alkaline earth metal oxides at zero pressure are presented in Table 5. The Young‟s modulus of BeO in its B3 phase is the highest, showing that it is the stiffest material among the considered oxide compounds. The pressure dependence of the Young‟s modulus (E) of MO (M=Be, Mg, Ca, Sr, Ba) is shown in Fig. 12; their values are found to increase with pressure. Poisson‟s ratio is related to the volume change during uniaxial deformation; the material being incompressible if its value is 0.5.The Poisson‟s ratio is the largest for the B1 phase in BeO and in the B3 phase for all the other oxides at zero pressure, (Table 4) showing that its volume changes is the least. For the B1 and B2 phases their values increase slowly with pressure predicting that the volume change decrease with pressure. Also the value of the Poisson‟s ratio is greater than or equal to 0.25 for ionic crystals and it is around 0.1 for covalent crystals. Next, the ductile nature of the oxides is analyzed by calculating the B/G ratio. The shear modulus G is related to its resistance to plastic deformation, whereas the bulk modulus refers to its resistance to fracture. A high B/G ratio is related with ductility, while a low value to the brittle nature of a material. If B/G>1.75, the material is ductile, otherwise brittle [73]. The computed values of B/G show that BeO in its B1 phase at zero pressure is more brittle than the others; while BaO is ductile at all pressures in the B1 phase and ductile in the B2 phase (Table 5). Upon compression, the ductile nature sets in, for MgO, CaO and SrO in their B1 and B2 phases (Table 6). The anisotropy A =2C44/ (C11+C12) is evaluated to get an insight of the crystal anisotropy. The value of A equals one, for an isotropic crystal and any value smaller or larger than one, indicates the anisotropy of the crystal .The magnitude of deviation from one, gives the amount of anisotropy possessed by the crystal. Our computed value of the anisotropy A 13

is found to soften (table 6), when the pressure is enhanced for the B4 and B1 phases of BeO, and for the B1 phase for all the other oxides; and it increases slowly with pressure for their B2 phase. Further the micro hardness of the oxides is determined using the expression [74]:

H

(1 2 ) E 6(1 )

(13) The hardness is found to increase slowly with pressure for BeO and MgO and for the B2 phase of BaO (Table 6); while in the others, there is an increase followed by a slight decrease in their values with pressure. The Debye temperature (θD) is an important parameter related to the thermal characteristics of materials, which correlates many physical properties of materials, such as specific heat, elastic constants and melting temperature. The Debye temperature is defined in terms of the mean sound velocity ν m and gives explicit information about the lattice vibrations and it is calculated using the equation [75]: (14) with

h / 2 , h is Planck‟s constant, kB is Boltzmann‟s constant, NA is the Avogadro‟s

number, ρ is density, M is molecular weight, n is the number of atoms in the molecule and 1/3

νm

1 2 3 νt3

1 ν l3

(15) where (16) and 14

νt

G ρ

1/2

(17) are the velocities of longitudinal and transverse sound waves respectively. The calculated density, the velocities and Debye temperature for BeO in the B4 phase and for MgO, CaO, SrO and BaO, in the B1 phase at zero pressure, together with the available experimental and theoretical values [65,75 – 78], are listed in Table 7. The computed Debye temperature of SrO is very close to the value reported by Gmelin et al.[78].We note that the velocities and the Debye temperature fall as one moves from BeO to BaO. The high value of the Debye temperature for BeO implies that its thermal conductivity and microhardness is more when compared with other oxides considered. The variation of velocities and Debye temperature with pressure are presented in Fig. 14. From the plot we found that the velocities and Debye temperature increases upon compression for the B4 phase of BeO and for the B1 phase for all the other oxides considered and an abrupt change occurs at their respective transition pressures. A similar trend is observed in the B2 phase except for transverse velocity of CaO.

