How many lattice points belong to a face centered cubic unit cell?
- 1
- 2
- 4
- 3
The Correct Option is C
Solution and Explanation
In a face-centred cubic (FCC) lattice, atoms are present at:
• 8 corners of the cube
• 6 face centres
Each corner atom is shared by 8 unit cells, so contribution per corner atom is:
\(\frac{1}{8}\)
Each face-centred atom is shared by 2 unit cells, so contribution per face atom is:
\(\frac{1}{2}\)
Now total number of lattice points in one FCC unit cell is:
\(8 \times \frac{1}{8} + 6 \times \frac{1}{2}\)
\(= 1 + 3 = 4\)
Hence, number of atoms per FCC unit cell = 4
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Concepts Used:
Unit Cells
The smallest portion of a crystal lattice which repeats in different directions to form the entire lattice is known as Unit cell.
The characteristics of a unit cell are:
- The dimensions are measured along the three edges, a, b and c. These edges can form different angles, they may be mutually perpendicular or may not.
- The angles held by the edges are α (between b and c) β (between a and c) and γ (between a and b).
Therefore, a unit cell is characterised by six parameters such as a, b, c and α, β, γ.
Types of Unit Cell:
Numerous unit cells together make a crystal lattice. Constituent particles like atoms, molecules are also present. Each lattice point is occupied by one such particle.
- Primitive Unit Cells: In a primitive unit cell constituent particles are present only on the corner positions of a unit cell.
- Centred Unit Cells: A centred unit cell contains one or more constituent particles which are present at positions besides the corners.
- Body-Centered Unit Cell: Such a unit cell contains one constituent particle (atom, molecule or ion) at its body-centre as well as its every corners.
- Face Centered Unit Cell: Such a unit cell contains one constituent particle present at the centre of each face, as well as its corners.
- End-Centred Unit Cells: In such a unit cell, one constituent particle is present at the centre of any two opposite faces, as well as its corners.