Reciprocal lattice and the metric tensor

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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

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

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

-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.

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