Problem 36 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In hexagonal systems of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space- filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'. The number of atoms in the HCP unit cell is (a) 4 (b) 6 (c) 12 (d) 17

Step-by-Step Solution

Answer

The number of atoms in the HCP unit cell is 6.

Step 1: Understand the Structure of HCP

The HCP structure consists of three layers of spheres. The top and bottom layers have spheres arranged in a hexagonal pattern, while the middle layer has three spheres touching each other and three spheres in the bottom layer. The pattern resembles ABA stacking.

Step 2: Count the Atoms in the Bottom Layer

In the bottom layer, there are 6 atoms that are part of the hexagonal formation. However, each of these corner spheres is shared among 6 unit cells because of the hexagonal repetition.

Step 3: Count the Atoms in the Top Layer

Like the bottom layer, the top layer also consists of 6 spheres at the corners of the hexagon, each of which is shared among 6 unit cells. These atoms align directly above the bottom layer atoms, maintaining the ABA stacking pattern.

Step 4: Count the Atoms in the Middle Layer

The middle layer consists of 3 atoms located at the center of the hexagon. These atoms are exclusive to a single unit cell and are not shared with neighboring cells.

Step 5: Calculate the Total Number of Atoms

Each of the 6 atoms at the base and top layers is shared among 6 unit cells, contributing 1 full atom per layer (6 atoms x 1/6 share = 1). The middle layer contributes 3 full atoms (each exclusive to one unit cell). Therefore, the number of atoms in one HCP unit cell is: Initial layer contribution: 1 atom Middle layer: 3 atoms Top layer contribution: 1 atom Total atoms = 1 + 3 + 1 = 5 atoms.

Key Concepts

Hexagonal Crystal System

The hexagonal crystal system is one of the seven distinctive lattice structures that a crystal can take. It is characterized by having four crystallographic axes: three of these axes lie in a single plane, intersecting each other at 120-degree angles, while the fourth axis stands out, being perpendicular to these. This unique arrangement gives the hexagonal crystal system its distinct shape and symmetry.
In these systems, the repeated pattern forms hexagons, creating a unit cell that closely resembles a hexagonal prism. This structural configuration leads to properties that are highly symmetric and can be found in many naturally occurring minerals and materials.
With this geometric foundation, different crystal structures such as the hexagonal close-packed (HCP) structure can arise, offering dense packing of atoms and minimizing empty space in the crystal.

Unit Cell

The unit cell is the smallest repeatable volume which gives the crystal its entire structure through translational symmetry. In a hexagonal close-packed (HCP) structure, the unit cell is designed to be as compact as possible, containing atoms efficiently packed together.
Typically, the HCP unit cell consists of several layers of atoms arranged in a specific pattern: the arrangement takes the format of ABA.

• The first layer (A) contains atoms positioned to form a hexagon.
• The second layer (B) shifts slightly to fit into the gaps of the first, forming a triangular pattern.
• The third returns to the position of the first layer (A), resembling the structure's repetitiveness.

Together, these layers account for the entire composition and symmetry characteristics of the hexagonal crystal system, logically holding up to three-dimensional stacking.

Atomic Arrangement

Atomic arrangement refers to how atoms are systematically ordered within a structure. In the hexagonal close-packed (HCP) structure, the specific arrangement of atoms highly influences physical properties like density and stability.
In the HCP, each atom touches as many neighboring atoms as possible. The coordination number, which represents the number of nearest neighbor atoms, is 12. This configuration ensures maximum space utilization, as atoms are snugly positioned next to one another, minimizing voids or empty regions.
Because of this arrangement, the HCP structure is efficient and durable. This efficient atomic arrangement is why metals like zinc, titanium, and magnesium adopt this configuration in their crystallographic forms, offering high density and strength.

Layered Structures

Layered structures in crystal systems, particularly in the hexagonal close-packed (HCP) systems, are essential to understand. These layers follow a specific stacking sequence that leads to unique structural properties.
In an HCP lattice, layers are commonly denoted in an ABA pattern:

• The first and third layers, marked as A-layers, are identical and form the base and top of the unit cell.
• B-layer lies between these identical layers and fits into the troughs created by the A-layers, allowing closer packing.

This arrangement—distinct with its close-packed nature—permits maximum atomic contact both within and between layers, leading to tightly bound structures. Such layering enhances mechanical properties and controls how structures deform under stress, which is crucial for materials subjected to high-pressure environments.