Close Packing In Solids In Three Dimensions

Crystalline solids exhibit a regular and repeating pattern of constituent particles. The diagrammatic representation of three-dimensional arrangements of constituent particles in a crystal, where each particle is depicted as a point in space is known as a crystal lattice. In a crystal lattice, the atoms are very closely packed, leaving very little space between them. Close packing in solids in three dimensions is discussed below:

• Three-dimensional close packing from two-dimensional square close-packed layers:

In this kind of close solid packing, the second layer is placed over the first layer in such a way that the spheres of the upper layer are exactly above those of the first layer. In other words, spheres of both the layers are perfectly aligned, horizontally and vertically. Let us name the arrangement of spheres in the first layer as ‘A’ type since all the layers have the same arrangement, the lattice can be observed to follow AAA…. type pattern. This kind of lattice is better known as a simple cubic lattice.

• Three-dimensional close packing from two-dimensional hexagonal close-packed layers:

The three-dimensional close-packed structure can be generated by placing layers one over the other.

Placing the second layer over the first layer:

In this kind of close packing, a second layer similar to the below layer is placed in such a way that the spheres of the second layer are placed in the depressions of the first layer. Since the spheres of the two layers are aligned differently, if the first layer is termed as ‘A; the second layer can be termed as ‘B’. We notice that a tetrahedral void is formed wherever a sphere of the second layer is above the void of the first layer (or vice versa). Whereas at other places, we observe that the triangular voids in the second layer are above the triangular voids in the first layer, such that the triangular shapes of these do not overlap. Such voids are known octahedral voids and are surrounded by six spheres.

We can easily calculate the number of these two types of voids. Let the number of close packed spheres be N, then:

The number of octahedral voids generated = N

The number of tetrahedral voids generated = 2N

Placing the third layer over the second layer:

There are two prominent ways in which the third layer can be placed over the second layer:

Covering Tetrahedral Voids

In this kind of close packing in solids, the spheres of the third layer are exactly aligned with those of the first layer. Thus, if we name the first layer as ‘A’ and second as ‘B’, then the third layer can again be named as ‘A’. This pattern can be written as ABAB ……. pattern. This structure is better known as a hexagonal close-packed (hcp) structure.

Covering Octahedral Voids

In this kind of close packing, the spheres of the third layer are not aligned with those of either the first or the second layer. Thus, if we name the first layer as ‘A’ and second as ‘B’, then the third layer will be named as ‘C’. This pattern of layers can be written as ABCABC……….. This crystal structure is called cubic close-packed (ccp) or face-centered cubic (fcc) structure.

In both of them, the coordination number is 12 as each sphere is in contact with twelve spheres.74% space in the crystal is filled in these kinds of close packing.

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