The number of atoms per unit cell in a simple cubic, face-centred cubic and body-centred cubic are....... respectively: (a) \(1,2,4\) (b) \(1,4,2\) (c) \(4,2,1\) (d) \(1,4,3\)

The simple cubic structure is one of the most fundamental and straightforward lattice arrangements in solid-state physics. In this structure, atoms are located at each corner of a cube-like a three-dimensional chessboard. Understanding how atoms contribute to the overall structure is important for grasping the concept of unit cells.

To visualize this, imagine cutting out a single cube from your big chessboard. You would notice that the corners are not exclusive to your single cube; rather, they are shared with adjacent cubes. With eight corners and each contributing only one-eighth of an atom to the unit cell, the total number of atoms is effectively one per unit cell.

• Corner atoms: Shared by 8 unit cells, each contributes 1/8th
• Total atoms per unit cell = 8 corners * (1/8 contribution per corner) = 1

Students often get confused, thinking that there should be eight atoms because of the eight corners. Remember, since these atoms are shared among several unit cells, only a fraction of each corner atom is part of any one unit cell.

The face-centered cubic structure, often abbreviated as FCC, is more complex and has a higher packing efficiency than the simple cubic structure. Here, besides the corner atoms, we have extra atoms on the center of each of the faces of the cube.

Let's dissect the contributions—the corners still present as in the simple cubic, each corner atom being a shared resource among eight surrounding unit cells. The face-centered atoms are unique in that only two unit cells share each atom on the faces, hence providing a half contribution. A cube has six faces, so we have six half-contributing atoms. Consequently, the total number of atoms per unit cell in a face-centered cubic structure is four.

• Corner atoms: 8 corners * (1/8 contribution per corner) = 1 atom
• Face atoms: 6 faces * (1/2 contribution per face atom) = 3 atoms
• Total atoms per unit cell = Corner atoms + Face atoms = 4

To avoid misconceptions, it's key to stress that each face atom is counted half because it's shared by only two unit cells—kind of like sharing a pizza between two friends.

Finally, we look at the body-centered cubic structure, abbreviated as BCC. What sets BCC apart from simple cubic and face-centered cubic structures is the additional atom located at the very center of the cube. This center atom is not shared with any other unit cells and so contributes fully to the total atom count.

The corner atoms continue to follow the same rule as in simple and face-centered cubes—each provides an eighth of an atom to our specific unit cell. Adding up the atoms from the corners and the one in the center, we find there are 2 atoms per unit cell in a body-centered cubic structure.

• Corner atoms: 8 corners * (1/8 contribution per corner) = 1 atom
• Central atom: 1 central atom fully inside the unit cell = 1 atom
• Total atoms per unit cell = Corner atoms + Central atom = 2

It's important for students to recognize that only the central atom is exclusive to the body-centered cubic unit cell, which is why it's counted as a whole atom, distinguishing the BCC from other cubic structures.