Breaking News

Reciprocal Lattice

The crystal, instead of being seen as different sets of  parallel  planes,  may be  represented  by  a normal  drawn perpendicular to each set of parallel planes from a common point as origin. The length of the normal is proportional to 1/dhkl .This length and direction of the normal is used to represent a set of parallel planes.If a point is placed at the end of each such normal ,an array of points is generated. Each point then represents a set of parallel equidistant lattice planes and hence, each point is represented by a set of Miller indices (hkl) of the crystal. This array of points is  known as the reciprocal lattice. The reciprocal lattice vector hkl has a direction same as the normal to the  dhkl planes and its magnitude is    1/dhkl .  
We see that the arrangements  of  the  points in  the  reciprocal lattice has the same  symmetry  as  the lattice points of the real crystal.

Relation between reciprocal lattice and X-ray diffractionThe concept of reciprocal lattice is very useful in X-ray crystallography. It was Ewald who developed the relation between the diffracted X-ray beams. The concept of reciprocal lattice is particularly helpful in understanding diffraction of X-rays by crystal planes. Let us rewrite the Bragg’s equation in reciprocal lattice as:    

 
Here, we have tried to relate the magnitude of the reciprocal lattice vector to diffraction angle and the wavelength of the X-ray. In order to see the geometric consequence of this equation, let us imagine a sphere of radius 1/λ = AO as shown in figure: 
 

Let AO also be the direction of the X-ray beam incident on the crystal plane at C, the center of the sphere. If θ is the Braggs angle, the reflected beam will strike the sphere (shown as a circle here) at the point P making an angle 2θ with the passing beam. It should be noted that the angle between the incident beam and AP  is θ. Since 

          2 sinθhkl/λ = 1/dhkl

we can see that 

           1/dhkl λ = OP. 

Thus, the point P is the reciprocal lattice point for the set of planes from which the X-ray beam is reflected. Also, since the Bragg’s diffraction conditions are satisfied, the diffracted beam will touch the sphere (circle in the picture) at point P. The same will be the case with other set of parallel planes except that they will strike the sphere at some other point which are the reciprocal lattice points for the respective set of planes. The three-dimensional sphere is called the sphere of reflection or Ewald sphere.

Designed By