Section 2.2 states

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch The reciprocal lattice and the Brillouin zone

Problem

Section 2.2 states, without proof, that applying the reciprocal-lattice construction a second time to b1,b2,b3\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3 returns a1,a2,a3\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3. Prove it for an orthorhombic lattice, whose primitive vectors are mutually perpendicular but of three different lengths, a1=ai^\mathbf{a}_1=a\hat{\mathbf{i}}, a2=bj^\mathbf{a}_2=b\hat{\mathbf{j}}, a3=ck^\mathbf{a}_3=c\hat{\mathbf{k}} with a≠b≠ca\ne b\ne c.

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