Question

2) (a) Calculate the equilibrium vacancy concentration (number of vacancies per m) for copper at 1000K given that copper has an FCC structure with a lattice parameter a 3.597 A and a vacancy formation energy Q,-0.9 eV. Boltzmann's constant is 8.61733x10 eV/K (b) Plot the vacancy fraction as function of temperature in the range 100-1100K)

Answer 1

N_V = N exp (- Q_V/ KT) where N_V rightarrow equilibrium vacancy concentration N rightarrow total no of atomic sites Q_V rightarrow Vacancy formation energy K rightarrow Boatman�s Constant T rightarrow temperature We are given, Q_V = 0.9 ev K = 8.61733 times 10^-5 ev/k T = 1000 K N = n/a^3 (concentration of atomic sites) Where n = 4 (No of atoms in unit all of F_cc a = 3.597 A^0 (lattice parameter) a^3 = (3.597 times 10^-10)^3 (Because 1 m = 10^+ 10 A^0) = 46.539 times 10^-30 m^3 Therefore N = 4/46.539 times 10^-30 m^-3 N = 0.0859 times 10^30 m^-3 now, exp (-QV/KT) = exp (-0.9/8.61733 times 10^-5 times 1000) = exp (-0.1044 times 10^2) = exp (-10.44) = 2.923 times 10^-5 N_V = 0.0859 times 10^30 times 2.923 times 10^-5 m^-3 \r\n N_V/N = exp (-Q_V/KT) Temperature NV/N (fraction) 100 0 200 2.09 times 10^-23 300 7.60 times 10^-16