Introduction to Crystallography and Mineral Crystal Systems
by Mike and Darcy Howard

Part 6: The Hexagonal System

Now we will consider the only crystal system that has 4 crystallographic axes! You will find that the Miller indices should actually be termed Bravais indices, but most people, probably out of habit, still call them Miller indices. Because there are 4 axes, there are 4 letters or numbers in the notation.

The forms of the hexagonal system are defined by the axial cross relationships. The hexagonal axes (fig. 6.1) consist of 4 axes, 3 of which are of equal length and in the same plane, as proposed by Bravais. These 3 axes, labeled a1, a2, and a3 have an angular relationship to each other of 120 degrees (between the + ends). At right angle (geometrical mathematicians say "normal") to the plane of the a axes is the c axis. Its length may vary from less than to greater than the length of any of the a axes. It will not equal the length of an a axis, however.

Note the orientation of the 4 axes and their + and - ends. If viewed vertically (down the c axis), the axes divide a circle into 6 equal parts and the axial notation reads (starting with a +) as +,-,+,-,+,-. The positive and negative ends alternating. In stating the indices of any face, four numbers (the Bravais symbol) must be given. In the Hermann-Mauguin symmetry notation, the first number refers to the principal axis of symmetry, which is coincident with c in this case. The second and third symbols, if present, refer to the symmetry elements parallel with and normal to the a1, a2, a3 crystallographic axes, respectively.

Now, surprise!! We find that the Hexagonal system has two divisions, based on symmetry. There are seven possible classes, all having 6-fold symmetry, in the Hexagonal division and five possible classes, all having 3-fold symmetry, in the Trigonal division. The general symbol for any form in the Hexagonal system is {hk-il}. The angular relation of the three horizontal axes (a1, a2, a3) shows that the algebraic sum of the indices h, k, i, is equal to 0.

The Hexagonal Division

Now, let's begin to consider the first class of the Hexagonal division. The Normal or Dihexagonal dipyramidalclass has 6-fold symmetry around the c or vertical axis. It also has 6 horizontal axes of 2-fold symmetry, 3 of which correspond to the 3 horizontal crystallographic axes and 3 which bisect the angles between the axes. It's Hermann- Mauguin notation is 6/m2/m2/m. Confused? Check out figure 6.2a and 6.2b which show the symmetry elements of this class, associated with axes and mirror planes.

6.2a & 6.2b symmetry elements
Rotational symmetry elements Symmetry planes

There are 7 possible forms which may be present in the Dihexagonal Dipyramidal class:

Form Number of Faces Miller Indices Form Expression
1. Base or basal pinacoid 2 (0001) open
2. First order prism 6 (10-10) open
3. Second order prism 6 (11-20) open
4. Dihexagonal prism 12 (hk-i0) example: (21-30) open
5. First order pyramid 12 (h0-hl) example: (10-11), (20-21) closed
6. Second order pyramid 12 (hh2hl) example: (11-22) closed
7. Dihexagonal dipyramid 24 (hk-il) example: (21-31) closed

See figures 6.3 through 6.8 (below) for what these forms look like.

First order hexagonal prism and c pinacoid Second order hexagonal prism and c pinacoid Dihexagonal prism and c pinacoid

First order hexagonal dipyramid Second order hexagonal dipyramid Dihexagonal dipyramid

The two faces of the Base, or basal pinacoid, are normal to the c axis and parallel to each other, and are generally denoted by the italic letter c. Their Miller indices are (0001) and (000-1).

The first and second order prisms cannot be distinguished from one another, as they each appear as a regular hexagonal prism with interfacial angles of 60 degrees, but when viewed down the c axis, as in figure 6.9, the relationships of the two forms to each other and to the a axes are readily noted.

The dihexagonal prism is a 12-sided prism bounded by 12 faces, each parallel to the vertical (c) axis. If you had both first and second order prisms equally expressed on the same crystal, you could not easily tell them apart from the dihexagonal form. See figure 6.5.

Corresponding to the 3 types of prisms are 3 types of pyramids. Note in the figures 6.6 and 6.7 on the previous page the similar shape, but difference in angular relation to the horizontal axes. The dihexagonal dipyramid is a double 12-sided pyramid (figure 6.8 ). The first order pyramid is labeled p. The second order pyramid is labeled s. The dihexagonal dipyramid is labeled v.

These forms look relatively simple until several of them are combined on a single crystal, then look out! You can even have several of the same form at different angles, thus 2 first order pyramids may be labeled p and u, respectively.

See figure 6.10 for a beryl crystal having all these forms displayed. Molybdenite and pyrrhotite also crystallize in this class.

The ditrigonal dipyramid{hk-il}has a 6-fold rotoinversion axis, which is chosen as c. We should note that -6 is equivalent to a 3-fold axis of rotation with a mirror plane normal to it. Three mirror planes intersect the vertical axis and are perpendicular to the 3 horizontal crystallographic axes. There are also 3 horizontal 2-fold axes of symmetry in the vertical mirror planes. The Herman-Mauguin notation is -6m2.