4. Conclusion In conclusion, the first principles calculations have been performed to investigate the structural, electronic and mechanical properties of alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) at zero and high pressures. The calculated ground state properties are in good agreement with the experimental and other available theoretical results. Our results suggest that BeO is stable in the wurtzite phase, while all the other four oxides are most stable in NaCl phase at zero pressure. A first order pressure-induced structural phase transition occurs from the hexagonal (B4) to NaCl (B1) phase in BeO at 112.1 GPa and from NaCl (B1) to CsCl (B2) phase 15

in MgO at 514.87 GPa, CaO at 61.27 GPa, SrO at 42 GPa and BaO at 14.5 GPa. The electronic band structure and density of states reveal that all the five alkaline earth metal oxides exhibit non-metallic nature at zero and elevated pressures. The energy band gap values, are found to increase in the order BeO > MgO > CaO >SrO > CaO. The charge density plots confirm the covalent nature of BeO and the ionic nature of the other oxides. The elastic properties of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) are analysed in detail at zero and high pressures. The computed elastic constants at zero pressure are in good agreement with the experimental and other theoretical values. The bulk modulus, Young‟s modulus, shear modulus and the hardness of these oxides are found to increase with pressure. The Debye temperatures of BeO, MgO, CaO, SrO and BaO computed using the elastic constants are 1234 K, 902K, 691K, 430K and 316K respectively. Upon compression the Debye temperature is found to increase slowly. We hope that this study of the influence of pressure on the physical properties of the alkaline earth oxides would be of great help to experimentalists in future.

Acknowledgement We thank our college management for their sustained support and encouragement.

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References [1] G.A.Slack and S.B.Austerman, J. Appl. Phys. 42 (1971) 4713. [2] B.Salamatinia, I.Hashemixadeh and Z.A.Ahamad, Iran J. Chem. Chem. Eng. 32 (2013) 113. [3] P.A.Tellex, J.R.Waldron, JOSA 45 (1955) 19. [4] Y.Tatekawa, K.Nakatani, H.Ishii, Surgery Today 26 (1996) 68. [5] J.A.Ober, D.E.Polyak, “Mineral Yearbook 2007: Strontium” [6] B.B.Karki, R.M.Wentzcovintch , S.de Gircoli and S.Baroni, Science 286 (1999) 1705. [7] P.Richet, H.K.Mao and P.M.Bell, J. Geophys. Res. 93 (1988) 15279. [8] V.A.Formichev, Sov. Phys. Solid State 13 (1971) 754. [9] B.Amrani, F.E.H.Hassan and H.Akbarzaden, J. Phys.: Condens. Matter 19 (2007) 436216. [10] T.H.Duffy, R.J.Hemley and H.K.Mao, Phys. Rev. Lett. 74 (1995) 1371. [11] K.J.Chang and M.L.Cohen, Phys. Rev. B 30 (1984) 4774. [12] M.J.Mehl, R.E.Cohen and H.Krakauer, J. Geophys. Res. 93 (1988) 8009. [13] D.G.Isaak, O.L.Anderson and T.Goto, Phys. Chem. Miner. 16 (1989) 704. [14] E.H.Bogardus, J. Appl. Phys. 36 (1965) 2504. [15] B.B.Karki, L.Stixrude, S.J.Clark, M.C.Warren, G.Acland and J.Crain, Am. Mineral. 82 (1997) 51. [16] D.Errandonea, D.Martinez-Garcia, R.Locomba-Perales, J.Ruiz-Fuertes and A.Segura, Appl. Phys. Lett. 89 (2006) 091913. [17] R.Locomba-Perales, D.Errandonea, J.Ruiz-Fuertes, A.Segura, J.Ruiz-Fuertes, P.Rodriguez-Hernandez, S.Radescu, J.Lopez-Solano, A.Mujica and A.Munoz, J. Appl. Phys. 110 (2011) 043703.

