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**CRYSTALLOGRAPHY**

Something is crystalline if the atoms or ions that compose it are arranged in a regular way (i.e, a crystal has internal order due to the periodic arrangement of atoms in three dimensions).

Gems are described as amorphous if they are non-crystalline.

Crystals characterized by well developed crystal faces (external surfaces) are described as euhedral . Crystals do not always show well developed crystal faces seen on euhedral examples.

A crystal is built up by arranging atoms and groups of atoms in regular patterns, for example at the corners of a cube or rectangular prism.

The basic arrangement of atoms that describes the crystal structure is identified. This is termed the unit cell.

Crystals must be charge balanced. This means that the amount of negative charge must be compensated by the same amount of positive charge.

Example:

*Al ^{3+} and O^{ 2-}*: These are combined as Al

*Hughes p. 50*: The unit cell is the fundamental building block of a crystal ie the smallest part which still possesses all the characteristics of the whole crystal before the discovery of x-ray crystallography, crystallographers determine unit cell dimensions by studying a crystal’s external shape(‘morphology’) and by measuring faces and face angles*Tulane*: The “lengths” of the various crystallographic axes are defined on the basis of the unit cell. When arrays of atoms or molecules are laid out in a space lattice we define a group of such atoms as the unit cell. This unit cell contains all the necessary points on the lattice that can be translated to repeat itself in an infinite array. In other words, the unit cell defines the basic building blocks of the crystal, and the entire crystal is made up of repeatedly translated unit cells.*Tulane*: In defining a unit cell for a crystal the choice is somewhat arbitrary. But the best choice is one where:- The edges of the unit cell should coincide with the symmetry of the lattice
- The edges of the unit cell should be related by the symmetry of the lattice
- The smallest possible cell that contains all elements should be chosen

- Crystal symmetry is a means of describing the repetition of the structural arrangement of atoms and bonds in a crystalline material.

- The axes of symmetry alone define the 7 crystal systems

*Axis of symmetry*: An imaginary lien positioned such that when the crystal structure is rotated around it, the characteristic profile appears identical two, three, four or six times during each complete rotation.

*Plane of symmetry*: An imaginary mirror plane through a crystal structure which divides it into 2 mirror-image halves.

*Centre of symmetry*: A crystal is said to possess a centre of symmetry when identical faces and edges occur on exactly opposite sides of a central point

- Crystals, and therefore minerals, have an ordered internal arrangement of atoms. This ordered arrangement shows symmetry, i.e. the atoms are arranged in a symmetrical fashion on a three dimensional network referred to as a lattice. When a crystal forms in an environment where there are no impediments to its growth, crystal faces form as smooth planar boundaries that make up the surface of the crystal. These crystal faces reflect the ordered internal arrangement of atoms and thus reflect the symmetry of the crystal lattice. To see this, let's first imagine a small 2 dimensional crystal composed of atoms in an ordered internal arrangement as shown below. Although all of the atoms in this lattice are the same, I have coloured one of them gray so we can keep track of its position.

- If we rotate the simple crystals by 90
^{o}notice that the lattice and crystal look exactly the same as what we started with. Rotate it another 90o and again its the same. Another 90o rotation again results in an identical crystal, and another 90o rotation returns the crystal to its original orientation. Thus, in 1 360o rotation, the crystal has repeated itself, or looks identical 4 times. We thus say that this object has 4-fold rotational symmetry.

Symmetry Operations and Elements

- A Symmetry operation is an operation that can be performed either physically or imaginatively that results in no change in the appearance of an object. Again it is emphasized that in crystals, the symmetry is internal, that is it is an ordered geometrical arrangement of atoms and molecules on the crystal lattice. But, since the internal symmetry is reflected in the external form of perfect crystals, we are going to concentrate on external symmetry, because this is what we can observe.
- There are 3 types of symmetry operations: rotation, reflection, and inversion. We will look at each of these in turn.
- Rotational Symmetry
- As illustrated above, if an object can be rotated about an axis and repeats itself every 90o of rotation then it is said to have an axis of 4-fold rotational symmetry. The axis along which the rotation is performed is an element of symmetry referred to as a rotation axis. The following types of rotational symmetry axes are possible in crystals.
- 1-Fold Rotation Axis - An object that requires rotation of a full 360o in order to restore it to its original appearance has no rotational symmetry. Since it repeats itself 1 time every 360o it is said to have a 1-fold axis of rotational symmetry.
- 2-fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2). Note that in these examples the axes we are referring to are imaginary lines that extend toward you perpendicular to the page or blackboard. A filled oval shape represents the point where the 2-fold rotation axis intersects the page.

