Calculate the number of spheres in the simple cubic, body-centered cubic, and face-centered cubic cells. Also, calculate the packing efficiency of each type of cell. simple cubic cells: spheres packing efficiency % body-centered cubic cells: spheres packing efficiency face-centered cubic cells: spheres packing efficiency % None
The Correct Answer and Explanation is:
Let’s break down the calculation of the number of spheres per unit cell and packing efficiency for each of the three types of cubic crystal systems:
1. Simple Cubic (SC) Cell
Number of spheres per unit cell:
- A simple cubic unit cell has 1 atom at each of its 8 corners.
- Each corner atom is shared among 8 adjacent unit cells.
- Contribution per corner atom = 1/8.
- Total atoms per unit cell = 8 corners × (1/8) = 1 atom.
Packing efficiency:
- Assume each sphere (atom) has radius r.
- The edge length a of the unit cell = 2r.
- Volume of unit cell = a³ = (2r)³ = 8r³.
- Volume of one sphere = (4/3)πr³.
- Packing efficiency = [Volume of sphere / Volume of unit cell] × 100
= [(4/3)πr³ / 8r³] × 100
= (π/6) × 100 ≈ 52.4%.
2. Body-Centered Cubic (BCC) Cell
Number of spheres per unit cell:
- 8 corner atoms × (1/8 each) = 1 atom.
- 1 atom completely inside the body = 1 atom.
- Total = 1 + 1 = 2 atoms.
Packing efficiency:
- In BCC, the atoms touch along the body diagonal.
- Body diagonal = √3 × a = 4r ⇒ a = 4r/√3.
- Volume of unit cell = a³ = (4r/√3)³ = 64r³ / 3√3.
- Total volume of 2 spheres = 2 × (4/3)πr³ = (8/3)πr³.
- Packing efficiency = [(8/3)πr³ / (64r³ / 3√3)] × 100
= (π√3 / 8) × 100 ≈ 68.0%.
3. Face-Centered Cubic (FCC) Cell
Number of spheres per unit cell:
- 8 corner atoms × (1/8) = 1 atom.
- 6 face atoms × (1/2) = 3 atoms.
- Total = 1 + 3 = 4 atoms.
Packing efficiency:
- Atoms touch along the face diagonal.
- Face diagonal = √2 × a = 4r ⇒ a = 4r/√2 = 2√2r.
- Volume of unit cell = a³ = (2√2r)³ = 16√2r³.
- Total volume of 4 spheres = 4 × (4/3)πr³ = (16/3)πr³.
- Packing efficiency = [(16/3)πr³ / 16√2r³] × 100
= (π / √2) × 100 ≈ 74.0%.
Summary Table:
| Lattice Type | Atoms per Cell | Packing Efficiency |
|---|---|---|
| Simple Cubic (SC) | 1 | ≈ 52.4% |
| Body-Centered Cubic | 2 | ≈ 68.0% |
| Face-Centered Cubic | 4 | ≈ 74.0% |
Explanation (300+ words):
Crystals form in highly ordered arrangements called unit cells, which are the smallest repeating units in a crystal lattice. In simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) structures, atoms are placed in different positions within the cube, affecting how densely packed they are—this is referred to as packing efficiency.
In an SC structure, atoms are only at the cube’s corners. Each corner atom is shared among 8 neighboring unit cells, so only 1/8 of it belongs to a single cell. Therefore, the unit cell contains a total of 1 atom. The atoms touch each other along the cube’s edges. Its low packing efficiency of about 52.4% means nearly half of the space is empty—this structure is rare in nature due to its inefficiency.
The BCC structure adds one atom in the center of the cube, in addition to the 8 corner atoms. The central atom is entirely within the unit cell. Atoms touch along the cube’s body diagonal, allowing more efficient packing. This yields 2 atoms per cell and a higher efficiency of about 68.0%. This arrangement is found in metals like iron and chromium.
The most efficient of the three is the FCC structure, which has atoms at each corner and at the center of each face. Atoms touch along face diagonals, and each face-centered atom contributes half to the unit cell. This results in 4 atoms per cell and the highest packing efficiency of about 74.0%. FCC structures are common in ductile metals like aluminum, copper, and gold due to their high density and close packing.
In conclusion, while SC is structurally simple, FCC is the most space-efficient and mechanically favorable structure.