This class is a 12-faced form with six faces above and 6 faces below the mirror plane that lies in the a1-a2-a3 axial plane. Figure 6.11a is the ditrigonal dipyramid form and figure 6.11b displays a drawing of benitoite, the only mineral described in this class.

The Hemimorphic (dihexagonal pyramid) class. This class differs from the above discussed classes in that it has no horizontal plane of symmetry and no horizontal axes of symmetry. There is no center of symmetry. Therefore, the Hermann-Mauguin notation is 6mm. The geometry of the prisms is the same. The basal plane is a pedion (remember a pedion differs from a pinacoid in that it is a single face) and the positive and negative pyramids of the 3 types. The difference may be readily noted on a form drawing of this class (fig. 6.12 ) when compared to figure 6.8 (two pages back).

Several minerals including zincite, wurtzite, and greenockite fall in this class (figs. 6.13a, b, & c).

In the Hexagonal Trapezohedral class, the symmetry axes are the same as the Normal (dihexagonal dipyramidal class discussed initially in this section), but mirror planes and the center of symmetry are not present. The Hermann-Mauguin notation is 622. Two enantiomorphic (mirror image) forms are present, each having 12 trapezium-shaped faces (figure 6.14).

Other forms, including pinacoid, hexagonal prisms, dipyramids, and dihexagonal prisms, may be present. Only 2 minerals are known to represent this crystal class: high (beta) quartz and kalsilite.

The Hexagonal Dipyramid class (figure 6.15) has only the vertical 6-fold axis of rotation and a symmetry plane perpendicular to it. The Hermann-Mauguin notation is 6/m. When this form is by itself, it appears to possess higher symmetry. However, in combination with other forms it reveals its low symmetry content.

The general forms of this class are positive and negative hexagonal dipyramids. These forms have 12 faces, 6 above and 6 below, and correspond in position to one-half of the faces of a dihexagonal dipyramid.

Other forms present may include pinacoid and prisms. The chief minerals crystallizing in this class are those of the apatite group.

The Trigonal Dipyramid possesses a 6-fold axis of rotary inversion, thus the Hermann- Mauguin notation of -6. This is equivalent of having a 3-fold axis of rotation and a symmetry plane normal to it (3/m). See figure 6.16.

Mathematically, this class may exist, but to date no mineral is known to crystallize with this form.

In the Hexagonal pyramid class, the vertical axis is one of 6-fold rotation. No other symmetry is present. Figure 6.17 is the hexagonal pyramid. The forms of this class are similar to those of the Hexagonal Dipyramid (discussed above), but because there is no horizontal mirror plane, different forms are present at the top and bottom of the crystal. The hexagonal pyramid has four 6-faced forms: upper positive, upper negative, lower positive, lower negative. Pedions, hexagonal pyramids and prisms may be present. Only rarely is the form development sufficient to place a crystal in this class. Nepheline is the most common representative of this class.

The Trigonal Division

Now we have worked through the first 7 classes in the Hexagonal System, all having some degree of 6-fold symmetry. Time to shed that 6-fold symmetry and look at the Trigonal Division of the Hexagonal System. Here, we will see that 3-fold symmetry rules.

Remember that prisms are open forms. In the trigonal division there are two distinctive sets of prisms to be concerned with. The first is called the trigonal prism. It consists of 3 equal-sized faces which are parallel to the c crystallographic axis and which form a 3-sided prism. You may think of it as one-half the faces of the first-order hexagonal prism.

In fact, the normal light-refracting 60 degree glass prism used in many physics lab workshops is this form, bounded on the end by the c pinacoid. There exists a second order prism, which on general appearance looks the same as the first order, but when other trigonal forms are present on the termination other than the c pinacoid, the two prisms may be readily distinguished, one from the other. The second order prism is rotated 60 degrees about the c axis when compared with the first order prism.

The second prism is the ditrigonal prism, which is a 6-sided open form. This form consists of 6 vertical faces arranged in sets of 2 faces.

Therefore the alternating edges are of differing character; especially noticable when viewed by looking down the c axis.

The differing angles between the 3 sets of faces are what distingish this form from the first order hexagonal prism.

The striations on the figure to the left are typical for natural trigonal crystals, like tourmaline. In the drawing, c is the pinacoid face and m the prism faces.

I think these forms are simple enough that we don't need any drawings to explain them, but look for them on figure 6.23 (below) - the tourmaline forms. They are given the normal prism notation of m and a.

Hexagonal-scalenohedral class. The first to consider are those forms with the symmetry - 3 2/m (Hermann-Mauguin notation). There are two principal forms in this class: the rhombohedron and the hexagonal scalenohedron.

In this class, the 3-fold rotoinversion axis is the vertical axis (c) and the three 2-fold rotation axes correspond to the three horizontal axes (a1, a2, a3).

There are 3 mirror planes bisecting the angles between the horizontal axes. See figure 6.18 to observe the axes and mirror planes for the rhombohedron.