This symbolism will be used for a 2-fold rotation axis throughout the lectures and in your text.

- 3-Fold Rotation Axis- Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (360/120 =3), and they will repeat 3 times in a 360o rotation. A filled triangle is used to symbolize the location of 3-fold rotation axis.
- 4-Fold Rotation Axis - If an object repeats itself after 90o of rotation, it will repeat 4 times in a 360o rotation, as illustrated previously. A filled square is used to symbolize the location of 4-fold axis of rotational symmetry.
- 6-Fold Rotation Axis - If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of rotational symmetry (360/60=6). A filled hexagon is used as the symbol for a 6-fold rotation axis.
- Although objects themselves may appear to have 5-fold, 7-fold, 8-fold, or higher-fold rotation axes, these are not possible in crystals. The reason is that the external shape of a crystal is based on a geometric arrangement of atoms. Note that if we try to combine objects with 5-fold and 8-fold apparent symmetry, that we cannot combine them in such a way that they completely fill space, as illustrated below.

- Mirror Symmetry
- A mirror symmetry operation is an imaginary operation that can be performed to reproduce an object. The operation is done by imagining that you cut the object in half, then place a mirror next to one of the halves of the object along the cut. If the reflection in the mirror reproduces the other half of the object, then the object is said to have mirror symmetry. The plane of the mirror is an element of symmetry referred to as a mirror plane, and is symbolized with the letter m. As an example, the human body is an object that approximates mirror symmetry, with the mirror plane cutting through the center of the head, the center of nose and down to the groin.
- The rectangles shown here have two planes of mirror symmetry. The rectangle on the left has a mirror plane that runs vertically on the page and is perpendicular to the page. The rectangle on the right has a mirror plane that runs horizontally and is perpendicular to the page. The dashed parts of the rectangles below show the part the rectangles that would be seen as a reflection in the mirror.
- The rectangles shown above have two planes of mirror symmetry. Three dimensional and more complex objects could have more. For example, the hexagon shown above, not only has a 6-fold rotation axis, but has 6 mirror planes.
- Note that a rectangle does not have mirror symmetry along the diagonal lines. If we cut the rectangle along a diagonal such as that labeled "m ???", as shown in the upper diagram, reflected the lower half in the mirror, then we would see what is shown by the dashed lines in lower diagram. Since this does not reproduce the original rectangle, the line "m???" does not represent a mirror plane.

Centre of Symmetry

- Another operation that can be performed is inversion through a point. In this operation lines are drawn from all points on the object through a point in the center of the object, called a symmetry center (symbolized with the letter "i"). The lines each have lengths that are equidistant from the original points. When the ends of the lines are connected, the original object is reproduced inverted from its original appearance. In the diagram shown here, only a few such lines are drawn for the small triangular face. The right hand diagram shows the object without the imaginary lines that reproduced the object.

- If an object has only a center of symmetry, we say that it has a 1 fold rotoinversion axis. Such an axis has the symbol , as shown in the right hand diagram above. Note that crystals that have a center of symmetry will exhibit the property that if you place it on a table there will be a face on the top of the crystal that will be parallel to the surface of the table and identical to the face resting on the table.

**The 32 Crystal Classes**

The 32 crystal classes represent the 32 possible combinations of symmetry operations. Each crystal class will have crystal faces that uniquely define the symmetry of the class. These faces, or groups of faces are called crystal forms. Note that you are not expected to memorize the crystal classes, their names, or the symmetry associated with each class. You will, however, be expected to determine the symmetry content of crystal models, after which you can consult the tables in your textbook, lab handouts, or lecture notes. All testing on this material in the lab will be open book.

In this lecture we will go over some of the crystal classes and their symmetry. I will not be able to cover all of the 32 classes. You will, however, see many of the 32 classes during your work in lab. I also want to point out that it is often not easy to draw a crystal of some classes where the symmetry can be represented without adding more symmetry or that can be easily seen in a two dimensional drawing.

· The crystallographic axes are imaginary lines we can draw within the crystal lattice. These will define a coordinate system within the crystal. For three dimensional space lattice we need three and in some cases four crystallographic axes that define directions within the crystal lattices. Depending on the symmetry of the lattice, the directions may or may not be perpendicular to one another, and the divisions along the coordinate axes may or may not be equal along the axes.

