Distinguish between the coordination number and the number of atoms per unit cell. Find both these numbers for; (i) a simple cubic lattice, (ii) a body centred cubic lattice and (iii) a face centered cubic lattice?

The coordination number is a key concept in solid state physics. It defines the number of nearest neighbor atoms to a given atom in a crystal lattice. Imagine standing in a room filled with friends. If you're at the center, your coordination number is the number of friends that are close enough to touch you. Coordination number is crucial because it affects the stability and properties of the crystal. For instance, a higher coordination number generally means a more stable, dense, and sometimes stronger structure.

A simple cubic lattice (also known as primitive cubic) is the most basic unit cell structure. In this arrangement:
* Each atom is positioned at the corners of the cube.
* The coordination number is 6, meaning each atom touches 6 other atoms.
* The number of atoms per unit cell is calculated by recognizing that each corner atom is shared by 8 unit cells. Hence, there is effectively 1 atom per unit cell: 8 corners x 1/8 atom per corner = 1 atom.

The body-centered cubic (BCC) lattice is a bit more complex. Here:
* In addition to the atoms at the corners of the cube, there is one atom completely inside the center of the cube.
* The coordination number is 8, so each atom touches 8 other atoms.
* For the number of atoms per unit cell, you have to add up the contributions: 1 atom at the center, plus 8 corner atoms shared among 8 unit cells. This gives a total of 2 atoms: 1 (from the center) + (8 corners x 1/8) = 2 atoms.

The face-centered cubic (FCC) lattice is even more densely packed. Here:
* Atoms are located at each corner and the centers of each face of the cube.
* The coordination number is 12, so each atom touches 12 other atoms.
* Calculating the number of atoms per unit cell involves recognizing that face atoms are shared with two unit cells and corner atoms are shared among 8 unit cells:
* There are 6 face atoms x 1/2 contribution each = 3 atoms.
* Plus, 8 corner atoms x 1/8 contribution each = 1 atom.
* This totals to 4 atoms per unit cell: 3 (from faces) + 1 (from corners) = 4 atoms.

The number of atoms per unit cell refers to the total number of atoms effectively contained in a single unit cell of a lattice. To understand this, think of how atoms are shared among neighboring unit cells:
* In a simple cubic lattice, atoms at the corners are shared by 8 unit cells, resulting in an effective count of 1 atom per unit cell.
* In a BCC lattice, with one atom at the center and corner atoms shared among 8 cells, the total is 2 atoms per unit cell.
* In an FCC lattice, face atoms shared with one other cell and corner atoms shared among 8 cells, results in 4 atoms per unit cell.
This concept is foundational for calculating the density and other properties of crystalline materials.