# Crystals: difference between basis and unit cell

• fisico30
In summary, a lattice is a set of points on which a basis can be placed, consisting of one or more atoms. The basis is different from the unit cell, which contains both translation information and the type of atoms to place. For high symmetry systems, there can be both a primitive and conventional unit cell, with the primitive unit cell being the smallest and the conventional unit cell showing symmetry better. In 2D, the primitive cell is the area described by two primitive vectors, while in FCC structures, there are an infinite number of possible primitive cells. In reciprocal space, the Brillouin zone perfectly represents the symmetry of the lattice. For more information, textbooks on solid state physics or the International Tables for Crystallography can be
fisico30
Hello Forum,

a lattice is a set of points. We can place a basis at each set of points.
The basis can be one atom or a group of atoms.

I thought that a translation of the basis would produce the whole crystal...
How is a basis different from the unit cell? Are they the same thing?

There is the primitive cell then...how does it compare to the basis?

thanks
fisico30

If I remember correctly from a long time ago. the basis is the set of atoms (one or more) to be placed on each point of the lattice, but without defining the actual lattice.

You can see that if the basis is a single atom, for example in a simple metal, then you can construct a whole load of different lattices from this basis. Indeed, there are FCC, BCC, HCP, and DHCP metals to name just a few.

The unit cell has both types of information. The translation information to build the lattice, and the basis, which atom to place where.

For high symmetry systems, there can be two types of unit cells. The primitive unit cell is the smallest unit cell from which you can construct the whole lattice. The conventional unit cell is larger, but shows the symmetry better.

For FCC, for example, the conventional unit cell is a cube that contains four primitive unit cells, as the face-centers are related by simple translation to the corner.

Hi M Quack,

thanks for the reply. I am not clear on your last example:

"For FCC, for example, the conventional unit cell is a cube that contains four primitive unit cells, as the face-centers are related by simple translation to the corner."

I get what a lattice and a basis is. In a 2D crystal there is an infinite number of independent primitive vectors. The weighted sum of these 2 vectors locate any point in the lattice...

So, in 2D, is the primitive cell the area described by these two primitive vectors?

For a FCC structure there is then an infinite number of possible primitive cells, one for each possible set of primitive vectors...is that correct?

The primitive unit cell is the smallest unit cell from which the lattice can be built. The choice is not unique, and I guess in principle it is infinite. In practice, you try to respect the symmetry of the crystal as best as you can. The volume of the primitive unit cell (or area in 2D) does not depend on this choice and is always the same.

The vectors that span the primitive unit cell are usually chosen to go from (0,0,0) to the nearest equivalent point on the lattice. For FCC that are the 3 face centers at (1/2,1/2,0), (0.,1/2,1/2) and (1/2,0,1/2).

In reciprocal space one chooses a different construction, the Brillouin zone. The definition is unique, there are no choices to make, and the BZ perfectly represents the symmetry of the (reciprocal) lattice. It often has a non-trivial shape, though.

If you are interested, look at one of the standard textbooks on solid state physics. Ashcroft and Mermin is one of the better ones. if you are interested in the exact mathematical definitions, then the International Tables for Crystallography are very good, but that's not where you will get an overview of what is going on.

Hello fisico30,

Thank you for bringing up this topic. The terms "basis" and "unit cell" are often used interchangeably, but they actually have distinct meanings in the study of crystals.

A basis refers to the atoms or molecules that make up the repeating pattern in a crystal. These atoms or molecules are placed at each point in the lattice to create the crystal structure. In other words, the basis is the building block of the crystal.

On the other hand, a unit cell is the smallest repeating unit of a crystal lattice. It is a three-dimensional shape that, when repeated in all directions, creates the entire crystal structure. The unit cell contains the basis, but it also includes the lattice points and the connections between them.

To further clarify, the basis is a localized concept, while the unit cell is a global concept. The basis only refers to the atoms or molecules at a specific point in the lattice, while the unit cell encompasses the entire crystal structure.

As for the primitive cell, it is the smallest possible unit cell that can be repeated to create the entire crystal structure. It is often used as a starting point for crystal structure analysis and can be thought of as the most basic unit cell that represents the crystal lattice.

I hope this explanation helps to clarify the difference between basis and unit cell. Both are essential concepts in the study of crystals and understanding their roles is crucial in understanding the properties and behavior of different crystal structures.

Best,

## 1. What is the difference between a basis and a unit cell in crystal structures?

A unit cell is the smallest repeating unit in a crystal lattice, while a basis is the arrangement of atoms, ions, or molecules within that unit cell. The basis defines the type and number of particles that make up the crystal, while the unit cell determines the overall symmetry and shape of the crystal.

## 2. How do the basis and unit cell work together in crystal structures?

The basis and unit cell work together to create a crystal lattice, which is the repeating pattern of particles in a crystal. The basis determines the type and arrangement of particles within the unit cell, while the unit cell dictates how these particles are arranged in space.

## 3. Can the basis and unit cell vary in different crystals?

Yes, the basis and unit cell can vary in different crystals. The basis depends on the type of particles present in the crystal, while the unit cell can vary depending on the crystal system and the arrangement of particles within the crystal.

## 4. How do basis and unit cell affect the properties of crystals?

The basis and unit cell play a crucial role in determining the properties of crystals. The arrangement of particles within the unit cell affects the crystal's symmetry, physical properties, and chemical properties. The type and number of particles in the basis also influence the crystal's properties, such as its color, density, and hardness.

## 5. What are some common crystal structures and their corresponding basis and unit cell?

Some common crystal structures include the simple cubic, body-centered cubic, and face-centered cubic. In the simple cubic structure, the basis consists of one particle at each corner of the unit cell, while in the body-centered cubic structure, there is an additional particle at the center of the unit cell. In the face-centered cubic structure, there are particles at each corner and at the center of each face of the unit cell. These structures have different unit cells, but the same basis of particles.

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