*Hughes:*Form refers to a specific type of crystal face. Possible forms depend on the shape of the unit cell*Tulane:*A crystal form is a set of crystal faces that are related to each other by symmetry*(One big chapter in Tulane material)*

Open Forms and Closed Forms

- A closed form is a set of crystal faces that completely enclose space. Thus, in crystal classes that contain closed forms, a crystal can be made up of a single form.

- An open form is one or more crystal faces that do not completely enclose space.
- Example 1. Pedions are single faced forms. Since there is only one face in the form a pedion cannot completely enclose space. Thus, a crystal that has only pedions, must have at least 3 different pedions to completely enclose space.

- Example 2. A prism is a 3 or more faced form wherein the crystal faces are all parallel to the same line. If the faces are all parallel then they cannot completely enclose space. Thus crystals that have prisms must also have at least one additional form in order to completely enclose space.

- Example 3. A dipyramid has at least 6 faces that meet in points at opposite ends of the crystal. These faces can completely enclose space, so a dipyramid is closed form. Although a crystal may be made up of a single dipyramid form, it may also have other forms present.

- In your textbook on pages 204 to 207, forms 1 through 18 are open forms, while forms 19 through 48 are closed forms.

- There are 48 possible forms that can be developed as the result of the 32 combinations of symmetry. We here discuss some, but not all of these forms.

Pedions

- A pedion is an open, one faced form. Pedions are the only forms that occur in the Pedial class (1). Since a pedion is not related to any other face by symmetry, each form symbol refers to a single face. For example the form {100} refers only to the face (100), and is different from the form {00} which refers only to the face (00). Note that while forms in the Pedial class are pedions, pedions may occur in other crystal classes.

Pinacoids

- A Pinacoid is an open 2-faced form made up of two parallel faces. In the crystal drawing shown here the form {111} is a pinacoid and consists of two faces, (111) and (). The form {100} is also a pinacoid consisting of the two faces (100) and (00). Similarly the form {010} is a pinacoid consisting of the two faces (010) and (00), and the form {001} is a two faced form consisting of the faces (001) and (00). In this case, note that at least three of the above forms are necessary to completely enclose space. While all forms in the Pinacoid class are pinacoids, pinacoids may occur in other crystal classes as well.

Domes

- Domes are 2- faced open forms where the 2 faces are related to one another by a mirror plane. In the crystal model shown here, the dark shaded faces belong to a dome. The vertical faces along the side of the model are pinacoids (2 parallel faces). The faces on the front and back of the model are not related to each other by symmetry, and are thus two different pedions.

Sphenoids

- Sphenoids are2 - faced open forms where the faces are related to each other by a 2-fold rotation axis and are not parallel to each other. The dark shaded triangular faces on the model shown here belong to a sphenoid. Pairs of similar vertical faces that cut the edges of the drawing are also pinacoids. The top and bottom faces, however, are two different pedions.

Prisms

- A prism is an open form consisting of three or more parallel faces. Depending on the symmetry, several different kinds of prisms are possible.
- Trigonal prism: 3 - faced form with all faces parallel to a 3 -fold rotation axis
- Ditrigonal prism: 6 - faced form with all 6 faces parallel to a 3-fold rotation axis. Note that the cross section of this form (shown to the right of the drawing) is not a hexagon, i.e. it does not have 6-fold rotational symmetry.
- Rhombic prism: 4 - faced form with all faces parallel to a line that is not a symmetry element. In the drawing to the right, the 4 shaded faces belong to a rhombic prism. The other faces in this model are pinacoids (the faces on the sides belong to a side pinacoid, and the faces on the top and bottom belong to a top/bottom pinacoid).
- Tetragonal prism: 4 - faced open form with all faces parallel to a 4-fold rotation axis or . The 4 side faces in this model make up the tetragonal prism. The top and bottom faces make up the a form called the top/bottom pinacoid.
- Ditetragonal prism: 8 - faced form with all faces parallel to a 4-fold rotation axis. In the drawing, the 8 vertical faces make up the ditetragonal prism.
- Hexagonal prism: 6 - faced form with all faces parallel to a 6-fold rotation axis. The 6 vertical faces in the drawing make up the hexagonal prism. Again the faces on top and bottom are the top/bottom pinacoid form.
- Dihexagonal prism: 12 - faced form with all faces parallel to a 6-fold rotation axis. Note that a horizontal cross-section of this model would have apparent 12-fold rotation symmetry. The dihexagonal prism is the result of mirror planes parallel to the 6-fold rotation axis.

