Size of Tetrahedral void

Size of Tetrahedral void

Relationship between Radius of Tetrahedral Void (r) and Radius of Sphere (R)

Tetrahedral Void and size of tetrahedral void

Let us consider an FCC unit cell, if we divide this unit cell into 8 equal units by passing an imaginary plane through the center of the X, Y and Z axis. Then every cubic unit has 1 tetrahedral void at the body center.

Size of Tetrahedral Void or Relationship between radius of tetrahedral void and radius of sphere

Consider edge length of cubic unit ‘a’ cm, then:

AB = a   ,   BC = a   ,   CD = a

First calculate the face diagonal of the cube:

AC2  =  AB2 + BC2   =   a2 + a2   =  2a2  

Now calculate the body diagonal of the cube:

AD2  =  AC2 + CD2  =  2a2 + a2   =  3a2

Now the ratio of body diagonal to face diagonal will be:

 \( \frac{AD}{AC}=\frac{\sqrt{3}.a}{\sqrt{2}.a}=\frac{\sqrt{3}}{\sqrt{2}} \)  ……….(i)

In term of Radius of atom (R) and Radius of tetrahedral void (r):

AC = 2R    ,    AD = 2R +2r

Now the ratio of body diagonal to face diagonal will be:

 \( \frac{AD}{AC}=\frac{2R+2r}{2R}=1+\frac{r}{R} \) ………..(ii)

Compare equation (ii) with (i):

\( 1+\frac{r}{R}=\frac{\sqrt{3}}{\sqrt{2}}~~~,~~~\frac{r}{R}=\frac{\sqrt{3}}{\sqrt{2}}-1 \)

\( \frac{r}{R}=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{2}}=\frac{1.732-1.414}{1.414} \)

\( \frac{r}{R}=0.225~~~~or~~~~r=0.225R \)

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