Introduction to Solid State Physics · Problem 2

WAVE DIFFRACTION AND THE RECIPROCAL LATTICE — Charles Kittel

Physics — Solid State · reciprocal-lattice-hexagonal

Problem

Hexagonal space lattice. The primitive translation vectors of the hexagonal space lattice may be taken as a1=(31/2a/2)x^+(a/2)y^  ;a2=−(31/2a/2)x^+(a/2)y^  ;a3=cz^  .\mathbf{a}_1 = (3^{1/2}a/2)\hat{\mathbf{x}} + (a/2)\hat{\mathbf{y}} \;; \quad \mathbf{a}_2 = -(3^{1/2}a/2)\hat{\mathbf{x}} + (a/2)\hat{\mathbf{y}} \;; \quad \mathbf{a}_3 = c\hat{\mathbf{z}} \;. (a) Show that the volume of the primitive cell is (31/2/2)a2c(3^{1/2}/2)a^2 c. (b) Show that the primitive translations of the reciprocal lattice are b1=(2π/31/2a)x^+(2π/a)y^  ;b2=−(2π/31/2a)x^+(2π/a)y^  ;b3=(2π/c)z^  ,\mathbf{b}_1 = (2\pi/3^{1/2}a)\hat{\mathbf{x}} + (2\pi/a)\hat{\mathbf{y}} \;; \quad \mathbf{b}_2 = -(2\pi/3^{1/2}a)\hat{\mathbf{x}} + (2\pi/a)\hat{\mathbf{y}} \;; \quad \mathbf{b}_3 = (2\pi/c)\hat{\mathbf{z}} \;, so that the lattice is its own reciprocal, but with a rotation of axes. (c) Describe and sketch the first Brillouin zone of the hexagonal space lattice.

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