1. Reciprocal lattice and the metric tensor
Concept of a metric and the dual space is known from the theory of relativity
-line element ds measuring the distance between 2 neighboring events in
space time reads
metric tensor
coordinate differentials
-in flat space time with coordinates
In 3D real space we can represent a vector
by its coordinates xi
according to
basis vectors

2. Changing the basis to
changes the coordinates
Matrix A and B are related according to
-quantities with a subscript transform like the basis vectors and are called covariant
-quantities with a superscript transform like the coordinates are called countervariant
Now we construct a new set of basis vectors, the countervariant basis, which is
identical to the basis of the reciprocal space
Consider the scalar product
metric tensor
where
-as we know from relativity

3. The new reciprocal basis reads
Let’s show that the
form really a set of basis vectors
coordinates with respect to the reciprocal basis
Note: in the lecture we introduced reciprocal basis vectors
so that
Application in solid state physics
-we have basis vectors (not necessarily orthogonal)
Metric tensor
Reciprocal lattice vectors

4. -We know from the conventional approach
As an example let’s consider the reciprocal lattice of the bcc lattice in real space
-We know from the conventional approach
bcc: a1=a(½, ½,-½), a2=a(-½, ½,½) and a3=a(½,- ½,½)
and
-Now we use the metric tensor
etc.