# Example Group (EN): Crystal structures

#1

One way to classify crystals is to look at their structure and to categorize them using seven different crystal families. In the following set of examples we will take a look at the basics of crystallography and we will introduce and explain each crystal system with the help of two examples.

To characterize the crystal’s structure we introduce a so called unit cell, which is basically a box containing one or more of the crystals’ atoms. There is one requirement though: We have to be able to rebuild the whole crystal stacking and aligning (identical) unit cells and the unit cell has to be as small as possible. The unit cell can be described by its side lengths a, b and c (also called lattice parameters) and the angles between them, alpha, beta and gamma. The atoms’ positions within the unit cell are often given using fractions of the side lengths as coordinates.

#2

1. Cubic crystal system
The unit cell of a cubic system represents a cube. All of its sides are equally long and the angles between them are 90°. An example for a cubic unit cell can be be seen here:

Picture by Bas Zoetekouw, published under BSD-License

Example 1:
Pyrite is is also known as “Fool’s Gold” due to it’s shiny, brass-like surface, yet chemically it has not much in common with real gold, as it is an iron sulfide with a cubic crystal structure.

Pyrite.wal (13.1 KB)

Example 2:
Galena, a naturally occurring form of lead sulfide, is also member of the cubic crystal family. It is mined as a source of lead, but may also contain great amounts of silver “impurities”.

Galena.wal (31.5 KB)

#3

2. Hexagonal crystal system
A hexagonal unit cell consists of two sides of the same and one of different length. Alpha and beta have values of 90°, gamma is 120°.

Using three of these unit cells (two of them turned by 120° and 240°) we can create a structure with a hexagonal base, as can be seen here:

In this picture the base of each unit cell is made up by two triangles.
A hexagonal unit cell is symmetric with respect to the lateral axis.

Example 1:
In this example we take a look at Beryl’s unit cell. Beryl is a rather common silicate containing beryllium and aluminum. There are several variations containing different impurities which change the crystal’s color, the better known of which are emerald or aquamarine. Beside these forms which are used as gems, beryl is also a source of the metal beryllium.

Beryl.wal (44.2 KB)

Example 2:
Here we can see the hexagonal structure generated by arranging four of beryl’s unit cells. Due to the symmetry discussed above, we can also reproduce a hexagonal structure by simply placing four unit cells next to each other instead of rotating two of them an placing them next to a third one.

Beryll-Hexagonal.wal (145.4 KB)

#4

3. Tetragonal crystal system
Just like a cubic unit cell’s, the tetragonal system’s angles are all 90°, but only two of its sides are of equal length, as can be seen here:

Example 1:
Crystobalite is a rare mineral with the same chemical formula as quartz (which is used in glass). Due to its distinct crystal structure it has different properties, though. Inspecting crystobalite’s unit cell can help understand why minerals with the same chemical formula but different structure (polymorphism) may perform differently. There are actually two different modifications of crystobalite, one of which has a tetragonal crystal structure as can be seen in this example.

Cristobalite.wal (8.4 KB)

Example 2:
Fluorapophyllite (aka Apophyllite-(KF)) belongs the group of apophyllites, which are another common representative of silicate minerals. This mineral commonly forms colorless crystals with tetragonal structure which are also used as gemstones.

Fluorapophyllite-(K).wal (56.1 KB)

#5

4. Trigonal crystals system
The trigonal crystal system is similar to the hexagonal system (two sides of equal length, alpha = beta = 90°, gamma = 120°), but its unit cell not symmetric. This can be seen in the following figure. Note, that there is no possibility to form a hexagonal “super cell”.

Left hand side: trigonal structure
Right hand side: hexagonal structure

Example 1:
Hematite (also haematite) contains more than 70% iron and can be found in enormous deposits, which makes it ideal as ore. It may also be used for jewelry or as a pigment.

Hematite.wal (20.5 KB)

Example 2:
Jarosite is an iron-sulfur-mineral, colored dark yellow or yellowish-brown. It has been used by painters and artist, but has been replaced by other pigments.

Jarosite.wal (37.5 KB)

#6

5. Orthorombic crystal system
The orthorombic unit cell has sides of three different lengths and all angles are 90°.

Example 1:
Hemimorphite is also a silicate mineral and often forms colorless/white needle-shaped clusters.

Hemimorphite.wal (22.8 KB)

Example 2:
Mozartite is a rare mineral of the orthorombic crystal family and is named after the Austrian composer Wolfgang Amadeus Mozart. It can be found in the form of short prismatic, red crystals.

Mozartite.wal (17.2 KB)

#7

6. Monoclinic crystal system
A monoclinic unit cell has three different side lengths, and only two angles of 90°.

Example 1:
Cryolite is an uncommon mineral which can be used to obtain aluminum.

Cryolite.wal (14.0 KB)

Example 2:
Uranophane or uranotile is a yellow, radioactive mineral. There are two different spatial modifications, one of which can be seen here (Uranophane-alpha)

Uranophane.wal (18.6 KB)

#8

7. Triclinic crystal system
In a triclinic unit cell all three angles are different and not equal to 90°. Usually alpha and beta refer to obtuse angles, gamma to the acute angle. Additionally a, b and c are different too.