Pyramids

- A pyramid is a 3, 4, 6, 8 or 12 faced open form where all faces in the form meet, or could meet if extended, at a point.
- Trigonal pyramid: 3-faced form where all faces are related by a 3-fold rotation axis.
- Ditrigonal pyramid: 6-faced form where all faces are related by a 3-fold rotation axis. Note that if viewed from above, the ditrigonal pyramid would not have a hexagonal shape; its cross section would look more like that of the trigonal prism discussed above.
- Rhombic pyramid: 4-faced form where the faces are related by mirror planes. In the drawing shown here the faces labeled "p" are the four faces of the rhombic pyramid. If extend, these 4 faces would meet at a point.
- Tetragonal pyramid: 4-faced form where the faces are related by a 4 axis. In the drawing the small triangular faces that cut the corners represent the tetragonal pyramid. Note that if extended, these 4 faces would meet at a point.
- Ditetragonal pyramid: 8-faced form where all faces are related by a 4 axis. In the drawing shown here, the upper 8 faces belong to the ditetragonal pyramid form. Note that the vertical faces belong to the ditetragonal prism.
- Hexagonal pyramid: 6-faced form where all faces are related by a 6 axis. If viewed from above, the hexagonal pyramid would have a hexagonal shape.
- Dihexagonal pyramid: 12-faced form where all faces are related by a 6-fold axis. This form results from mirror planes that are parallel to the 6-fold axis.

Dipyramids

- Dipyramids are closed forms consisting of 6, 8, 12, 16, or 24 faces. Dipyramids are pyramids that are reflected across a mirror plane. Thus, they occur in crystal classes that have a mirror plane perpendicular to a rotation or rotoinversion axis.
- Trigonal dipyramid: 6-faced form with faces related by a 3-fold axis with a perpendicular mirror plane. In this drawing, all six faces belong to the trigonal-dipyramid.
- Ditrigonal -dipyramid: 12-faced form with faces related by a 3-fold axis with a perpendicular mirror plane. If viewed from above, the crystal will not have a hexagonal shape, rather it would appear similar to the horizontal cross-section of the ditrigonal prism, discussed above.
- Rhombic dipyramid: 8-faced form with faces related by a combinations of 2-fold axes and mirror planes. The drawing to the right shows 2 rhombic dipyramids. One has the form symbol {111} and consists of the four larger faces shown plus four equivalent faces on the back of the model. The other one has the form symbol {113} and consists of the 4 smaller faces shown plus the four on the back.
- Tetragonal dipyramid: 8-faced form with faces related by a 4-fold axis with a perpendicular mirror plane. The drawing shows the 8-faced tetragonal dipyramid. Also shown is the 4-faced tetragonal prism, and the 2-faced top/bottom pinacoid.
- Ditetragonal dipyramid: 16-faced form with faces related by a 4-fold axis with a perpendicular mirror plane. The ditetragonal dipyramid is shown here. Note the vertical faces belong to a ditetragonal prism.
- Hexagonal dipyramid: 12-faced form with faces related by a 6-fold axis with a perpendicular mirror plane. The vertical faces in this model make up a hexagonal prism.
- Dihexagonal dipyramid: 24-faced form with faces related by a 6-fold axis with a perpendicular mirror plane.

Trapezohedrons

- Trapezohedron are closed 6, 8, or 12 faced forms, with 3, 4, or 6 upper faces offset from 3, 4, or 6 lower faces. The trapezohedron results from 3-, 4-, or 6-fold axes combined with a perpendicular 2-fold axis. An example of a tetragonal trapezohedron is shown in the drawing to the right. Other examples are shown in your textbook.

Scalenohedrons

- A scalenohedron is a closed form with 8 or 12 faces. In ideally developed faces each of the faces is a scalene triangle. In the model, note the presence of the 3-fold rotoinversion axis perpendicular to the 3 2-fold axes.

Rhombohedrons

- A rhombohedron is 6-faced closed form wherein 3 faces on top are offset by 3 identical upside down faces on the bottom, as a result of a 3-fold rotoinversion axis. Rhombohedrons can also result from a 3-fold axis with perpendicular 2-fold axes. Rhombohedrons only occur in the crystal classes 2/m , 32, and .

Disphenoids

- A disphenoid is a closed form consisting of 4 faces. These are only present in the orthorhombic system (class 222) and the tetragonal system (class )
- The rest of the forms all occur in the isometric system, and thus have either four 3-fold axes or four axes. Only some of the more common isometric forms will be discussed here.

Hexahedron

- A hexahedron is the same as a cube. 3-fold axes are perpendicular to the face of the cube, and four axes run through the corners of the cube. Note that the form symbol for a hexahedron is {100}, and it consists of the following 6 faces: (100), (010), (001), (00), (00), and (00).

