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

Part 4: The Tetragonal System

So you didn't get enough punishment in Article 3 and are back for more. Don't say we didn't warn you that thinking about all this crystallography stuff is addictive and will warp your sense of priorities! You either love these articles by now or are totally masochistic. The lack of geometrical reasoning necessary to understand crystallography and symmetry is what drove a lot of college wannabe geologists into the College of Business! Let's begin...

Our discussion of the TETRAGONAL SYSTEM starts by examining the tetragonal axial cross and comparing it to the isometric axial cross (Article 3). Remember that in the isometric system all 3 axes were the same length and at right angles to each other. In the tetragonal system, we retain the same angular relationships, but vary the length of the vertical axis, allowing it to be either longer or shorter than the other two. We then relabel the vertical axis as c, retaining the same positive and negative orientation of this axis (see fig. 4.1a and 4.1b)

As to the Hermann-Mauguin notation for the tetragonal system, the first part of the notation (4 or -4) refers to the c axis and the second or third parts refer to the a1 and a2 axes and diagonal symmetry elements, in that order. The tetragonal prism and pyramid forms have the symmetry notation 4/m2/m2/m.

First, I want to consider the tetragonal prisms. There are 3 of these open forms consisting of the 1st order, 2nd order, and ditetragonal prisms. Because they are not closed forms, in our figures we will add a simple pinacoid termination, designated as c. The pinacoid form intersects only the c axis, so its Miller indices notation is {001}. It is a simple open 2-faced form.

The first order prism is a form having 4 faces that are parallel to the c axis and having each face intersect the a1 and a2 axes at the same distance (unity). These faces are designated by the letter m (given with Miller indices in fig. 4.2a and by m in fig. 4.2c) and the form symbol is {110}. The second order prism is essentially identical to the first order prism, but rotated about the c axis to where the faces are parallel to one of the a axes (fig. 4.2b), thus being perpendicular to the other a axis. The faces of the second order prism are designated as a and their form symbol is {100}.

It becomes apparent that the faces of both prisms are identical, and their letter designation is only dependent on how they are oriented to the two a axes. When these forms are combined (fig. 4.2c), then you may readily see their relationships, one to the other. If each form is equally developed, the result is an eight-sided prism. In this instance, we must remember that this apparent shape is the combination of two distinct forms. The third prism form is the ditetragonal prism (fig. 4.3, the common {210} form). It may easily be confused with the combination form of the first and second order prisms, especially if they are equally developed. But compare the orientation of the ditetragonal prism to the a axes in relation to the combination form. What you should do is envision looking down the c axis of the ditetragonal prism and the combined 1st and 2nd order tetragonal prisms, then you will see the similarity.

The ditetragonal prism {210} would closely approximate the combined prism forms, and with natural malformations, could be indistinguishable one from the other. When examining a natural crystal surface, features, such as orientation of striations, growth or etch pits, may be different on the two prisms of the combined form, whereas with the ditetragonal prism all these features will have the same orientation. The ditetragonal prism has the symbol (hk0).

The blue lines indicating the a axes are projected additionally on the top and bottom of this shaded drawing, so you can understand the perspective of this eight sided "stop sign" form. Another form in the tetragonal system is the dipyramid and -- yes, you guessed it -- there are 3 types of dipyramids. They correspond to the three types of prisms just described. The name dipyramid is given to a closed form whose plane intersects all three axes (this is true in all crystal systems but the isometric).

We do not allow this form to intersect the c axis at the same length as the a axes, because we already defined that form as an octahedron in the isometric system. So it can intersect at either a longer or shorter distance along the c axis than the length of the a axes. Note the orientation to the axial cross (fig. 4.4, the common {111} form). We designate the faces of the first order dipyramid as p. The second order dipyramid has the basic shape as the form of the first order dipyramid, differing only in its orientation to the axial cross (fig. 4.5, the common {011} form). The second order dipyramid faces are designated by the letter e.

Zircon is a wonderful mineral to observe both the tetragonal dipyramid and tetragonal prism faces on. In fact, you might be surprised at the variation of the length of the c axis in zircon crystals from different localities. Zircon may vary from short stocky nearly equidimensional crystals to almost acicular and have the same basic forms. Before we discuss the 3rd dipyramid form, you need to look at the various drawings in Figure 4.6 to realize the variety of what may be produced by combining these simple tetragonal forms. In figure 4.6c, the faces designated as u represent another 1st order dipyramid with a different angle of intersection with the vertical axis.

Now to the 3rd dipyramid form, the ditetragonal dipyramid. Yes, it's a closed termination form having 16 faces (fig. 4.7). Think of this form as a double 8-sided pyramid whose 16 similar faces meet the 3 axes at unequal distances. The general symbol is {hkl}. This form is rarely dominant, but is common enough as a subordinant form on zircon to be nicknamed a zirconoid. Anatase may also have this form expressed. The ditetragonal prism is often combined with the 1st order prism. In figure 4.7, although the prism is not present and therefore is simply at the junction of the two faces, we have marked its position if it had been expressed by an arrow and the letter m.

The next forms in this system to consider have the Hermann-Mauguin symmetry notation of -42m. These closed forms include the tetragonal scalenohedron (AKA rhombic scalenohedron) and the disphenoid (AKA tetragonal tetrahedron). Important to remember with both these forms is the existence of a 4-fold axis of rotary inversion.

