Question

please answer b,c,d

Draw FCC structure By considering the packing of atoms along the diagonal of one face of the cubic structure, find the relationship between the atom radius r and the lattice parameter a. Calculate the number of atoms per lattice (N) for the FCC structure. Calculate the atomic packing factor (APF) for the FCC structure.

Answer 1

FCC Coordination number is the number of nearest neighbor to a particular atom in the crystal coordination number 12 total 4 atoms per unit cell In the FCC lattice each atom is in contact with 12 neighbor atoms. FCC coordination number Z 12 4 R For example, the face centered atom in the front face is in contact with four corner atoms and four other face-centered atoms behind it (two sides, top and bottom) and is also touching four face-centered atoms of the unit cell in front of it APF 0.74 the closest packing possible of spherical atoms cubic closest-packed Face centered cubic unit cell A face centered cubic unit cell contains atoms at all the corners of the crystal lattice and at the center of all the faces of the cube. The atom present at the face-center is shared between two adjacent unit cells and only of each atom belongs to a unit cell Thus in a face centered cubic unit cell: a) 8 cornersx per coner atom 8 x 1 atom b) 6 face-centered atoms x atom per unit cell 3 atoms Therefore the total number of atoms in a unit cell 4 atoms

Atomic packing factor for FCC

Let a be the A the side length of the unit cell of FCC lattice and R the diameter of the atoms.

The FCC unit cell is formed by 4 atoms:
- 8 times one eighth of an atom at the corners of the cube
- 4 times a half of an atom at the center of the of the faces.

At the faces the atoms at the corners and the center atom touch, so that the perfectly fill the face. Hence the length of the face diagonal is
D = R + 2R + R = 4R
From Pythagorean theorem you get
A² + A² = D²
=>
A = √8 · R = √2 · 2·R

The volume of the cube cell is
Vc = A³ = √2 · 16·R
The volume of the atoms in the cell is
Va = 4 · (4·π·R³ /3) = 16·π·R³ /3

The packing density is
p = Va / Vc
= (16·π·R³ /3) / (√2 · 16·R)
= π / (3·√2)
= 0.74