I Questions Regarding Primitive Unit Cell

[Slimy0233]

TL;DR Summary

Is the effective number of lattice points inside a primitive unit cell always 1?
My professor says so.

Questions regarding Primitive Unit Cell (and what I think the answer are, correct me if I am wrong)

1. Can there be more than one Primitive Unit cells for the same crystal?
yes, Wigner Seitz cell always will exist. There can be other primitive Unit cells along with Wigner Seitz too. But will there always exist two primitive unit vectors? One Wigner Seitz and another just a normal unit cell?

Also Wiki says: "In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone."

A square is a parallelogram and so is a rectangle, the above paragraph seems to be saying that if the every Bravais lattice would have a parallellepiped and a Wigner Seitz as their possible primitive unit cells. So, does that mean, there will be at least two primitive unit cells for every crystal there is?2. Does the Primitive Unit Cell only have lattice points at the corners?
Not necessarily, but the effective lattice points just has to be 1. (not sure)

3. Can a crystal not have a Primitive Unit Cell? (as the smallest unit cell is not primitive)

4. Are all primitive unit cells one of the bravais lattices?

5. Is the effective number of Lattice points in a primitive unit cell always 1?
I think so and Wikipedia article confirms this.
Effective number of lattice points in a primitive unit cell is always 1 and can't be anything else.

I have had great confusion especially with point 5 as some websites are providing information which are misleading

 

[nasu]

The image in the link contains an error. The pink, rectangular cell covering two squares is not primitive. It has two nodes (the four in the corners participating with 1/4 and the two on the vertical sied participating with 1/2).

 

[Slimy0233]

The image in the link contains an error. The pink, rectangular cell covering two squares is not primitive. It has two nodes (the four in the corners participating with 1/4 and the two on the vertical sied participating with 1/2).

I thought that too. Maybe people who don't know the basics of solid state physics have no business writing about it.

 

[GeologistInDisguise]

TL;DR Summary: Is the effective number of lattice points inside a primitive unit cell always 1?
My professor says so.

3. Can a crystal not have a Primitive Unit Cell? (as the smallest unit cell is not primitive)

I think you may be slightly confused on the difference between a bravais lattice and a unit cell. A primitive bravais lattice is a lattice that has atoms only at the corners. You can look at this as having 1/4 of an atom at each corner, therefore only 1 total atom or you can view it as a single atom with some attached translations that describe the location of the next unit cells.

A primitive unit cell is the smallest volume repeat unit of a crystal structure. That does not mean it is a primitive bravais lattice. The primitive unit cell of NaCl for example is a face centered bravais lattic in the Cubic crystal system.

 

[nasu]

The FCC cell is not primitive. It is the "conventional" unit cell. The primitive cell of the FCC is some slanted paralleipiped (the red one in the image).