**Geometry of the reciprocal lattice.**

The basis set of the reciprocal lattice vectors are defined by the equation:

**a_{i} ^{.} b_{j} = 2pd **

Where the **a**'s are the basis vectors of the direct lattice. The **b**'s are then called the **fundamental vectors of the reciprocal lattice**).

Using the above equation in two dimensions we have:

**a _{1} ^{.} b_{1} = 2p**,

**a _{1} ^{.} b_{2} = **0,

From these equations we see that **b _{1}** must be perpendicular to

If **a _{1 }** and

where n

b_{1} = **2p **/a_{1}, b_{2} = **2p **/a_{2}

Now look at the bottom figures of the same image (above). A general two-dimensional lattice is shown there. If the angle between **a**_{1 }and **a**_{2} is q then the angle between **b _{1} **and

b_{1} = (**2p **/a_{1}) ^{.} cos(90^{0} - q) b_{2} = (**2p **/a_{2}) ^{.} cos (90^{0} - q)

In **three dimensions** we can define the **fundamental reciprocal lattice vectors as:**

**b _{1} **=

** b _{2} **=

**b _{3} **=

The term in the numerator is a vector perpendicular to both **a _{2} **and

** a _{1} ^{.} b_{1} **=

The vector product **a _{2} x a_{3} **is perpendicular to both

**a _{2} ^{.} b_{1} **=

**a _{3} ^{.} b_{1} **=

Because the scalar product of the two vectors at right angles depends on cos 90^{o}
which is zero.

In three dimensions the reciprocal lattice is made up from points connected to some given point defined as the origin by the general vectors

**G = **n_{1}**b _{1}** + n

By analogy with the two-dimensional case, we can see that the fundamental vectors for
a three-dimensional simple cubic reciprocal lattice lay in the same direction as those of the direct lattice and have the absolute values given by b_{i} = **2p **/a_{i}. And the fundamental vectors for a general three-dimensional reciprocal lattice has a different angle with the vectors of the direct lattice if the direct lattice is not an orthogonal one.