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
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-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
metric tensor
where
-as we know from relativity
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The new reciprocal basis reads
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
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As an example lets consider the reciprocal
lattice of the bcc lattice in real space
-We know from the conventional approach
bcc a1a(½, ½,-½), a2a(-½, ½,½) and a3a(½,-
½,½)
and
-Now we use the metric tensor
etc.