Question

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

Answer 1

The unit cell, or building block of a crystal, is the smallest repeating unit in the crystal lattice. The identical unit cells are defined in such a way that they occupy the available area without overlapping. A crystal lattice is a three-dimensional arrangement of atoms, molecules, or ions within a crystal. It consists of a large number of unit cells. Every lattice point is occupied by one of the three component particles.

Complete answer:
A particle's contribution is always determined by the number of unit cells with which it is shared, which is determined by its location. A cubic unit cell's edge is shared by four distinct unit cells. A crystal lattice is formed by the joining of several unit cells. An atom, a molecule, or an ion occupy each lattice point. There are three sorts of unit cells based on these arrangements.
Cubic unit cell (primitive): Atoms may be found in all four corners of the unit cell.
Atoms are present at the unit cell's corners as well as the body center in a body-centered cubic unit cell.
Atoms are present in the unit cell's corners and face centers in a face-centered cubic unit cell.
We can compute the contribution of each atom to each of the above-mentioned unit cells, as well as the total number of atoms present in each of them.
The contribution of each atom to that unit cell is determined by the number of unit cells with which it is shared. If it is present in only one unit cell, such as the body-centered atom, its contribution is 1, if it is shared by two unit cells, it contributes $\dfrac{1}{2}$, if it is shared by four unit cells, it contributes $\dfrac{1}{4}$, and so on. Because atoms at the body's core aren't fully present in a single unit cell, their contribution will be equal to one.
For atoms at the unit cell's corners: When numerous unit cells are joined to form a crystal lattice, the corners of eight unit cells come together at one location. It would appear as four unit cells above it and four unit cells below it, sharing the corner in the middle. As a result, one atom in the corner is shared by 8 unit cells and therefore contributes $\dfrac{1}{8}$ of the total.
I) An atom in a cubic unit cell's corner is shared by eight neighboring unit cells. As a result, one unit cell shares one-eighth of the atom.
(ii) An atom in the cubic unit cell's body center is not shared with its neighboring unit cell. As a result, the atom solely belongs to the unit cell in which it is found, and its contribution to the unit cell is 1.

Note:
We multiply the contributions of atoms at various lattice locations by the amount of such atoms to get the total number of atoms in a unit cell, and then add them all together. For instance, in a basic cubic cell with atoms at all four corners, the total number of atoms is 8 corners x $\dfrac{1}{8}$ = 1 atom. In the case of a bcc lattice, it will be = $8(\dfrac{1}{8})+\text{ }6\text{ (}\dfrac{1}{2})=\text{ }4$ atoms, and so on.