Practice Set - 4
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MSE303: Electronic and Magnetic Properties of Materials Practice Problems – Set 4 Topics covered: Ionic Conductivity 1. Show that for metal-deficient non-stoichiometric ionic oxide (MO), concentration of holes is proportional to partial pressure of oxygen as ! ∝ ! ! ! ! for both metal-deficient and oxygen-excess cases. 2. LiF has a Schottky formation energy of 2.6 eV and a bandgap of 12 eV. Write defect reaction for Schottky defect formation for LiF. Estimate the relative concentrations of ionic and electronic defects at 500 ℃ and determine which are dominant on an absolute concentration basis. (assume 3. Uranium dioxide, UO 2 is of the fluorite structure. Thy oxygen ions occupy tetrahedral sites in an FCC lattice of uranium ions, leaving octahedral interstitial sites empty. Some relevant defect formation energies and other useful information are as follows: ! × ⇌ ! ∙∙ + ! !! ∆ = 3.0 ! × ⇌ ! !!!! + ! ∙∙∙∙ ∆ = 9.5 2 ! × + ! × ⇌ ! !!!! + ! ∙∙ + ! ↑ + ! ∙∙ ∆ = 6.4 ℰ ! = 5.2 (a) What are the predominant intrinsic ionic defects in stoichiometric UO 2 at 1600 ℃? Calculate their concentrations. (b) Calculate the intrinsic electron and hole concentration at the same temperature. (Assume that the bandgap decreases with temperature at the rate of ~1 meV/K and effective densities of states in conduction and valence bands is 1x10 19 /cm 3 .) (c) UO 2 is easily rendered extrinsic by reduction to UO 2-x or oxidation to UO 2+x at high temperatures. Write defect reactions for reduction and oxidation of UO 2 . How does defect formation depend on oxygen partial pressure in environment? (d) Write full electroneutrality (charge neutrality) condition for UO 2 at 1600 ℃. 4. Figure below shows the diffusion coefficient of Na in NaCl containing a small amount of CdCl 2 solute, plotted against 1/T. Two linear regions are shown, for which the slopes are given (in units of dergrees K). Answer the following: (a) Write defect reaction for CdCl 2 in NaCl. (b) Explain the existence of intrinsic and extrinsic regimes. (c) Write an expression for the diffusion coefficient of sodium in intrinsic and extrinsic regimes, respectively. (d) Determine the activation energy for vacancy migration from data. (e) Determine the Schottky defect formation enthalpy. (f) Using the data detrermine the concentration of Cd in this sample (g) Calculate the ionic conductivity due to sodium vacancies at 550 ℃ . (h) NaCl has a bandgap of 7.3 eV. Will this sample be an ionic or electronic conductor in the temperature range shown? Explain?
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Transcript of Practice Set - 4
MSE303: Electronic and Magnetic Properties of Materials Practice Problems – Set 4
Topics covered: Ionic Conductivity
1. Show that for metal-deficient non-stoichiometric ionic oxide (MO), concentration of holes is proportional to partial pressure of oxygen as
𝑛! ∝ 𝑝!!! !
for both metal-deficient and oxygen-excess cases.
2. LiF has a Schottky formation energy of 2.6 eV and a bandgap of 12 eV. Write defect reaction for Schottky defect formation for LiF. Estimate the relative concentrations of ionic and electronic defects at 500 ℃ and determine which are dominant on an absolute concentration basis. (assume
3. Uranium dioxide, UO2 is of the fluorite structure. Thy oxygen ions occupy tetrahedral sites in an FCC lattice of uranium ions, leaving octahedral interstitial sites empty. Some relevant defect formation energies and other useful information are as follows:
𝑂!× ⇌ 𝑉!∙∙ + 𝑂!!! ∆𝐻 = 3.0 𝑒𝑉 𝑈!× ⇌ 𝑉!!!!! + 𝑈!∙∙∙∙ ∆𝐻 = 9.5 𝑒𝑉 2𝑂!× + 𝑈!× ⇌ 𝑉!!!!! + 𝑈!∙∙ + 𝑂! ↑ +𝑉!∙∙ ∆𝐻 = 6.4 𝑒𝑉
ℰ! = 5.2 𝑒𝑉 (a) What are the predominant intrinsic ionic defects in stoichiometric UO2 at 1600 ℃? Calculate
their concentrations. (b) Calculate the intrinsic electron and hole concentration at the same temperature. (Assume that the
bandgap decreases with temperature at the rate of ~1 meV/K and effective densities of states in conduction and valence bands is 1x1019 /cm3.)
(c) UO2 is easily rendered extrinsic by reduction to UO2-x or oxidation to UO2+x at high temperatures. Write defect reactions for reduction and oxidation of UO2. How does defect formation depend on oxygen partial pressure in environment?
(d) Write full electroneutrality (charge neutrality) condition for UO2 at 1600 ℃.
4. Figure below shows the diffusion coefficient of Na in NaCl containing a small amount of CdCl2 solute, plotted against 1/T. Two linear regions are shown, for which the slopes are given (in units of dergrees K). Answer the following: (a) Write defect reaction for CdCl2 in NaCl. (b) Explain the existence of intrinsic and extrinsic regimes. (c) Write an expression for the diffusion coefficient of sodium in intrinsic and extrinsic regimes,
respectively. (d) Determine the activation energy for vacancy migration from data. (e) Determine the Schottky defect formation enthalpy. (f) Using the data detrermine the concentration of Cd in this sample (g) Calculate the ionic conductivity due to sodium vacancies at 550 ℃ . (h) NaCl has a bandgap of 7.3 eV. Will this sample be an ionic or electronic conductor in the
temperature range shown? Explain?
5. i. Write the Schottky defect reaction for TiO2 and then calculate the equilibrium oxygen vacancy
concentration in TiO2 at 1400°C given that enthalpy of defect formation is 5.2 eV. You can neglect the entropy of defect formation.
ii. Calculate the ionic conductivity at 1400oC assuming oxygen vacancy diffusion as the main mechanism for ionic conductivity.
iii. Calculate electronic conductivity at 1400oC iv. Calculate transference number for electronic and ionic conductivities.
[Eg = 3.2 eV, 𝜇! = 𝜇! = 0.1 cm2/V.s @ 1400oC, me=0.33 mo, mh=0.77mo, (Nc.Nv)1/2 =1.75 x1015. T3/2 cm-3. Diffusivity of oxygen vacancies is given as 2.2x10-7(m2/s) x exp (-100 kJ/RT), Density of TiO2 is 4 g/cc, molecular weight is 80 g/mol. Mobility of ionic species can be described using Einstein relation 𝜇! = 𝑍!𝑒𝐷! 𝑘!𝑇]
intrinsic
