THÔNG TIN TÀI LIỆU
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION Le Thu Lam DIFFUSIONAL AND ELECTRICAL PROPERTIES OF OXIDE MATERIALS WITH FLUORITE STRUCTURE Speciality: Theoretical and Mathematical Physics Classification: 9.44.01.03 SUMMARY OF PhD THESIS Ha Nội, 2020 The work was completed at Hanoi National University of Education Science supervisor: Assoc.Prof.Dr NGUYEN THANH HAI Assoc.Prof.Dr BUI DUC TINH The first reviewer: Assoc.Prof.Dr Nguyen Hong Quang Institute of Physics The second reviewer: Assoc.Prof.Dr Nguyen Nhu Dat Duy Tan University The third reviewer: Assoc.Prof.Dr Nguyen Thi Hoa University of Communications and Transport The thesis will be defended at Hanoi National University of Education on The thesis is available at: Library Library INTRODUCTION Rationale Nowadays, solid oxide fuel cell (SOFC) can be seen as a renewable energy source which has the most potential To improve performance and commercialize of SOFC, diffusion and electrical properties of fluorite structure materials need to be investigated Most of the previous theoretical studies concerning the diffusion and electrical properties of the bulk material with fluorite structure are based on the simple theory of thermal lattice vibrations Diffusion coefficient and ionic conductivity of undoped material with the thermal oxygen vacancy were calculated by statistical moment method (SMM) In the doped materials, the most oxygen vacancy are generated by dopant and they requires a available calculation models Moreover, there is a lack of theoretical methods related the oxide thin film with fluorite structure and the experimental methods give the conflicting results about the effect of thickness on the conductivity ionic of the thin films For those reasons mentioned above, we decided to choose the topic:“Diffusional and electrical properties of oxide materials with fluorite structure” Purpose, object and scope of research The purpose of thesis is to study the influence of temperature, pressure and dopant on the diffusion coefficient and ionic conductivity of the bulk material as CeO2 , c-ZrO2 , YDC and YSZ For YDC and YSZ thin films, we find the thickness dependence of the lattice constant, diffusion coefficient and ionic conductivity The roles of grain boundary and substrate are canceled in our study Research method In the thesis, the diffusion and electrical properties of the fluorite structure materials are investigated using SMM including the anharmonicity effects of thermal lattice Scientific and practical senses The results related the vacancy migration paths, the vacancydopant associations, the influences of temperature, pressure, dopant and thickness on the diffusion coefficient and ionic conductivity provide the important information about the diffusion and electrical properties of the fluorite structure materials The obtained results of dopant and thickness dependences of the ionic conductivity could be applied to make the electrolytes with high ionic conductivity used in SOFC New contribution of thesis The thesis has established a new theoretical model using SMM to investigate the diffusional and electrical properties of fluorite structure materials from simple to complex In comparison with the last model for CeO2 , the advantage of this model is finding the preference migration path of oxygen vacancy Some of calculated results related bulk materials are in better agreement with the experimental results compared to the other theoretical results The calculated results of thin film supplement to the experimental results The obtained results related the oxide thin film with fluorite structure supplement the experimental investigation Thesis outline In addition to the introduction, the conclusions, the references, the thesis content is divided into main chapters CHAPTER OVERVIEW OF DIFFUSIONAL AND ELECTRICAL PROPERTIES OF FLUORITE STRUCTURE MATERIALS 1.1 Fluorite structure materials Ceria (CeO2 ) and cubic phase of zirconia (c-ZrO2 ) with the open fluorite structure allow the oxygen vacancy to migrate through the lattice with relative ease However, the number of oxygen vacancy is small due