Closed Packed Structure Part -1

Closed Packed structure

 In solids, the constituent particles are close-packed, leaving the minimum vacant space. Let us consider the constituent particles as identical hard spheres and build up the three-dimensional structure in three steps.

 Close Packing in One Dimension

There is only one way of arranging spheres in a one-dimensional close packed structure, that is to arrange them in a row and touching each other 

In this arrangement, each sphere is in contact with two of its neighbours. The number of nearest neighbours of a particle is called its coordination number. Thus, in one dimensional close packed arrangement, the coordination number is 2.

 

Close Packing in Two Dimensions

Two-dimensional close packed structure can be generated by stacking (placing) the rows of close packed spheres. This can be done in two different ways.

Square close packing

(i) The second row may be placed in contact with the first one such that the spheres of the second row are exactly above those of the first row. The spheres of the two rows are aligned horizontally as well as vertically. If we call the first row as ‘A’ type row, the second row being exactly the same as the first one, is also of ‘A’ type. Similarly, we may place more rows to obtain AAA type of arrangement as shown in Fig. In this arrangement, each sphere is in contact with four of its neighbours. Thus, the two dimensional coordination number is 4. Also, if the centres of these 4 immediate neighbouring spheres are joined, a square is formed. Hence this packing is called square close packing in two dimensions.

 

Hexagonal close packing

The second row may be placed above the first one in a staggered manner such that its spheres fit in the depressions of the first row. If the arrangement of spheres in the first row is called ‘A’ type, the one in the second row is different and may be called ‘B’ type. When the third row is placed adjacent to the second in staggered manner, its spheres are aligned with those of the first layer. Hence this layer is also of ‘A’ type. The spheres of similarly placed fourth row will be aligned with those of the second row (‘B’ type). Hence this arrangement is of ABAB type. In this arrangement there is less free space and this packing is more efficient than the square close packing. Each sphere is in contact with six of its neighbours and the two dimensional coordination number is 6. The centres of these six spheres are at the corners of a regular hexagon (Fig. ) hence this packing is called two dimensional hexagonal close-packing. It can be seen in Figure  that in this layer there are some voids (empty spaces). These are triangular in shape. The triangular voids are of two different types. In one row, the apex of the triangles are pointing upwards and in the next layer downwards.

 

Close Packing in Three Dimensions

 All real structures are three dimensional structures. They can be obtained by stacking two dimensional layers one above the other. In the last Section, we discussed close packing in two dimensions which can be of two types; square close-packed and hexagonal close-packed. Let us see what types of three dimensional close packing can be obtained from these.

 (i) Three-dimensional close packing forms two-dimensional square close-packed layers: While placing the second square close-packed layer above the first we follow the same rule that was followed when one row was placed adjacent to the other. The second layer is placed over the first layer such that the spheres of the upper layer are exactly above those of the first layer. In this arrangement spheres of both the layers are perfectly aligned horizontally as well as vertically as shown in Fig.. Similarly, we may place more layers one above the other. If the arrangement of spheres in the first layer is called ‘A’ type, all the layers have the same arrangement. Thus this lattice has AAA.... type pattern. The lattice thus generated is the simple cubic lattice, and its unit cell is the primitive cubic unit cell 

 

(ii) Three dimensional close packing from two dimensional hexagonal close packed layers: Three dimensional close packed structure can be generated by placing layers one over the other.