17

[18] J.Ruiz-Fuertes, S.Lopez-Moreno, J.Lopez-Solano, D.Errandonea, A.Segura, R.LocombaPerales, A.Munoz, S.Radescu, P.Rodriguez, M.Gospodinov, L.L.Nagomaya and C.Y.Tu, Phys. Rev. B 86 (2012) 125202. [19] R.Jeanloz, T.J.Ahrens, H.K.Mao and P.M.Bell, Science 206 (1979) 829. [20] G.Kalpana, B.Palanivel and M.Rajagopalan, Phys. Rev. B 52 (1995) 4. [21] M.J.Mehl, R.J.Hemley and L.L.Boyer, Phys. Rev. B 33 (1986) 8685. [22] S.Speziale, S.R.Shieh and T.S.Duffy, J. Geophy. Res.B 111 (2006) 02203. [23] T.Tsuchiya and K.Kawamura, J. Chem. Phys. 114 (2001) 10086. [24] A.S.Rao and R.J.Kearney, Phys. Status Solidi B 95 (1979) 243. [25] O.E.Taurian, M.Springborg and N.E.Christensen, Solid State Commun. 55 (1985) 351. [26] A.Hasegawa and A.Yanase, J. Phys. C 13 (1980) 1995. [27] R.G.Amorim, M.V.Alves and J.P.Rino, Comput. Mater. Sci. 37 (2006) 349. [28] B.Ghebouli, M.A.Ghebouli, M.Fatmi and M.Benkerri, Mater. Sci. Semicond. Process. 13 (2010) 92. [29] P.Cortona and A.V.Monteleone, J. Phys.: Condens. Matter 8 (1996) 8983. [30] G.Kresse, J.Hafner, Phys. Rev. B 47 (1993) 558. [31] G.Kresse, J.Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [32] J.P.Perdew and A.Zunger, Phys. Rev. B 23 (1981) 5048. [33] J.P.Perdew, S.Burke, Phys. Rev. B 54 (2004) 16533. [34] J.P.Perdew, S.Burke and M.Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [35] H.J.Monkhorst and J.D.Pack, Phys. Rev. B 13 (1976) 5188. [36] D.Groh, R.Pandey, M.B.Sahariah, E.Amzallag, I.Baraille and M.Rerat, J. Phys. Chem. Solids 70 (2009) 789. [37] Y.Duan, L.Qin, G.Tang and L.Shi, Eur. Phys. J. B 66 (2008) 201. 18

[38] R.M.Hazen and L.W.Finger, J. Appl. Phys. 59 (1986) 3728. [39] X.F.Fan, H.D.Sun, Z.X.Shen, J.L.Kuo and Y.M.Lu, J. Phys.: Condens. Matter 20 (2008) 235221. [40] R.W.J.Wycoff, Crystal Structure; Vol 1, Inter-science Publishers (John Wiley), New York, (1965). [41] J.Yamashita and S.Asano, J. Phys. Soc. Japan 52 (1983) 3506. [42] M.P.Habas, R. Dovesi and A.Lichanot, J. Phys.: Condens. Matter 10 (1998) 6897. [43] M.Causa, R.Dovesi, C.Pisani and C.Roetti, Phys. Rev. B 33 (1986) 1308. [44] B.B.Karki, J.Crain, J. Geophys. Res.103 (1998)12405. [45] O.L.Anderson and P.Andreatch Jr., J. Amer. Ceram. Soc. 49 (1966 )404. [46] I.Jackson and Niesler; High Pressure Research in Geophysics, Edited by S.Akimoto and M.H. Manghnani; Centre for Academic Publishing; Tokyo, (1982). [47] M.S.T.Bukowinsky, Geophys. Res. Lett. 12 (1985) 536. [48] P.R.Son and R.A.Bartels, J. Phys. Chem. Solids 33 (1972) 819. [49] M.S.Vassiliou, and T.J.Ahrens, Geophys. Res. Lett. 8 (1981) 729. [50] J.P.Mammone, H.K.Mao and P.M.Bell, J. Geophys .Res. Lett. 8 (1981)140. [51] Y.Sato, R.Jeanloz, J. Geophys. Res. 86 (1981) 11773. [52] L.G.Liu and W.A.Basset, J. Geophys. Res.77 (1972) 4934. [53] L.G.Liu, J. Appl. Phys. 42 (1971) 3702. [54] H.Zang and M.S.T.Bukowinsky, Phys. Rev. B 44 (1991) 2495. [55] A.P.Lukirshu and I.A.Brytov, Sov. Phys. Solid State 6 (1964) 33. [56] S.Pantelides, D.J.Michish and A.B.Kunz, Phys .Rev. B 10 (1974) 5203. [57] R.C.Whited, C.J.Flaten and W.C.Walker, Solid State Commun. 13 (1973) 1903. [58] O.Madelung, Semiconductors: Data Handbook: 3rd Edn. (Springer, Berlin, 2004). 19