The general form {hk-il} is a hexagonal scalenohedron(figure 6.19). The primary difference in the rhombohedron and this form is that with a rhombohedral form, you have 3 rhombohedral faces above and 3 rhombohedral faces below the center of the crystal.

In a scalenohedron, each of the rhombohedral faces becomes 2 scalene triangles by dividing the rhombohedron from upper to lower corners with a line. Therefore, you have 6 faces on top and 6 faces below, the scalenohedron being a 12-faced form. These forms are illustrated in figure 6.20.

With this form, you can have both positive {h0-hl} and negative {0h-hl} forms for the rhombohedron...

and positive {hk-il} and negative {kh-il} forms for the scalenohedron.

To further complicate matters, the rhombohedron and scalenohedron, as forms, often combine with forms present in higher hexagonal symmetry classes. Thus, you may find them in combination with hexagonal prisms, hexagonal dipyramid, and pinacoid forms.

Calcite is the most common, well crystallized, and collectible mineral with these forms. See figure 6.21 for some crystallization forms of calcite. Several other minerals, such as chabazite and corundum, commonly show form combinations.

On the last 3 drawings in figure 6.21, see if YOU can name the faces present. I have already given the notation in the first 5 figures. Email me with your answer, and I'll tell you if you are right!

The next crystal class to consider is the Ditrigonal pyramid. The vertical axis is a 3-fold rotation axis and 3 mirror planes intersect in this axis. The Hermann-Mauguin notation is 3m, 3 referring to the vertical axis and m referring to three planes normal to the three horizontal axes (a1,a2,a3). These 3 mirror planes intersect in the vertical 3-fold axis.

The general form {hk-il} is a ditrigonal pyramidform. There are 4 possible ditrigonal pyramids, with the indices {hk-il}, {kh-il}, {hk-i-l}, and {kh-i-l}.

The forms are similar to the hexagonal-scalenohedral form discussed previously, but contain only half the number of faces owing to the missing 2-fold rotation axes. So crystals in this class have different forms on the top of the crystal than on the bottom. Figure 6.22 shows the ditrigonal pyramid.

Figure 6.23 shows 2 tourmaline crystals, the most common mineral crystallizing in this class, which display 3m symmetry.

This form may be combined with pedions, hexagonal prisms and pyramids, trigonal pyramids, trigonal prisms, and ditrigonal prisms to yield some complicated, though interesting, forms.

We now have come to the Trigonal-trapezohedral class. The 4 axial directions are occupied by the rotation axes. The vertical axis is an axis of 3-fold rotation and the 3 horizontal axes have 2-fold symmetry.

This is similar to those in class -32/m (hexagonal-scalenohedron), but the planes of symmetry are missing. There are 4 trigonal trapezohedrons, each composed of 6 trapezium-shaped faces. Their Miller indices are: {hk-il}, {i-k-hl}, {kh-il}, and {-ki-hl}. These forms correspond to 2 enantiomorphic pairs, each with a right and left form (one pair illustrated in figure 6.24).

Other forms which may be present include pinacoid, trigonal prisms, hexagonal prism, ditrigonal prisms, and rhombohedrons.

Quartz is the common mineral which crystallizes in this class, but only rarely is the trapezohedral face (s) displayed. When it is, it is a simple matter to determine if the crystal is right- or left-handed in form (figure 6.25).

Cinnabar also crystallizes in this class.

The Rhombohedral class has a 3-fold axis of rotoinversion, which is equivalent to a 3-fold axis of rotation and a center of symmetry. The general form is {hk-il} and the Hermann- Mauguin notation is -3.

This form is tricky because unless other forms are present, its true symmetry will not be apparent. The pinacoid {0001} and the hexagonal prisms may be present.

Dolomite and ilmenite are the two most common minerals crystallizing in this class. See figure 6.26.

Now we arrive at the final class in the Hexagonal system. The Trigonal pyramid has one 3-fold axis of rotation as its sole element of symmetry. See figure 6.27. There are, however, 8 trigonal pyramids of the general form {hk-il}, four above and four below. Each of these correspond to 3 faces of the dihexagonal dipyramid (discussed above). In addition to this, it is possible that there may be trigonal pyramids above and equivalent, but independent, pyramids below. Only when several trigonal pyramids are in combination with one another is the true symmetry revealed.

It appears that only one mineral, a rare species called gratonite, belongs to this class and it has not been studied sufficiently to remove all doubt in some crystallographer's minds.

All crystals in the Hexagonal system are oriented so that the negative end of the a3 axis (see again figure 6.1) is considered to be 0 degrees for plotting purposes. This becomes important when looking at the distribution of rhombohedral forms and determining if they are + or -.

I suggest you read page 88 of the Manual of Mineralogy - after J. D. Dana by Klein and Hurlbut (20th edition) if you wish further detail.

WOW! We have now wrapped up the Hexagonal system. I hope you are not feeling too hexed by all this discussion. If so, let's just lose that old feeling and prepare yourself to become even less symmetric as we move to the next Crystal System -- Monoclinic.

Part 7: The Monoclinic System

Index to Crystallography and Mineral Crystal Systems

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