Temperature Dependence of Elastic Constants of Alkaline Earth Oxides

The elastic properties of alkaline earth oxides (AEOs) under high temperature are discussed within the framework of many body Lundqvist potential incorporating the contribution of thermal phonon pressure. The short-range repulsive interaction is taken up to the second nearest neighbors. The derived expressions are used to compute the values of second-order elastic constants (SOECs) and bulk modulus of alkaline earth oxides at different temperatures (300◦K–2000◦K). The results are found to be satisfactory and are in agreement with available experimental and the theoretical results.


Introduction
The general theory for the thermoelastic behaviour of ionic solid was first of all given by Wallace [1]. Srivastava [2,3] and Varshney [4] have applied it to the solids crystallizing in NaCl and CsCl structure. Kumar and coworker [5][6][7] have analyzed the variation of elastic constant with temperature for MgO and CaO using different approaches modifying the Suzuki relation of equation of state [8]. However, their results [5][6][7] are better than Suzuki's results but are much different from the experimental results. Any equation of state contains the bulk modulus and its derivatives as dependent parameter. The results obtained by Kumar and coworkers [5][6][7] show the worse agreement between the computed values of bulk modulus at different temperatures with their experimental results in case of MgO and CaO. This clearly indicates that modification improves some values of the elastic constants at different temperatures but shows poor agreement of bulk modulus at different temperature. It may be due to the reason that they have not considered the higher order terms that is, anharmonic terms, in the expansion of the logarithmic series of volume change. But the elastic properties are directly related to the lattice potential which consists of the various interaction systems.
In the present study, we have used many body Lundqvist potentials [13][14][15] incorporating thermal phonon pressure to study the second-order elastic constant of all alkaline earth oxides solid, that is, MgO, CaO, SrO, and BaO solid at different temperatures (up to 2000 • K). This potential has been recently used [11,16,17] to explain the elastic properties of alkaline earth oxides (AEOs) and rare gas solids (RGSs) at different pressures. The potential has been suitably modify to take into consideration the changes in the lattice parameter due to the increase in temperature. Now we have derived the relation of second-order elastic constants by modifying the lattice parameter. The results predicted by the present theory are discussed and compared with available experimental and others theoretical results.

Theory
The total lattice energy of diatomic solid [14] is regarded as where ∈ (k) is the valency of k type ion, ∈ the magnitude of ∈ (k), and r(l , kk ) the interionic separation between ion 0 k at origin to ion l k . The first term represents Coulomb energy, the second the overlap repulsive energy coupling the neighbours and the third term the three body  potential. The function f is related to the overlap integrals of free-ion one electron wave functions and is assumed significant only for nearest neighbours and is related to overlap integrals reported by Hafmeister and Flygare [18]. However, in the present study, we have chosen this function f and its derivatives as disposable parameters at r = a, the nearest-neighbour distance. On simplification, the above lattice energy per unit cell φ(r) for alkaline earth oxide solids can be expressed as where α m (= 1.7476) is the Madelung constant for NaCl structure. N and N are the numbers of the nearest and next 4 SRX Materials Science to nearest neighbours. φ 1 (r) and φ 2 (r) are the short range potentials between nearest neighbours and next nearest neighbours related to Lundqvist potential as follows: where b and ρ are the Born repulsive parameters. r + and r − are the characteristic length for the positive and negative ions. β i j are Pauling coefficients [19] defined as follows: where ∈ i and ∈ j are the valencies of two ions. p i and p j are the number of outermost electrons in ions i and j.
Considering that the vibrational energy of the solid is a purely temperature dependent function as dΦ dr r=a = −P + TβK (5) and using the above lattice potential functions (1) and (3), the expressions for second-order elastic constant at any temperature T for alkaline earth oxides are derived as where A 1 , B 1 , A 2 , and B 2 are short range force constants defined as The equilibrium condition (dφ(r))/dr = 0 is written as where β is the coefficient of volume thermal expansion, a is the nearest neighbour distance, K T is the bulk modulus, and

Results and Discussion
In order to compute the values of elastic constants and bulk modulus with the help of the above described theory, one requires f (r), adf (r)/dr, A 1 , B 1 , A 2 , and B 2 . Out of these the first four parameters are evaluated with the help of (6, 11) at T = 300 K using the input data from  [5,7] and very close to the experimental results [9]. Similarly the computed values of C 12 of MgO, CaO, SrO, and BaO are decreasing slowly with the increase of temperature and are much better than the earlier computed values [5,7] in case of MgO and CaO. The computed values of C 44 are close to the experimental values [9] up to 1200 K and are better than the earlier reported values [5][6][7] in case of CaO. Our calculated values of C 44 are not so good in case of MgO. This may be due to the reason that many body interactions do not show  [20]. Thus these AEOs show systematic variation in the values of SOE constant either with increase of pressure [9,20] or with increase of temperature as compared to the values of these constants at ambient condition [10,11]. On the other hand the values of bulk modulus are in excellent agreement with the experimental values [9] at different temperatures and are better than the earlier study which are based on the equation of state. The values of bulk modulus are decreasing at different temperatures as we move from lighter to heavier alkaline earth oxides. On the other hand, the value of bulk modulus is decreasing with the increase of temperature in all cases. The reason for this may be that the constant C 11 is a longitudinal elastic constant (like Young's modulus) relating longitudinal stress and longitudinal strain. A longitudinal force causes the lattice to stretch. This causes a change in volume of the crystal lattice. It is well known that the internal energy is the function of volume (V ) and temperature (T). The temperature dependence is both explicit via the Boltzmann factor and implicit via the static lattice energy and normal mode frequency. Both static lattice energy and normal mode frequency are the function of the volume (V ), which heavily depend on temperature. Thus the constant C 11 depends on the temperature. On the other hand, the constants C 44 and C 12 are shearing constant relating shearing stress to shearing strain. Shearing is caused by tangential force, and the lattice constant is unaffected by the shearing, and consequently the C 44 and C 12 depend less on temperature. This agreement demonstrates the validity of the contribution of many body interactions considered in developing the present theory.
The various results predicted in the present paper regarding the variation of the elastic constants with temperature for SrO and BaO up to 1800 K will be useful in analyzing the experimental data perhaps which are not yet available. Therefore the present study may be useful to study the elastic and thermal properties of alkaline earth chalcogenides and in geophysics.