3.4 The shape of the unit cell and the crystal systems

Fortunately, there are only seven qualitatively different shapes that a parallelepiped can adopt, and hence the unit cell of all minerals can have only seven qualitatively different shapes. These shapes give rise to seven basic crystal systems characterised by the following properties:

Cubic

a = b = c

a = b = g = 90°

Hexagonal

a = b ≠ c

a = b = 90°, g = 120°

Tetragonal

a = b ≠ c

a = b = g = 90°

Trigonal

a = b = c

a = b = g < 120°, 90°

Orthorhombic

a ≠ b ≠ c

a = b = g = 90°

Monoclinic

a ≠ b ≠ c

a = g = 90°, b > 90°

Triclinic

a ≠ b ≠ c

ab g

 

Shapes of single crystals

The shape of single crystals grown in an artificial environment independent of any other crystals is dictated by the crystallographic structure, with crystal faces in the developing crystallite favouring lattice orientations with the highest density of lattice points. In face centred cubic (fcc) crystals the faces which have the highest density of lattice points are {1,1,1} followed by {1,0,0} where the curly brackets {} signifies the set of faces of that form. Thus {1,1,1} includes the faces (1,1,1), (1,-1,1), (1,-1,-1), (1,1,-1), (-1,1,1), (-1-1,1), (-1,1,-1) and (-1,-1,-1). Crystals with all faces of the form {1,1,1}, (e.g. spinel-group minerals such as magnetite) typically form octahedra. In body centred cubic (bcc) crystals the faces with the highest density of lattice points are {1,1,0} followed by {1,0,0}.

Schematic illustration of faces with the highest density of lattice points in fcc and bcc crystals, respectively.

 

3.5 Symmetry in crystals

Each of the seven crystal systems is characterised by different symmetries. By inspection of the above table we can see that there must be a general decrease in the symmetry from the cubic system down through to the triclinic system. Consequently, the triclinic system is said to have lower symmetry than the cubic system and, conversely, the cubic system higher symmetry than the triclinic system.. In order to understand the basic symmetry elements in crystallography we will initially consider examples of the symmetry elements in a cube (a = b = c, a = b = g = 90° ). The basic symmetry elements are:

Rotation axes of symmetry:

Rotation axes of symmetry are axes which upon rotation reproduce the exact configuration of the crystal. An n-fold rotation axis of symmetry repeats the structure n times in one complete 360° rotation. In crystals rotation axes can be sixfold (termed hexad), fourfold (tetrad), threefold (triad), twofold (diad), or onefold (monad). The monad is a trivial since it merely states that upon rotation through 360° the crystal returns to the initial position. A cube contains a number of different tetrads, triads and diads (try and determine the total number in each case):

Centre of symmetry

If every atom in a crystal stucture with coordinates x, y and z is duplicated by an atom at x, y and z, the structure is said to possess a centre of symmetry .

Mirror planes:

Planes which divide crystals into mirror ../Images are termed mirror planes.

Inversion axes of symmetry:

A symmetry type involving a rotation about a line plus an inversion through a point (on the line) is known as an inversion axis of symmetry. In any crystal the operation of an inversion axes of symmetry can always be achieved by a combination of the other symmetry operators in that crystal.

As already stated each of the seven crystal systems have characteristic symmetries. Recognition of these symmetry elements allows us to classify any crystal into the appropriate class. For example the presence of a tetrad indicates either cubic or tetragonal; more than one tetrad and it must be cubic (however, the presence of a tetrad is not the characteristic symmetry element of the cubic system which is, rather, the presence of three triads). Hexads are diagnostic of the hexagonal system, while traids preclude triclinic and monoclinic systems. The triclinic system has no rotational axes of symmetry, except of course the trivial monad. The characteristic symmetry elements in each of the seven groups are listed below:

Cubic

Three triads

Hexagonal

One hexad (// z)

Tetragonal

One tetrad (// z)

Trigonal

One triad (// [111])

Orthorhombic

Three perpendicular diads (// x, y and z)

Monoclinic

One diad (// y)

Triclinic

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