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. K  R  2m . Hence, if k  k  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:     K  s1b  s2b2  s3b3

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

   a2  a3 b1  2    a1  a2  a3     a3  a1 b2  2    a1  a2  a3     a1  a2 b3  2    a1  a2  a3 

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 k  k  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:    a1  axˆ a2  ayˆ a3  azˆ

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

 2  2  2 b  xˆ b  yˆ b  zˆ 1 a 2 a 3 a which is a simple cubic lattice itself with a lattice constant 2/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:

 a  a  a a  yˆ  zˆ a  zˆ  xˆ a  xˆ  yˆ 1 2 2 2 3 2

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

 4 1  4 1  4 1 b  yˆ  zˆ  xˆ b  zˆ  xˆ  yˆ b  xˆ  yˆ  zˆ 1 a 2 2 a 2 3 a 2

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

Face-Centered Cubic Lattice Body-Centered Cubic Reciprocal Lattice Nonprimative Translation Vectors and Multiatom Basis

If the crystal of interest has a basis of more than one atom or if you do not use the primative lattice vectors in the equations to obtain b1, b2 and b3 but instead use nonprimative lattice vectors and hence have a basis that includes more than one atom, corrections have to be made in the reciprocal lattice that you obtain. This correction is made by calculating the structure factor. The case of the diamond structure is done in the homework and the simple case of the B.C.C. structure considered as a S.C. structure with a basis of two atoms is considered below.