6. Arrangement of the atom / particles of the solids in three dimensions

Arrangement of the atom / particles of the solids in three dimensions

Now having gained a knowledge of some of the terms, let us study how the different arrangements in space are brought about.

Firstly we will focus our attention on the solids containing only one type of lattice points.

The solids which contain only one type of lattice points are:

• metallic solids (eg. Iron)
• molecular solids (eg. dry ice)
• covalent network solids (eg. diamond)

(Ionic solids do not fall into this category as they contain more than one type of particles,they will be studied in the later parts of the chapter)

All the atoms or particles of the solids will be represented by solid spheres, each of radius ‘r’.

We will be taking these spheres of radius ‘r’ and explore how we can arrange these in three dimensions.Firstly we will begin with arrangement in one dimension.

Arrangement in 1-D : In one dimension it is possible to arrange the spheres in two possible ways.

1.     

Not Stable [because the potential energy of the system is not minimum]

2.          a = 2r

          Coordination number = 2

1-D close packing stable arrangement

This is the predominant way of packing in one dimension and as such most of the space lattices will  show such an arrangement in one dimension along the planes of close packing.

Arrangement in two dimension :  

In two dimensions also there are two ways of packing the spheres(in moving from one dimension to two dimensions it can be imagined that the two dimensional array will be made up of 1-D closed pack arrays / lines  which are stacked one on top of other.

1. Square packing : If the one dimensional arrays are placed on top of one another, we get the square packing in two dimensions.

One sphere will be in contact with 4 other spheres.

 area of square = a² = 4r²                               

area of atoms in the square =

 fraction of area occupied by spheres =  = 78%

2. Hexagonal close packing : (in 2-D) If in a two dimensional arrangement,every one dimensional array is placed in the cavity of the just preceding array, we get the hexagonal packing in two dimensions.

 area of hexagon =   

area of atoms =  

fraction of area occupied = 91%

As is evident from the above calculations, the spheres are in closer contact in the hexagonal arrangement, hence the hexagonal arrangement is considered to be a better way of packing as compared to the  square packing.

Arrangement in three Dimensions :

1. Simple cubical arrangement in three dimensions :

(will be made up of 2-D sheets arranged one over other)

The simple cubical packing is obtained by arranging the square pack sheets of two dimension one over the other such that spheres of the second sheet are exactly (vertically) above the spheres of first sheet.

(Note that , hence crystal thus formed will belong to the cubic crystal class, and as the lattice points are only at the corners, hence the unit cell will be simple, therefore what we get is the simple cubic)

(i) Relation between ‘a’ and ‘r’

 a = 2r (because atoms along the edge are touching each other)

(ii) Effective no. of atoms per unit cell :

 (Z) =  × 8 = 1

(iii) Packing fraction :

Packing efficiency =

   =  = 0.52  (or 52%)

(Note : This is not a very efficient way of packing as the packing fraction is very low)

(iv)  Coordination Number :

It is defined as the number of atoms touching any one particular atom. For simple cubic, coordination number = 6.

(v)  Density of unit cell : It is the ratio of mass of the spheres present in unit cell and total volume of unit cell.

 Density of the unit cell =

     Þ    

Where Z = no. of atoms in a unit cell

M/N_A = mass of a single  atom

M = molar mass

N_A = Avogadro number (6.023 × 10²³)

2. Body centred cubic :

The body centred cubic is a unique way of packing, as the 2D-arrays that can be imagined to constitute the space lattice are themselves formed in a unique way. The lattice points in the 2D array do not touch each other. The spheres start touching each other only upon moving from  2D to 3D, i.e when the 2D arrays are placed on top of each other in such a fashion that the spheres of the next plane are into the cavities of the first plane of spheres.The third plane of spheres is then exactly identical to the first plane of spheres.

(i)  a  ¹ 2r (as atoms along the edge are not touching each other)

they touch along  the body diagonal, hence a = 4r.

(ii) Effective number of atoms (Z) = 1 +  × 8 = 2.

(iii) Packing fraction =  =  0.68 = 68%

(iv) Coordination No.= 8

 (the sphere at the body centre will be touching the spheres at the eight corners)

(v) Density =  =   (Q Z = 2)

3.  Close packing in three dimensions :

(These are made up of two dimensional hexagonally packed sheets) In second layer we have two kinds of voids.                            

(i) Voids of second layer below which there are spheres of first layer (all voids of type ‘a’).

(ii) Voids of second layer below which there are voids of first layer (all voids of type ‘b’).

For third layer, we have two possibilities.

 (A) If spheres of III^rd layer are placed in voids of II^nd layer below which there are spheres of I^st layer (voids of type ’a’) then I^st layer and  III^rd layer are identical so this is called AB–AB pattern repeat or hexagonal close packing)

Hcp unit cell : a = 2r = b

(i)  Calculation of ‘c’ :

For the estimation of ‘c’, consider the spheres marked 1,2,3,4 in the unit cell as shown.These four spheres form a regular tetrahedron. The length of the perpendicular from ‘4’ to the equilateral triangle 1-2-3 will be equal to c/2.

  cos 30º =

Apply pythagoras theorem.

x² + (c/2)² = a²   Þ   c =  4r

Volume of the hexagon = Area of base x Height.

(ii)  Effective no. of atoms per unit cell :

Z =  × (no. of atoms at corner) +  × (no. of atoms at face centres) + 1 × (no. of atoms inside the body)

          =  × 12  +  × 2 + 1 × 3 = 2 + 1 + 3 = 6

Packing fraction : = 0.74 = 74%

Coordination No. : C.N. = 12

Density (Q Z = 6)

(B) ABC-ABC arrangement :

If the third  layer spheres are placed in those  voids of second layer under which there are voids of the first layer of spheres (voids of type ‘b’), then the first and the third layer of spheres will not be identical.Such an arrangement will lead to an ABC-ABC type of arrangement.It is also known as the cubical close packing (ccp) or also as the Face Centred Cubic structure (FCC).

(a = tetrahedral voids)

(b = octahedral voids)

In the ABC – ABC pattern, the spheres of 4th layer are vertically above the spheres of Ist layer then these consecutive layers are different from each other, fourth layer will be idential to first layer so it will be called ABC – ABC repeat pattern.It is also called the ccp (cubical close packing) because a cubical type of unit cell is used for the study of this arrangement.

(i) Relation between ‘a’ and ‘r’ :

  a  ¹ 2r

   (as the spheres touch along the face diagonal)

(ii) Effective no. of atoms :

   

(iii) Packing fraction :  =  = 0.74 = 74%        

(iv) Coordination number : 12      

(v) Density

Note : In close packings, whenever two consecutive layers are of different kinds(FCC, HCP) then packing efficiency will always be 74%