Having dispensed with the hexagonal system in article 6, we are ready to resume our task of the removal of symmetry from 3-axis systems. Consider the axial cross, consisting of the a, b, and c axes (each of unequal length), of the Monoclinic System (fig. 7.1). In all previous 3-axes systems, we considered what happens when we vary one or more of the axial lengths, retaining the axial angles at 90 degrees to each other. But in the Monoclinic System, we will look at what happens when we have 3 axes of unequal length and vary the angle off of 90 degrees between two of the axes. Obviously, we must again lose some symmetry!

The axes are designated as follows: the inclined axis is a and slopes out of the paper towards the viewer, the vertical axis is c, and the remaining axis which is at right angle to the plane of the a and c axes is b. When properly oriented, the inclined axis a slopes toward the observer, b is horizontal and c is vertical. Both b and c axes are in the plane of the paper.

In Figure 7.1, the angle between c and b remains 90 degrees and the angle (^) between c and a is the one we will vary. Its called beta and is represented by the Greek letter in the axial figure. For most monoclinic crystals, the ^ beta is greater than 90 degrees, but in some rare instances, the angle may be 90 degrees.

When this occurs, the monoclinic symmetry is not readily apparent from the morphology. The 2-fold rotation axis (the direction perpendicular to the mirror plane) is usually taken as the b axis. Then the a axis is inclined downward toward the front in the figure. Calculations of axial ratios in orthogonal crystal systems (where all the axes are perpendicular to each other) are relatively easy, but become quite tedious in systems with one or more inclined axes.

I suggest an advanced mineralogy text, not an introductory one, if you ever get involved in something like this. Not even your standard mineralogy texts these days give the formulae to do these calculations. Aside from the axial constants necessary to describe minerals in the monoclinic system, the ^ beta must also be given. Given this situation, you might wish to look up this information for orthoclase in a standard mineralogy textbook, like Klein and Hurlbutís Manual of Mineralogy after E. S. Dana. You will find that for orthoclase a:b:c = 0.663:1: 0.559. ^beta = 115 degrees, 50 minutes.

Cleavage is important to consider in this system. If there is a good pinacoidal cleavage parallel to the b axis (as in the mineral orthoclase), then it is usually called the basal cleavage. In the monoclinic pyroxenes and amphiboles, where there are 2 equivalent cleavage directions, they are usually considered to be vertical prismatic cleavages.

There are only 3 symmetry classes to consider in the monoclinic system: 2/m, m, and 2.

In the 2/m symmetry class, however, there are 2 types of forms, pinacoids and prisms. Remember that a pinacoid form consists of 2 parallel faces (open form).

The a pinacoid is also called the front (used to be called the orthopinacoid), the b is called the side pinacoid (used to be called the clinopinacoid), and the c is termed the basal pinacoid.

There are 2 additional pinacoids with the general form notations of {h0l} and {-h0l}. The presence of one of these forms does not necessitate the presence of the other one.

These 3 pinacoids together form the diametrical prism (fig. 7.2), which is the analogue of the cube in the isometric system. To further confuse the issue, most newer textbooks call the pinacoid form a parallelohedron. So we have 3 names in recent literature for the same thing.

Let's first look at a drawing to show you where the mirror plane is and the orientation of the 2-fold rotational axis (fig. 7.3).

As described above, the b axis is the 3-fold rotation axis.

The 4-faced prism {hkl} is the general form. A monoclinic prism is shown in Figure 7.4. The general form can occur as two independent prisms {hkl} and {-hkl}. There are also {0kl} and {hk0} prisms. The {0kl} prism intersects the b and c axes and is parallel to the a axis.

Here is the fun part. The only form in the 2/m class which is fixed by making the b axis the axis of 2-fold rotation is the b pinacoid {010}. Either of the other 2 axis may be chosen as c or a!

As an example, the {100} pinacoid, the {001} pinacoid, and the {h0l} pinacoids may be converted into each other by simply rotating their orientation about the b axis! Corollary to this situation, the prisms may be interchanged in the same manner. We now need to look at some illustrations of some relatively common monoclinic minerals. In these drawings you should recognize the letter notation where a, b, and c are the pinacoid forms (the diametrical prism, remember?); m is the unit prism and z is a prism; o, u, v, and s are pyramids; p, x, and y are orthodomes; and n is a clinodome.

Figures 7.5a, b, and c are common forms for the mineral orthoclase and 7.5d is a common form for selenite (gypsum). Many common minerals crystallize in this symmetry class, including azurite, clinopyroxene and clinoamphibole groups, datolite, epidote, gypsum, malachite, orthoclase, realgar, titanite, spodumene, and talc.

The second monoclinic symmetry class is m and represents a single vertical mirror plane (010) that includes the c and a crystallographic axes.

A dome is the general form {hkl} in this class (fig. 7.6) and is a 2-faced figure that is symmetrical across a mirror plane. There are 2 possible orientations of the dome, {hkl} and {-hkl).

The form {010} is a pinacoid, but all the faces on the other side of the mirror plane are pedions. These include {100}, {- 100}, {00-1), and {h0l}. