Reciprocal lattice : defination, properties, construction

      Many properties of crystalline solid or substances are different in directions. W.H miller explain the concept of miller indices to explain planes and directions of a crystal.
     In real practice there are many set of planes with different orientation and spacings and it becomes difficult to visualise the slopes of planes. Since all parallel planes have a common normal and similar indices, so miller indices of plane can not explain completely.
      Therefore to understand many crystallography problems we consider the normal to be set of planes rather than the planes themselves. The orientation of plane is understood by the direction of normal. So to understand this we use the concept of reciprocal lattice.
    We will introduce reciprocal Space where the crystal structure will be represented by a different frame work. Here the dimension of quantities are L'(-1) . This will help us to understand easily.
Reciprocal lattice is also known as K-space, Fourier space  momentum space.

Defination of reciprocal lattice 

         The reciprocal lattice in Fourier space is an infinite periodic three-dimension array of points whose spacing are inversely proportional to the distances between planes in the direct lattice. 
     The concept of reciprocal lattice can be explained by two important properties of the crystal planes :
         1. Slope of the planes 
         2. Interplanner spacing 
Example of reciprocal lattice 

Time crystal 

       Time crystals are the systems where the role of space in ordinary crystalline crystal is replaced by time. 
         In time crystals atoms are show periodicity in time. The time crystals important point is we replace space by time. 
  A time crystal also show rigid pattern as we can found in regular crystal. But here the pattern are repeating in times in stead of space. 

Construction of reciprocal lattice 

    In construction of reciprocal we use the concept of normals. 
A reciprocal lattice from a direct lattice can be construed by these steps : 
1. Take any point as an origin in direct lattice. 
2. Draw normal to Evey points from origin in direct lattice. 
3. Set the length of every normal drawn equal to the reciprocal of the interplanner spacing (1/d(hkl)) of set of parallel planes (h K l) it represent.
4. Place a point at the end if each normal. 
         The assembly of these points represent a lattice array which is called as reciprocal lattice
This represent a tabulation of 
1. The normals to all the direct lattice planes 
2. Their interplanner spacing 
Example of reciprocal lattice 
This is just a pic , here we are not discussing in details about this. 
Reciprocal lattice
 A points in direct lattice is represented by translational vector T as 
      T= n1a+n2b+n3c
Where n1, n2 ,n3 are integer. 
Similarly, the reciprocal lattice vector can be represented by 
      G= h A + k B + l C 
G vector is known as reciprocal lattice 
Vector A, B and C are primitive translational vectors.
h, k and l are integer. ( these are correspond to parallel planes having miller indices (h, k, l). 
So crystal can be explained by 
    - crystal lattice 
    - reciprocal lattice 

  The crystal lattice is a lattice in real or ordinary space whereas reciprocal lattice is a lattice in the associated reciprocal Space. 
An important relation :
        G. T = 2(pi)×integer 
Where G vector and T vector  are reciprocal  lattice and translational vector. 

Properties of reciprocal lattice 

 1. The reciprocal lattice is a lattice in K space or Fourier space. Direct lattice is a lattice in ordinary space. 
2. The primitive vectors in reciprocal Space have dimension of L inverse.
3. The reciprocal of reciprocal lattice is direct lattice. 
4. The volume of reciprocal lattice is inversely proportional to the volume of unit cell of direct lattice. 
5. Each point in reciprocal lattice is represent a set of parallel planes of crystal lattice. 
6. Each vector of reciprocal lattice is normal to a set of parallel planes of direct lattice. 
7. When we rotate a crystal, both direct and reciprocal lattice rotate.

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