Explain how much portion of an atom located at (i) corner and (ii) body- centre of a cubic unit cell is part of its neighbouring unit cell.

To understand how much of an atom located at different positions within a cubic unit cell contributes to neighboring unit cells, we need to consider the geometry of the unit cell and how atoms are shared among them. Let's break this down for both the corner and body-centered positions.

Atoms at the Corners of a Cubic Unit Cell

In a cubic unit cell, there are atoms located at each of the eight corners. Each corner atom is shared among eight adjacent unit cells. This means that when you look at one corner atom, only a fraction of it belongs to the unit cell you are examining.

• Since each corner atom is shared by eight unit cells, the contribution of one corner atom to a single unit cell is:
• Contribution from one corner atom: 1/8
• Therefore, if you sum up the contributions from all eight corner atoms in a unit cell, you get:
• Total contribution from corner atoms: 8 x (1/8) = 1 atom

This means that while there are eight corner atoms, they collectively contribute the equivalent of one whole atom to the unit cell.

Atoms at the Body Center of a Cubic Unit Cell

Now, let’s consider the atom located at the body center of the cubic unit cell. This atom is entirely contained within the unit cell and is not shared with any neighboring cells.

• Since the body-centered atom is completely inside the unit cell, its contribution is:
• Contribution from the body-centered atom: 1

This means that the body-centered atom contributes fully to the unit cell without sharing with any adjacent unit cells.

Summary of Contributions

To summarize:

• An atom at the corner of a cubic unit cell contributes 1/8 of its volume to that unit cell, and with eight corners, this totals to 1 whole atom.
• An atom at the body center contributes 1 whole atom to the unit cell.

This understanding is crucial in crystallography and materials science, as it helps in calculating the density and packing efficiency of different crystal structures. By knowing how atoms are shared, we can better understand the properties of materials at the atomic level.