Materials Science and Engineering: 2nd year, 2011:  CRYSTALLOGRAPHY   

Classwork 4: Crystal Structures  1. What is the coordination number in the: (i) (ii) (iii)

face-centred cubic (fcc) structure, hexagonal closed-packed(hcp) structure, body-centred cubic (bcc) structure?

Answers 1. i) 12 ii) 12 iii) 8

2. How many atoms per unit cell are there in the: (i) (ii) (iii)

face-centred cubic structure, hexagonal closed-packed structure, body-centred cubic structure?

Answers 2. i) 8 ∗ 1 8 6 ∗ 1 6 ii) 4 ∗ 1 4 1 2 iii) 8 ∗ 1 8

1

4

2

3. Calculate the packing fraction for the structure: (i) (ii)

face-centred cubic structure and body-centred cubic

Answers 3. i) (Packing fraction) = (Volume occupied) / (Volume total). Volume occupied = 4 ∗ 4 3 ∗ ∗ Volume total = with 2 4 (atoms touching along face diagonal) so 4 Packing fraction = 4 ∗ 3 ∗ ∗ √8 74% ii) Volume occupied = 2 ∗ 4 3 ∗ ∗ with 3 4 Volume total =

(atoms touching along body diagonal) so

/ Packing fraction = 2 ∗ 4 3 ∗ ∗ Note bcc is less close packed than fcc

√

68%

√8

4/√3

4. For the:(i) face-centred cubic structure, (ii) body-centred cubic structure, Identify the number and type of indices, , of the close-packed directions. Assuming the atoms are spheres in contact, derive an expression for the radius, r, of an atom in terms of the lattice parameter, a, for each of the two crystal structures. Answers 4. i) , which is 6 directions (or 12 if negatives considered). ∗ or /2√2. And as shown before, r = 1 √8 ii) which is 4 directions (or 8 if negatives considered). And as shown before, r = √3 4 ∗ .  

5. What are the Miller indices for the close-packed planes in the face-centred cubic structure? How many planes of this type are there in the structure? Calculate the fractional area occupied by atoms in these planes. Answers 5. The type of plane is the {111}. There is 4 planes of this type in the structure (or 8 if negatives considered).. Fractional area close packed plane: 2r 2r

120°

= (NAtoms in unit cell ∗ Area of an Atom) / (unit cell) =

∗

= =

° .√ ⁄

.√

90.7%

6. Draw a section of one of the closest packed planes in the body-centred cubic structure. State the Miller indices of these planes and the number of planes of this type in the structure on your drawing. Indicate and give the indices [uvw] of the close-packed directions. Answers 6.

It is a plane of the type {110}. (in red) There are 6 planes of this type in the structure (12 with megatives). The close-packed directions are [111]. (in blue)

7. For an ideal hexagonal close-packed structure, the atoms within the close-packed planes and the atoms in the adjacent planes are in contact. Hence, calculate the ideal axial ratio, c/a, for the structure. Answers 7. The ideal axial ratio (c/a) for a hcp system can be calculated considering non-interacting hard spheres model. If the sphere radius is r, the the lattice parameters a=b and c can be written in terms of r.

These two relationships can be solved for the ideal axial ratio c/a. 2

4

3

4 3

3

1.633

4

4

8. Metallic crystals deform by slip on close-packed planes, along close-packed directions. Considering the number and nature of the close-packed planes and directions in the three metallic structures, which structure would you expect to deform most easily and which least easily. Think about the common usage of the following metal whose structures are given below: fcc: Cu, Al, Ag, Au bcc: Fe hcp: Zn Answers 8. In hcp there is only one close packed plane with three close packed directions, this type of structure is only deformable as single crystals. In poly crystals the close packed planes are not aligned and deformation cannot be transferred from one crystal to the other. This structure deforms the least easily. Both bcc and fcc have a multitude of cp planes and directions so are deformable in polycrystals and the deformation in one crystal can be transferred to another (one needs a minimum of 5 cp systems (combination of plane and direction for this) However the cp planes in bcc are less close packed than those in hcp, which makes deformation somewhat more difficult in bcc. For this reason slip in bcc is more affected by temperature.