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STUDY OF INTERSTITIAL VOIDS:

Packing sphere in any pattern in any fashion generate some empty space between them these space (gaps) are known as voids they are named according to the orientation of spheres constituting voids.
(A)  VOID IN TWO DIMENSIONS:
(1) Triangular Void: The triangular voids are found in the planes of the close packed structures, whenever three spheres are in contact in such a fashion. Coordination number of trigonal void is three.
Where, R= Radius of the sphere, r = maximum radius of a sphere that can be placed inside the void.
(2) Square void: The square voids are found in the planes of the close packed structures, whenever three spheres are in contact in such a fashion. The coordination number of square void is four.
(B) VOIDS IN THREE DIMENSIONS:
(3) Cubical void: The cubical void is generally not found in closed packed structures, but is generated as a result of distortions arising from the occupancy of voids by larger particles. Cubical voids found in simple cubic unit cell and it is 3-dimensional and has coordination number eight (8).
(C) VOIDS IN CLOSE PACKING IN 3D (HCP AND CCP):
 In hcp as well as ccp only 74% of the available space is occupied by spheres. The remaining space is vacant and constitutes interstitial voids or interstices or holes. These are of two types
(1)  Tetrahedral voids     (2) Octahedral voids
(1) Tetrahedral voids: In close packing arrangement, each sphere in the second layer rests on the hollow (triangular void) in three touching spheres in the first layer. The centres of theses four spheres are at the corners of a regular tetrahedral. The vacant space between these four touching spheres is called tetrahedral void. In a close packing, the number of tetrahedral void is double the number of spheres, so there are two tetrahedral voids for each sphere 
Radius of the tetrahedral void relative to the radius of the sphere is 0.225

In a multi layered  close  packed structure , there is a tetrahedral hole above and below each atom hence there is twice as many tetrahedral holes as there are in close packed atoms.
DERIVATION OF RELATION: Derivation of the relationship between the radius (r) of the octahedral void and the radius (R) of the atoms in close packing. 
A sphere into the octahedral void is shown in the diagram. A sphere above and a sphere below this small sphere have not been shown in the figure. ABC is a right angled triangle. The centre of void is A.

(2) Octahedral voids: As already discussed the spheres in the second layer rest on the triangular voids in the first layer. However, one half of the triangular voids in the first layer are occupied by spheres in the second layer while the other half remains unoccupied. The triangular voids ‘b’ in the first layer are overlapped by the triangular voids in the second layer. The interstitial void, formed by combination of two triangular voids of the first and second layer is called octahedral void because this is enclosed between six spheres centres of which occupy corners of a regular octahedron 
In close packing, the number of octahedral voids is equal to the number of spheres. Thus, there is only one octahedral void associated with each sphere. Radius of the octahedral void in relation to the radius of the sphere is 0.414
DERIVATION OF RELATION: Derivation of the relationship between radius (r) of the tetrahedral void and the radius (R) of the atoms in close packing: To simplify calculations, a tetrahedral void may be represented in a cube as shown in the figure. In which there spheres form the triangular base, the fourth lies at the top and the sphere occupies the tetrahedral void. 
RADIUS RATIO RULES IN IONIC SOLIDS:
The structure of many ionic solids can be accounted by considering the relative sizes of the cation and anion, and their relative numbers. By simple calculations, we can work out as how
Many ions of a given size can be in contact with a smaller ion. Thus, we can predict the coordination number from the relative size of the ions.
Following conditions must be satisfied simultaneously during the stacking of ions of different sizes in an ionic crystal:
(1) An anion and a cation are assumed to be hard spheres always touching each other.
(2) Anions generally will not touch but may be close enough to be in contact with one another in a limiting situation.
(3) A cation should surround itself with as many anions as possible. Each ion tends to surround itself with as many ions of opposite sign as possible to reduce the potential energy. This tendency promotes the formation of close-packed structures.
RADIUS RATIO AND COORDINATION NUMBER:
LOCATING TETRAHEDRAL AND OCTAHEDRAL VOIDS:
(1) The close packed structures have both octahedral and tetrahedral voids. In a ccp structure, there is 1 octahedral void in the centre of the body and 12 octahedral void on the edges. Each one of which is common to four other unit cells. Thus, in cubic close packed structure.
Octahedral voids in the centre of the cube =1
Effective number of octahedral voids located at the 12 edge of = 12*1/4=3
Total number of octahedral voids = 4
(2) In ccp structure, there are 8 tetrahedral voids. In close packed structure, there are eight spheres in the corners of the unit cell and each sphere is in contact with three groups giving rise to eight tetrahedral voids
TETRAHEDRAL VOIDS LOCATION:
(3) Circles labelled T represent the centers of the tetrahedral interstices in the ccp arrangement of anions.  The unit cell "owns" 8 tetrahedral sites.
OCTAHEDRAL VOIDS LOCATION:

Circles labelled O represent centers of the octahedral interstices in the ccp arrangement of anions (FCC unit cell).  The cell "owns" 4 octahedral sites.

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