Question

Calculate the radii of octahedral voids in terms of the radius of the spheres forming close-packed structures.

Answer 1

An octahedral void is generated when six nearest neighbours (atoms) are positioned in an octahedral arrangement and mutually touch. If the constituent particles are supposed to be spheres, four spheres form a plane, one above and one below it. Assume the radius of each sphere is R and the octahedral void is r.

A sphere with radius r fits snugly into the octahedral void shown in the figure. The drawing shows a shaded sphere sitting in the nothingness. The spheres above and below the plane are not shown in the figure. Using the right-angled triangle ECD, we have

CD = `sqrt (CE^2 + ED^2`   ...(i)

From the figure, it is clear that

CD = R + R

= 2R

EC = ED

= R + r

Putting the values in eq. (i), we have

`2R = sqrt((R + r)^2 + (R + r)^2)`

`2 R = sqrt 2 (R + r)`

`sqrt 2 R = R + r`

`r = (sqrt 2 - 1)R`

`r/R = (sqrt 2 - 1)`

`r/R = 1.414 - 1`

`r/R = 0.414`

A particle with a radius smaller than 0.414 times that of the constituent particle can only be inserted in an octahedral vacuum without disrupting the crystal lattice (close-packed structure).