EENS 2110 
Mineralogy 
Tulane University 
Prof. Stephen A. Nelson 
Crystal Form, Zones, Crystal Habit 

Crystal Forms As stated at the end of the last lecture, the next step is to use the Miller Index notation to designate crystal forms. A crystal form is a set of crystal faces that are related to each other by symmetry. To designate a crystal form (which could imply many faces) we use the Miller Index, or MillerBravais Index notation enclosing the indices in curly braces, i.e. {101} or {111} Such notation is called a form symbol. An important point to note is that a form refers to a face or set of faces that have the same arrangement of atoms. Thus, the number of faces in a form depends on the symmetry of the crystal. General Forms and Special Forms A general form is a form in a particular crystal class that contains faces that intersect all crystallographic axes at different lengths. It has the form symbol {hkl} All other forms that may be present are called special forms. In the monoclinic, triclinic, and orthorhombic crystal systems, the form {111} is a general form because in these systems faces of this form will intersect the a, b, and c axes at different lengths because the unit lengths are different on each axis. In crystals of higher symmetry, where two or more of the axes have equal length, a general form must intersect the equal length axes at different multiples of the unit length. Thus in the tetragonal system the form {121} is a general form. In the isometric system a general form would have to be something like {123}. 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.
In your textbook on pages 139 to 142, 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 2faced 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 2fold 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. 







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. 







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. 







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 6fold axes combined with a perpendicular 2fold 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 3fold rotoinversion axis perpendicular to the 3 2fold axes. 
Rhombohedrons
A rhombohedron is 6faced closed form wherein 3 faces on top are offset by 3 identical upside down faces on the bottom, as a result of a 3fold rotoinversion axis. Rhombohedrons can also result from a 3fold axis with perpendicular 2fold 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 3fold 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. 4fold 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:

Octahedron An octahedron is an 8 faced form that results form three 4fold axes with perpendicular mirror planes. The octahedron has the form symbol {111}and consists of the following 8 faces:
Note that four 3fold axes are present that are perpendicular to the triangular faces of the octahedron (these 3fold axes are not shown in the drawing). 
Dodecahedron A dodecahedron is a closed 12faced 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 24faced 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 12faced 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 3fold 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 12faced form that occurs in the crystal class 2/m. Note that there are no 4fold 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 4fold axes. 
Tetartoid
Tetartoids are general forms in the tetartoidal class (23) which only has 3fold axes and 2fold axes with no mirror planes. 
In class we will fill in the following table in order to help you better understand the relationship between form and crystal faces. The assignment will be to determine for each form listed across the top of the table the number of faces in that form, the name of the form, and the number of cleavage directions that the form symbol would imply for each of the crystal classes listed in the lefthand column. Before we can do this, however, we need to review how we define the crystallographic axes in relation to the elements of symmetry in each of the crystal systems. Triclinic  Since this class has such low symmetry there are no constraints on the axes, but the most pronounced face should be taken as parallel to the c axis. Monoclinic  The 2 fold axis is the b axis, or if only a mirror plane is present, the b axis is perpendicular to the mirror plane. 
Orthorhombic  The current convention is to take the longest axis as b, the intermediate axis is a, and the shortest axis is c. An older convention was to take the c axis as the longest, the b axis intermediate, and the a axis as the shortest. Tetragonal  The c axis is either the 4 fold rotation axis or the rotoinversion axis. Hexagonal  The c axis is the 6fold, 3fold, axis, or . Isometric  The equal length a axes are either the 3 4fold rotation axes, rotoinversion axes, or, in cases where no 4 or axes are present, the 3 2fold axes.

Symmetry 
{010} 
{001} 

#Faces  Form  #Cleavage Directions  #Faces  Form  #Cleavage Directions  
1  
2  
2/m  
2/m2/m2/m  
4/m2/m2/m  
4/m 2/m 
Symmetry 
{110} 
{111}  
#Faces  Form  #Cleavage Directions  #Faces  Form  #Cleavage Directions  
1  
2  
2/m  
2/m2/m2/m  
4/m2/m2/m  
4/m 2/m 
A zone is defined as a group of crystal faces that intersect in parallel edges. Since the edges will all be parallel to a line, we can define that the direction of the line using a notation similar to Miller Indices. This notation is called the zone symbol. The zone symbol looks like a Miller Index, but is enclosed in square brackets, i.e. [uvw]. For a group of faces in the same zone, we can determine the zone symbol for all nonhexagonal minerals by choosing 2 nonparallel faces (hkl) and (pqr).

To do so, we write the Miller Index for each face twice, one face directly beneath the other, as shown below. The first and last numbers in each line are discarded. Then we apply the following formula to determine the indices in the zone symbol. u = k*r  l*q, v = l*p  h*r, and w = h*q  k*p 
For example, faces (110) and (010) are not parallel to each other. The zone symbol for these faces (and any other faces that lie in the same zone) is determined by writing 110 twice and then immediately below, writing 010 twice. Applying the formula above gives the zone symbol for this zone as [001]. 
Note that this zone symbol implies a line that is perpendicular to the face with the same index. In other words, [001] is a line perpendicular to the face (001). It can thus be used as a symbol for a line. In this case, the line is the c crystallographic axis. 
Zone symbols, therefore are often used to denote directions through crystals. Being able to specify directions in crystals is important because many properties of minerals depend on direction. These are called vectorial properties. 
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:
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:

Crystal Habit 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 wellformed 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.


Examples of questions on this material that could be asked on an exam
