UNIT CELL: PROPERTIES, NETWORK CONSTANTS AND TYPES - CHEMISTRY - 2025

The unit cell is an imaginary space or region that represents the minimum expression of a whole; that in the case of chemistry, the whole would be a crystal composed of atoms, ions or molecules, which are arranged following a structural pattern.

Examples that embody this concept can be found in everyday life. For this, it is necessary to pay attention to objects or surfaces that exhibit a certain repetitive order of their elements. Some mosaics, bas-reliefs, coffered ceilings, sheets and wallpapers, can encompass in general terms what is understood by unit cell.

Paper unit cells of cats and goats. Source: Hanna Petruschat (WMDE).

To illustrate it more clearly, we have the image above that could be used as a wallpaper. In it cats and goats appear with two alternative senses; cats are upright or upside down, and goats are lying down facing up or down.

These cats and goats establish a repetitive structural sequence. To build the whole paper, it would be enough to reproduce the unit cell across the surface a sufficient number of times, using translational movements.

Possible unit cells are represented by the blue, green and red boxes. Any of these three could be used to get the role; but, it is necessary to move them imaginatively along the surface to find out if they reproduce the same sequence observed in the image.

Starting with the red box, it would be appreciated that if three columns (of cats and goats) were moved to the left, two goats would no longer appear at the bottom but only one. Therefore, it would lead to another sequence and cannot be considered as a unit cell.

Whereas if they imaginatively moved the two boxes, blue and green, the same sequence of the paper would be obtained. Both are unit cells; however, the blue box obeys the definition more, since it is smaller than the green box.

Unit Cell Properties

Its own definition, in addition to the example just explained, clarifies several of its properties:

-If they move in space, regardless of the direction, the solid or complete crystal will be obtained. This is because, as mentioned with cats and goats, they reproduce the structural sequence; which is equal to the spatial distribution of the repeating units.

-They must be as small as possible (or occupy little volume) compared to other possible cell options.

-They are usually symmetrical. Also, its symmetry is literally reflected in the crystals of the compound; if the unit cell of a salt is cubic, its crystals will be cubic. However, there are crystal structures that are described as unit cells with distorted geometries.

-They contain repetitive units, which can be replaced by points, which in turn make up what is known as a lattice in three dimensions. In the previous example the cats and goats represent the lattice points, seen from a higher plane; that is, two dimensions.

Number of repeating units

The repeating units or lattice points of the unit cells maintain the same proportion of the solid particles.

If you count the number of cats and goats within the blue box, you will have two cats and goats. The same goes for the green box, and the red box too (even if it is already known that it is not a unit cell).

Suppose for example that cats and goats are G and C atoms, respectively (a strange animal weld). Since the ratio of G to C is 2: 2 or 1: 1 in the blue box, it can be safely expected that the solid will have the formula GC (or CG).

When the solid has more or less compact structures, as happens with salts, metals, oxides, sulphides and alloys, in unit cells there are no whole repetitive units; that is, there are portions or parts of them, which add up to one or two units.

This is not the case for GC. If so, the blue box would “split” the cats and goats in two (1 / 2G and 1 / 2C) or four parts (1 / 4G and 1 / 4C). In the next sections it will be seen that in these unit cells the reticular points are conveniently divided in this and other ways.

What network constants define a unit cell?

The unit cells in the GC example are two-dimensional; however, this does not apply to real models that consider all three dimensions. Thus, the squares or parallelograms, are transformed into parallelepipeds. Now, the term "cell" makes more sense.

The dimensions of these cells or parallelepipeds depend on how long their respective sides and angles are.

In the lower image we have the lower rear corner of the parallelepiped, composed of the sides a, b and c, and the angles α, β and γ.

Parameters of a unit cell. Source: Gabriel Bolívar.

As can be seen, a is slightly longer than b and c. In the center there is a dotted circle to indicate the angles α, β and γ, between ac, cb and ba, respectively. For each unit cell these parameters have constant values, and define its symmetry and that of the rest of the crystal.