[59] R.Pandey, J.E.Jaffe and A.B.Kunz, Phys. Rev. B 43 (1991) 9228. [60] L.F.Mattheiss, Phys. Rev. B 5 (1972) 290. [61]Micheal P.Marder, Condensed Matter Physics; 2nd Edn., A John Wiley & Sons, INC., Publications (2010) 295. [62] H.Oda, O.L.Anderson, D.G.Isaak and I.Suzuki, Phys. Chem. Miner. 19 (1992) 96. [63] P.Chang and E.K.Graham, J. Phys. Chem. Solids 38 (1977) 1355. [64] A.Lichanot, Solid State Commun. 116 (2000) 543. [65] Y.D.Guo, X.L.Cheng, L.P.Zhou, Z.J.Liu and X.D.Yang, Physica B 373 (2006) 334. [66] H.Baltache,R.Khetana, M.Sahnoun, M.Driz, B.Abbar and B.Bouhafs, Physica B 344 (2004) 334. [67] W.Voigt, Lehrbuch de Kristallphysik (Terubner, Leipzig, 1928). [68] A.Reuss, Z.Angew, Math. Mech. 9 (1929) 49. [69] R.Hill, Proc. Phys. Soc., London, Sec. A 65 (1952) 349. [70] S.P.Singh, Seema Gupta and S.C. Goyal, Physica B 391 (2007) 307. [71] G.V.Sin‟ko, A.Smirnov, J. Phys.: Condens. Matter 14 (2002) 6989. [72] K.A.Matori, M.H.M.Zaid, H.A.A.Sidek, M.K.Halimah, Z.A.Wahab and M.G.M.Sabri, Int. J. Phys. Sci. 5 (2010) 2212. [73] S.F.Pugh, Philos. Mag. 45 (1954) 823. [74] A.M. Ibrahim, Nucl. Instrum. Meth. B 34 (1988) 135. [75] O.L.Anderson, J. Phys. Chem. Solids 24 (1963) 909. [76] G.K.White and O.L.Anderson, J. Appl. Phys. 37 (1996) 430. [77] E.Gmelin and Z.Naturforsch 24a (1969) 1794. [78] E.Gmelin and Z.Naturforsch 25a (1969) 887.

20

FIGURE CAPTIONS Figure 1. Unit cell of the different phases of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba). Figure 2. Total energy (in eV) versus cell volume of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba). Figure 3. Cohesive energies of the alkaline earth metals (M) and the corresponding oxides (MO) in their stable structure Figure 4. Enthalpy versus pressure curve of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba). Figure 5. Brillouin zones of the NaCl (B1) and wurtzite (B4) with relevant symmetry points. Figure 6. Electronic band structure of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba): (a) zero pressure: (b) high pressure. Figure 7. Variation of band gap with pressure less than their respective transition pressures for the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) Figure 8. Total density of states (DOS) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at zero pressure in the stable NaCl structure. Figure 9. Partial density of states (DOS) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at zero pressure in the stable NaCl structure. Figure 10. Total density of states (DOS) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at high pressure in the stable B1 structure. Figure 11. Charge density distribution of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at zero pressure in the stable B1 structure. Figure 12. Pressure dependence of elastic moduli of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba).

21

Figure 13. Pressure dependence of stability criteria of the alkaline earth metal oxides: MgO, CaO, SrO, BaO. Figure 14. Pressure dependence of longitudinal velocity ν l (m/s), transverse velocity ν t (m/s), average velocity ν m (m/s) and Debye temperature (K) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba).

