Show that the radius of the particle that can be placed in a tetrahedral void of an hcp or a ccp structure of particles, without disturbing the close-packed structure, should not exceed 0.225 times the radius of the packed particles.

Explanation

In an hcp or ccp structure, the void that is surrounded by 4 nearest neighbour particles occupying the 4 corners of a regular tetrahedron is called a tetrahedral void. 4 particles surrounding a tetrahedral void are in contact with one another. A tetrahedral void is created when 4 particles are placed at alternate corners of a cube as shown in the figure, Suppose, the radius of each packed particle = R & that of tetrahedral void = r.

As the particles are in contact, face diagonal, AE = R + R = 2R
If the edge length of the cube be a, then it is clear from the figure that the face diagonal, AE = `sqrt(a^(2)+a^(2))=asqrt2`
`therefore" "2R=asqrt2or,a=Rsqrt2`
It is also clear from the figure that the body diagonal, `AD=sqrt(AE^(2)+ED^(2))=sqrt((asqrt2)^(2)+a^(2))=asqrt3=Rsqrt2xxsqrt3=Rsqrt6`
If a particle with radius 'r' is placed in the tetrahedral void,
AF = `1/2xx` body diagonal = `1/2xxAD=1/2xxRsqrt6=Rsqrt(3/2)=R+r`
or, `R(sqrt(3/2)-1)=ror,Rxx0.225=r`
Therefore, radius of the particle that can be placed in a tetrahedtral void without disturbing the close-packed structure should be equal to 0.225 times the radius of packed particles.