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
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