TABLE CAPTIONS

Table 1 Calculated lattice parameters a (Å), Cohesive energy Ecoh (eV), Bulk modulus B0 (GPa) and its pressure derivative B0' of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) for four different structures. Table 2 Transition pressure from B4 to B1 in GPa for BeO and from B1 to B2 in GPa for the other alkaline earth metal oxides (MgO, CaO, SrO, BaO) with experimental and other theoretical values. Table 3 Valence band width (VBW) and band gap (Eg) in eV of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba) at zero pressure. Table 4 Calculated linear and quadratic pressure coefficients a (eV/GPa) and b (eV/GPa2) of Eg(P) = Eg(0) + aP + bP2 of the alkaline earth metal oxides MO (M=Be, Mg, Ca, Sr, Ba). Table 5 Calculated elastic constants C11 ,C12 , C44 (GPa),Bulk modulus Bo (GPa), Shear modulus G(GPa), Young‟s modulus E (GPa), Poisson‟s ratio ν, B/G ratio, Zener isotropy (A) and micro hardness parameter Hv (GPa) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) for the four phases at zero pressure. Table 6 Calculated B/G ratio, Poisson‟s ratio ν, micro hardness parameter Hv (GPa) and Zener isotropy (A) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at zero pressure and high pressure (GPa). Table 7 Calculated Density ρ (g/cm3), longitudinal velocity vl (m/s), transverse velocity vt (m/s), average velocity vm (m/s) and Debye temperature θD (K) of the alkaline earth metal oxides MO (M= Be, Mg, Ca, Sr, Ba) at zero pressure.

22

Research Highlights ► Structural, electronic and elastic properties of MO (M= Mg, Ca, Sr, Ba) are analyzed. ► A pressure induced structural phase transition is predicted under high pressure. ► Electronic structures reveal non-metallic behavior at normal and high pressure. ► The electronic band gap values are found to increase in the order MgO>CaO>SrO>CaO. ► The charge density plots and the Poisson‟s ratio values confirm their ionic nature.

23

Table 1 BeO NaCl CsCl ZB WZ

MgO NaCl

CsCl ZB WZ CaO NaCl

CsCl ZB WZ SrO NaCl

CsCl ZB WZ BaO NaCl CsCl ZB WZ

a

c

u

Ecoh

B0

B0’

3.65 3.651a, 3.633b, 3.622c 2.359 2.331b 3.817 3.828a, 3.803b,3.763c 2.71 2.714a,2.703b, 2.676c 2.698d

-

-

9.46

-

-

7.012

231 232a,265c 219

3.90 3.9c 3.80

-

-

8.15

4.398 4.413a,4.36b, 4.349c,4.376d

0.375 0.3773a,0.379b 0.377c,0.378d

10.42

200 203a,229c 205 206a,229c

3.91 3.8c 3.81 4.1c

4.241 4.21e, 4.22f,4.19g 4.2h,4.17c

-

-

9.8

4.0 4.15j, 3.92h 4.1c

2.648 2.6h, 2.57m 4.6 4.538c 3.269 3.259c,3.278n

-

-

7.15

-

-

8.47

4.9035 4.99c,4.873n

0.379 0.395c,0.404n

9.54

167 162i, 160j 182h, 167.6k 172l,164g 151 181m 146 142c 126 120c

4.843 4.81e, 4.86h 4.838o,4.761c

-

-

8.6

4.2 4.26p, 4.9q 4.44h,3.6c

2.933 2.95h 5.27 5.178c 3.921 3.8939n

-

-

7.64

-

-

8.31

5.0973 4.6755n

0.381 0.4997n

8.32

107 110p, 112q 120h, 133.8k 129l,120c 128 122h 87 83c 112

5.21 5.16e, 5.22h,5.093c

-

-

8.13

3.14 3.14h 5.61 5.501c 4.30

-

-

7.31

-

-

7.54

5.16

0.389

7.91

82 90.6r, 101h 109.8l,97c 91 132h 67 67c 88

4.1 4.09h 4.5c 4.0 3.95h 4.0 3.9c 4.0

5.604 5.52e, 5.65h,5.489c 3.367 3.39h 5.996 5.842c 4.369

-

-

8.56

-

-

7.89

-

-

8.46

6.5535

0.389

8.37

75 72.2p, 81h,80c 87 92h 57 59c 56

4.1 4.21h,4.7c 3.8 4.07h 3.9 5.2c 4.0

a

3.7 4.0h 3.4 4.1c 4.0 3.9c

4.0 4.05h 3.6 3.3c 3.8

Ref[9];bRef[36];cRef[37]; dRef[38]-Exp.; eRef[40]-Exp.; fRef[41];gRef[11];hRef[42]; iRef[45]-Expt;jRef[46]-Exp.; k Ref[20]; lRef[12];mRef[43]; nRef[39]; oRef[44]; pRef[7]-Expt.;qRef[47]-Expt; rRef[48].

