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
