Question

The number of atoms per unit cell in a simple cubic, face centered and body centered cubic are ……………. respectively.
\[A.1,4,2\]
\[B.4,1,2\]
\[C.2,4,1\]
\[D.4,8,2\]

Answer 1

Hint: A unit cell is defined as the smallest repeating unit of a large crystal which has the same geometry. In order to find the number of atoms per unit cell first we have to identify the lattice structure and its corresponding unit cell.

Complete step by step answer:
In a simple cubic lattice, eight atoms or components are present at eight corners of the crystal lattice. The atom in the corner is shared by eight adjacent cubes, so the contribution of each atom will be \[\dfrac{1}{8}\] atom per unit cell.
Thus in simple cubic the number of atoms per unit cell =\[8X\dfrac{1}{8} = 1\].
In face centered cubic lattice, eight atoms or components are present at eight corners of the crystal lattice and six atoms are present at the faces. The atom in the corner is shared by eight adjacent cubes and the atoms in faces are shared by two adjacent cubes. So the contribution of each atom will be \[\dfrac{1}{8}\] atom per unit cell for atoms in corner and \[\dfrac{1}{2}\] atom per unit cell for atoms in faces.
Thus the number of atoms in face centered cubic =\[8X\dfrac{1}{8} + 6X\dfrac{1}{2} = 1 + 3 = 4\].
In a body centered cubic lattice, eight atoms or components are present at eight corners of the crystal lattice and one atom is present at the centre of the unit cell. The atom in the corner is shared by eight adjacent cubes and the atom in the centre is not shared by two adjacent cubes. So the contribution of each atom will be \[\dfrac{1}{8}\] atom per unit cell for atoms in corner and one atom per unit cell for atom in centre.
Thus the number of atoms in body centered cubic =\[8X\dfrac{1}{8} + 1x1 = 1 + 1 = 2\].
Therefore in a simple cubic, face centered and body centered cubic the numbers of atoms per unit cell are \[1\], \[4\] and \[2\] respectively.
Option (A) is correct.
Note:
The presence of atoms at corners, faces and centre can be identified by the lattice name. As a simple cubic has only eight atoms at corners, face centered has atoms in every face (six faces has six atoms), body centered has one atom at the centre. The sharing of each atom with adjacent has to be considered for evaluating the atoms per unit cell.