Solid State

In a body-centered cubic (BCC) lattice, there is one atom at each corner of the cube and one atom at the center of the cube. The atoms at the corners and the body center touch each other along the body diagonal.

Let the edge length of the cube be and the radius of the sphere (atom) be .

The body diagonal of the cube has a length of .

Along this diagonal, the atoms are arranged as:
Corner atom → Body-center atom → Opposite corner atom.

Hence, the total distance along the diagonal equals the sum of their diameters:

Therefore, the relation between the radius and the edge length is:

The type of cubic lattice can be identified using the following formula: where: - = Number of atoms per unit cell,
- = Density ,
- = Avogadro’s number ,
- = Edge length ,
- = Atomic mass .
Substituting the values into the formula:
After solving, we obtain , which corresponds to a Face-Centered Cubic (FCC) lattice.

Density of a crystalline solid is calculated as the mass per unit volume. For a unit cell with edge length , the volume is given by .
Number of Atoms per Unit Cell: represents the number of formula units per unit cell. For instance, equals 1 in a simple cubic lattice, 2 in a body-centered cubic (BCC) lattice, and 4 in a face-centered cubic (FCC) lattice.
Mass of the Unit Cell: The mass of one formula unit is the molar mass divided by Avogadro's number . The mass of the unit cell then is:
Volume of the Unit Cell:
Density Calculation: Density is mass divided by volume. For the crystalline solid, it is calculated as: Simplifying this: The formula ties together molar mass, density, and unit cell dimensions:
Details included:
- = Number of formula units per unit cell,
- = Molar mass,
- = Avogadro’s number ,
- = Edge length of the unit cell.