**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 s_{1}, s_{2},
s_{3} are any integers and the reciprocal lattice basis vectors **b**_{1},
**b**_{2}, **b**_{3} can be obtained by the following
equations:

_{}

_{}

_{}

where **a**_{1},
**a**_{2}, **a**_{3} 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