38

Table 2

Present

BeO

MgO

CaO

SrO

BaO

112.1

514.9

61.3

42

14.5

> 200a

63b

36c

9d

Expt.

14.5e 107.4f

Others

515g

55h

31.7i

21h

1050j

55.7i

35k

17.4l

a

Ref[49];bRef[50];cRef[51];dRef[52];eRef[53];fRef[9]; gRef[12];hRef[21]; i Ref[20];j Ref[11]; kRef[54]; lRef[42]

39

Table 3

Compound

Quantity

Present

Expt

Others

BeO

VBW Eg VBW Eg VBW Eg VBW Eg

19.1 7.2 5.00 4.43 3.92 4.16 2.55 4.01

20.5a 10.6 6.5c 7.83f

7.44b 4.8d,7e

5.7m,5.22g

VBW Eg

2.42 2.27

4.28g

MgO CaO SrO

BaO a

Ref[55 ]; b Ref[9]; c Ref[54 ]; d Ref[11]; Ref[37]; kRef[26 ];l Ref[25 ]; m Ref[24 ].

e

6.93g

Ref[56 ];

j

40

f

3.43h,1.5i 3.45j 1.72k,2.38l 2.31h,3.9k 3.07j 1.72j

Ref[ 57]; g Ref[58 ]; h Ref[59 ]; i Ref[50 ];