 

(a) Placing second layer over the first layer 

  • A two-dimensional hexagonal close-packed layer 'A' is taken and a similar layer 'B' is placed over it such that the spheres of the second layer are placed in the depressions of the first layer.
  • The spheres of the two layers are aligned differently, hence they are called A and B layers respectively.
  • All the triangular voids of the first layer are not covered by the spheres of the second layer, resulting in the formation of tetrahedral voids wherever a sphere of the second layer is above the void of the first layer.
  • Tetrahedral voids are called so because a tetrahedron is formed when the centers of four spheres are joined. They are marked as 'T' in Fig. and are shown separately in Fig.
  • The number of tetrahedral voids formed is twice the number of close-packed spheres in the layer. Hence, if the number of close-packed spheres is N, the number of tetrahedral voids generated is 2N.
  • In some places, the triangular voids of the second layer are aligned with those of the first layer, without overlapping. The triangular voids of the second layer have their apex pointing in the opposite direction to that of the first layer. These voids are surrounded by six spheres and are called octahedral voids. In Fig., these voids have been marked as ‘O’.
  • The number of octahedral voids depends on the number of close-packed spheres in the lattice. Let the number of close-packed spheres be N.
  • The number of octahedral voids generated in a lattice of N spheres is N.
  • Similarly, the number of tetrahedral voids generated in a lattice of N spheres is 2N.

(b) Placing third layer over the second layer 

  • When a third layer is placed over the second layer, there are two possibilities.
  • The first possibility is that the spheres of the third layer cover the tetrahedral voids of the second layer.
  • In this case, the spheres of the third layer are exactly aligned with those of the first layer.
  • This pattern of spheres is repeated in alternate layers, which is often written as ABAB.... pattern.
  • This structure is called a hexagonal close-packed (hcp) structure.
  • The hcp structure is found in many metals like magnesium and zinc.

(ii) Covering Octahedral Voids: 

  •  When the third layer is placed over the second layer, there are two possibilities.
  • The first possibility is that the tetrahedral voids of the second layer may be covered by the spheres of the third layer.
  • In this case, the spheres of the third layer are exactly aligned with those of the first layer. Thus, the pattern of spheres is repeated in alternate layers. This pattern is often written as ABAB ....... pattern.
  • This structure is called a hexagonal close-packed (hcp) structure. Metals such as magnesium and zinc have this type of arrangement of atoms.
  • The second possibility is that the third layer may be placed above the second layer in a manner such that its spheres cover the octahedral voids.
  • When placed in this manner, the spheres of the third layer are not aligned with those of either the first or the second layer. This arrangement is called a 'C' type.
  • Only when the fourth layer is placed, its spheres are aligned with those of the first layer. This pattern of layers is often written as ABCABC ...........
  • This structure is called a cubic close-packed (ccp) or face-centred cubic (fcc) structure. Metals such as copper and silver crystallise in this structure.

 

Both these types of close packing are highly efficient and 74% space in the crystal is filled. In either of them, each sphere is in contact with twelve spheres. Thus, the coordination number is 12 in either of these two structures. 

  

 Tetrahedral voids

 In solids with close packing of atoms, there are some empty spaces that exist between the atoms. These empty spaces are called voids. When two-dimensional close-packed layers are stacked on top of each other, some of these voids are created. One such void is the tetrahedral void.

Tetrahedral voids are the voids formed when two-dimensional close-packed layers of atoms are stacked over one another such that the spheres of the second layer occupy the depressions in the first layer. As a result, the triangular voids that remain unoccupied by the spheres of the second layer are called tetrahedral voids. These voids have a tetrahedral shape, and when four atoms of the third layer are placed around these voids, they form a tetrahedron.

 

The tetrahedral voids are characterized by their radius, which is typically around 0.155 times the radius of the spheres that form the layers. Tetrahedral voids are commonly found in ionic solids, where they are occupied by smaller ions like Li+, Na+, K+ or Mg2+.

 

An example of a crystal with tetrahedral voids is NaCl. In this crystal structure, Na+ ions occupy the octahedral voids, while Cl- ions occupy the face-centred cubic lattice. Each Cl- ion is surrounded by six Na+ ions, which form an octahedral arrangement around the Cl- ion. The Na+ ions, in turn, occupy the tetrahedral voids formed by the Cl- ions, and each Na+ ion is surrounded by four Cl- ions, which form a tetrahedral arrangement around the Na+ ion.

