Presentation on theme: "Reciprocal lattice and the metric tensor"— Presentation transcript:
1Reciprocal 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 inspace time readsmetric tensorcoordinate differentials-in flat space time with coordinatesIn 3D real space we can represent a vectorby its coordinates xiaccording tobasis vectors
2Changing the basis tochanges the coordinatesMatrix 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 countervariantNow we construct a new set of basis vectors, the countervariant basis, which isidentical to the basis of the reciprocal spaceConsider the scalar productmetric tensorwhere-as we know from relativity
3The new reciprocal basis reads Let’s show that theform really a set of basis vectorscoordinates with respect to the reciprocal basisNote: in the lecture we introduced reciprocal basis vectorsso thatApplication in solid state physics-we have basis vectors (not necessarily orthogonal)Metric tensorReciprocal 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 approachbcc: a1=a(½, ½,-½), a2=a(-½, ½,½) and a3=a(½,- ½,½)and-Now we use the metric tensoretc.