In the class, we learned that the energy cost of creating a vacancy in a crystal (εv) is balanced by the increase in configurational entropy. However, the amount of entropy gained by placing a single vacancy in the crystal increases with the number of atomic sites in the crystal. Therefore, there may be a minimum number of atomic sites (N) that are needed to create sufficient entropy to stabilize a vacancy. (i) Equate the vacancy formation energy with the configurational entropy associated with adding a single vacancy to a crystal with N sites, and solve for N as a function of εv and T. (ii) Using the equation above, graph ln(N) v. 1/T for vacancy formation energies of 0.5 eV, 1.0 eV, 2.0 eV, and 3.0 eV. These four curves can be plotted on the same graph. (iii) Assume a cube shaped crystal with atomic sites at the vertices of a primitive cubic lattice with a lattice spacing of 0.2 nm. Under what conditions can a crystal with edge length of 100 nm be perfect? Under what conditions can a 1 micron crystal be perfect?
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