Understanding 3D Si Dispersion Relations & Reciprocal Lattice Vectors

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SUMMARY

This discussion clarifies the concepts of 3D silicon (Si) dispersion relations and reciprocal lattice vectors. The first Brillouin Zone (BZ) for a diamond lattice extends from -π/a to π/a, confirming that its size is 2π/a. The center point, known as the Γ point, corresponds to a pseudomomentum of k=0. The reciprocal lattice vector, which connects equivalent points in adjacent cells, is indeed 2π/a, as each cell spans this width.

PREREQUISITES
  • Understanding of 3D silicon (Si) crystal structures
  • Familiarity with Brillouin Zones (BZ) in solid-state physics
  • Knowledge of phonon dispersion relations
  • Basic concepts of reciprocal lattice vectors
NEXT STEPS
  • Study the properties of phonon dispersion relations in silicon
  • Explore the mathematical derivation of reciprocal lattice vectors
  • Learn about the significance of the Γ point in crystal momentum
  • Investigate the implications of Brillouin Zone boundaries on electronic properties
USEFUL FOR

Physicists, materials scientists, and students studying solid-state physics, particularly those focusing on crystal structures and phonon behavior in materials like silicon.

jacare
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I am trying to understand 3D Si dispersion relations and reciprocal lattice vectors. My confusion is that when I look at dispersion relations the wave vector typically is normalized from 0 to 1 by a/2pi. I thought the edge of the first BZ was pi/a. Is this correct or is it 2pi/a for a diamond lattice? Also, the reciprocal lattice would be 2Npi/a where N is an integer correct?
 
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Which dispersion relations are you lookong at, energy or phonon?
 
i am looking at phonon dispersion relations
 
The first edge of the BZ is at pi/a. The first BZ extends in both directions, from -pi/a to pi/a. So the size of the BZ is 2pi/a. The center point (often called the \Gamma point) is where the pseudomomentum k=0. The reciprocal lattice vector gets you from the Gamma point in one cell to the Gamma pt in another cell (or any other equivalent point). So this has to be 2pi/a because each cell is that wide.
 

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