What is the lowest and highest number of shared and unshared atoms in a unit cell of a FCC structure and a simple cubic structure?

Answer 1

Your question, if I understand it correctly, asks to explain the reasoning behind the coordination number, or number of adjacent atoms, of an atom in a simple cubic structure versus an atom in a face centered cubic structure (FCC).

Before we proceed, I would like to clarify one thing:

A unit cell of a simple cubic crystal has 1 atom, while a unit cell of FCC crystal has 4 atoms. This may be a little counterintuitive at first, but consider how the atoms are shared. For the simple cubic structure, there are eight individual atoms - one at each corner of the cube. The unit cell, however, has to share each atom with the 8 other adjacent cells. Thus a unit cell gets 8 atoms only 1/8 of the time, hence 8*(1/8) = 1 atom per simple cubic unit cell. Similarly, FCC has the 1 atom from simple cubic, plus half of the 6 atoms on each of it's faces. Thus, FCC has 4 atoms per unit cell.

On to the main question. In short, given a homogeneous, perfect crystal the coordination numbers of all the atoms are the same. All atoms are shared equal with their neighbors.

For the simple cubic case this is easy to see. Like the simple gumdrop creations of second graders, you can start at any gumdrop to make the creation. Any corner is the same relative to its neighbors as any other corner. For FCC the same is true.

Answer 2

In a face-centered cubic (FCC) structure, the lowest number of shared atoms in a unit cell is 4, and the highest number is 8. For unshared atoms, the lowest number is 0, and the highest is 4. In a simple cubic structure, the lowest number of shared atoms in a unit cell is 1, and the highest number is 8. For unshared atoms, the lowest number is 0, and the highest is 1.