1. Reciprocal lattice of BCC is FCC lattice
Solid state physics G

2. Primitive basis vectors of the BCC
Primitive translation vectors of the bcc 𝑎 1 = 1 2 𝑎 −𝑥+𝑦+𝑧 𝑎 2 = 1 2 𝑎 𝑥−𝑦+𝑧 𝑎 3 = 1 2 𝑎(𝑥+𝑦−𝑧) Reciprocal lattice 𝑏 1 =2π 𝑎 2 x 𝑎 3 𝑎 1 . 𝑎 2 x 𝑎 3 𝑏 2 =2π 𝑎 3 x 𝑎 1 𝑎 1 . 𝑎 2 x 𝑎 3 𝑏 3 =2π 𝑎 1 x 𝑎 2 𝑎 1 . 𝑎 2 x 𝑎 3
Primitive basis vectors of the BCC

3. BCC reciprocal lattice vectors
𝑎 2 x 𝑎 3 = 1 2 𝑎 2 𝑥 𝑦 𝑧 1 − −1 = 1 4 𝑎 2 2𝑦+2𝑧 𝑎 1 x 𝑎 2 = 1 2 𝑎 2 𝑥 𝑦 𝑧 − −1 1 = 1 4 𝑎 2 2𝑥+2𝑦 𝑎 3 x 𝑎 1 = 1 2 𝑎 2 𝑥 𝑦 𝑧 1 1 −1 −1 1 1 = 1 4 𝑎 2 2𝑥+2𝑧 𝑎 1 . 𝑎 2 x 𝑎 3 = 1 2 𝑎 −𝑥+𝑦+𝑧 1 4 𝑎 2 2𝑦+2𝑧 = 1 2 𝑎 3 𝑏 1 =2𝜋 𝑎 2 x 𝑎 3 𝑎 1 . 𝑎 2 x 𝑎 3 =2𝜋 1 4 𝑎 2 2𝑦+2𝑧 1 2 𝑎 3 = 2π 𝑎 𝑦+𝑧 𝑏 2 =2𝜋 𝑎 3 x 𝑎 1 𝑎 1 . 𝑎 2 x 𝑎 3 =2𝜋 1 4 𝑎 2 2𝑥+2𝑧 1 2 𝑎 3 = 2π 𝑎 𝑥+𝑧 𝑏 3 =2𝜋 𝑎 1 x 𝑎 2 𝑎 1 . 𝑎 2 x 𝑎 3 =2𝜋 1 4 𝑎 2 2𝑥+2𝑦 1 2 𝑎 3 = 2π 𝑎 𝑥+𝑦
Primitive basis vectors of the FCC lattice
BCC reciprocal lattice vectors
FCC primitive vectors
Since 2 and a is constant, these BCC reciprocal lattice vectors are just FCC primitive lattice vectors