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** 1. Introduction**

The discipline of crystallography has developed a descriptive terminology which is applied to crystals and crystal features in order to describe their structure, symmetry, and shape. This terminology defines the crystal lattice which provides a mineral with its ordered internal structure. It also describes various types of symmetry. By considering what type of symmetry a mineral species possesses, the species may be categorized as a member of one of six crystal systems and one of thirty-two crystal classes.

The concept of

A

A specified motif which is translated linearly and repeated many times will produce a

The number of possible lattices is limited. In the plane only five different lattices may be produced by translation. The French crystallographer Auguste Bravais (1811-1863) established that in three-dimensional space only fourteen different lattices may be constructed. These fourteen different lattice structures are thus termed the

**symmetry**describes the periodic repetition of structural features. Two general types of symmetry exist. These include**translational symmetry**and**point symmetry**. Translational symmetry describes the periodic repetition of a motif across a length or through an area or volume. Point symmetry, on the other hand, describes the periodic repetition of a motif around a point.**Reflection, rotation, inversion, and rotoinversion are all point symmetry operations.**A

**reflection**occurs when a motif on one side of a plane passing through the center of a crystal is the mirror image of a motif which appears on the other side of the plane. The motif is said to be reflected across the mirror plane which divides the crystal.**Rotational symmetry**arises when a structural element is rotated a fixed number of degrees about a central point before it is repeated. If a crystal possesses**inversion symmetry**, then every line drawn through the center of the crystal will connect two identical features on opposite sides of the crystal.**Rotoinversion**is a compound symmetry operation which is produced by performing a rotation followed by an inversion.A specified motif which is translated linearly and repeated many times will produce a

**lattice**. A lattice is an array of points which define a repeated spatial entity called a**unit cell**. The unit cell of a lattice is the smallest unit which can be repeated in three dimensions in order to construct the lattice. The corners of the unit cell serve as points which are repeated to form the lattice array; these points are termed**lattice points**.The number of possible lattices is limited. In the plane only five different lattices may be produced by translation. The French crystallographer Auguste Bravais (1811-1863) established that in three-dimensional space only fourteen different lattices may be constructed. These fourteen different lattice structures are thus termed the

**Bravais lattices**.
The reflection, rotation, inversion, and rotoinversion symmetry operations may be combined in a variety of different ways. There are thirty-two possible unique combinations of symmetry operations. Minerals possessing the different combinations are therefore categorized as members of thirty-two

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A simple crystal may consist of only a single crystal form. A more complicated crystal may be a combination of several different forms. The crystal forms of the five non-isometric crystal systems are the monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome and sphenoid, disphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron and tetrahedron. Fifteen different forms are possible within the isometric system.

Each crystal class is a member of one of six

**crystal classes**; each crystal class corresponds to a unique set of symmetry operations. Each of the crystal classes is named according to the variant of a**crystal form**which it displays. Each crystal class is grouped as one of the six different**crystal systems**according to which characteristic symmetry operation it possesses.A

**crystal form**is a set of planar faces which are geometrically equivalent and whose spatial positions are related to one another by a specified set of symmetry operations. If one face of a crystal form is defined, the specified set of point symmetry operations will determine all of the other faces of the crystal form.A simple crystal may consist of only a single crystal form. A more complicated crystal may be a combination of several different forms. The crystal forms of the five non-isometric crystal systems are the monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome and sphenoid, disphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron and tetrahedron. Fifteen different forms are possible within the isometric system.

Each crystal class is a member of one of six

**crystal systems**. These systems include the isometric, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic systems. The hexagonal crystal system is further broken down into the hexagonal and rhombohedral divisions. Every crystal of a certain crystal system will share a characteristic symmetry element with the other members of its system. The crystal system of a mineral species may sometimes be determined visually by examining a particularly well-formed crystal of the species.### 2. Symmetry and Lattices

Crystals possess a regular, repetitive internal structure. The concept of

Two general types of symmetry exist. These consist of

**symmetry**describes the repetition of structural features. Crystals therefore possess symmetry, and much of the discipline of crystallography is concerned with describing and cataloging different types of symmetry.Two general types of symmetry exist. These consist of

**translational symmetry**and**point symmetry**. Translational symmetry describes the periodic repetition of a structural feature across a length or through an area or volume. Point symmetry, on the other hand, describes the periodic repetition of a structural feature around a point. Reflection, rotation, and inversion are all point symmetries.###
__Lattices__

The concept of a

The

The number of possible lattices is limited. In the plane only five different lattices may be produced by translation. One of these lattices possesses a square unit cell while another possesses a rectangular unit cell. The third possible planar lattice possesses a centered rectangular unit cell, which contains a lattice point in the center as well as lattice points on the corners. The unit cell of the fourth possible planar lattice is a parallelogram, and that of the final planar lattice is a hexagonal unit cell which may alternately be considered a rhombus.

**lattice**is directly related to the idea of translational symmetry. A lattice is a network or array composed of single motif which has been translated and repeated at fixed intervals throughout space. For example, a square which is translated and repeated many times across the plane will produce a planar square lattice.The

**unit cell**of a lattice is the smallest unit which can be repeated in three dimensions in order to construct the lattice. In a crystal, the unit cell consists of a specific group of atoms which are bonded to one another in a set geometrical arrangement. This unit and its constituent atoms are then repeated over and over in order to construct the crystal lattice. The surroundings in any given direction of one corner of a unit cell must be identical to the surroundings in the same direction of all the other corners. The corners of the unit cell therefore serve as points which are repeated to form a lattice array; these points are termed**lattice points**. The vectors which connect a straight line of equivalent lattice points and delineate the edges of the unit cell are known as the**crystallographic axes**.The number of possible lattices is limited. In the plane only five different lattices may be produced by translation. One of these lattices possesses a square unit cell while another possesses a rectangular unit cell. The third possible planar lattice possesses a centered rectangular unit cell, which contains a lattice point in the center as well as lattice points on the corners. The unit cell of the fourth possible planar lattice is a parallelogram, and that of the final planar lattice is a hexagonal unit cell which may alternately be considered a rhombus.

