Calculate the number \((n)\) of atoms contained within (a) cubic cell, (b) a body centred cubic cell, (c) a face centred cubic cell.

The term 'simple cubic cell' refers to one of the simplest forms of crystal lattice structures. In a simple cubic cell, atoms are arranged in a way that they form a cube with one atom at each corner. But one crucial point to remember is that each corner atom is not unique to just one cube. Instead, it's shared with seven other neighboring cubes.

To understand this visually, think of one of those shared corner atoms as the tip of an iceberg. You only see a tiny portion peeking out of your cube, while the rest extends into the neighboring cubes. Because of this social sharing of atoms, each corner atom contributes only 1/8th to the total count within a unit cell. With eight corners in play, 8 multiplied by 1/8 gives us our magic number: precisely 1 atom per simple cubic cell.

A body-centered cubic (BCC) cell takes the simplicity of the cubic structure a step further. Just like the simple cubic, there's an atom at each of the eight corners. But, as the name suggests, this time there's an added bonus - an extra atom cozily positioned right at the center of the cell.

While the corner atoms continue their communal lifestyle, each contributing 1/8 of an atom to the total count, the body-centered atom doesn't share its personal space at all. It's a full atom that belongs entirely to the cell it's in. When we pull out our calculators, we combine these shared atoms with our wholly-owned central atom to find that a BCC cell contains 2 full atoms. This makes the body-centered cubic cell distinctively different from its simple cubic cousin.

The face-centered cubic (FCC) cell is a bit like an evolution in the cubic family, offering even more complexity and efficiency in structure. It not only has atoms at all eight corners but also boasts an atom smack in the middle of each of the six faces of the cube.

These face atoms are social butterflies, but unlike their corner counterparts, they only have to split their time with one other cube, each offering a 1/2 share. With six faces contributing half an atom each, they add up to 3 atoms. Add in the corner contributions from earlier, and we land at a total of 4 atoms for the face-centered cubic cell. The FCC structure is particularly well-loved in the material science community for its high packing density and unique properties.

Atom counting in unit cells isn't just about tallying up atoms; it's about understanding how they share space in a solid. Each type of cubic cell adopts a distinct configuration, with atoms positioned at strategic points of sharing and ownership.

Remember, when you count atoms, consider their contributions: corner atoms divide their allegiance between eight unit cells, giving just 1/8 to the count, while face-centered atoms are only split in half, contributing 1/2. The whole atoms, like those found at the body center in a BCC cell, count as 1, as they're exclusive to their cube. Recognizing these relationships and how they piecemeal together is the essential skill of unit cell atom counting. Through it, we gain insights into the inherent packing of these crystalline structures which directly influence the properties of materials.