to the high vacancy formation energy Hence, they are not good ionic materials Yttria (Y2 O3 ) doped with CeO2 (YDC) and c-ZrO2 (YSZ) stabilize the cubic phase of c-ZrO2 down to room temperatures and increase the vacancy concentration The migration of oxygen vacancy occurs via the site exchange with O2− ions at the opposite neighbour sites However, the vacancy migration is blocked by the vacancy-vacancy interactions and vacancy-dopant associations Thin films have a high surface-to-volume ration and large concentration of grain boundary The vacancy formation and migration at the surface and grain boundary effect strongly on the vacancy formation and migration in the whole thin film Due to the high ionic conductivity, these materials are widely applied as the useful electrolytes in solid oxide fuel cell (SOFC) 1.2 The main research methods and results The diffusional and electrical properties of the fluorite structure materials are investigated by the theoretical methods (molecular dynamics simulation, density functional theory, Monte-Carlo simulations, ) and the experimental method (sputtering, chemical vapor deposition, pulse laser deposition, ) For the bulk materials, the previous results showed that the oxygen vacancy in defect clusters is located at the 1NN sites relative to Y3+ ion in YDC and at the 2NN sites relative to Y3+ ion in YSZ The oxygen vacancy migrates dominantly along the direction and the presence of dopant in the cation barriers blocks the vacancy migration Remarkable, the ionic conductivity depends linearly on the dopant concentration For CeO2 thin film, the charged particle is mainly electron But the current in doped thin films (YDC and YSZ) is predominantly carried by the oxygen vacancy The measured results of ionic conductivity depend strongly on the substrate, measurement and fabrication methods The enhancement of ionic conductivity with the decreasing thickness was observed for YDC thin film For YSZ thin film, the experiments showed the conflicting results about the effects of substrate and thickness on the ionic conductivity The statistical moment method The statistical moment method (SMM) investigate the physical properties of crystal including the anharmonicity effects of thermal lattice vibrations Based on statistical operator ρˆ, the authors established the general formula of moments to determine high level moments from low level moments The expressions related mechanical, thermal, electrical properties are derived in closed analytic forms in terms of the power moments of the atomic displacements The calculated results are in good agreement with the other theoretical and experimental results CHAPTER INVESTIGATE OF DIFFUSIONAL AND ELECTRICAL PROPERITES OF CERIA AND ZIRCONIA 2.1 Anharmonicity vibration and Helmholtz free energy 2.1.1 Anharmonicity vibration The expression of ionic interaction potential in RO2 system [96] U= NR ϕR i0 (|ri + ui |) + i NO ϕO i0 (|ri + ui |) (2.1) i The displacements of ions in RO2 system are determined as y0R ≈ 2γR θ2 AR , 3kR y0O ≈ 2γO θ2 βO AO − + 3γO KO 3KO (2.22) 1+ θ2 6γO KO 2βO βO kO 2γO θ − (2.28) (x cothx − 1) − O O 27γO kO γO 3kO 2.1.2 Helmholtz free energy The Helmholtz free energy of ions in RO2 system are given by [96,98] ΨR ≈ U0R + ΨR + 3NR θ2 R 2γ1R R γ (X ) − a + R 2 kR 2θ3 aR R (γ ) XR − (γ1R )2 + 2γ1R γ2R (1 + XR ) , kR (2.39) O θ 2γ1 O O ≈ U0O + ΨO + 3NO γ2 XO − a1 + kO + ΨO + 2θ3 aO O O O O (γ2 ) XO − (γ1 ) + 2γ1 γ2 (1 + XO ) (kO ) + + 3NO − θβ 6KO γO kO θ2 β −1 + KO KO 2γO aO 3KO βO aO βO kO aO βO 1 + + (XO − 1) 6KO kO 9KO 9KO − (2.42) Eqs (2.39) and (2.42) enable us to calculate the Helmholtz free energy of RO2 system Ψ = CR ΨR + CO ΨO − T SC (2.46) 2.1.3 Equation of states At temperature T = K, the equation of states can be written as [97] P v = −a CR ωR ∂kR ∂uO ωO ∂kO ∂uR 0 + + CO + ∂a 4kR ∂a ∂a 4kO ∂a (2.49) 2.2 Electrical and diffusion theories 2.2.1 Diffusion coefficient and ionic conductivity The diffusion coefficient and ionic conductivity are given by [2,3,92-94] D = r12 n1 f σ = ωO exp 2π Svf kB exp − (Ze)2 ωO kB T a3 r1 n1 f 2π exp Svf kB Ea kB T , f exp − kgBvT T (2.59) exp − Ea kB T (2.64) with Ea is the vacancy activation energy Ea = Ef + Em (2.60) in which Ef and Em are the vacancy formation and migration energies, respectively 2.2.2 Vacancy activation energy 2.2.2.1 Vacancy formation energy The vacancy formation energy can be written as ∆Ψ = − Ef va va + CO (NO − 1)ψO − CR NR ψR lt lt CR NR ψR + CO (NO − 1)ψO , ≈ ∆Ψ + (2.68) ∗min + ψ ∗max CO ψ O O lt − CO ψO + T Svf + P ∆V, (2.72) with the free energies are determined via the average interaction potentials of an ion in RO2 system having an oxygen vacancy NO − uva = bO−O ϕ∗O−O + bO−R ϕ∗O−R , (2.80) O i i0 i i0 NO − i uva = R NO − NR i bO−R ϕ∗O−R + i i0 i bR−R ϕ∗R−R i i0 (2.81) i 2.2.2.2 The vacancy migration energy The vacancy migration energy Em is given by Em = Ψva −Ψyn va +P ∆V, (2.82) yn with Ψyn va is determined by the average interaction potentials uR 2− hopping to and uyn O of an ion in RO2 system that has an ion O the saddle point A ϕB R−O − ϕR−O , (2.86) NR B A UOO−O + NR uva R−O + ϕR−O − ϕR−O − ∆uO−O + ∆uO−O NO − (2.98) va uyn = uva R−R + uR−O + R uyn = O 11 + 2θ3 aY1 Y Y (γ ) a1 XY − (γ1Y )2 + 2γ1Y γ2Y aY1 (1 + XY ) kY4 (3.22) In Eq (3.22), the free energies of R4+ , Y3+ and O2− ions are determined by the average interaction potentials of an ion [118] + 1− biO−R ϕ∗O−R i0 uO = i bR−R ϕ∗R−R + 1− i i0 uR = i uY = NR N −1 i Nva 2N − Nva 2N − NY − −R biY −R ϕ∗Y + i0 N −1 , (3.36) ϕ∗O−O bO−O i0 i i bR−O ϕ∗R−O , (3.37) i i0 i −O bYi −O ϕ∗Y (3.38) i0 i 3.2 Diffusion coefficient and ionic conductivity The diffusion coefficient and ionic conductivity are given by D = r12 n1 f ωO exp 2π Svass kB (Ze)2 ωO kB T a3 r1 n1 f 2π exp σ = exp − Svass kB T Ea kB T , (3.41) ass exp − kgBv T exp − Ea kB T (3.42) with [38,39,51] Ea = Eass + Em (3.40) in which Eass is the vacancy-dopant association energy 3.2.1 Vacancy-dopant association energy The expression of vacancy-dopant association energy Eass is given by Eass = − ΨRNR YNY ONO + ΨRNR −2 YNY +2 ONO −1 − ΨRNR −1 YNY +1 ONO + ΨRNR −1 YNY +1 ONO −1 + T Svass + P ∆V, (3.46) 12 with the free energies is determined by Eq (3.21) with Eqs (3.36) – (3.38), ∆V is the volume change of system between the associated state and isolated state of defect pair 3.2.2 Vacancy migration energy The vacancy migration energy Em can be written as [49-51,54,55] Em = Ψyn − Ψ0 + P ∆V, (3.71) with Ψyn is determined by the average interaction potentials of an ion R4+ , Y3+ and O2− yn yn B uB R = uR + ∆uR , uY = uY + ∆uY , (3.72) O−R , + ∆uO−O + ∆uO−Y uB O = uO + ∆uO O O (3.89) yn O−R depend strongly on the and ∆uO−Y in which ∆uyn O R , ∆uY , ∆uO cation configurations around site A and saddle point B 3.3 Results and discussion 3.3.2 Vacancy activation energy a Vacancy - dopant association energy Table 3.2 Vacancy - dopant association energies at the 1NN and 2NN sites in YDC and YSZ Eass (eV) YDC YSZ Method SMM DFT [38] SMM MD [51] MD [121] DFT [49] 1NN -0.2971 -0.086 -0.2080 -0.28 0.18 -0.2988 2NN 0.48352 0.1055 -0.2798 -0.45 -0.26 -0.3531 The results of vacancy-dopant association energy at 1NN and 2NN sites enable us to evaluate the vacancy distribution around 13 dopant (Table 3.2) For YDC, the dopant traps the oxygen vacancy at the 1NN site and repels it from the 2NN site but for YSZ, the oxygen vacancy locates mainly at the 2NN site relative to the dopant Fig 3.4 The dopant concentration dependence of the vacancy-dopant association energy for YDC (a) and YSZ (b) at various temperatures Figure 3.4 shows that the vacancy-dopant association energy decrease with the increasing of dopant concentration The reducing of association energy lead to the consequence as the number of vacancy increases quickly with an increase in dopant concentration b Vacancy migration energy The calculated results of vacancy migration energy through three cation barriers R4+ - R4+ , R4+ - Y3+ and Y3+ - Y3+ show that the presence of Y3+ ions in the cation barrier blocks the vacancy migration due to the formation of R4+ - Y3+ and Y3+ - Y3+ cation barriers with high barrier (Table 3.4) The vacancy diffusion process occurs mainly pass the R4+ - R4+ barrier and contributes dominantly on the vacancy diffusion process in the