Question

In a face centered unit cell,the number of nearest neighbours?

• A. 8
• B. 12
• C. 6
• D. 14

Answer 1

The correct option is B 12

There are $12$ nearest neighbours to a given lattice point in a FCC lattice. We say that the coordination number of a lattice point in the FCC lattice is 12. The distance of the nearest lattice points in terms of the lattice parameter (i.e., edge length of the cubic unit cell) is $\frac{a}{\sqrt{2}}$.

One way one can get this is as follows. Consider the lattice point at the centre of the top face of an FCC unit cell. The four corners of this face are nearest neighbours to the central lattice point. The centres of four vertical faces are another nearest lattice points. And in a 3D packing a unit cell will be sitting on the top of our unit cell. The centres of four vertical faces of this top cube is also nearest to our central lattice point. This gives 12 nearest neighbours: 4 corners of the horizontal face, four centres of the four vertical faces of the bottom cube and four centres of the vertical faces of the top cube.

The nearest neighbours define a polyhedron which is called a coordination polyhedron. The 12 nearest neighbours of an FCC lattice define a polyhedron called cuboctahedron. This can be imagined as a polyhedron formed by the centres of 12 edges of a cube.