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

Part 8: The Triclinic System

You must be glad to get to the last system's article! I know I am. In our overall examination of 3-axes systems, this one is relatively short and only moderately difficult to understand due to the lack of symmetry.

So let's start, as we have with all the other systems, by looking at the axial cross of the Triclinic System (fig. 8.1). In this figure, we see that all 3 axes (a, b, and c) are unequal in length to each other and that there are no axial angles of 90 degrees. In the Monoclinic System, at least we had a and b axes at right angles, but here we have lost even that!

Note that the angle beta is still between the c and a axes, but we now have 2 additional angles to define, neither of which are equal to 90 degrees. One angle is termed alpha and is defined as the angle between the c and b axes and the second is gamma which is defined as the angle between a and b. Now, we must have some accepted conventions or rules to follow to orient a triclinic crystal, or we will always be in a state of confusion with other folks over just the orientation.

Remember, in the orientation of any crystal, you also are determining the position of the 3 crystallographic axes. So, the rules are: 1) the most pronounced zone should be vertical and therefore the axis in this zone becomes the c; 2) the {001}form (basal pinacoid) should slope forward and to the right; and 3) select two forms in the vertical zone, one will be the {100} and the other will be the {010}. Now, the direction of the a axis is determined by the intersection of {101} and {001} and the direction of the b axis is determined by the intersection of {100} and {001}. Once this is done, the a axis should be shorter than the b axis so that the convention becomes c < a < b. The axial distances and the 3 angles, alpha, beta, and gamma, can be calculated only with considerable difficulty. As in the Monoclinic system, the b axis length is defined as unity (1). The crystallography information concerning a triclinic mineral will include the following (an example): a:b:c = 0.972: 1 : 0.778; alpha = 102 degrees 41 minutes, beta = 98 degrees 09 minutes, gamma = 88 degrees 08 minutes.

In the triclinic system, we have two symmetry classes. The first we will consider is the -1 (Hermann-Mauguin notation). In this class, there is a 1-fold axis of symmetry, the equivalent of a center of symmetry or inversion.

Figure 8.2 shows a triclinic pinacoid (or parallelohedron). This class is termed the pinacoidal class after its general form {hkl}. So all the forms present are pinacoids and therefore consist of two identical and parallel faces.

When you orient a triclinic crystal, the Miller indices of the pinacoid determine its position. There are 3 pinacoids.

Remember pinacoids intersect one axis and are parallel to the other 2 (in 3 axes systems). So let's start by looking at the -1 symmetry. This is a one-fold axis of rotoinversion, which may be viewed as the same as having a center of symmetry.

Figure 8.3 shows a triclinic pinacoid, also called a parallelohedron. This class is referred to as the pinacoidal class, due to its {hkl} form. With -1 symmetry, all forms are pinacoids so they consist of 2 identical parallel faces. Once a triclinic crystal is oriented, then the Miller indices of the pinacoid establish its position.

Figure 8.3 Triclinic pinacoids, or parallelohedrons

There are 3 general types of pinacoids: those that intersect only one crystallographic axis, those that intersect 2 axes, and those that intersect all 3 axes. The first type are the pinacoids {100}, {010}, and {001}. The {100} is the front pinacoid and intersects the a axis, the {010} is the side or b pinacoid and intersects the b axis, and the {001} is the c or basal pinacoid and intersects the c axis. All of these forms are by convention based on the + end of the axis.

The second type of pinacoid is termed the {0kl}, {h0l}, and {hk0} pinacoids, respectively. The {0kl} pinacoid is parallel to the a axis and therefore intersects the b and c axes. It may be positive {0kl} or negative {0-kl}. The {h0l} pinacoid is parallel to the b axis and intersects the a and c axes. It may be positive {h0l} or negative {-h0l}. Finally, the {hk0} pinacoid is parallel to the c axis and intersects the a and b axes. It may be positive {hk0} or negative {h-k0}.

The third type of pinacoid is the {hkl}. There exist positive right {hkl}, positive left {h- kl}, negative right {-hkl}, and negative left {-h-kl}. Each of these 2-faced forms may exist independently of the others. Figure 8.3 shows some of the pinacoidal forms in this class. A number of minerals crystallize in the -1 class including plagioclase feldspar pectolite, microcline, and wollastonite. The second symmetry class of the triclinic system is the 1, which is equivalent to no symmetry! It is a single face termed a pedion and the class called the pedial class after its {hkl} form. Because the form consists of a single face, each pedion or monohedron stands by itself. Rare is the mineral that crystallizes in this class, axinite being an example.

We have now finished our discussion of the Crystal Systems and their geometrical and symmetry relationships. I can hardly believe it! If you feel like pursuing the subject of symmetry further, go to Article 9 for my summary remarks and some suggested additional references and articles.

Part 9: Conclusion and Further Reading

Index to Crystallography and Mineral Crystal Systems

Table of Contents

Bob Keller