crystal lattice Figure 3.5 shows that the migration energy increases with the increasing dopant concentration This dependence arises from an increase of the dopant concentration and leads to a greater fraction 14 Table 3.4 The vacancy migration energy pass three cation barriers Em (eV) YDC YSZ Method R4+ - R4+ R4+ - Y3+ Y3+ - Y3+ SMM 0,2334 0,7295 1,0521 DFT [38] 0,48 0,533 0,8 DFT+MC [122] 0,52 0,57 0,82 SMM 0,3625 1,0528 1,5091 DFT+MC [40] 0,58 1,29 1,86 DFT [49] 0,2 1,19 1,23 of cation barriers R4+ - Y3+ and Y3+ - Y3+ with high migration energies Figure 3.5 The dependence of the vacancy migration energies Em on the dopant concentration in YDC (a) YSZ (b) at various temperatures c Vacancy activation energy At low dopant concentration, the number of high energy barriers R4+ - R4+ , R4+ - Y3+ is small and the activation energy is nearly equal that for the vacancy migration across the R4+ - R4+ barrier As dopant concentration increases, the oxygen-vacancy exchange through the R4+ - Y3+ , Y3+ - Y3+ edges rather than R4+ - R4+ barrier can be expected to increase Therefore, the calculated activation energy increases with the increasing dopant concentration (Fig 3.6) 15 Figure 3.6 The dopant concentration dependence of the vacancy activation energy Ea for YDC (a) at 773 K and YSZ (b) at 1000 K 3.3.3 Diffusion coefficient and ionic conductivity a Vacancy diffusion coefficient Fig 3.7 shows that the diffusion coefficient increases with the increasing temperature and it decreases with an increase in dopant concentration The dopant concentration dependence generates from the influence of cation barrier on the vacancy-oxygen ion exchange The increase of dopant concentration promotes the number of the cation barriers with high energy and that prevents the vacancy migration process Figure 3.7 The diffusion coefficient D is as a function of temperature inverse (1/T ) at the various dopant concentrations in YDC (a) YSZ (b) The diffusion coefficient decreases strongly with an increase in 16 Figure 3.8 The pressure dependence of diffusion coefficient D in YDC (a) YSZ (b) at various dopant concentrations pressure (Fig 3.8) The shrinked lattice crystal blocks the vacancy migration and increases the vacancy-dopant association energy b Ionic conductivity The ionic conductivity increases with the dopant concentration but after reaching to the maximum value, it reduces fast with further increase in the dopant concentration (Fig 3.9) The vacancydopant associations and the cation barriers with high energy are the reasons for the nonlinear dependence of the ionic conductivity on the dopant concentration Figure 3.9 The dependence of ionic conductivity on the dopant concentration in YDC (a) at 1073 K and YSZ (b) at 873 K and 973 K The mobility of oxygen vacancy is enhanced due to temperature 17 and the diffusion process occurs more easier Consequently, the ionic conductivity increases with the temperature (Fig 3.10) Figure 3.10 The dependence of ionic conductivity vs 1/T at the various dopant concentrations for YDC (a) and YSZ (b) Figure 3.11 presents the dependence of ionic conductivities of YDC and YSZ on pressure P at various dopant concentrations x = 0,1; x = 0,2; x = 0,3 The shrinked crystal lattice at high pressure blocks the diffusion processes and decreases the ionic conductivities We predict that the pressure dependence of ionic conductivity in YSZ is larger than that in YDC Figure 3.11 The dependence of ionic conductivity on pressure P in YDC (a) and YSZ (b) at dopant concentrations as x = 0,1; x = 0,2; x = 0,3 18 CHAPTER INVESTIGATE OF DIFFUSIONAL AND ELECTRICAL PROPERTIES OF YTTRIA-DOPED CERIA AND YTTRIA-STABILIZED ZIRCONIA THIN FILMS One divides thin film into n crystal layers that consist of two external layers and (n-2) internal layers We assume that the interaction potentials between ions in the internal layers is similar to those in the bulk material The next calculation will be performed for these types of layers 4.1 Anharmonicity vibration and Helmholtz free energy 4.1.1 Anharmonicity vibration a The internal layers The displacement expressions of R4+ , Y3+ and O2− ions have the