# Crystal Structure | Reciprocal Lattice

## Reciprocal Lattice

Defination – In a crystal, there exists many set of planes with different orientations and spacing. These planes can cause diffraction. If we draw normals to all set of planes, from a common origin, then end points of normals a lattice which is called reciprocal lattice.

### The general procedure for locating the reciprocal lattice points

(i) A lattice point is taken as common origin

(ii) From the common origin, draw a normal to each plane.

(iii) Place a point on the normal to each plane (h,k,l) at a distance from the origin equal to 1/dhkl

(iv) Such points from a periodic array called reciprocal lattice.

### Vector development of the reciprocal lattice

Let the crystallographic axes are a, b and c respectively of a primitive unit cell. Height of this cell = d100.

Volume of this call, V = area × d100

So,  1/d100 = area/volume

In vector notation, the normal to a plane is represented by the unit vector n̂, then

Firom figure,

Similarly,

These three vectors are choosen as the three reciprocal axes for defining three dimensional reciprocal lattice. Which are defined as –

We know,

Similarly,

### Show that the volume of the unit cell of the reciprocal lattice is inversely proportional to the volume of a unit cell of direct lattice

The volume of direct lattice =

The volume of reciprocal lattice =

We have to prove :

Now,