In solids the constituent particles are arranged in such a way that there is minimum space between them. Therefore, their structure is called Close Packed Structure. If the constituent particles are treated as homogeneous hard, solid spheres, we can form three-dimensional, compact complex structures in the following three steps.

## Close Packed in one Dimension

The spheres are placed in a row touching each other to form a smooth compaction in one dimension, as shown in the figure. In this type of Close Packed Structure, each sphere touches two adjacent spheres. The number of adjacent spheres that a sphere touches is called its coordination number. Hence, the coordination number of spheres in a One Dimensional, Close Packing is 2.

## Close Packed in two Dimension

Rows of one dimensional spheres as described above are arranged to form two dimensional Close Packed. This arrangement can be done in two ways. Due to which (i) square close packed and (ii) hexagonal close packed structures are formed.

### Square close packed in two Dimension

The second row is placed near the first row in such a way that the circles’s center of both the rows are horizontally and vertically in the same line. If the first row is called an A type row, then the second row will also be A row due to the same being the first. Similarly AAAA …. type of arrangement can be obtained by arranging multiple rows. As shown in Fig.

In this arrangement four other spheres are adjacent to each of spheres. They touch each other. Hence, the combination number of spheres is 4. If the centers of adjacent four spares are added, a square is obtained. Therefore, this class is called a Square close packed structure.

### Hexagonal close packing in two Dimension

The second row is placed near the first row in this way. So that the balls of second row fall into the depression (trough) made by the balls of first row. The third row is placed in the same way as the first row in such a way that the centers ofir circles are in one line. If the first row is called A then the second row is called B, And the third row being the same as the first would be called A, thus ABABA. The structure is obtained as shown in the figure.

In this arrangement each sphere is in contact with six adjacent spheres. If the centers of six spheres are joined by straight lines, then the shape of a hexagon is obtained. Therefore, this structure is called hexagonal close packed structure. In this case the free space is less than the square free packaging, so this arrangement is more efficient.

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## Close Packing in three dimension

We have seen in the second step on making Close Packed structures that there are two types of two-dimensional shapes of spheres. (i) square close packed and (ii) hexagonal close packed. Now a three dimensional structure can be obtained by stacking these two dimensional layers one on top of other. Now we will see how many types of three dimensional structures can be obtained from these two dimensional layers.

### Three Dimensional close packing from two dimentional square close packing layers

To achieve this type of compaction, the second layer is placed on the first layer in such a way that the shells of top layer are just above the spheres of first layer. In this type of arrangement, the centers of spheres of both the layers lie in a line horizontally and vertically. The other layers can also be placed in the same way. If the arrangement of spheres of first layer is called A type arrangement, then the arrangement of second and third and other layers will also be of A type. In this way AAAA .. type of structure is formed in stereoscopic, which is shown in Fig.

Thus formed lattice is the normal cubic lattice and its unit cell is the primitive cubic unit cell.

### Three Dimensional close packing from Hexagonal close packing in two Dimension

In order to form a three-dimensional, close packed structure from two-dimensional hexagonal close packed layers, these layers are placed on each other as follows.

#### 1. lay the second layer on the first layer

The second layer is placed on the first layer in such a way that the shells of second layer fall into the depression formed by the shells of first layer. Thus the centers of spheres of first and second layers will not be on the same line. Hence, both the layers represent different positions. If the first layer is represented by A, then the second layer will be represented by B. As shown in Fig. Each layer of two-dimensional hexagonal close packed has some gaps (empty spaces) which are triangular in shape. There are two types of triangular vacancies. In one row the top of triangle is up and in the other row is down. A tetrahedral viod is formed whenever a sphere of second layer lies above the gap in the first layer. When the centers ofse four spheres are joined, a rhombus is formed. That’s why this vacancy is called a tetrahedral vacancy. These are represented by t in the figure. The spacing shown by O in the figure is on top of triangular spacing of first layer and the top of second is on the bottom. O spaces are made up of six circles. The octahedral shape is formed by joining the centers ofse six spheres. Hence, these voids are called octahedral voids. The tetrahedral and octahedral vacancies are shown separately in the figure.

#### 2. lay the third layer on the second layer

The third layer can be placed over the second layer in two ways.

