In the formation of crystals, the constituent particles (atoms, ions or molecules) get closely packed together. The closely packed arrangement is that in which maximum available space is occupied. This corresponds to a state of maximum density. The closer the packing, the greater is the stability of the packed system.
(1) Close packing in two dimensions : The two possible arrangement of close packing in two dimensions.
(i) Square close packing : In which the spheres in the adjacent row lie just one over the other and show a horizontal as well as vertical alignment and form square. In this arrangement each sphere is in contact with four spheres.
(ii) Hexagonal close packing : In which the spheres in every second row are seated in the depression between the spheres of first row. The spheres in the third row are vertically aligned with spheres in first row. The similar pattern is noticed throughout the crystal structure. In this arrangement each sphere is in contact with six other spheres.
(2) Close packing in three dimensions : In order to develop three dimensional close packing, let us retain the hexagonal close packing in the first layer. For close packing, each spheres in the second layer rests in the hollow at the centre of three touching spheres in the layer as shown in figure. The spheres in the first layer are shown by solid lines while those in second layer are shown by broken lines. It may be noted that only half of the triangular voids in the first layer are occupied by spheres in the second layer (i.e., either b or c). The unoccupied hollows or voids in the first layer are indicated by (c) in figure.
There are two alternative ways in which species in third layer can be arranged over the second layer,
(i) Hexagonal close packing : The third layer lies vertically above the first and the spheres in third layer rest in one set of hollows on the top of the second layer. This arrangement is called ABAB ?. type and 74% of the available space is occupied by spheres. This arrangement is found in Be, Mg, Zn, Cd, Sc, Y, Ti, Zr, Tc, Ru.
(ii) Cubic close packing : The third layer is different from the first and the spheres in the third layer lie on the other set of hollows marked ?C? in the first layer. This arrangement is called ABCABC?.. type and in this also 74%
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Limiting Radius ratios and Structure
(2) Tetrahedral void : A tetrahedral void is developed when triangular voids (made by three spheres in one layer touching each other) have contact with one sphere either in the upper layer or in the lower layer.
Tetrahedral void
The number of tetrahedral voids is double the number of spheres in the crystal structure.
\[\frac{r}{R}=0.225\]
where, r is the radius of the tetrahedral void or atom occupying tetrahedral void.
R is the radius of spheres forming tetrahedral void.
(3) Octahedral void : This type of void is surrounded by six closely packed spheres, i.e. it is formed by six spheres.
The number of octahedral voids is equal to the number of spheres.
\[\frac{r}{R}=\,0.414\]
(4) Cubic void : This type of void is formed between 8 closely packed spheres which occupy all the eight corner of cube.
\[\frac{r}{R}=0.732\]
The decreasing order of the size of the various voids is,
Cubic > Octahedral > Tetrahedral > Trigonal
