If the radius of the tetrahedral void is r and radius of the atoms in close packing is R, derive the relation between r and R?

‚ÄčA tetrahedral void may be represented in a cube. In which there spheres form the triangular base, the fourth lies at the top and the sphere occupies the tetrahedral void. 

Let the length of the side of the cube = a
Radius of sphere = R
Radius of void = r


From right angled triangle ABC, face diagonal

  AB = AC2+ BC2 = a2+ a2 = 2a

As spheres A and B are actually touching each other, face diagonal AB = 2R
therefore,  2R = 2a 
  R = 1/√2 a      ....(1)


 Again from the right angled triangle ABD

  AD= AB2+ BD2= 2a2+ a2 = 3a

But as small sphere that is void touches other spheres, evidently body diagonal AD = 2(R + r)
therefore, 2(R + r) = 
3a
or R + r = 3/2 a ....(2)
 
Dividing equation (2) by equation (1) 
(R + r)/R = 32a / 12a = 32
1 + r/R =
32 = 1.225
r/R = 1.225 – 1 = 0.225

 r = 0.225 R

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root R=r

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