- The spheres of third layer should be placed just above the tetrahedral gaps of second layer. In this situation, the spheres of third layer come just above the spheres of first layer. That is, their centers lie in a line. Thus the third layer also becomes (A) the same as the first layer. In this way, the alternating layers are repeated in sequence. So became ABABAB….. pattern. It is shown in Fig. This structure is called hexagonal close packed (hcp) structure. This type of structure is found in metals like Mg Lin etc.

2. Another way to place the third layer over the second is as follows. The third layer is placed on the second layer in such a way that its spheres are just above the octahedral voids and cover them completely. Thus the spheres of third layer do not lie in the same line as the spheres of first or second layer. If this arrangement is called C type arrangement. The fourth layer is similar to the first layer (A). Thus ABCABC… Order is made. This arrangement is shown in the figure.

This structure is called cubic close packing structure (ccp) or face centered cubic (fcc) structure. Metals like Cu, Ag etc. crystallize in this type of structure. Both the hexagonal close packed (hcp) and the cubic-close-packed (ccp) structures have a high degree of compaction. and fills 74% of crystal space. In both, each sphere is in contact with 12 other spheres, so their coordination number is 12.

## Interstital vids or Interstetial Sites

Some gaps are left in the close packing of spares. These spaces are called interstitial holes or interstitial voids. For example, in the hcp and ccp structures of spheres, 26% of space remains vacant. These spaces are called holes, voids. These site or voide are of two types.

### Tetrahedral site or voide

This hole is made up of four identical spheres. In this, three spheres are in a plane touching each other and the fourth sphere is placed above or below the hole made by them. The centers ofse four spheres form the four corners of a rhombus and at the center of this tetrahedron there is an empty space between the four circles, which is called a tetrahedral hole. The size of tetrahedral hole r = 0.225R, where R = radius of sphere and r is the radius of tetrahedral hole.

#### Finding the relationship between the radius r of a tetrahedral hole and the radius R of a sphere

In the figure, a tetrahedral hole is shown with a black circle.

In Fig. AB = Face diagonal and

AD = Body diagonal.

The edge of cube = a.

Let the radius of hole r and radius of spare is R.

AB = √AC² + BC² = √2a

The spheres touch each other at points A and B.

Therefore

AB = 2R

ie

2R = √2 a

According to right angled triangle ABD

AD = √AB² + BD²

= √2a²+a²

AD = √3 a

But AD = 2r+2R

Therefore 2r+2R = √3a

r+R = √3a/2

since a = √2R

Therefore r+R = √3×√2R/2

r = (√3×√2R/2) – R

r = R[(√3/√2) – 1]

r = R[0.225]

either r/R = 0.225

#### symptoms of tetrahedral hole

The radius of hole (r) is much smaller than the radius of sphere (R).

Total number of tetrahedral holes in a structure is the 2x of spares’s number.

The coordination number (covalent number) of a tetrahedral hole is 4.

The shape of a tetrahedral hole is not tetrahedral, but the spheres from which the hole is formed are located in a tetrahedral arrangement.

### Octahedral sites or voids

This hole is made by touching each other of six spheres. Four ofse spheres touching each other in a plane are placed in such a way that their centers are at the four corners of square. A circle is placed on the upper hole ofse four circles and a circle is kept on the lower hole. There is an octahedral hole for each sphere.

Size of octahedral hole (r) = 0.414R

Here R is the radius of sphere.

#### To find the relation between the octahedral hole’ s radius (r) and the sphere’ s radius (R).

The cross-section of an octahedral is shown in the figure. Here the black circle represents the octahedral hole. The centers of spheres A, B, C, D form a square. whose each side = a.

AC= √AB² + BC² = √a² +a² = √2a

AC = 2r + 2R

2r + 2R = √2a

AB = 2R या a = 2R.

2r + 2R = √2.2R

r = R (√2 – 1)

r/R = 1.414 – 1 = 0.414

r = 0.414R

#### Symptoms of octahedral holes

- The hole’s radius (r) is less than that sphere’s radius (R).
- The total number of octahedral holes in a structure is equal to the number of spheres.
- The coordinate number of an octahedral hole is 6.
- The shape of an octahedral hole is not octahedral, but an octahedral shape is formed by joining the centers of six spheres (which form an octahedral hole).

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