Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

Chemistry Class 12

Class 12 : Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

The document Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev is a part of the Class 12 Course Chemistry Class 12.
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What is a Crystal?

Crystalline solid consists of a large number of small units, called crystals, each of which possesses a definite geometric shape bounded by plane faces. The crystals of a given substance produced under a definite set of conditions are always of the same shape.

➤ Unit Cells

In this topic, we would be studying certain properties of a solid which depend only on the constituents of the solid and the pattern of arrangement of these constituents.

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRevA unit cell is the shortest portion of a lattice

To study these properties, it would be convenient to take up a small amount of the solid as possible because this would ensure that we deal with only the minimum number of atoms or ions.

This smallest amount of the solid whose properties resemble the properties of the entire solid irrespective of the amount taken is called a unit cell. 

It is the smallest repeating unit of the solid. Any amount of the solid can be constructed by simply putting as many unit cells as required.

➤ Characteristics of Unit Cell

The following characteristics define a unit cell:

  • A unit cell has three edges a, b, and c, and three angles α, β, and γ between the respective edges.
  • The a, b and c may or may not be mutually perpendicular.
  • The angle between edge b and c is α, a and c is β and of between a and b is γ.


Simple Cubic (SC) Unit Cell

A simple cubic unit cell consists of eight corner atoms. In a simple cubic unit cell, a corner atom touches with another corner atom. The simple cubic unit cell is shown below.Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRevSimple cubic unit cell of a crystal lattice

➤ Number of Atoms in a Simple Cubic Unit Cell
A simple cubic unit cell consists of 8 corner atoms. Each and every corner atom is shared by eight adjacent unit cells. Therefore, one corner atom contributes 1/8th of its parts to one unit cell. Since, there are eight corner atoms in a unit cell, the total number of atoms is (1/8)* 8=1. Therefore, the number of atoms in a simple cubic unit cell is one.

➤ Atomic Radius
In a simple cubic lattice, a corner atom touches with another corner atom.
Therefore, 2r=a. So, the atomic radius of an atom in a simple cubic unit cell is a/2.

➤ Coordination Number
Consider a corner atom in a simple cubic unit cell. It has four nearest neighbours in its own plane. In a lower plane, it has one more nearest neighbour and in an upper plane, it has one more nearest neighbour. Therefore, the total number of nearest neighbour is six.

➤ Packing Density
The packing density of a simple cubic unit cell is calculated as follows:
The unit cell Number of atoms
pDoc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
Substituting, r=a/2, we get,
Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
The packing density of a simple cubic unit cell is 0.52. It means, 52% of the volume of the unit cell is occupied by atoms and the remaining 48% volume is vacant.

Question 1:The volume occupied by atoms in a simple cubic unit cell is (edge length = a):

Body-Centered Cubic Unit Cell

A Body-Centered unit cell is a unit cell in which the same atoms are present at all the corners and also at the center of the unit cell and are not present anywhere else. This unit cell is created by placing four atoms that are not touching each other. 

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRevBody-centered cubic unit cell

Then we place an atom on top of these four. Again, four spheres eclipsing the first layer are placed on top of this. The effective number of atoms in a Body-Centered Cubic Unit Cell is 2 (One from all the corners and one at the center of the unit cell). 

Moreover, since in BCC the body-centered atom touches the top four and the bottom four atoms, the length of the body diagonal (√3a) is equal to 4r.

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRevThe packing fraction in this case is  = Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev 
 VF ≈ 0.32

Question 2:In a body centered cubic cell, an atom at the body center is shared by:

Face Centered Cubic (FCC) Unit Cell

In an fcc unit cell, the same atoms are present at all the corners of the cube and are also present at the center of each square face and are not present anywhere else. The effective number of atoms in fcc is 4 (one from all the corners, 3 from all the face centers since each face-centered atom is shared by two cubes).

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRevFace centered cubic unit cellSince, here each face-centered atom touches the four corner atoms, the face diagonal of the cube Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev is equal to 4r.

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev.
VF ≈ 0.26

Question 3:The coordination number of an atom in a fcc lattice is _________.

Hexagonal Primitive Unit Cell

Each corner atom would be common to 6 other unit cells, therefore their contribution to one unit cell would be 1/6. Therefore, the total number of atoms present per unit cell effectively is 6.

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

In figure ABCD is the base of the hexagonal unit cell AD=AB=a. The sphere in the next layer has its center F vertically above E and it touches the three spheres whose centers are A, B, and D.
Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

Hexagonal close-packing

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
Hence,
Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
The height of the unit cell (h) = Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
The area of the base is equal to the area of six equilateral triangles, = Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev  .
The volume of the unit cell = Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev.
Therefore,
Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev  
VF ≈ 0.26


The Density of Crystal Lattice

The density of crystal lattice is the same as the density of the unit cell which is calculated as
Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev 
= Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev

Doc: Crystal Lattice and Unit cells Class 12 Notes | EduRev
Here,
Z = no. of atoms present in one unit cell  
m = mass of a single atom  
Mass of an atom present in the unit cell = m/NA

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