similar forms to those of bulk materials b The external layers The displacements of R4+ , Y3+ and O2− ions are determined by the expression as R,Y y0ng ng 2γR,Y θ = ng KR,Y 1 − ng AR,Y − ng 2γR,Y θ ng kR,Y ng βR,Y ng 3γR,Y + ng KR,Y ng xng R,Y cothxR,Y 1 + −1 − ng γR,Y ng KR,Y ng βR,Y ng ng 27γR,Y kR,Y θ2 ng ng βR,Y kR,Y , ng γR,Y (4.32) 19 O y0ng ng 2γO θ ng βO ng A − ng + ng ng O 3γ K KO O O ng 2γO θ − xng cothxng O −1 ng 3 k O O = 1+ − ng γO ng KO 2 θ ng ng βO kO ng (4.33) γO 2βO 27γO kO 4.1.2 Helmholtz free energy a The internal layers The expression of the Helmholtz free energy of the internal layers has the similar forms to that of bulk materials b The external layers The Helmholtz free energy of the internal layers is given by ng R ng ∗ng Ψng = Ψng RO2−x/2 − NY u0ng + ΨY − T SC , (4.41) with the Helmholtz free energies of R4+ , Y3+ and O2− are determined by the general formula as 2γ1ng θ2 a1ng + γ2ng X − ng (k ) ΨR,Y,O = U0ng + Ψ0ng + 3N ng ng + 2θ3 a1ng 2 (γ2ng ) Xng − (γ1ng ) + 2γ1ng γ2ng (1 + Xng ) ng (k ) + 3Nng − θβng 6Kng γng βng aR,O (Kng ) + kng θ2 βng −1 + Kng Kng βng kng a1ng (Kng ) + 2γng aR,O 1ng (Kng )3 βng (Xng − 1) 6Kng kng − (4.47) 4.2 Lattice constant, diffusion coefficient and ionic conductivity a Lattice constant The expression of lattice constant can be written as amm = (n − 2)atr + 2ang , n (4.56) + 20 with the value of atr is equal to that of lattice constant in the bulk material b Diffusion coefficient and ionic conductivity The diffusion coefficient and ionic conductivity are given by Dmm = σmm = (n − 2)Dtr + 2Dng , n (n − 2)σtr + 2σng , n (4.57) (4.58) with the values of Dtr and σtr are equal those of the diffusion coefficient and ionic conductivity in the bulk materials, Dng and σng are determined by the expression (3.41) and (3.42) The method to calculate Dng and σng is similar to the method used for the bulk materials YDC and YSZ in chapter 4.3 Results and discussion 4.3.1 Lattice constant Figure 4.2 The dependence of lattice constant on the number of layers (a) and dopant concentration (b) for YDC thin film A discontinuity in periodic crystalline structure at the external layers changes the interaction potential of ions and the anharmonicity effect in these layers These changes increase distances between ions and the anharmonicity vibrations of crystal lattice 21 Consequently, the calculated lattice constant of the external layers is larger than that of the internal layers The Figs 4.2 and 4.3 show that the lattice constants of YDC and YSZ thin films decrease with an increase in the thickness and the dopant concentration dependence is similar with that in bulk materials Figure 4.3 The dependence of lattice constant on the thickness (a) at T = 650 C and dopant concentration (b) at T = 773 K for YSZ thin film The experimental results [141,149] are presented for comparison 4.3.2 Diffusion coefficient and ionic conductivity The calculated results show that the diffusion coefficient and ionic conductivity of the external layers are enhanced approximately three order of magnitude than those of the internal layers In comparison to the internal layers, the vacancy diffusion in the external layers become much easier and the concentration of mobile vacancy is enhanced At the small thickness, the diffusion coefficient and ionic conductivity of the external layers influence mainly on the whole thin film The role of external layers decreases with the increasing thickness Consequently, the diffusion coefficient and ionic conductivity of YDC thin film decrease with an increase of thickness (Fig 4.4) Our results show that the diffusion coefficient and ionic conductivity of YDC thin films approach the values of bulk 22 when the thickness is about some µm Figure 4.4 The thickness dependence of the diffusion coefficient (a) and ionic conductivity (b) of YDC thin film Figure 4.5 The dependence of ionic conductivity on the thickness (a) and temperature (b) for YSZ thin film The calculated ionic conductivity of YSZ thin film is enhanced approximately three order of magnitude compared that of bulk material and it decreases quickly with the increasing thickness (Fig 