Question

If the radius of the octahedral void is r and radius of the atoms in close-packing is R, derive relation between r and R.

Answer 1

Let take r and R are the radii of the octahedral site and atoms respectively,
then use Pythagoras theorem we get
(2R)2 =(r+R )2 + (r+R)2
4R2 = 2(r+R )2
divide by 2 we get
2R2 =(r+R)2
take root both sides we get
R/2 =r+R
R/2-R=r
R=r(/2 -1)
value of /2 =1.414
r=R (1.414-1)
R=0.414 R

Answer 2

Using Pythagoras theorem we get;

(2R)2 = (R+r)2 +(R+r)2;

4R2   = 2(R+r)2;

We divide by 2;

2R2  = (R+r)2;

Take root both side we get;

R√2 =R+r;

R√2 – R = r;

r  = R(√2–1);

and √2= 1.414;

r  = R(1.414–1) ;

r  = 0.414 R