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3. __Crystal Systems__

Every crystal class which belongs to a certain crystal system will share a characteristic symmetry element with the other members of its system. For example, all crystals of the isometric system possess four 3-fold axes of symmetry which proceed diagonally from corner to corner through the center of the cubic unit cell. In contrast, all crystals of the hexagonal division of the hexagonal system possess a single six-fold axis of rotation.

In addition to the characteristic symmetry element, a crystal class may possess other symmetry elements which are not necessarily present in all members of the same system. The crystal class which possesses the highest possible symmetry or the highest number of symmetry elements within each system is termed the holomorphic class of the system. For example, crystals of the holomorphic class of the isometric system possess inversion symmetry, three 4-fold axes of rotational symmetry, the characteristic set of four 3-fold axes of rotational symmetry which is indicative of the isometric crystal system, six 2-fold axes of rotational symmetry, and nine different mirror planes. In contrast, a crystal which is not a member of the holomorphic class yet still belongs to the isometric system may possess only three 2-fold axes of rotational symmetry and the characteristic four 3-fold axes of rotational symmetry.

**The crystal system of a mineral species may sometimes be determined in the field by visually examining a particularly well-formed crystal of the species.**

**Isometric**

The isometric crystal system is also known as the cubic system. The crystallographic axes used in this system are of equal length and are mutually perpendicular, occurring at right angles to one another.All crystals of the isometric system possess four 3-fold axes of symmetry, each of which proceeds diagonally from corner to corner through the center of the cubic unit cell. Crystals of the isometric system may also demonstrate up to three separate 4-fold axes of rotational symmetry. These axes, if present, proceed from the center of each face through the origin to the center of the opposite face and correspond to the crystallographic axes. Furthermore crystals of the isometric system may possess six 2-fold axes of symmetry which extend from the center of each edge of the crystal through the origin to the center of the opposite edge. Minerals of this system may demonstrate up to nine different mirror planes. Examples of minerals which crystallize in the isometric system are halite, magnetite, and garnet. Minerals of this system tend to produce crystals of equidimensional or equant habit.

**Hexagonal**

Minerals of the hexagonal crystal system are referred to three crystallographic axes which intersect at 120° and a fourth which is perpendicular to the other three. This fourth axis is usually depicted vertically. The hexagonal crystal system is divided into the hexagonal and rhombohedral or trigonal divisions. All crystals of the hexagonal division possess a single 6-fold axis of rotation. In addition to the single 6-fold axis of rotation, crystals of the hexagonal division may possess up to six 2-fold axes of rotation. They may demonstrate a center of inversion symmetry and up to seven mirror planes. Crystals of the trigonal division all possess a single 3-fold axis of rotation rather than the 6-fold axis of the hexagonal division. Crystals of this division may possess up to three 2-fold axes of rotation and may demonstrate a center of inversion and up to three mirror planes.Minerals species which crystallize in the hexagonal division are apatite, beryl, and high quartz. Minerals of this division tend to produce hexagonal prisms and pyramids. Example species which crystallize in the rhombohedral division are calcite, dolomite, low quartz, and tourmaline. Such minerals tend to produce rhombohedra and triangular prisms.

**Tetragonal**

Minerals of the tetragonal crystal system are referred to three mutually perpendicular axes. The two horizontal axes are of equal length, while the vertical axis is of different length and may be either shorter or longer than the other two. Minerals of this system all possess a single 4-fold symmetry axis. They may possess up to four 2-fold axes of rotation, a center of inversion, and up to five mirror planes.Mineral species which crystallize in the tetragonal crystal system are zircon and cassiterite. These minerals tend to produce short crystals of prismatic habit.

**Orthorhombic**

Minerals of the orthorhombic crystal system are referred to three mutually perpendicular axes, each of which is of a different length than the others.Crystals of this system uniformly possess three 2-fold rotation axes and/or three mirror planes. The holomorphic class demonstrates three 2-fold symmetry axes and three mirror planes as well as a center of inversion. Other classes may demonstrate three 2-fold axes of rotation or one 2-fold rotation axis and two mirror planes. Species which belong to the orthorhombic system are olivine and barite. Crystals of this system tend to be of prismatic, tabular, or acicular habit.

**Monoclinic**

Crystals of the monoclinic system are referred to three unequal axes. Two of these axes are inclined toward each other at an oblique angle; these are usually depicted vertically. The third axis is perpendicular to the other two. The two vertical axes therefore do not intersect one another at right angles, although both are perpendicular to the horizontal axis.Monoclinic crystals demonstrate a single 2-fold rotation axis and/or a single mirror plane. The holomorphic class possesses the single 2-fold rotation axis, a mirror plane, and a center of symmetry. Other classes display just the 2-fold rotation axis or just the mirror plane.Mineral species which adhere to the monoclinic crystal system include pyroxene, amphibole, orthoclase, azurite, and malachite, among many others. The minerals of the monoclinic system tend to produce long prisms.

**Triclinic**

Crystals of the triclinic system are referred to three unequal axes, all of which intersect at oblique angles. None of the axes are perpendicular to any other axis.Crystals of the triclinic system may be said to possess only a 1-fold symmetry axis, which is equivalent to possessing no symmetry at all. Crystals of this system possess no mirror planes. The holomorphic class demonstrates a center of inversion symmetry. Mineral species of the triclinic class include plagioclase and axinite; these species tend to be of tabular habit.