Question

What are the number of atoms per unit cell and the number of nearest neighbours in a body centered cubic structure? (a) 4,12 (b) 1,6 (c) 2,8 (d) 2,5

Short answer

The number of atoms per unit cell in a body centered cubic structure is 2, and the number of nearest neighbours is 8.

Step-by-step solution

Number of Atoms per Unit Cell

In a body-centered cubic (bcc) structure, there is one atom at each of the eight corners of the cube and one atom at the center. Each corner atom is shared by eight adjacent unit cells, so each corner contributes 1/8 of an atom to the unit cell. For the eight corners, this results in 1 atom total (8 corners * 1/8 contribution per corner). The atom at the center is entirely within the unit cell and contributes 1 whole atom. Thus, the total number of atoms per unit cell in a bcc structure is 1 (from corners) + 1 (from center) = 2.

Number of Nearest Neighbours

In a bcc structure, the number of nearest neighbours or the coordination number is the number of closest atoms to any given atom. For the body-centered atom, the nearest neighbours are the atoms at the corners of the cube. Since there are 8 corners, the body-centered atom has 8 nearest neighbours. Conversely, each corner atom also has 8 nearest neighbours formed by the body-centered atoms of the surrounding unit cells. Hence, the number of nearest neighbours in a bcc structure is 8.

Key concepts

Coordination Number

Understanding the coordination number in a crystal lattice is vital for comprehending various properties of materials. In the context of a body-centered cubic (bcc) structure, the coordination number refers to the number of immediate neighboring atoms that surround a particular atom.

Each atom in a bcc lattice is in direct contact with eight other atoms. This means that the central atom in the cube touches all eight corner atoms. Simultaneously, each corner atom is equally in contact with the same number of body-centered atoms from the adjacent unit cells. Therefore, the coordination number is a key indicator of the tightness of packing within a crystal structure and influences properties such as density, melting points, and the manner in which atoms interact with each other.

Unit Cell in Crystal Structures

A unit cell is the smallest repeating unit of a crystal lattice that retains the overall symmetry of the entire crystal. It is like a building block from which the whole crystal can be constructed through translation in three-dimensional space.

When considering the body-centered cubic structure, the unit cell is defined by atoms at each of its eight corners and an additional atom at the very center. This bcc unit cell is particularly special because it exhibits a high packing efficiency while not being the tightest possible arrangement, offering a unique combination of space utilization and structural stability. The bcc unit cell illustrates how basic repeating units determine the macroscopic properties of solids, playing an essential role in material science and engineering.

Nearest Neighbours in Crystal Lattice

In a crystal lattice, the nearest neighbours are the closest atoms to a given atom without counting the atom itself. These neighbors have the most significant influence on an atom's behavior and properties within the material.

For the body-centered cubic (bcc) structure, each atom's nearest neighbours can be visualized by focusing on its position within the unit cell. The unique feature of the bcc structure is that its atoms are equidistant from their nearest neighbors, which are either the corner atoms for the center atom or the center atom for the corner atoms. This effectively contributes to the stability and strength of materials with a bcc structure. By recognizing the nearest neighbours' arrangement, scientists and engineers can predict how materials will respond to external forces and changes in temperature.