Question

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

Short answer

The number of atoms per unit cell in a simple cubic structure is 1, and the number of nearest neighbours is 6. Therefore, the correct answer is (a) 1,6.

Step-by-step solution

Analyze the Simple Cubic Structure

In a simple cubic (sc) lattice, there is one atom at each corner of the cube. Since each corner atom is shared by eight adjacent cubes, only 1/8th of each corner atom is within one unit cell.

Calculate the Number of Atoms Per Unit Cell

In the simple cubic structure, there are 8 corner atoms per unit cell. Each corner atom contributes 1/8th to the unit cell. Hence, the total number of atoms per unit cell is 8 * (1/8) = 1.

Determine the Number of Nearest Neighbours

In a simple cubic lattice, each atom has 6 nearest neighbours: one on each face of the cube. There is one neighbour in the center of each of the six faces adjacent to each corner.

Key concepts

Number of Atoms per Unit Cell

Understanding the basic building blocks of crystalline solids is essential for grasping various physical properties such as conductivity, strength, and density. One such fundamental concept is the number of atoms per unit cell in a crystalline structure. In the case of a simple cubic structure, calculations hinge on the fact that each corner of the cube is shared among eight neighboring unit cells. This implies that an individual unit cell only 'owns' an eighth of each atom that resides on its corners.

Since a simple cubic cell has eight corners, and assuming each corner is occupied by an atom, we multiply the eight corners by the fractional ownership of 1/8 per atom, resulting in a total of exactly one atom per unit cell. This is compactly expressed by the equation: \[ \text{Number of atoms per unit cell} = 8 \cdot \frac{1}{8} = 1 \]. Thus, option (a) reflects the correct number of atoms per unit cell for a simple cubic lattice.

Number of Nearest Neighbours

Neighbouring atoms have a direct impact on an atom's stability and bonding characteristics, making the concept of 'nearest neighbours' critical in materials science. In simple cubic structures, each atom can be visualized as being surrounded by six immediate neighbours. This arrangement is a direct consequence of the cube's geometry. Each of the cube's faces is shared with an adjacent unit cell, and it is in the center of each of these faces that we find one of the six nearest neighbours.

To put it simply, imagine an atom positioned at the corner of the cube. Directly adjacent to this atom, at the center of each face of the cube, is another atom. There are six faces to a cube; thus, each corner atom has a direct relationship with six other atoms, one on each face. This atomic arrangement is succinctly described as having six nearest neighbours, which corresponds to the first part of option (a) from our multiple-choice answers.

Cubic Lattice

The cubic lattice is a cornerstone of crystallography and materials science, representing one of the simplest and most symmetric three-dimensional arrangements of points in space. It is a type of Bravais lattice, which is a classification system used to categorize crystal structures based on their geometric arrangement.

In a cubic lattice, points (which can represent atoms, ions, or molecules) are arranged at the corners of a cube. This arrangement leads to a highly symmetrical structure, where each axis is of equal length, and the angles between them are all 90 degrees. There are three main types of cubic lattices: simple cubic, body-centered cubic, and face-centered cubic. Each type has its own unique arrangement of points and resultant properties.

The 'simple cubic' variety, as discussed here, is characterized by points at the cube's corners only. Its simplicity provides an excellent model for understanding more complex crystal structures. It forms the foundational geometry from which other cubic lattices can be derived and understood. The close-packed versions, like face-centered cubic or body-centered cubic, contain additional points either at the center of each face or in the center of the cube, respectively.