Applying some imagination again, the image parameters would define a cube-like cell stretched out on its edge a. Thus, unit cells arise with different lengths and angles of their edges, which can also be classified into various types.

Types

The 14 Bravais networks and the seven basic crystal systems. Source: The original uploader was Angrense at Portuguese Wikipedia.

Note to begin with in the upper image the dotted lines within the unit cells: they indicate the lower rear angle, as just explained. The following question can be asked, where are the lattice points or repeating units? Although they give the wrong impression that the cells are empty, the answer lies at their vertices.

These cells are generated or chosen in such a way that the repetitive units (grayish points of the image) are located at their vertices. Depending on the values ​​of the parameters established in the previous section, constant for each unit cell, seven crystal systems are derived.

Each crystal system has its own unit cell; the second defines the first. In the upper image there are seven boxes, corresponding to the seven crystal systems; or in a more summarized way, crystalline networks. Thus, for example, a cubic unit cell corresponds to one of the crystal systems that defines a cubic crystal lattice.

According to the image, the crystalline systems or networks are:

-Cubic

-Tetragonal

-Orthorhombic

-Hexagonal

-Monoclinic

-Triclinic

-Trigonal

And within these crystalline systems arise others that make up the fourteen Bravais networks; that among all the crystalline networks, they are the most basic.

Cubic

In a cube all its sides and angles are equal. Therefore, in this unit cell the following is true:

α = β = γ = 90º

There are three cubic unit cells: simple or primitive, body-centered (bcc), and face-centered (fcc). The differences lie in how the points are distributed (atoms, ions or molecules) and in the number of them.

Which of these cells is the most compact? The one whose volume is more occupied by points: the cubic one centered on the faces. Note that if we substituted the dots for the cats and goats from the beginning, they would not be confined to a single cell; they would belong and would be shared by several. Again, it would be portions of G or C.

Number of units

If cats or goats were at the vertices, they would be shared by 8 unit cells; that is, each cell would have 1/8 of G or C. Join or imagine 8 cubes, in two columns of two rows each, to visualize it.

If cats or goats were on the faces, they would only be shared by 2 unit cells. To see it, just put two cubes together.

On the other hand, if the cat or the goat were in the center of the cube, they would only belong to a single unit cell; The same happens with the boxes in the main image, when the concept was addressed.

Having said the above, within a simple cubic unit cell there is a unit or reticular point, since it has 8 vertices (1/8 x 8 = 1). For the cubic cell centered in the body there are: 8 vertices, which is equal to one atom, and a point or unit in the center; therefore, there are two units.

And for the face-centered cubic cell there are: 8 vertices (1) and six faces, where half of each point or unit is shared (1/2 x 6 = 3); therefore, it has four units.

Tetragonal

Similar comments can be made regarding the unit cell for the tetragonal system. Its structural parameters are the following:

α = β = γ = 90º

Orthorhombic

The parameters for the orthorhombic cell are:

α = β = γ = 90º

Monoclinic

The parameters for the monoclinic cell are:

α = γ = 90 °; β ≠ 90º

Triclinic

The parameters for the triclinic cell are:

α ≠ β ≠ γ ≠ 90º

Hexagonal

The parameters for the hexagonal cell are:

α = β = 90 °; γ ≠ 120º

The cell actually constitutes one third of a hexagonal prism.

Trigonal

And finally, the parameters for the trigonal cell are:

α = β = γ ≠ 90º

References

1. Whitten, Davis, Peck & Stanley. (2008). Chemistry. (8th ed.). CENGAGE Learning P 474-477.
2. Shiver & Atkins. (2008). Inorganic chemistry. (Fourth edition). Mc Graw Hill.
3. Wikipedia. (2019). Primitive cell. Recovered from: en.wikipedia.org
4. Bryan Stephanie. (2019). Unit Cell: Lattice Parameters & Cubic Structures. Study. Recovered from: study.com
5. Academic Resource Center. (sf). Crystal structures.. Illinois Institute of Technology. Recovered from: web.iit.edu
6. Belford Robert. (February 7, 2019). Crystal lattices and unit cells. Chemistry Libretexts. Recovered from: chem.libretexts.org