The tetragonal disphenoid exists as both a positive and negative form. It has only 4 faces (fig. 4.8a). Both forms may be expressed on a single crystal (fig. 4.8b). The faces are designated by the letter p for the positive form and p1 for the negative form. This form differs from the tetrahedron of the isometric system in that the vertical axis is not the same length as the other two axes. The only common mineral in this class is chalcopyrite. Any mineral thought to be in this class must have very accurate interfacial angle measurements made to prove it is tetragonal and not isometric.

The tetragonal scalenohedron (fig. 4.9) is rare by itself, but is often expressed with other forms on chalcopyrite and stannite. It may be derived from the disphenoid form of this system by drawing a line from one corner of each disphenoid face to the center of the line joining the two opposite corners, and raising two faces from the resulting division. Thus, from a 4-faced disphenoid form, we derive an 8-faced form. If you are still having trouble visualizing the form in figure 4.9, you might try thinking of it as the combination of 4 classic diamond - shaped kites, every other one in an upside down orientation! This form really was a problem for my illustrator to draw!

An open form in this system is the ditetragonal pyramid, whose general notation is {hkl} (fig. 4.10). This form has no symmetry plane in relation to the 2 horizontal a axes. The symmetry notation is 4mm. Two orientations of this form in relation to the a axes exist, one noted as {hhl} and the other as {h0l}. Along with the ditetragonal pyramid may be an open single-faced form termed a pedion, having a Miller indices of {001}. The pedion will be a single face perpendicular to the c axis that "cuts off" the sharp termination of the ditetragonal pyramid. There are upper and lower forms for both the ditetragonal pyramid and the pedion, the upper being considered positive and the lower negative (just like the orientation of the c axis).

The ditetragonal pyramid looks like one half of the ditetragonal dipyramid, but on a well-formed example is present on only one end of the c axis! This form is rarely dominant, usually being subordinant to other common prism and dipyramidal forms. Diaboleite is the only mineral known to represent this crystal class. It is interesting to note that although the mineral diaboleite was first described in 1923, it was not until 1941 that crystallographers had comprehensively investigated its forms, allowing the recognition of this form. In literature earlier than 1941, you will find the note that no mineral is known to exist in this crystal class.

The tetragonal trapezohedron is the next form to consider. It is a closed form consisting of 8 trapezohedral faces, which correspond to half the faces of the ditetragonal dipyramid. Its symmetry notation is 422, having a 4-fold rotational axis parallel to the c axis and 2 2-fold axes at right angles to the c axis. Missing are a center of symmetry and any mirror planes. There exists right- and left-handed forms (fig. 4.11). Only phosgenite represents this crystal class.

In a simple form drawing (designated as e in figs. 4.12a and 4.12b), the tetragonal dipyramid appears to have a higher symmetry than 4/m, but when viewed as displayed on an actual crystal of scheelite (blue faces on fig. 4.12b), the true symmetry is revealed.Minerals possibly expressing this closed crystal form, aside from scheelite,include powellite, fergusonite, and members of the scapolite group.

Our next form is an interesting one in that it possesses only a 4-fold axis of rotary inversion corresponding to the c axis. Its symmetry notation is -4. The closed form of this tetragonal disphenoid (AKA tetragonal tetrahedron) possesses only 4 faces, which are isoceles triangles (fig. 4.13).

Without other modifying forms, like the pinacoid and tetragonal prisms, the form will appear to have two vertical symmetry planes present, giving it the symmetry of -42m (like the disphenoid we discussed above). Only one mineral - cahnite - is known to represent this class.

We have now reached our final form in the tetragonal system. Although it looks simple, it, like the last form, has very low symmetry. The tetragonal pyramid (AKA hemihedral hemimorphic) is an open form with only a 4-fold axis of rotation corresponding to the c axis (fig. 4.14). The term hemimorphic sounds fancy, but is simply a short way of saying that it appears that only half a form is displayed! No center of symmetry or mirror planes exist in this class. It has both upper {hkl} and lower {hk-l) forms, each having right- and left-hand variations. Two other tetragonal pyramids have the general form notation of {hhl} and {0kl}, depending on their form orientation to the axial cross. Wulfenite is the only mineral species to represent this form, although its crystals do not always show the difference between the pyramidal faces, above and below, to characterize distinct complimentary forms.

Well now! That was a little tedious, but certainly not that difficult. Maybe you are beginning to feel more comfortable with crystallographer's terminology. We hope you now understand that by simply stretching or compressing the vertical axis of the axial cross we had used previously in the isometric system, we defined the tetragonal system. Then, by examining the presence or absence of the various symmetry elements (mirror planes, axes of rotation, and center of symmetry), we were able to describe all possible crystal forms in the tetragonal system. Many crystallographers prefer to tackle the hexagonal system next because it has its corollaries in the tetragonal system, but we would rather play around and vary the length of yet another axis of our axial cross and see what comes of it in Article 5 - the Orthorhombic system.

So until that time, consider the symmetrical world around you and don't be afraid to look at your own mirror image!


Part 5: The Orthorhombic System

Index to Crystallography and Mineral Crystal Systems

Table of Contents

Bob Keller