# Location of Octahedral and Tetrahedral Voids in Unit Cell

#### In FCC unit cell

##### Location of Octahedral voids

One octahedral void is present at the body center of the cube and 12 octahedral voids are present on the center of the 12 edges of the cube. But each void on the edge center is shared by 4-unit cells. Hence, its contribution in the unit cell = 1/4

Therefore, the effective number of octahedral voids in the ccp or fcc structure \( =1+12\times \frac{1}{4}=1+3=4 \) voids

##### Location of Tetrahedral voids

The tetrahedral voids are found to be present on the body diagonals, two on each body diagonal at one-fourth of the distance from each end.

The 8 tetrahedral voids present in the ccp arises from the fact that there are 8 spheres at the corners of the unit cell and each sphere at the corner touches three spheres present on the face-centers of the three joining faces, each give rise to one tetrahedral void.

Thus, in ccp, total number of voids per unit cell = 8 (tetrahedral) + 4 (Octahedral) = 12.

Similarly, in hcp, total number of voids per unit cell = 12 (tetrahedral) + 6 (Octahedral) = 18.

###### Note: (Applicable only for FCC and HCP)

Let the number of closed packed spheres be N, then:

The number of tetrahedral voids generated = 2N

The number of octahedral voids generated = N

Total number of tetrahedral and octahedral voids = 3N

#### In HCP arrangement

##### Location of Octahedral voids

A total of 6 octahedral voids are located within the body of the HCP arrangement. 3 voids are between the first and second layer while 3 voids are between the second and third layer. All voids are located near to 3 alternate faces.

Therefore, the effective number of octahedral voids in the hcp arrangement **= 3 × 2 = 6 octahedral voids**

##### Location of Tetrahedral voids

- 8 tetrahedral voids are located within the body of the HCP arrangement. 4 voids are between the first and second layer while 4 voids are between the second and third layer.
- 4 × 2 = 8 voids (6 voids are located near to 3 alternate faces while 2 voids are along the central axis.)

- There are 4 tetrahedral voids present at the edge lengths representing the height of the hexagonal unit cell. 2 tetrahedral voids are along the each edge length. The contribution of each void is 1/3 because the length of each edge is shared with 3 unit cells.
- 6 × 2 × 1/3 = 4 voids

Therefore, the effective number of tetrahedral voids in the hcp arrangement **= 8 + 4 = 12 tetrahedral voids**

#### In BCC unit cell

##### Location of Octahedral voids

- 6 octahedral voids are present on the center of the 6 faces of the cube. Each void on the face center is shared by 2 unit cells. Hence, each void on face center contribute 1/2 part.
- 6 × 1/2 = 3 voids

- 12 octahedral voids are present on the centers of the 12 edges of the cube. But each void on the edge center is shared by 4-unit cells. Hence, each void on edge center contribute 1/4 parts
- 12 × 1/4 = 3 voids

Therefore, the effective number of octahedral voids in the BCC unit cell **= 3 + 3 = 6 voids**

##### Location of Tetrahedral voids

24 tetrahedral voids are present on the 6 faces of the cube. 4 voids on each face. Each void is shared by 2 unit cells. Hence the voids on face of the cube contribute 1/2 part.

Therefore, the effective number of tetrahedral voids in the BCC unit cell **= 24 × 1/2 = 12 voids**

#### In Simple or Primitive unit cell

There are no octahedral and tetrahedral voids in a simple unit cell. It consists of a cubic void, which are formed in the center of the 8 sphere. Cubic voids are larger than octahedral and tetrahedral voids.

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