Classification Equilibrium Concentration in Elemental Crystals Point Defects in Ionic Crystals Introduction to diffusion Ficks Laws Concentration Profile Atomic Mechanisms Temperature Dependence,Chapter 4 Point Defects and Diffusion in Solids,2,Cubic Lattice Structures,body diagonal:,The closest packed plane defined by 3 face diagonals,nearest-neighbor distance? coordiation numer?,3,The 14 Bravais Lattices,abc =90,abc =90 90,abc 90,Monoclinic,Triclinic,a=bc =90,a=b=c =90,a=b=c 90,(also called Trigonal),4,The hexagonal close-packed crystal structure,Zr, Mg, Zn, Cd,5,6,The Miller Indices,Principal planes in the cubic lattice,7,8,9,d(hkl) = a/(h2+k2+l2)1/2,The Interplanar Distance,a,For a cubic system:,10,Five fundamental thermodynamic properties:,Three derived thermodynamic properties:,Basic Thermodynamic properties,* Intensive properties (lower case, independent of quantity), the rest are extensive properties,*,*,v=V/n specific volume an intensive property,11,The heat capacities (also called specific heats):,The coefficient of thermal expansion and the coefficient of compressibility :,12,2. First Law of Thermodynamics,The First law of thermodynamics is an empirical observation, never refuted, that the change in the internal energy of a closed system resulting from addition of heat and performance of work is given by: U = Q W where U = U(final) - U(initial) = change in system internal energy Q = heat added to the system W = work done by the system,U + Usurr = 0,13,3. Second Law of Thermodynamics,S+Ssurr 0,In any spontaneous process the entropy will either increase or remain the same.,13,4. Third Law of Thermodynamics,As the temperature of a system approaches to absolute zero, its entropy tends to a constant value S0 which is independent of pressure, state of aggregation, etc. Thus, for a given process: S = 0 at T = 0K,14,At equilibrium, the Gibbs free energy of a system with constant T and p is a minimum. dGT,p = 0,5. Single Component Phase Equilibria,The Thermodynamic Equilibrium:,15,6. Gibbs Phase Rule: F = C + 2 P,Components (C): distinct chemical species whose quantities can be independently varied. Phases (P) are regions of a system in which all properties are uniform and are distinct from other regions in the same system. Degrees of freedom (F) are the number of system variables (e.g., properties, composition) that can be independently specified without changing the phase(s).,Phase Diagram of a Pure Substance and The Gibbs Phase Rule,16,Free Energy vs. T across Tm for a Solid and a Liquid,Classification of Point Defects,Point Defects in an Elemental Crystal,Two point defects, vacancies and self-interstitials are intrinsic to the material, meaning that they form spontaneously in the lattice without any external intervention.,Frenkel pair,Interstitials in the bcc lattice,self interstitials interstitial impurities,a,Split interstitial,octahedral site,tetrahedral site,Formation of a self interstitial (more difficult),Formation of a vacancy,Equilibrium Concentrations of Point Defects in Elemental Crystals,G= H TS,H = EPD + pV,EPD - energy required to create the point defect from the perfect lattice.,pV - work involved in the change in system volume as the atom is moved between the interior site and the surface.,The criterion of chemical equilibrium is the minimization of the Gibbs free energy of the system at constant temperature and pressure. The entropy of mixing is responsible for the spontaneous existence of point defects at equilibrium; the magnitude of the positive energy of point defect formation, EPD, governs the concentration of these species at thermal equilibrium.,S = SPD + Smix,k - Boltzmann constant 1.381x10-23 J/K or 8.63x10-5 eV/K,G(0) - the free energy of the perfect lattice of N atoms V - the energy of formation of a single vacancy (roughly equals to the heat of vaporization per atom) - the atomic volume in the crystal lattice,The Gibbs free energy of the system containing vacancies:,The equilibrium concentration of thermal vacancies approximately:,Example: The formation energy of vacancies in copper is 100 kJ/mole. At 1300 K (which is just below the melting point) what fraction of the lattice sites are empty? Using these values in the above equation, the site fraction of vacancies is xV = 10-4. This value is too small to influence properties such as the density, but even smaller values of xV are critical in determining the transport property of self-diffusion.,p/kT 0.25, 0.8 Sv 0,The equilibrium concentration of thermal self-interstitials:,i - the energy of formation of a single self-interstitial,For copper, the interstitial formation energy is i 300 kJ/mole, which is about three times larger than the energy required to create a vacancy in this metal. As a result, the equilibrium concentration of interstitials is very much smaller than that of vacancies (by 8 orders of