# 12.1: Crystall Lattices and Unit Cells

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

A unit cell is the most basic and least volume consuming repeating structure of any solid. It is used to visually simplify the crystalline patterns solids arrange themselves in. When the unit cell repeats itself, the network is called a lattice.

## Introduction

The work of Auguste Bravais in the early 19th century revealed that there are only fourteen different lattice structures (often referred to as Bravais lattices). These fourteen different structures are derived from seven crystal systems, which indicate the different shapes a unit cell take and four types of lattices, which tells how the atoms are arranged within the unit. The kind of cell one would pick for any solid would be dependent upon how the latices are arranged on top of one another. A method called X-ray Diffraction is used to determine how the crystal is arranged. X-ray Diffraction consists of a X-ray beam being fired at a solid, and from the diffraction of the beams calculated by Bragg's Law the configuration can be determined.

The unit cell has a number of shapes, depending on the angles between the cell edges and the relative lengths of the edges. It is the basic building block of a crystal with a special arrangement of atoms. The unit cell of a crystal can be completely specified by three vectors, a, b, c that form the edges of a parallelepiped. A crystal structure and symmetry is also considered very important because it takes a role in finding a cleavage, an electronic band structure, and an optical property. There are seven crystal systems that atoms can pack together to produce 3D space lattice.

The unit cell is generally chosen so as to contain only complete complement of the asymmetric units. In a geometrical arrangement, the crystal systems are made of three set of ($$a$$, $$b$$, $$c$$), which are coincident with unit cell edges and lengths. The lengths of a, b, c are called the unit cell dimensions, and their directions define the major crystallographic axes. A unit cell can be defined by specifying a, b, c, or alternately by specifying the lengths |a|, |b|, |c| and the angles between the vectors, $$\alpha$$, $$\beta$$, and $$\gamma$$ as shown in Fig. 1.1. Unit cell cannot have lower or higher symmetry than the aggregate of asymmetric units. There are seven crystal systems and particular kind of unit cell chosen will be determined by the type of symmetry elements relating the asymmetric units in the cell.

The unit cell is chosen to contain only one complete complement of the asymmetric units, which is called primitive (P). Unit cells that contain an asymmetric unit greater than one set are called centered or nonprimitive unit cells. The additional asymmetric unit sets are related to the first simple fractions of unit cells edges. For example, (1/2, 1/2, 1/2) for the body centered cell $$I$$ and (1/2,1/2, 0) for the single-face-centered cell $$C$$. The units can be completely specified by three vectors (a, b, c) and the lengths of vectors in angstroms are called the unit cell dimensions. Vectors directions are defined the major crystallographic axes. Unit cell can also be defied by specifying the lengths (|a|, |b|, |c|) and the angles between the vectors ($$\alpha$$, $$\beta$$, and $$\gamma$$) as shown in Fig.1.1.

Table 1: includes the allowable unit cell types found in crystals and their distinguishing characteristics.
Crystal System Types of Lattices (number of particles) Description of Cell
Cubic Simple or Primitive (1) Face centered (4) Body centered (2) Quadratic prism that is equilateral and equiangular (i.e. a cube)
Tetragonal Simple or Primitive (1) Body centered (2) Quadratic prism that has two equal edges and one different sized edge, and is equiangular. (i.e. a rectangular prism with a square base)
Orthorhombic Simple or Primitive (1) Face centered (4) Body centered (2) End or Base centered (2) Quadratic prism that has no equal edges, and is equiangular. (i.e. a rectangular prism without any square base faces)
Monoclinic Simple or Primitive (1) End or Base centered (2) Quadratic prism that has no equal edges, two edges at 90 degrees of each other, and one thats not at 90. (i.e. a parallelogram extended to some distant not equal to the width of the base)
Rhombohedral Simple or Primitive (1) Quadratic prism that has equal length edges, but one slanted side (i.e. a slanted cube)
Triclinic Simple or Primitive (1) Quadratic prism with no equal length edges, two unequal angles and one angle at 90 degrees.
Hexagonal Simple or Primitive (1) Hexagonal prism
Note: Edges refers to all parallel edges. And all parallel edges are equal to each other.

This table describes the fourteen different kind of unit cells available. As you can see not every crystal systems can have all the different types of lattices.

## Volumes

Calculating the volume for a unit cell is the same as calculating the volume for any prism - base area multiplied by height. The equations for each different crystal system are as follow:

Crystal System Volume =
Cubic abc
Tetragonal abc
Orthorhombic abc
Monoclinic abc sin(?), where ? is the acute non 90 degree angle.
Rhombohedral abc sin(60°)
Triclinic abc ((1- cos²? - cos²? - cos²?) + 2(cos(?) cos(?) cos(?))½
Hexagonal abc sin(60°)
Note: a, b, & c are represent the edges. ?, ?, & ? are the angles.

Most calculations involving unit cells can be solved with the formula: density = Mass/Volume. Then in addition to the obvious three the number of particles per cell can also be calculated by the density/molar mass.

## Density - Particles

It can easily be seen that not all the particles are complete in the unit cell form. For the fractional particles in unit cell, its corner particles will always sum to one whole particle, its face particles (for face centered lattices) will sum to three whole particles, and for the base particles will sum to one.

## References

1. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. "Crystal Structures" General Chemistry: Principles & Modern Applications, ninth Edition. New Jersey: Pearson Education, Inc., 2007. 501-508.
2. "The Solid State" TutorVista.com. 29 May 2008.
3. "Unit Cell Dimensions" WebMineral. David Barthelmy. 2005. 29 May 2008.
4. Alexander Mcpherson. Introduction To MacroMolecular Crystallography. Irvine:University of California; 2003.

12.1: Crystall Lattices and Unit Cells is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.