Derivation Radius of Tetrahedral and Octahedral Voids.

Dear Student,

There is no derivation of radius of tetrahedral and octahedral void. Derivation of the relationship between radius of the octahedral void and tetrahedral void  with radius of atoms in close packing is shown below :
For octagedral :

Consider triangle ABCBC2=AB2 + AC2(2R)2=(R+r)2+(R+r)2(2R)2=2(R+r)2or, 2R2=(R+r)2or, 2R=R+ror, r=2R-R=R(1.414-1)=0.414 R

For tetrahedral void:

Let the length of the side of the cube=aFrom right angled triangle ABC, face diagonalAB=AC2+BC2=a2+a2=2aAs spheres A and B are actually touching each other, face diagonal AB=2RSo, 2R=2aor, R=12a  ....(i)From right angled trainage ABDADAB2+BD2=2a2+a2=3aBut as small sphere touches other spheres, body diagonal AD=2(R+r)or, 2(R+r)=3aor,R+r=32a ...(ii)Dividing (ii) by (i)R+rR=32×aa2=32or, 1+rR=32=1.225or, rR=1.225-1=0.225or, r=0.225 R

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