Octahedral and Tetrahedral Voids

( Voids in FCC unit cell)
(Effective number of octahedral holes = 4, Effective number of tetrahedral holes = 8)

The close packing system involves two kinds of voids – tetrahedral voids and octahedral voids. The former has four spheres adjacent to it while the latter has six spheres adjacent to it. These voids are only found in either fcc or Hexagonal Primitive unit cells.

Let us first consider a fcc unit cell (as shown in the figure 8(a).). Not all the atoms of the unit cell are shown (for convenience).

Let us assume that there is an atom (different from the one that forms the fcc) at the center of an edge.

Let it be big enough to touch one of the corner atoms of the fcc. In that case, it can be easily understood that it would also touch six other atoms (as shown) at the same distance.

Such voids in an fcc unit cell in which if we place an atom it would be in contact with six spheres at equal distance (in the form of an octahedron) are called octahedral voids (as seen in fig. )

On calculation, it can be found out that a fcc unit cell has four octahedral voids effectively. The number of effective octahedral voids in a unit cell is equal to the effective number of atoms in that unit cell.

Let us again consider a fcc unit cell. If we assume that one of its corners is an origin, we can locate a point having coordinates (1/4 , 1/4 , 1/4 ).

If we place an atom (different from the ones that form the fcc) at this point and if it is big enough to touch the corner atom, then it would also touch three other atoms (as shown in the fig. 8(b) which are at the face centers of all those faces which meet at that corner.

Moreover, it would touch all these atoms at the corners of a regular tetrahedron. Such voids are called tetrahedral voids (as seen in fig. )

Since there are eight corners, there are eight tetrahedral voids in a fcc unit cell. We can see the tetrahedral voids in another way.

Let us assume that eight cubes of the same size make a bigger cube as shown in the fig. 8(c). Then the centers of these eight small cubes would behave as tetrahedral voids for the bigger cube (if it were face centered).

The number of tetrahedral voids is double then the number of octahedral voids. Therefore, the number of tetrahedral voids in hcp is 12.

Also Read :

→ Unit Cells & Crystal
→Bravais Lattices
→Close Packing of Spheres
→ Octahedral and Tetrahedral Voids
→ Radius Ratio Rules
→ Classification of Ionic Structures
→ Imperfections in a Crystal

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