SEMI-EMPIRICAL PREDICTIONS FOR HARDNESS OF RARE EARTH PYROCHLORES; HIGH-PERMITTIVITY DIELECTRICS AND THERMAL BARRIER COATING MATERIALS

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INTRODUCTION
Pyrochlore crystals have been a rapidly growing area of research for scientific community due to their unique technological applications such as exotic magnetism, low thermal conductivity (thermal barrier coatings on diesel engines), anti-erosion of Ag, nuclear waste management, catalysis, sensors, oxide fuel cell, lasers, and in various electronic devices due to unique crystal structure; and the broad range of the study of physical, electronic, mechanical, optical and thermal properties [1][2][3][4].Oxides with Pyrochlore structure having rare earth (A) elements with the common formula    (A 3+ = La 3+ , Ce 3+ , Pr 3+ , Nd 3+ , Sm 3+ , Eu 3+ , Gd 3+ , Tb 3+ , Dy 3+ , Ho 3+ , Er 3+ , Tm 3+ , Yb 3+ , Lu 3+ and Y 3+ and B 4+ = Sn 4+ , Ti 4+ , Zr 4+ , Hf 4+ , Ru 4+ , Ir 4+ etc), possess a different kind of the real-world uses since they have great melting points, large thermal expansion coefficient, high structural stability and low thermal conductivity [5][6][7][8][9].Their complicated chemical structures make them susceptible to being quickly accommodated faults [10].Pyrochlores are therefore a very promising candidate for usage as high-permittivity dielectrics [7], thermal barrier coatings (TBCs), solid-state electrolytes [12], nuclear wastage hosts [13] and catalysis [14].Due to the numerous features of pyrochlores, research is being done to maximize the effectiveness of each attribute while maintaining the chemical and physical durability as well as the thermal consistency of pyrochlores.Liu et al. [13] carefully investigated La 2 R 2 O 7 (R = Ge, Ti, Sn, Zr, Hf) pyrochlores and noticed less heat conduction.It is too revealed why the La-O weak bonds affect the thermal and mechanical characteristics of these La 2 R 2 O 7 pyrochlores.Using density functional theory, the electronic structure, thermal and mechanical properties of Th-doped Gd 2 Zr 2 O 7 compounds studied by Zhao et al [15].Feng et al. [4] predicted the elastic properties of rare earth stannate pyrochlores through density functional theory and ultrasonic resonance method.