Octahedron

- An octahedron is an 8 faced form that results form three 4-fold axes with perpendicular mirror planes. The octahedron has the form symbol {111}and consists of the following 8 faces: (111), (), (11), (1), (1), (1), (11), and (11). Note that four 3-fold axes are present that are perpendicular to the triangular faces of the octahedron (these 3-fold axes are not shown in the drawing).

Dodecahedron

- A dodecahedron is a closed 12-faced form. Dodecahedrons can be formed by cutting off the edges of a cube. The form symbol for a dodecahedron is {110}. As an exercise, you figure out the Miller Indices for these 12 faces.

Tetrahexahedron

- The tetrahexahedron is a 24-faced form with a general form symbol of {0hl} This means that all faces are parallel to one of the a axes, and intersect the other 2 axes at different lengths.

Trapezohedron

- An isometric trapezohedron is a 12-faced closed form with the general form symbol {hhl}. This means that all faces intersect two of the a axes at equal length and intersect the third a axis at a different length.

Tetrahedron

- The tetrahedron occurs in the class 3m and has the form symbol {111}(the form shown in the drawing) or {11} (2 different forms are possible). It is a four faced form that results form three axes and four 3-fold axes (not shown in the drawing).

Gyroid

- A gyroid is a form in the class 432 (note no mirror planes)

Pyritohedron

- The pyritohedron is a 12-faced form that occurs in the crystal class 2/m. Note that there are no 4-fold axes in this class. The possible forms are {h0l} or {0kl} and each of the faces that make up the form have 5 sides.

Diploid

- The diploid is the general form {hkl} for the diploidal class (2/m). Again there are no 4-fold axes.

Tetartoid

- Tetartoids are general forms in the tetartoidal class (23) which only has 3-fold axes and 2-fold axes with no mirror planes.

In nature perfect crystals are rare. The faces that develop on a crystal depend on the space available for the crystals to grow. If crystals grow into one another or in a restricted environment, it is possible that no well-formed crystal faces will be developed. However, crystals sometimes develop certain forms more commonly than others, although the symmetry may not be readily apparent from these common forms. The term used to describe general shape of a crystal is *habit.*

Some common crystal habits are as follows (discussed previously):

Individual Crystals

*Cubic*- cube shapes*Octahedral*- shaped like octahedrons, as described above.*Tabular*- rectangular shapes.*Equant*- a term used to describe minerals that have all of their boundaries of- approximately equal length.
*Acicular*- long, slender crystals.*Prismatic*- abundance of prism faces.*Bladed*- like a wedge or knife blade.

Groups of Distinct Crystals

*Dendritic*- tree-like growths.*Reticulated -*lattice-like groups of slender crystals.*Radiated -*radiating groups of crystals.*Fibrous*- elongated clusters of fibers.*Botryoidal*- smooth bulbous or globular shapes.*Globular*- radiating individual crystals that form spherical groups.*Drusy*- small crystals that cover a surface.*Stellated*- radiating individuals that form a star-like shape.

Some minerals characteristically show one or more of these habits, so habit can sometimes be a powerful diagnostic tool.

*Tulane: Crystal form, Zones and habit p. 12*

Although a crystal structure is an ordered arrangement of atoms on a lattice, as we have seen, the order may be different along different directions in the crystal. Thus, some properties of crystals depend on direction. These are called vectorial properties, and can be divided into two categories: continuous and discontinuous.

Continuous Vectorial Properties

Continuous vectorial properties depend on direction, but along any given the direction the property is the same. Some of the continuous vectorial properties are:

Hardness - In some minerals there is a difference in hardness in different directions in the crystal. Examples: Kyanite, Biotite, Muscovite. This can become an important identifying property and/or may lead to confusion about the hardness if one is not aware of the directional dependence.

Velocity of Light (Refractive Index) - For all minerals except those in the isometric system, the velocity of light is different as the light travels along different directions in the crystal. We will use this directional dependence of light velocity as an important tool in the second half of the course. Refractive Index is defined as the velocity of light in a material divided by the velocity of light in a vacuum. Because the velocity of light depends on direction, the refractive index will also depend on direction.

Thermal Conductivity - The ability of a material to conduct heat is called thermal conductivity. Like light, heat can be conducted at different rates along different directions in crystals.

Electrical Conductivity- The ability of a material to allow the passage of electrons is called electrical conductivity, which is also directionally dependent except in isometric crystals.

Thermal Expansion - How much the crystal lattice expands as it is heated is referred to as thermal expansion. Some crystals expand more in one direction than in others, thus thermal expansion is a vectorial property.

