⬢ Packing of Solids ⬢

🔍 Introduction

Packing of solids refers to the arrangement of atoms, ions, or molecules in a three-dimensional structure. The efficiency of packing determines many physical properties of materials including density, stability, and coordination number.

📊 Types of Packing

1️⃣ Close Packing of Spheres

In close packing, spheres are arranged to minimize empty space and maximize packing efficiency. There are two main types:

Step 1: Simple Square Packing (2D)

● ● ● ●
● ● ● ●
● ● ● ●
● ● ● ●

Spheres arranged in a simple square pattern

Packing Efficiency: 78.54%

Step 2: Hexagonal Close Packing (2D)

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Spheres arranged in a hexagonal pattern (more efficient)

Packing Efficiency: 90.69%

🏗️ Three-Dimensional Packing

Cubic Close Packing (CCP) or Face-Centered Cubic (FCC)

Step 1: First Layer (A)

Arrange spheres in hexagonal close packing pattern

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This forms the base layer with maximum 2D efficiency

Step 2: Second Layer (B)

Place spheres in the tetrahedral holes of layer A

○ ○ ○
● ● ● ●
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○ = Second layer spheres, ● = First layer spheres

Step 3: Third Layer (C)

Place spheres in positions different from both A and B layers

◐ ◐ ◐
○ ○ ○
● ● ● ●

◐ = Third layer, ○ = Second layer, ● = First layer

Sequence: ABCABC...

Hexagonal Close Packing (HCP)

Alternative Third Layer Arrangement

Third layer placed directly above the first layer (A)

● ● ● ●
○ ○ ○
● ● ● ●

Sequence: ABABAB...

This creates hexagonal symmetry

🕳️ Types of Holes in Close Packing

1️⃣ Tetrahedral Holes

Formed by 4 spheres arranged tetrahedrally

Number of tetrahedral holes = 2N
(where N = number of spheres)

Radius ratio: r_hole/r_sphere = 0.225

2️⃣ Octahedral Holes

Formed by 6 spheres arranged octahedrally

Number of octahedral holes = N
(where N = number of spheres)

Radius ratio: r_hole/r_sphere = 0.414

📋 Comparison Table

Property	CCP (FCC)	HCP	Simple Cubic
Packing Efficiency	74%	74%	52.4%
Coordination Number	12	12	6
Layer Sequence	ABCABC...	ABABAB...	AAAA...
Tetrahedral Holes	8 per unit cell	12 per unit cell	None
Octahedral Holes	4 per unit cell	6 per unit cell	None
Examples	Cu, Ag, Au	Zn, Cd, Mg	Po (α-form)

🔢 Mathematical Relationships

Packing Efficiency Calculation

Packing Efficiency = (Volume occupied by spheres / Total volume) × 100%

For FCC Structure:

Packing Efficiency = (π√2)/6 × 100% = 74.05%

Radius Relationships:

FCC: a = 2√2 × r

HCP: a = 2r, c = 2√(2/3) × a

Where: a = lattice parameter, r = atomic radius, c = height of unit cell

🎯 Key Points to Remember

Important Concepts:

• Coordination Number: Number of nearest neighbors around each sphere

• Packing Efficiency: Percentage of space filled by spheres

• Tetrahedral Holes: Smaller holes formed by 4 spheres

• Octahedral Holes: Larger holes formed by 6 spheres

• Both CCP and HCP have the same packing efficiency (74%)

• Close packing maximizes density and minimizes potential energy

🔬 Applications

Understanding packing of solids is crucial for:

• Crystallography: Determining crystal structures

• Materials Science: Designing new materials with specific properties

• Ionic Compounds: Predicting structures based on radius ratios

• Metallurgy: Understanding metallic bonding and properties

🌟 Remember: The beauty of close packing lies in achieving maximum efficiency with minimum energy! 🌟