A B O  
, here A & B are metallic cations these can be either divalent and pentavalent or trivalent and tetravalent, respectively [16].Rare earth pyrochlores    display cubic symmetry, space group Fd-3m, No. (227) with eight formula units per unit cell, wherein one-eighth of the anions are vacant to ensure charge neutrality and the cations form a face-centered cubic (FCC) array [17].A-type cations occupy the 16d (0.5, 0.5, 0.5) sites and the tetravalent atom B-type of cations occupy the 16c (0, 0, 0) sites.Oxygen atom is situated at two locations: O1 is located on the 8b (0.375, 0.375, 0.375) site and O2 on the 48f (x, 0.125, 0.125) site [18] respectively.Lattice of atoms is completely explained in terms of structure parameter x defining the position of O (48f) atoms and the lattice size a.The x parameter determines the equilibrium position of O (48f) when, 0.3125 ≤  ≥ 0.375, and affects the local coordination around the A 3+ and B 4+ cations [17].In the ideal pyrochlore structure with x = 0.3125 A-atoms and the B-atoms are coordinated at distorted cubes and regular octahedral, respectively.As the value of x approaches 0.375, the polyhedral at B site transform into tetragonal anti-prisms, while polyhedral of the rare earth elements at A-site become ordinary cubes.This arrangement of atoms is stable when, 1.46 ≤   ≥ 1.78 ⁄ ( and  denote the radii of A 3+ and B 4+ cations, respectively) [19], while   ⁄ < 1.46 results in a stable fluorite structure.In configuration, the bigger A 3+ cation is surrounded by two O (8b) and six O (48f) atoms and the smaller B 4+ cation is surrounded by six O (48f) atoms in a distorted octahedron.Here, the 48f oxygen atom is coordinated to two A 3+ and two B 4+ sites, while the 8b oxygen atom is contained within an A 3+ cationformed tetrahedron.
In particular, hardness is one of the most important physical parameters that characterizes the mechanical properties of materials.Vicker's micro-hardness has been shown correlated nicely with the strength and bulk modulus in numerous materials [20,21].On the basis of chemical bonds and effect of valence d-electrons, the hardness of covalent compounds studied by Guo et al. [22].Jhi and other workers [23] have developed the correlation between valence electrons concentrations and hardness for semiconducting materials.Recently, Yang et al. [24] have studied the physical parameters of La2B2O7 (B =Zr, Sn, Hf, Ge) pyrochlores using First-principles techniques.Feng et al. [25] have reported the electronic structure, mechanical properties and thermal conductivity of Ln 2 Zr 2 O 7 (Ln =La, Pr, Nd, Sm, Eu, Gd) pyrochlores employing the local-density approximation (LDA) of spin polarized scheme +U calculations and next to it verified by experimental results and confirm that studied compounds are promising thermal barrier coating (TBC) materials.However, studies of Ln 2 B 2 O 7 (B = Hf, Ge, Ru, Ir) with pyrochlore structures are still insufficient.
Recently, theoretical studies rely on empirical relationships becomes a crucial component of crystallography study.The empirical relationships have always been evolved as a uncomplicated, ready to use, and to produce improved outcomes for physical parameters.Empirical relationships are most often accepted as the preferred technique for computational solid-state investigations.In earlier researches, author and co-workers have been reported the structural, mechanical and electronic properties of zinc-blende and rock-salt structured binary and ternary tetrahedral chalcopyrite semiconductors using various methods successfully [26][27][28][29][30][31][32][33][34].Even though empirical relations frequently don't produce incredibly accurate answers for each unique substance, they can nevertheless be extremely helpful.To the best of my knowledge, mechanical property, particularly Vicker's hardness, has not been studied theoretically using POT of solids in rare earth pyrochlore oxides, till date.Therefore, to bridge this gap, it would be interesting to offer a different rationale for the vicker's hardness of cubic-structured pyrochlore oxides.Herein, a model obtained from POT of solids is being put forward to investigate the mechanical properties of rare earth pyrochlore oxides.It is now widely accepted that a metal's plasmon energy changes [28][29][30][31][32][33][34][35], when it combines chemically to create a compound, conduction electrons in a metal are collectively excited to form plasma of energy ℏω .This energy depends upon the density of conduction band.Hence the plasmon energy is affected by the density of conduction electrons as well as effective number of valence electrons, both of which change in the formation of compound.
In present work, the relationship between vicker's hardness and plasmon energy for the cubic structured pyrochlore oxides    (A = La→Lu, Y & B = Sn, Ti, Zr, Hf, Ge, Ru, Ir) has been discussed employing the plasma oscillation theory (POT) of valence electrons in solids.In investigation of hardness, effect of valence d-electrons of the metal has been taken into account and the obtained values of vicker's hardness is in compliance with available experimental data as compared to previous theoretical investigations on the same.This model divulges vicker's hardness unprecedentedly and accurately for many pyrochlores.

THEORETICAL FORMALISM OF HARDNESS
The hardness as a characteristic of pyrochlore crystals has been proved a crucial physical parameter, which determines the bond strengths in these crystals.The term hardness refers to stiffness or temper, resistance to scratching, abrasion, or cutting.The elastic and plastic deformation characteristics of solid materials are usually related to hardness.The metal handbook defines the hardness; the resistance of a material to plastic deformation is measured by its hardness.A true scientific definition of hardness has yet to come.This is due to the fact that hardness is influenced by a variety of factors.As a result, a number of different hardness measurement methods have been evolved.Hardness is a mechanical property that delineates not only the state of material under consideration, but also some deeper basic features of the material, such as the chemical bonding character.Many theoretical approaches to determining the hardness of solids have been presented.Among them Vicker's and Knoop hardness scales are most often used to measure this mechanical parameter.However, it is not remarkable that only the resultant force per unit area could play a key role in determining the hardness of the materials.Since, the hardness of the metal and compound is determined by the energy of the quanta in plasma oscillations of the valence electrons and is given as [35][36], From the classical consideration, we get the frequency of plasmon oscillations as ω = 4πN e m ⁄ .
Hence the amount of energy given to plasmon becomes ℏω = ℏ 4πN e m ⁄ . ( Here ħω p is the plasmon energy, N e -effective no. of free electrons per unit volume participating in plasma oscillations; e-electronic charge and m-mass of an electron.For a compound, the plasmon energy in Eq. ( 1) may also be defined as [36] ℏω = 28.8Zσ W ⁄ . ( Eq. ( 2) is valid for free electrons and up to first approximation.It can be employed for insulators and semiconductors.During last four decades, Kumar [37] has explored the relationships for electronic properties of chalcopyrite solids in term of plasmon energy.Gupta et al [34] and Verma et al [38] have established a relationships between the mechanical properties and melting point, inter-atomic distance and iconicity of the compounds.Verma et al. [38] have developed an empirical relation between micro-hardness and bulk modulus for tetragonal structured chalcopyrite solids as follows where K is crystal structure dependent constant with values of 0.50 and 0.59 for I − III− VI and II − IV− V chalcopyrites, respectively.Previously, many researchers have studied the static, dynamical, optical, mechanical and electronic properties of binary and ternary chalcopyrites semiconductors, and rare earth chalcogenides using empirical models based on plasmon oscillation theory of solids [28][29][30][31][32][33][34].As per above study, the mechanical properties of pyrochlores have a fundamental relationship between hardness H and plasmon energy ℏ .
Physical parameters are obviously important for device development, but no one has demonstrated meaningful results on Vicker's hardness using various approaches.As, this modulus is much valance electrons dependent.Hence, we tried a new empirical approach to come across accurate results close to experimental values using plasmon energy of compound.
The experimental vicker's hardness H v [GPa] of these pyrochlores shows linearity when plotted a graph with respect to plasmon energy (ħω p ) 3/4 and presented in Figure 2.2.It is noteworthy that the method of determining bulk modulus by this mechanism creates a difference from experimental techniques, and this could develop errors.
It is macro and micro changes in the compounds characteristic which leads to mechanical changes; here, it is plasmon energy of valance electrons that defines this macro and micro change in the compound.As a result, a linear regression line is plotted for the vicker's hardness, which results in an expression of vicker's hardness [GPa] as; Here constants P and Q, and regression coefficient R are presented in Table 1.In this table probability parameter S shows the significance of regression as null i.e. there is no correlation.A detailed study of elastic properties such as hardness of complex structured pyrochlores, covalent crystals and binary solids has been given elsewhere [34,35] and will not be discussed here.The properties of A 2 B 2 O 7 pyrochlore compounds have been thoroughly studied and some of these materials have been noticed for practical applications [4][5][6][7][8][9][10].The information about elastic properties of pyrochlores such as hardness is rather incomplete and experimental data found only for few pyrochlores.This study suggests the elastic properties in more accurate manner.