Compressibility - Compressibility is a measure of how the lattice is reduced as atoms are pushed closer together under pressure. Some directions in crystals may be more compressible than others.

Discontinuous Vectorial Properties

Discontinuous vectorial properties pertain only to certain directions or planes within a crystal. For these kinds of properties, intermediate directions may have no value of the property. Among the discontinuous vectorial properties are:

Cleavage - Cleavage is defined as a plane within the lattice along which breakage occurs more easily than along other directions. A cleavage direction develops along zones of weakness in the crystal lattice. Cleavage is discontinuous because it only occurs along certain planes.

Growth Rate - Growth rate is defined as the rate at which atoms can be added to the crystal. In some directions fewer atoms must be added to the crystal than in other directions, and thus some directions may allow for faster growth than others.

Solution Rate - Solution rate is the rate at which a solid can be dissolved in a solvent. In this case it depends on how tightly bonded the atoms are in the crystal structure, and this usually depends on direction.

Characterized by a single 4-fold or 4-fold rotoinversion axis.

Tetragonal-pyramidal Class, 4, Symmetry content - 1A4

Since this class has a single 4-fold axis and no mirror planes, there are no pyramid faces on the bottom of the crystal. Wulfinite is the only mineral known to crystallize in this class.

Tetragonal-disphenoidal Class, , Symmetry content - 14

With only a single 4-fold rotoinversion axis, the disphenoid faces consist of two identical faces on top, and two identical faces on the bottom, offset by 90o. Note that there are no mirror planes in this class. Only one rare mineral is known to form crystals of this class.

Tetragonal-dipyramidal Class, 4/m, Symmetry content - 1A4, 1m, i

This class has a single 4-fold axis perpendicular to a mirror plane. This results in 4 pyramid faces on top that are reflected across the mirror plane to form 4 identical faces on the bottom of the crystal. Scheelite and scapolite are the only common minerals in this class.

Tetragonal-trapezohedral Class, 422, Symmetry content - 1A4, 4A2

This class has a 4 fold axis perpendicular to 4 2-fold axes. There are no mirror planes. Only one rare mineral belongs to this class.

Ditetragonal-pyramidal Class, 4mm, Symmetry content - 1A4, 4m

This class has a single 4-fold axis and 4 mirror planes. The mirror planes are not shown in the diagram, but would cut through the edges and center of the faces shown. Note that the ditetragonal pyramid is a set of 8 faces that form a pyramid on the top of the crystal. Only one rare mineral forms in the crystal class.

Tetragonal-scalenohedral Class, 2m, Symmetry Content - 14, 2A2, 2m

This class has a 4-fold rotoinversion axis that is perpendicular to 2 2-fold rotation axes. The 2 mirror planes a parallel to the and are at 45o to the 2-fold axes. Chalcopyrite and stannite are the only common minerals with crystals in this class.

Ditetragonal-dipyramidal Class, 4/m2/m2/m, Symmetry content - 1A4, 4A2, 5m, i

This class has the most symmetry of the tetragonal system. It has a single 4-fold axis that is perpendicular to 4 2-fold axes. All of the 2-fold axes are perpendicular to mirror planes. Another mirror plane is perpendicular to the 4-fold axis. The mirror planes are not shown in the diagram, but would cut through all of the vertical edges and through the center of the pyramid faces. The fifth mirror plane is the horizontal plane. Note the ditetragonal-dipyramid consists of the 8 pyramid faces on the top and the 8 pyramid faces on the bottom.

Common minerals that occur with this symmetry are anatase, cassiterite, apophyllite, zircon, and vesuvianite.

Characterized by having only two fold axes or a 2-fold axis and 2 mirror planes.

Rhombic -disphenoidal Class, 222, Symmetry content - 3A2

In this class there are 3 2-fold axis and no mirror planes. The 2-fold axes are all perpendicular to each other. The disphenoid faces that define this group consist of 2 faces on top of the crystal and 2 faces on the bottom of the crystal that are offset from each other by 90o. Epsomite is the most common rare mineral of this class.

Rhombic-pyramidal Class, 2mm (mm2), Symmetry content - 1A2, 2m

This class has two perpendicular mirror planes and a single 2-fold rotation axis. Because it has not center of symmetry, the faces on the top of the crystal do not occur on the bottom. A pyramid, is a set of 3 or more identical faces that intersect at a point. In the case of the rhombic pyramid, these would be 4 identical faces, labeled p, in the diagram.

Hemimorphite is the most common mineral with this symmetry.

