Can the volume of primitive unit cell and unit cell be different?

[Ahsan Khan]

Hi all,

I read The unit cell is the smallest structure that repeats itself by translation through the crystal.
Some says premitive unit cells contains atoms only at the corners while a unit cell may contain extra atoms in between(like bcc or fcc).

At one place I found this:

For each crystal structure there is a conventional unit cell, which is the smallest unit that has the full symmetry of the crystal (see below). However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space. This primitive unit cell does not, however, display all the symmetries inherent in the crystal. A Wigner-Seitz cell is a particular kind of primitive cell which has the same symmetry as the lattice.



I am confused what does they mean by premitive unit cell and unit cell. If the only difference between them is of the extra atoms. Then for a given crystal the volume of a premitive unit cell and a unit cell must be same. Or there be anything smaller than unit cell, but above what if found says "However, the convention unit cell is not always the smallest possible choice."Please explain.

Also at many place people say a premitive unit cell contains only one lattice point. I am unable to comprehend this because of a fact for example I know in simple cubic 8 lattice points(not one) occupied by 8 atoms.

Any help will be highly appreciated.

 

[Alkim]

The "primitive unit cell" is the one that allows you to reproduce the whole crystal structure, given you know the space group and the atomic positions of the different types of atoms, while if you have the "unit cell" you don´t need any additional information to reproduce the crystal structure.
Example:

The chemical formula of the mineral baddeleyite is ZrO2. The primitive cell consists of 3 atoms:

Atom x y z
Zr 0.27580 0.04110 0.20820
O 0.07030 0.33590 0.34060
O' 0.44230 0.75490 0.47890

Note that there are two crystallografical types of oxygen, although chemically it is the same element. If you want to obtain the complete structure of the crystal you just need to know its symmetry type, defined by the space group. In this case it is P21/c (number 17). Knowing this symmetry, the positions of all atoms within the unit cell can be determined by symmetrical operations defined by so called Wyckoff symbols (see e.g. http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-table?from=getwp). The unit cell itself consists of 4 formula units, i.e. 4 Zr atoms and 8 O atoms, see e.g. http://webmineral.com/data/Baddeleyite.shtml#.UvqUavlkiSo

Hope this helps.

 

[Ahsan Khan]

Thanks Alkim what you said helped me, kindly let me confirm that I got it correctly by commenting on my conclusion- " The (space each type of cell occupies) volume of primitive unit cell and unit cell is the same."

The "primitive unit cell" is the one that allows you to reproduce the whole crystal structure, given you know the space group and the atomic positions of the different types of atoms, while if you have the "unit cell" you don´t need any additional information to reproduce the crystal structure.
...
.....

Hope this helps.

I will also like to the meaning of this:

Also at many place people say a premitive unit cell contains only one lattice point. I am unable to comprehend this because of a fact for example I know in simple cubic 8 lattice points(not one) occupied by 8 atoms.

Any help will be highly appreciated.

 

[Alkim]

Excuse me, I think I just confused you more. I think this should solve your doubts: http://en.wikipedia.org/wiki/Bravais_lattice

 

[alphy]

I can provide some clarification on the concept of primitive unit cell and unit cell in crystal structures. The unit cell is a fundamental concept in crystallography, representing the repeating unit of a crystal lattice. It is essentially a three-dimensional box that contains all the information about the crystal structure, including the arrangement of atoms and their relative positions. The unit cell can be described by its dimensions, which can be different for different crystal structures.

A primitive unit cell is a specific type of unit cell that is the smallest possible unit that can be tiled to form the entire crystal structure. It contains the minimum number of atoms needed to fill the space and does not display all the symmetries of the crystal. This means that there may be smaller unit cells that can be constructed, but they may not fill the space completely or may not have the same symmetry as the crystal.

On the other hand, a conventional unit cell is the smallest unit that has the full symmetry of the crystal. This means that it may contain extra atoms in between the lattice points, such as in the case of bcc or fcc structures. The volume of a primitive unit cell and a conventional unit cell may be different because of this extra space occupied by the additional atoms in the conventional unit cell.

In terms of the number of lattice points, a primitive unit cell may contain only one lattice point, but this does not mean that there is only one atom in the unit cell. As you mentioned, a simple cubic structure has 8 atoms occupying 8 lattice points, but a primitive unit cell for this structure would only contain one lattice point. This is because the other 7 lattice points are occupied by the same atom, which represents the symmetry of the crystal structure.

In summary, the volume of a primitive unit cell and a conventional unit cell may be different due to the extra atoms present in the conventional unit cell. The concept of a primitive unit cell is important in understanding the basic building block of a crystal structure, while the conventional unit cell provides a more complete representation of the crystal's symmetry.