RESULT AND DISCUSSION
In the present study, it is shown that empirical relationship exists for pyrochlores and this can be successfully employed to investigate the hardness from their plasmon energy as a key parameter.
The investigated values of hardness of these materials have been given in Table 2 and weighed against the available experimental data and other such theoretical investigations reported so far.Vicker's hardness of these crystals exhibits a linear relationship when plotted against their Plasmon energy ћωp (in eV), which are calculated using the relation ( 2) and falls on a straight line.From Fig. 2.1, it is obvious that the hardness follows rising trends in these compounds with their increasing plasma energy.
It can be seen that the values of hardness computed from this proposed empirical relation is in better conformity with experimental data in comparison to the values reported by earlier workers.As an illustration the results obtained for H v (in GPa) vary from experimental values by the amount as follows:  Cal.
[ Hence, it is clear from these results that the proposed empirical model is quite reasonable and can give us full insight for investigation and predicting the elastic properties of these materials.

CONCLUSION
According to above results and discussions attained by employing the extended empirical relation, plasma energy of the valence electrons is key parameter for investigating the mechanical/elastic properties of pyrochlore structured solids.In the current investigation, estimates varied somewhat from experimental data by 0.20% to 13.67%.The obtained results express a continuous tendency and are commensurate with the already revealed results reported so far, this confirms the viability of the suggested method.The predicted values of studied compounds are cited in the Table 2. Hence it is clear that such theory can be easily extended to pyrochlore crystals and excellent agreement was found between estimated and the experimental data as compared to other such reported theoretical investigations, previously.In this manner, this model is pioneer in determining the hardness of most of the compounds with the plasmon energy, which assists to determine other crucial properties for technological applications.Furthermore, it is found that in these crystals (investigated), hardness of A2B2O7 crystals shows a linear relationship, when plotted against plasma energy of their valance electrons.Hence, it makes sense to state that such empirical model can be easily unfolded to perovskite structured solids to determine their elastic properties; the work on the project is ongoing, and it will be reported in upcoming articles.

Figure 2 . 2 .
Figure 2.2.Experimental Hv [GPa] versus Plasmon energy for pyrochlore compounds.This straight line exhibits a linear relationship obtained by regression analysis of data.Experimental data are taken from[4,25].

Table 1 .
Ternary tetrahedral semiconductors data gives the following linear regression result.
1.43% for La 2 Sn 2 O 7 , 0.20% for Nd 2 Sn 2 O 7 , 0.86% for Sm 2 Sn 2 O 7 , 1.44% for Gd 2 Sn 2 O 7 , 3.61% for Er 2 Sn 2 O 7 , 4.50% for La 2 Zr 2 O 7 , 1.67% for Pr 2 Zr 2 O 7 , 7.33% for Nd 2 Zr 2 O 7 , 5.77% for Sm 2 Zr 2 O 7 and 13.67% for Eu 2 Zr 2 O 7 .A few higher deviations of hardness values in these compounds are owing to larger bond lengths.With increasing bond length, plasmon energy decreases, which widen the variance of these results.Another reason for greater deviations of hardness values is unit cells mismatch between substrate and deposited films.Explicitly hardness of ternary compounds is more sensitive with plasmon energy as observed in calculated values.