### The concentration of vacancies in the cube

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1. Considering a situation where one gram atom of aluminum, in the form of a cube, is heated to a temperature that is 5 degrees below the melting point of aluminum, answer the following questions:

(a) What is the concentration of vacancies in the cube? Assume that the energy of formation of vacancies in aluminum is 0.5 eV.
(b) Imagine that the vacancies in (a) are retained at a low temperature by quenching. During subsequent annealing, the vacancies cluster together to form circular stacking faults 10 nm in diameter. Calculate the number of stacking faults.

(c) Explain why does it cost energy to form vacancies in solids?

2. A MgO single crystal in the form of a cube is deformed in compression. As a result of deformation, the [1-10] (110) and [110] (1-10) slip systems are activated. During deformation, dislocations on the two slip systems intersect each other, resulting in the formation of jogs; one of the jogs is edge in character and 1 nm long.

(a) Calculate the energy of the jog, given that the lattice parameter of MgO is 0.5 nm and the shear modulus is 1011 Pa.

(b) Could you thermally activate the formation of the jog in (a)? Explain your answer.

(c) Sketch the extra half plane of the edge jog in (a).

3. V3Si, a superconductor, crystallizes in the A15 structure. The V-V atoms are covalently bonded, whereas the Si-Si atoms are metallically bonded.

(a) Sketch the unit cell of V3Si (b) What are possible slip systems? Give reasons for your choices.

4. (a) Explain the strengthening effects of twins in FCC crystals.

(b) A material has a certain fraction of the (111) twinned volume. In one sample the twins are thicker, whereas they are thinner in the other sample. Which sample would you prefer for strengthening? Give reasons for your choice.

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