Rhombic-dipyramidal Class, 2/m2/m2/m, Symmetry content - 3A2, 3m, i

This class has 3 perpendicular 2-fold axes that are perpendicular to 3 mirror planes. The dipyramid faces consist of 4 identical faces on top and 4 identical faces on the bottom that are related to each other by reflection across the horizontal mirror plane or by rotation about the horizontal 2-fold axes.

The most common minerals in this class are andalusite, anthophyllite, aragonite, barite, cordierite, olivine, sillimanite, stibnite, sulfur, and topaz.

Characterized by having only mirror plane(s) or a single 2-fold axis.

Sphenoidal Class, 2, Symmetry content - 1A2

In this class there is a single 2-fold rotation axis. Faces related by a 2-fold axis are called sphenoids, thus this is the sphenoidal class. Only rare minerals belong to this class.

Domatic Class, m, Symmetry content - 1m

This class has a single mirror plane. Faces related by a mirror plane are called domes, thus this is the domatic class. Only 2 rare minerals crystallize in this class.

Prismatic Class, 2/m. Symmetry content - 1A2, m, i

This class has a single 2-fold axis perpendicular to a single mirror plane. This class has pinacoid faces and prism faces. A prism is defined as 3 or more identical faces that are all parallel to the same line. In the prismatic class, these prisms consist of 4 identical faces, 2 of which are shown in the diagram on the front of the crystal. The other two are on the back side of the crystal.

The most common minerals that occur in the prismatic class are the micas (biotite and muscovite), azurite, chlorite, clinopyroxenes, epidote, gypsum, malachite, kaolinite, orthoclase, and talc.

Characterized by only 1-fold or 1-fold rotoinversion axis

Pedial Class, 1, Symmetry content - none

In this class there is no symmetry, so all crystal faces are unique and are not related to each other by symmetry. Such faces are called Pedions, thus this is the Pedial Class. Only a few rare minerals are in this class.

Pinacoidal Class, , Symmetry content - i

Since in this class there is only a center of symmetry, pairs of faces are related to each other through the center. Such faces are called pinacoids, thus this is the pinacoidal class. Among the common minerals with pinacoidal crystals are: microcline (K-feldspar), plagioclase, turquoise, and wollastonite.

- Under certain conditions, crystals may form symmetrical intergrowths, known as twins, in which the crystal lattice of one part bears a definite crystallographic relationship to that of the other section.

**Types of Twinning**

**Contact twins:**have a planar composition surface separating two individual crystals. These are usually defined by a twin law that expresses a twin plane(i.e. an added mirror plane). Contact twins can also appear as repeated or multiple twins:- If the composition’s surfaces are parallel to one another, they are called
**polysynthetic twins**. - If the composition’s surfaces are not parallel to one another they are called
**cyclical twins**. **Penetration twins:**have an irregular composition surface separating two individual crystals. These are defined by a twin centre or twin axis.

**Origin of Twinning**

- Twin crystals may be classified into one of three categories according to the origin:
**Growth twins:**Formed via a mistake during growth, where atoms take up incorrect positions on the growing crystal. This results in a change in growth direction, and formation of a twin crystal.**Transformation twins:**These are ‘secondary twins’ which occur in a pre-existing crystal. They occur when a crystal formed at high temperatures is cooled and subsequently rearranges its structure from the high-temperature form. This is common in feldspars, but is not found in corundum.**Glide or deformation twins:**Also a type of secondary twinning, these result from a deformation of the crystal due to mechanical stress. If the stress produces slippage(gliding) of atoms on a small scale, twins may result; large slippage may cause slippage without twinning, eventually resulting in fracturing.- This type of twinning is exemplified by calcite, where pressure from a razor blade on the rhombohedron edge produces glide twins right before one’s eyes. It is also common in corundum, and typically occurs repeatedly throughout a crystal. (Hughes p52)

Metamict minerals are minerals whose crystal structure has been partially destroyed by radiation from contained radioactive elements. The breakdown of the crystal structure results from bombardment of a particles emitted by the decay of U and Th radioactive isotopes.

The mineral zircon (ZrSiO_{4}) often has U and Th atoms substituting for Zr in the crystals structure. Since U and Th have radioactive isotopes, Zircon is often seen to occur in various stages of metamictization.

By definition, a mineral has to have an ordered atomic arrangement, or crystalline structure. There are some Earth materials that fit all other parts of the definition of a mineral, yet do not have a crystalline structure. Such compounds are termed ** amorphous **(without form).