Table 4

Compound

a

bx10-5

BeO

0.044

0.0

MgO

0.037

-3.0

CaO

0.001

-9.0

SrO

0.002

0.0

BaO

0.004

0.0

41

Table 5

BeO NaCl CsCl ZB WZ MgO NaCl

CsCl ZB WZ CaO NaCl

CsCl ZB WZ SrO NaCl

CsCl ZB WZ BaO NaCl

C11

C12

C44

C13

C33

B0

G

E

ν

B/G

A

H

292.2 403.9a 875.1

211 217.4a -120.5

300.1 299.7a -198.7

-

-

238.1

160.7

351.9

0.253

1.69

1.19

23.1

-

-

211.4

-

-

-

-

-

-

342.8 374.4a 432.1 432.5a

149.6 195.3a 120.7 135.8a

211.9 221.9a 155 131.3a

-

-

213.9

224.5

373.8

0.209

1.38

0.86

30

84.6 99a

482.5 474.1a

214

155.3

289

0.208

0.54

0.56

35.5

297 297b 338c 299d 326e 270f 314.7a 568.9 777.4g 163.2 159a 206.7 194.6a

99.6 95b 91c 96.4d 108e 73f 93.5a -71.9 -112.6g 108.8 133.3a 106.1 104.5a

151.9 156b 118c 157.1d 188e 127f 158.7a -59.6 -57.8g 126.9 74a 48.7 590a

-

-

165.5 171c 163.6d 182e 139f

127.9

305

0.252

1.29

0.76

26.2

-

-

-

-

-

-

-

-

-

-

141.7 163c 126.9

87.02

212.2

0.40

1.458

0.933

5.04

102.6 99.9a

154.8 119.6a

129.1

49.3

127.6

0.33

2.08

0.311

10.7

227.2 221b 274c 235.5h 235e 206f 228.9a 384

64.7 57b 54c 54.6h 61e 50f 60.5a -6.2

87.3 80b 53c 82.1h 95e 66f 78.9a 4.1

-

-

118.8 128c 116.1h 121e 102f

84.88

205.6

0.221

1.399

0.598

15.65

-

-

80.5

197

-0.01

1.535

0.021

33.82

101.6 93.7a 106.9

71.8 75.8a 70.6

42.5 38.5a 18.2

-

-

123.6 132.8c 81.73

31.46

83.64

0.414

2.597

0.49

1.695

58.2

60.3

60.2

18.2

49.4

0.363

3.2

0.205

3.5

167.2 174b 223c 183.2i 186e 171f 197.2a 290.6

47.8 47b 46c 47.1i 49e 34f 51.6a -1.2

54 56b 87c 57.7i 68e 49f 55.5a -4.8

-

-

87.6 105c 90.6i 96e 80f

56.28

139

0.222

1.556

0.502

10.54

-

-

23.6

65.6

-0.01

4.06

-0.03

11.06

71.29 77.7a 69.8

59.59 66.2a 50.1

26.26 26.8a 9.8

-

-

96.06 115.3c 63.49

18.09

49.33

0.455

3.509

0.401

0.505

44.9

89.6

56.2

9.8

27.5

0.417

4.57

0.16

1.3

136.1 174j

46.4 49j

39.9 34j

-

-

76.3 74j

41.88

106.2

0.255

1.821

0.437

6.91

42

133f 33f 31f 66f a a a 147.7 47 36.3 CsCl 421.8 58.7 70.2 179.7 114.7 283.7 0.122 1.566 0.292 377.7g -22.9g 22.89g ZB 65.58 48.7 20.32 54.32 15.56 42.61 0.426 3.491 0.355 64.4a 55.3a 17a WZ 64.5 44.8 9.6 42.1 50.6 47.2 9.6 27 0.404 4.34 0.18 a Ref[37]; bRef[54]-expt.; cRef[66]; dRef[13]-expt.; eRef[64]; fRef[21]; gRef[67]; hRef[62]-expt. ; iRef[48]expt; jRef[63]-expt.

43

31.46 0.737 1.45

Table 6

BeO

B4

B1

MgO

B1

B2

CaO

B1

B2

SrO

B1

B2

BaO

B1

B2

P 0 40 80 112.13

B/G 0.544867 0.845301 1.149486 1.279218

ν 0.20787 0.25936 0.29395 0.312713

Hv 35.53134 41.63649 42.95220 41.80763

A 0.560879 0.459435 0.392585 0.347748

112.13 140 200 0 100 200 300 400 500 514.87 514.87 600 700 0 20 40 60 61.27 61.27 80 100 0 20 40 42 42 60 100 0 10 14.6 14.6 20 50

1.282064 1.266267 1.252969 1.293809 1.464663 1.681656 1.859548 2.020092 2.161331 2.182963 1.848689 1.816474 1.806352 1.384356 1.65528 1.956684 2.274881 2.005565 2.507258 1.983345 2.219533 1.558383 2.049852 2.609452 2.646407 1.881822 1.811533 1.793513 1.822124 2.198239 2.38525 1.668675 1.680671 1.706239

0.190479 0.187422 0.184802 0.192713 0.221913 0.25186 0.271989 0.287544 0.299572 0.301295 0.270855 0.267421 0.266321 0.208911 0.248568 0.281661 0.308298 0.286224 0.32398 0.284174 0.304142 0.23569 0.290197 0.330093 0.3322 0.274282 0.266885 0.26491 0.268031 0.302494 0.316081 0.250251 0.251738 0.254851

120.7900 136.1663 165.8867 26.18898 49.15377 60.43284 68.09281 73.38641 77.66946 78.11991 113.5107 133.8431 154.4334 16.4478 17.99132 15.42369 15.76768 19.82598 13.74439 22.46515 21.96271 9.904954 10.52163 9.376151 9.284147 17.30694 24.12963 29.11677 6.475691 6.570826 6.393022 24.22644 26.03081 30.364

0.761595 0.725217 0.666804 0.766515 0.388462 0.267059 0.203751 0.162471 0.133222 0.129364 0.322544 0.35378 0.379489 0.602076 0.366812 0.239564 0.186486 0.22942 0.177955 0.300771 0.280088 0.502326 0.241714 0.129155 0.123746 0.280546 0.351592 0.393119 0.438356 0.26393 0.216182 0.351206 0.358297 0.373134

44

Table 7

a

Compound BeO MgO

ρ 2.8 3.0 3.63a

νl 11361 9826 10654a

νt 7485 6115 6741a

νm 7254 6740 7416a

CaO

3.2

8394

5085

5619

SrO

4.9 4.82a

5743 6647a

3385 4043a

3751 4466a

BaO

5.7

4765

2672

2973

Ref-[67]; b Ref- [76]; c Ref- [77] ; d Ref- [78]

45

θD 1234 902 1205a 940b 691 605c 430 797a 446d 316 370d