Primitive vectors and the FBZ

In summary, the conversation discusses the uniqueness of primitive cells and their relation to the construction of the first Brillouin zone using the Wigner-Seitz method. It is explained that while primitive cells may not be unique, the first Brillouin zone is unique and can be constructed using any choice of primitive vectors. The Wigner-Seitz method ensures the uniqueness of the Brillouin zones.
  • #1
Niles
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Hi

We know that there is no unique primitive cell, meaning there is no unique choice of primitive vectors. Now, when we find our reciprocal primitive vectors, then we can construct the first Brillouin zone (FBZ) by using the Wigner-Seitz method.

But we know that primitive vectors are not unique, so if we construct the FBZ by using these vectors (via the Wigner-Seitz method), then how do we know that we have found the unique FBZ?


Niles.
 
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  • #2
Primitive cells not being unique simply means that you can choose different cells to represent a particular lattice. It does not mean that a specific cell is necessarily not unique. There is only one first Brillouin zone, but it is not the only choice of unit cell for the reciprocal lattice.

EDIT: It's the Wigner-Seitz method that makes the Brioullin zones (1st, 2nd, ...) unique.
 
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1. What are primitive vectors and how are they related to the FBZ?

Primitive vectors are the shortest set of linearly independent vectors that define the unit cell of a crystal lattice. They are related to the first Brillouin zone (FBZ) because the FBZ is constructed from the reciprocal lattice vectors, which are defined as the inverse of the primitive vectors.

2. How do primitive vectors affect the symmetry of a crystal lattice?

The primitive vectors determine the symmetry of a crystal lattice because they define the shape and orientation of the unit cell. This, in turn, determines the symmetry operations that are possible within the lattice.

3. Can a crystal lattice have more than one set of primitive vectors?

No, a crystal lattice can only have one set of primitive vectors. This is because the primitive vectors must be linearly independent and any additional vectors would not be necessary to define the lattice unit cell.

4. How are primitive vectors calculated or determined experimentally?

Primitive vectors can be calculated or determined experimentally by analyzing the crystal structure and identifying the shortest set of linearly independent vectors that define the unit cell. They can also be obtained from experimental methods such as X-ray diffraction.

5. What is the significance of the first Brillouin zone in the study of crystal lattices?

The first Brillouin zone is significant because it represents the smallest region of reciprocal space that contains all possible wave vectors for a given crystal lattice. It is used in the study of electronic band structures and phonon dispersion relations in materials, providing valuable information about their properties and behaviors.

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