4.5) This dependence of the ionic conductivity on the thickness are in good agreement with the most of experimental results [8688,141,150-153] Our study for YDC and YSZ thin films shows that the interaction potential and anharmonicity effect in the external layers are the reasons for the enhanced ionic conductivity of thin 23 films compared to that of the bulk materials However, the calculated results are significant differences with the experimental results [72,151] There are two reasons for this feature: (i) The roles of substrate and grain boundary are ignored in the calculations (ii) The experimental results depend strongly on the substrate type [72-75], the thin film preparation techniques [154] and the measurement methods [155,156] 24 CONCLUSION The thesis has achieved the main results as follow: Based on the expressions of Helmholtz free energy, the thesis have performed the model to calculate the vacancy activation energy in the materials with fluorite oxide structure The diffusion coefficient and ionic conductivity can be derived The thesis has performed the expressions for the anharmonicity effect and Helmholtz free energy of ions in the external layer of YDC and YSZ thin films The diffusion coefficient and ionic conductivity in these layers can be derived The calculated results related the bulk materials show the preferred migration path of oxygen vacancy, the distribution of oxygen vacancy and the influence of dopant on the oxygen vacancy migration The dependences of diffusion coefficient and ionic conductivity on the temperature, dopant concentration and pressure are evaluated in detail The calculated results for thin films YDC and YSZ evaluated the important role of external layers on the diffusional and electrical properties of those thin films The lattice constants, diffusion coefficients and ionic conductivities increase with the decreasing in thickness of thin films The obtained results include the anharmonicity effects of thermal lattice vibrations and the role of the vacancy distribution around dopant on the diffusional and electrical properties of these materials This calculation model could be extended to study those properties of CeO2 and c-ZrO2 with others dopants and mineral materials with perovskite structure 25 LIST OF THE PUBLISHED WORKS RELATED TO THE THESIS V.V Hung, L.T Lam (2018), Investigate the vacancy migration energy in ZrO2 by statistical moment method, HNUE Journal of Science 63 (3), 56 V.V Hung, L.T Lam (2018), Investigate the vacancy diffusion in ZrO2 by statistical moment method, HNUE Journal of Science 63 (3) 34 L.T Lam, V.V Hung (2019), Investigation of oxygen vacancy migration energy in yttrium doped cerium, IOP Conf Series 1274 (012004), L.T Lam, V.V Hung (2019), Effects of temperature and dopant concentration on oxygen vacancy diffusion coefficient of yttria-stabilized zirconia, IOP Conf Series 1274 (012005), L.T Lam, V.V Hung, B.D Tinh (2019), Investigation of electrical properties of Yttria- doped Ceria and YttriaStabilized Zirconia by statistical moment method, Journal of the Korean Physical Society 75 (4), 293 L.T Lam, V.V Hung, N.T Hai (2019), Effect of temperature on electrical properties of Yttria-doped Ceria and Yttriastabilized Zirconia, HNUE Journal of Science, HNUE Journal of Science 64 (6), 68 L.T Lam, V.V Hung, N.T Hai (2019), Study of oxygen vacancy diffusion in Yttria-doped Ceria and Yttria-stabilized Zirconia by statistical moment method, Communications in Physics 29 (3), 263 ... (Xng − 1) 6Kng kng − (4.47) 4.2 Lattice constant, diffusion coefficient and ionic conductivity a Lattice constant The expression of lattice constant can be written as amm = (n − 2)atr + 2ang ,... ions and the anharmonicity effect in these layers These changes increase distances between ions and the anharmonicity vibrations of crystal lattice 21 Consequently, the calculated lattice constant... diffusion and electrical properties of the fluorite structure materials are investigated using SMM including the anharmonicity effects of thermal lattice Scientific and practical senses The results
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