Consider the CuAu crystal structure. It can be described as a simple cubic latti

Question

Consider the CuAu crystal structure. It can be described as a simple cubic lattice with a basis of Cu (0, 0, 0), Cu (1 2, 1 2, 0), Au (1 2, 0, 1 2), and Au (0, 1 2, 1 2). (a) How many atoms of each type are there per unit cell? (b) Draw the unit cell for CuAu. Use a different symbol for each type of atom. Provide a legend indicating which symbol represents which type of atom. (c) Give an alternative lattice and basis representation for CuAu for which one atom of the basis is Au (0, 0, 0). (d) A related crystal structure is that of Cu3Au. This unit cell is similar to the face-centered cubic unit cell with Au at the corners of the unit cell and Cu at all of the face-centered positions. Describe this structure as a lattice and a basis. (e) The Cu3Au crystal structure is similar to the FCC crystal structure, but it does not have the face-centered cubic lattice. Explain briefly why this is the case

Explanation / Answer

Based on symmetry of the lattice, the environment at 0,0,0, shouldbe the same at 0,0,1 and 1,00, etc.

Thus, there is a Cu at the corner (0,0,0), (0,0,1), (1,0,0),(0,1,0), (1,1,0), (1,1,1) (Note because these Cu are at the corner,they are shared by eight other unit cells, so these are 1/8 Cuatom; I know this may be difficult to visualize without a diagram,so PM if you need more help)

Also, there is a Cu at the (0.5,0.5,0) and the (0.5,0.5,1). Theseare at the center of the top and base face of the unit cell; theyare each shared between two unit cells. Thus these are 1/2 Cu,effectively.

Net count of Cu = corner + face = 8*(1/8)+2*(1/2) = 2

For Au, (0.5,0,0.5), (0,0.5,0.5), (0.5,1,0.5), and (1,0.5,0.5) atthe center of horizontal faces. These four Au are each shared 1/2because they are at faces.

Thus, Au = 4*(1/2) = 2

Thus there is 2 Au, and 2 Cu

You would have expected this 1:1 ratio based on thestoichiometry.

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