Physical Chemistry – The Solid State – Cubic Crystals – Face Corners, Edge ,Diagonal, center- part-5.
Physical Chemistry – The Solid State – Cubic Crystals – Face Corners, Edge ,Diagonal, center- part-5.
Contents [hide]
- 0.1 Physical Chemistry – The Solid State (Cubic Crystals) – Part 5
- 0.2 1. Types of Cubic Crystals
- 0.3 2. Face, Corners, Edge, Diagonal, and Center in Cubic Crystals
- 0.4 (A) Corners
- 0.5 (B) Faces
- 0.6 (C) Edges
- 0.7 (D) Diagonal
- 0.8 (E) Center
- 0.9 3. Summary Table of Atoms in a Unit Cell
- 0.10 Key Takeaways
- 0.11 Physical Chemistry – The Solid State – Cubic Crystals – Face Corners, Edge ,Diagonal, center- part-5.
- 0.12 Solid State Chemistry
- 0.13 UNIT 5 SOLID STATE
- 0.14 Solid state physics Unit 1 Crystal Structure
- 0.15 SOLID STATE
- 1
Solid State – Cubic Crystals (Part 5): Face, Corners, Edges, Diagonals, Center
Physical Chemistry – The Solid State (Cubic Crystals) – Part 5
In solid-state chemistry, cubic crystals are a key concept in understanding crystal structures. Let’s break down the important elements: Face, Corners, Edges, Diagonal, and Center in cubic crystals.
1. Types of Cubic Crystals
Cubic crystals exist in three main forms:
Simple Cubic (SC) – Atoms at corners only
Body-Centered Cubic (BCC) – Atoms at corners + one atom at the center
Face-Centered Cubic (FCC) – Atoms at corners + one atom at the center of each face
Important: The number of atoms per unit cell differs for each type:
SC: 1 atom
BCC: 2 atoms
FCC: 4 atoms
2. Face, Corners, Edge, Diagonal, and Center in Cubic Crystals
(A) Corners
- In a cubic unit cell, there are 8 corners, each occupied by an atom.
- But each corner atom is shared among 8 unit cells → Contribution per unit cell = 1/8
Total corner atoms in a unit cell =
8×18=18 \times \frac{1}{8} = 1
(B) Faces
- A cube has 6 faces, and in FCC, each face contains 1 atom at its center.
- Each face-centered atom is shared between 2 unit cells → Contribution per unit cell = 1/2
Total face atoms in FCC unit cell =
6×12=36 \times \frac{1}{2} = 3
(C) Edges
- A cube has 12 edges, and sometimes atoms are located at edge centers.
- Each edge-centered atom is shared between 4 unit cells → Contribution per unit cell = 1/4
Total edge atoms =
12×14=312 \times \frac{1}{4} = 3
(D) Diagonal
- Face Diagonal (dₑ): Connects opposite corners of a face. de=2adₑ = \sqrt{2}a
- Body Diagonal (d_b): Connects opposite corners of the entire cube. db=3ad_b = \sqrt{3}a
Where a = edge length of the cube
Used in calculating atomic radius (r)
SC: r=a2r = \frac{a}{2} BCC: r=3a4r = \frac{\sqrt{3}a}{4} FCC: r=2a4r = \frac{\sqrt{2}a}{4}
(E) Center
- In BCC, there is one atom exactly at the center of the cube.
- This atom is completely inside the unit cell → Full contribution
Total atoms in BCC unit cell =
1(corner atoms)+1(body-centered atom)=21 (\text{corner atoms}) + 1 (\text{body-centered atom}) = 2
3. Summary Table of Atoms in a Unit Cell
| Structure | Corner Atoms | Face Atoms | Edge Atoms | Center Atom | Total Atoms |
|---|---|---|---|---|---|
| SC | 8 × (1/8) = 1 | 0 | 0 | 0 | 1 |
| BCC | 8 × (1/8) = 1 | 0 | 0 | 1 | 2 |
| FCC | 8 × (1/8) = 1 | 6 × (1/2) = 3 | 0 | 0 | 4 |
Key Takeaways
Corner atoms contribute 1 per unit cell (shared among 8). Face atoms contribute 3 in FCC (shared among 2 per face). Edge atoms contribute 3 in edge-centered structures (shared among 4). Center atom is fully inside the unit cell in BCC. Diagonals help in finding atomic radius and packing efficiency.
Want practice questions or more explanations? Let me know!
Physical Chemistry – The Solid State – Cubic Crystals – Face Corners, Edge ,Diagonal, center- part-5.
Solid State Chemistry
UNIT 5 SOLID STATE
Solid state physics Unit 1 Crystal Structure
SOLID STATE
It looks like you’re referring to Physical Chemistry – The Solid State topic, specifically focusing on Cubic Crystals, including aspects like Face Corners, Edge, Diagonal, and Center. Let me explain these concepts clearly as part of a “Part-5” style summary. This can serve as a revision or focused note.
Solid State – Cubic Crystals (Part 5): Face, Corners, Edges, Diagonals, Center
Types of Cubic Unit Cells
- Simple Cubic (SC)
- Body-Centered Cubic (BCC)
- Face-Centered Cubic (FCC)
Atoms at Different Positions
| Position in Unit Cell | Contribution per Unit Cell | Number of Such Atoms | Net Atoms Contributed |
|---|---|---|---|
| Corner | 1/8 | 8 | 8 × 1/8 = 1 |
| Face center | 1/2 | 6 | 6 × 1/2 = 3 |
| Edge center | 1/4 | 12 | 12 × 1/4 = 3 |
| Body center | 1 | 1 | 1 × 1 = 1 |
Diagonal Lengths in Cubic Cells
- Edge Length: Let it be aa
- Face Diagonal (on a face):
Face Diagonal=2a\text{Face Diagonal} = \sqrt{2}a
- Body Diagonal (across the cube):
Body Diagonal=3a\text{Body Diagonal} = \sqrt{3}a
Atomic Radius Relation (r) with Edge Length (a)
| Lattice Type | Relation between rr and aa | No. of Atoms per Unit Cell |
|---|---|---|
| SC | r=a2r = \frac{a}{2} | 1 |
| BCC | r=3a4r = \frac{\sqrt{3}a}{4} | 2 |
| FCC | r=2a4r = \frac{\sqrt{2}a}{4} | 4 |
Packing Efficiency
| Type | Efficiency (%) | Explanation |
|---|---|---|
| SC | ~52.4% | Least efficient |
| BCC | ~68% | Moderately efficient |
| FCC | ~74% | Most efficient cubic structure |
Key Takeaways
- Corners contribute least (1/8 per corner).
- Face-centered atoms are shared between 2 unit cells (1/2 contribution).
- Edge-centered atoms are shared among 4 unit cells (1/4 contribution).
- Body-centered atoms are wholly inside one unit cell (full contribution).
- Atomic radius relations are derived from geometrical considerations of diagonals.
Let me know if you want numerical problems, derivations, or visual diagrams related to this part.
