Prove that the reciprocal lattice of a Bravais Lattice is... Say that the original Lattice is described by R =...
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Prove that the reciprocal lattice of a Bravais Lattice is... Say that the original Lattice is described by R =...
Prove that the reciprocal lattice of a Bravais Lattice is also a Bravais Lattice Say that the original Lattice is described by R = n1 â1 + n2 â2 + n3 â3 , and its reciprocal lattice by K = k1 b̂1 + k2 b̂2 + k3 b̂3 , such that âi · b̂j = 2πδij (definition of a reciprocal lattice). This means, obviously, that eiR·K = 1. 2π (n1 k1 + n2 k2 + n3 k3 ) = 2πn (1) this has to hold for all n1 , n2 , n3 and therefore ki must be whole numbers. This means that K is a Bravais lattice. The reciprocal lattice of a Bravais Lattice is also a Bravais lattice!! 1 What is the reciprocal lattice of the following lattices a. Simple Cubic b. Body Centered Cubic. What is the volume of the primitive cell for these reciprocal lattices? 1. ~a1 = ax̂, ~a2 = aŷ, ~b1 = 2π ~a2 ×~a3 = ~a1 ·~a2 ×~a3 ~a3 = aẑ 2π x̂, a ~b2 = 2π ŷ, a ~b3 = 2π ẑ. a Meaning, the reciprocal lattice of an SC lattice a × a × a, is a simple cubic lattice 2π a × 2π a × 2. ~a1 = ax̂, 2π a ~a2 = aŷ, ~b1 = 2π (x̂ a − ẑ) ~b2 = 2π (ŷ a − ẑ) ~b3 = 4π ẑ a ~a3 = a2 (x̂ + ŷ + ẑ) an FCC lattice V 1 = b1 · b2 × b3 = 2 · 2π 3 a 1