The Solid State

2

Step 1: A BCC unit cell has atoms at all 8 corners and 1 atom at the center of the cube. Step 2: Contribution from corner atoms = 8 × (1/8) = 1. Step 3: The atom at the body center is entirely within the unit cell and is not shared with any other unit cell. So its contribution = 1. Step 4: Total atoms per unit cell = 1 (from corners) + 1 (body center) = 2. Step 5: Examples of BCC metals include iron (Fe) and chromium (Cr).

4

Step 1: An FCC unit cell has atoms at all 8 corners and at the center of all 6 faces. Step 2: Contribution from corner atoms = 8 × (1/8) = 1. Step 3: Each face-center atom is shared by 2 unit cells, so contribution = 1/2 per face. Step 4: Contribution from face-center atoms = 6 × (1/2) = 3. Step 5: Total atoms per unit cell = 1 + 3 = 4. Copper, gold, and silver crystallize in FCC structure.

Face centered cubic / CCP (74%)

Step 1: Packing efficiency is the percentage of total space in a unit cell that is actually occupied by atoms. Step 2: Simple cubic packing efficiency = π/6 × 100 = 52.4% (lots of empty space). Step 3: BCC packing efficiency = √3π/8 × 100 = 68% (more efficient). Step 4: FCC (or CCP and HCP) packing efficiency = π/(3√2) × 100 = 74% (most efficient of the three). Step 5: The FCC/CCP arrangement wastes the least space, making it the most efficient packing among cubic lattices.

d = (z × M) / (a³ × Nₐ)

Step 1: Density = Mass / Volume. Step 2: Volume of unit cell = a³ (where a is edge length). Step 3: Mass of unit cell = z × M / Nₐ, where z = number of atoms per unit cell, M = molar mass, Nₐ = Avogadro's number (6.022 × 10²³). Step 4: Combining these: d = (z × M) / (a³ × Nₐ). Step 5: This formula is very useful — if you know the density and molar mass, you can find z and identify the type of cubic unit cell.