Hexagonal Close Packing Formula 

Hexagonal Close Packing (HCP) is one of the most efficient ways atoms can be packed in a crystal lattice. It is a highly ordered structure found in many metals and is a crucial concept in solid state chemistry and crystal structures. Understanding the hexagonal close packing formula helps JEE aspirants solve various questions related to unit cell calculations, packing efficiency, and number of atoms per unit cell.

1.0Importance of HCP in Chemistry and Material Science

HCP is vital in understanding the physical properties of metals and ionic compounds. Many elements like magnesium, titanium, and zinc crystallize in HCP structures. The hexagonal close packing formula allows students to calculate important parameters, such as the number of atoms in the unit cell, atomic radius, and nearest-neighbour distances, essential for various JEE Chemistry questions.

2.0HCP Structure: Detailed Explanation

What is Hexagonal Close Packing?

Hexagonal Close Packing refers to the arrangement where spheres (atoms, ions, or molecules) are packed closely together to maximize space utilization. It forms a hexagonal lattice where each atom is surrounded by 12 others, creating a densely packed structure.

Layer Arrangement in HCP

HCP follows an ABAB... pattern:

• Layer A: The first layer of spheres is arranged in a hexagonal pattern.
• Layer B: The second layer sits in the depressions of the first, but does not cover all the depressions.
• Layer A repeated: The third layer is directly above the first.

This alternating arrangement is the hallmark of HCP and is different from the cubic close packing (CCP) or face-centered cubic (FCC) arrangement, which follows an ABCABC... pattern.

Visualization of HCP Unit Cell

The HCP unit cell is prismatic (not cubic) and contains atoms at the corners and faces of the hexagonal prism. This visualization is frequently asked in JEE Chemistry exams.

3.0Derivation of Hexagonal Close Packing Formula

The HCP formula calculates the number of atoms per unit cell and related parameters.

Step 1: Identify Contributions of Atoms

• Corner atoms: 12 corners, each shared by 6 unit cells → 12 × 1/6 = 2 atoms
• Face-centred atoms on hexagonal faces: 2 faces, each shared by 2 cells → 2 × 1/2 = 1 atom
• Interior atoms (middle layer): 3 atoms entirely within the unit cell → 3 atoms

Total atoms per HCP unit cell:

n = 2 + 1 + 3 = 6 atoms

Step 2: Volume of the HCP Unit Cell

The volume of a hexagonal unit cell is:

$V_{c}ell=\frac{3\sqrt{3}}{2}{a}^{2}c$

where a = base edge, c = height of unit cell.

Step 3: Atomic Radius and Unit Cell Parameters

In HCP, the relationship between atomic radius (r) and lattice parameters:

a=2r

$c=1.633a\approx 3.266r$

Step 4: Packing Efficiency Formula

Volume of 6 atoms in unit cell:

Packing efficiency:

$PE=\frac{V_{\text{atoms}}}{V}\times 100\%=\frac{8\pi r^3}{24\sqrt{2}{r}^3}\times 100\%=\frac{\pi }{3\sqrt{2}}\times 100\%\approx 74.05\%$

Thus, the HCP formula provides a precise method to calculate packing efficiency, number of atoms, and density.

4.0Calculations Involving HCP: Packing Efficiency and Number of Atoms

Calculating Number of Atoms in HCP

As derived, the HCP unit cell contains 6 atoms.

Calculating Packing Efficiency

For JEE Chemistry, remember:

• Packing efficiency of HCP = 74%

Calculation of Density

Density formula for HCP:

$\text{Density}=\frac{\text{Number of atoms in cell}\times \text{Atomic mass}}{\text{Volume of unit cell}\times N_A}$
Where:

• ( $N_A$ ) = Avogadro’s number
• Volume and number of atoms are calculated as above.

Coordination Number

Coordination number in HCP = 12 (each atom is in contact with 12 other atoms).

5.0Key Properties of HCP Arrangement

• Closest Packing: One of the most efficient ways to pack spheres.
• Coordination Number: 12, indicating high stability.
• Atomic Arrangement: ABAB pattern.
• Examples: Magnesium, zinc, cadmium, beryllium.

6.0Applications and Examples of HCP

Metals with HCP Structure

Common metals crystallizing in HCP:

• Magnesium (Mg)
• Zinc (Zn)
• Titanium (Ti)
• Cobalt (Co)

Relevance in Material Science

HCP structure determines:

• Metallic properties (ductility, malleability)
• Physical characteristics (hardness, density)

7.0Solved Problems

Problem 1:Calculate the total number of atoms in a hexagonal close-packed (HCP) unit cell.

Solution:
HCP unit cell contains:

• Corner atoms: 12 corners × 1/6 contribution per unit cell = 2 atoms
• Face-centered atoms (top & bottom hexagonal faces): 2 faces × 1/2 contribution per unit cell = 1 atom
• Interior atoms (middle layer): 3 atoms entirely inside the cell

Total atoms per unit cell:

n = 2 + 1 +3 = 6 atoms

Problem 2: Given the atomic radius r, calculate the packing efficiency of HCP.
Solution:

1. Volume of one atom: $V_{\text{atom}}=\frac{4}{3}\pi r^3$
2. Total volume of atoms in unit cell: $V_{\text{atoms}}=6\times \frac{4}{3}\pi r^3=8\pi r^3$
3. Unit cell volume: $V_{\text{cell}}=\frac{3\sqrt{3}}{2}{a}^{2}c,\text{ where }a=2r,c=1.633a$

$V_{\text{cell}}=\frac{3\sqrt{3}}{2}(2r{)}^2(1.633\cdot 2r)=21.44r^3$

Packing efficiency:

$\text{PE}=\frac{V_{\text{atoms}}}{V}\times 100\%=\frac{8\pi r^3}{24\sqrt{2}{r}^3}\times 100\%\approx 74\%$