**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 |
a ≠ b ≠ 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 |
- |