Tetrahedral voids are also found in molecular solids, where they are occupied by small molecules like H2O or NH3.

In summary, tetrahedral voids are the voids formed when two-dimensional close-packed layers of atoms are stacked over one another. They have a tetrahedral shape and are typically occupied by smaller ions or molecules.

 

Octahedral voids

 Octahedral voids are the empty spaces present between the spheres in a close-packed structure that are surrounded by six spheres arranged in an octahedral shape. These voids are formed by the alignment of the second layer of spheres over the triangular voids of the first layer.

The shape of an octahedral void is similar to a regular octahedron, which is a polyhedron with eight faces, each of which is an equilateral triangle. The center of the octahedral void is equidistant from the centers of the six surrounding spheres, and the distance between the spheres is approximately the same as the diameter of the sphere.

The number of octahedral voids present in a close-packed structure is equal to the number of close-packed spheres present in it. In other words, the ratio of the number of octahedral voids to the number of close-packed spheres is 1:1.

Octahedral voids play an important role in determining the properties of materials. For example, the presence of octahedral voids in metals can affect their mechanical and electrical properties. In some cases, the presence of impurities in the octahedral voids can lead to the formation of different phases or alloys.

 

 Cubic closed packed structure

Cubic close packing (ccp), also known as face-centered cubic (fcc) structure, is a type of crystal structure in which the constituent particles (usually atoms or ions) are arranged in a tightly packed pattern. In this structure, each atom has 12 nearest neighbors arranged in the form of a regular octahedron. The coordination number of this structure is 12.

The cubic close packing structure can be thought of as a three-dimensional stack of closely packed spheres, where each layer of spheres is placed in the depressions of the layer below it. The spheres in the second layer occupy the tetrahedral voids formed by the spheres in the first layer, and the spheres in the third layer occupy the octahedral voids formed by the spheres in the first and second layers. This arrangement of spheres results in a close packing of the constituent particles, with a packing efficiency of 74%.

This structure is often represented as an ABCABC.... pattern, where A, B, and C represent the positions of the atoms in the crystal lattice. Metals such as copper, silver, gold, and aluminum crystallize in this structure, making it one of the most common crystal structures in nature.

The cubic close packing structure has a number of important physical properties, including high strength, ductility, and electrical conductivity. These properties are a result of the close packing of the constituent particles and the strong bonding between them.

 Coordination number

In chemistry, coordination number is defined as the number of atoms, ions, or molecules that a central atom or ion is bonded to. The coordination number is an important factor in determining the structure, stability, and properties of a compound.

In a simple example, the coordination number of an atom in a crystal lattice is determined by counting the number of nearest neighbours that it has. For example, in a face-centered cubic (fcc) structure, each atom is in contact with 12 nearest neighbours, so the coordination number is 12. Similarly, in a body-centered cubic (bcc) structure, each atom is in contact with 8 nearest neighbours, so the coordination number is 8.

The coordination number is also important in understanding the properties of coordination compounds, which are compounds that contain a central metal ion or atom bonded to other molecules or ions. The coordination number of the central metal ion determines the geometry of the compound, which in turn affects its reactivity, solubility, and other properties.

In summary, the coordination number is a fundamental concept in chemistry that is used to describe the bonding and structure of molecules, ions, and solids.

 Formula of the solid compound

If A compound is formed by two elements M and N. The element N forms ccp and atoms of M occupy 1/3rd of tetrahedral voids. then the formula of compound is 

The ccp lattice is formed by the atoms of the element N.

Here, the number of tetrahedral voids generated is equal to twice the number of atoms of the element N.

According to the question, the atoms of element M occupy 1/3rd of the tetrahedral voids.

Therefore, the number of atoms of M is equal to 2x1/3 = 2/3rd of the number of atoms of N.

Therefore, ratio of the number of atoms of M to that of N is M: N =2/3 :1 = 2 : 3

Thus, the formula of the compound is M2N3.