Reciprocal Lattice

 

            A reciprocal lattice of a crystal structure is a set of vectors denoted by K that satisfies the constructive interference condition, i.e.  .  Hence, if  was equal to any vector in the set K, then there would be constructive interference and a peak in the x-ray intensity.  It can be shown that the set of K obeying this requirement can be written as:

 

where s1, s2, s3 are any integers and the reciprocal lattice basis vectors b1, b2, b3 can be obtained by the following equations:

 

 

where a1, a2, a3 are the primitive lattice vectors describing the structure of the crystal.  This result is very useful because you are able to calculate the acceptable values of  directly from the structure of the crystal!  Let us look at some important examples.

 

 

 

 

Reciprocal Lattice of a Simple Cubic Lattice

 

A simple cubic lattice of lattice constant denoted by a has primitive lattice vectors given by:

 

 

Then using the equations above to obtain the reciprocal lattice basis vectors, we obtain:

 

 

which is a simple cubic lattice itself with a lattice constant 2p/a.

 

 

 

Normal Space

 

k-space or Reciprocal Space

 
 

 


                

 

 

A face centered cubic lattice of lattice constant denoted by a has primitive lattice vectors given by:

 

 

Then using the equations above to obtain the reciprocal lattice basis vectors, we obtain:

 

 

Which are the primitive basis vectors of the body-centered cubic lattice with a lattice constant 4p/a.

Face-Centered Cubic Lattice

 

Body-Centered Cubic Reciprocal Lattice