Crystal Directions: Finding Basis Vectors

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SUMMARY

The discussion focuses on identifying basis vectors for the Wurtzite and Zincblende crystal structures, specifically in relation to the directions [1 1 1] and [0 0 0 1]. The user clarifies that the [0 0 0 1] direction corresponds to three vectors at 120-degree angles in the plane, with one vector aligned along the c-axis of the hexagonal crystal. The user concludes that the notation [1 1 1] refers to the Zincblende structure, while [0 0 0 1] pertains to the Wurtzite structure, indicating the representation of the same direction in different basis systems.

PREREQUISITES
  • Understanding of crystal structures, specifically Wurtzite and Zincblende.
  • Knowledge of basis vectors in crystallography.
  • Familiarity with Miller indices notation.
  • Concept of angles between vectors in a crystal lattice.
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  • Research the properties of Wurtzite and Zincblende crystal structures.
  • Study the concept of Miller indices and their application in crystallography.
  • Learn about the geometric interpretation of basis vectors in crystal lattices.
  • Explore the relationship between different crystal orientations and their basis representations.
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Researchers, materials scientists, and students in crystallography or solid-state physics who are studying crystal structures and their representations.

aaaa202
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I found the following picture of the Wurtzite and Zincblende structure as viewed from a specific direction. I could do with some help on which basis vectors are chosen for either the [1 1 1] or [0 0 0 1]. Now I am pretty sure that the last one refers to a basis where we have three vectors separated by a 120degree angle in the plane and then one vector along the c-axis of the hexagonal crystal. But I don't see what other system of basis vectors you would choose to make [1 1 1] match [0 0 0 1]. Because that's what [1 1 1]/[0 0 0 1] means here right? - that they are reprsenting the direction on the picture in two different bases.
 

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I think they simply mean direction [111] in case of zincblende and [0001] in the case of wurtzite.
 

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