Some of these amorphous compounds are called mineraloids. These usually form at low temperatures and pressures during the process of chemical weathering and form mammillary, botryoidal, and stalactitic masses with widely varying chemical compositions. Limonite [FeO.(OH).nH_{2}O] and allophane ( a hydrous aluminum silicate) are good examples.

Polymorphism means many forms or shapes. It is when minerals have the same chemical composition but different crystal structures resulting in different minerals. Diamond and graphite are both composed of carbon but obviously different minerals with vastly different physical and optical properties. Graphite is the soft pencil lead and used as a lubricant, while diamond is a hard gemstone and used as an abrasive for cutting in its industrial applications.

Polymorphism means "many forms". In mineralogy it means that a single chemical

composition can exist with two or more different crystal structures. As we will see when we look more closely at crystal structures, if a crystal is subjected to different pressures and temperatures, the arrangement of atoms depends on the sizes of the atoms, and the sizes change with temperature and pressure. In general, as pressure increases the volume of a crystal will decrease and a point may be reached where a more compact crystal structure is more stable. The crystal structure will then change to that of the more stable structure, and a different mineral will be in existence. Similarly, if the temperature is increased, the atoms on the crystal structure will tend to vibrate more and increase their effective size. In this case, a point may be reached where a less compact crystal structure is more stable. When the crystal structure changes to the more stable structure a different mineral will form.

The change that takes place between crystal structures of the same chemical compound are called *polymorphic transformations.*

Isomorphism means one or the same form or shape. Isomorphic minerals have the same geometric structural arrangement, but with different atoms or ions in the sites that results in different minerals. Minerals with the same anion belong to isostructural groups, such as garnet and spinel groups. These groups have the same structural configuration but a wide diversity in chemical composition.

Pseudomorphism means false form or shape. It is when a mineral crystal's chemical composition and/or crystal structure is changed, but the external form is preserved. An example is in the alteration of azurite crystals to malachite, and the replacement of wood by chalcedony to produce petrified wood. A common misconception is that tiger's-eye and hawk's-eye (blue tiger's-eye) are quartz pseudomorphs after the asbestos mineral crocidolite, that is the quartz replaced the asbestos fibers. This has recently been shown to be a false assumption. Heaney and Fisher (2003) have a new interpretation of the origin of tiger's-eye in that quartz crystal growth is synchronous with the crocidolite through a crack-seal vein-filling process (p. 323).

** Pseudomorphism **is the existence of a mineral that has the appearance of another mineral. Pseudomorph means false form. Pseudomorphism occurs when a mineral is altered in such a way that its internal structure and chemical composition is changed but its external form is preserved. Three mechanisms of pseudomorphism can be defined:

**Substitution**. In this mechanism chemical constituents are simultaneously removed and replaced by other chemical constituents during alteration. An example is the replacement of wood fibers by quartz to form petrified wood that has the outward appearance of the original wood, but is composed of quartz. Another example is the alteration of fluorite which forms isometric crystals and is sometimes replaced by quartz during alteration. The resulting quartz crystals look isometric, and are said to be pseudomorphed after fluorite.**Encrustation**. If during the alteration process a thin crust of a new mineral forms on the surface of a preexisting mineral, then the preexisting mineral is removed, leaving the crust behind, we say that pseudomorphism has resulted from encrustation. In this case the thin crust of the new mineral will have casts of the form of the original mineral.**Alteration**. If only partial removal of the original mineral and only partial replacement by the new mineral has taken place, then it is possible to have a the space once occupied entirely by the original mineral be partially composed of the new mineral. This results for example in serpentine pseudomorphed after olivine or pyroxene, anhydrite (CaSO_{4}) pseudomorphed after gypsum (CaSO_{4}.2H_{2}O), limonite [FeO.(OH).nH_{2}O] after pyrite (FeS_{2}), and anglesite (PbSO_{4}) after galena (PbS).

Webster, ** Practical Gemmology**, Lessons 1,2

Read, ** Gemmology**, Chapters 1,2,3

Hurlbut, C. S., & Kammerling, R. C. (1991), ** Gemology**, Chapters 1,2, NY: John Wiley & Sons, Inc.

Gemmological Association of Great Britain, ** FGA Foundation course materials, **Chapters 1,2

*GIA Colored Stones course materials*

** CGA Preliminary course materials, **Chapters 1,2

Schumann, Walter. (1997) ** Gemstones of the World, **NY: Sterling Publishing Co, Inc.

Harvey, Anne (1981**). Jewels**. London: Bellew & Higton Publishers Limited.

Kunz, G. F. (1971). ** The curious lore of precious stones**. NY: Dover.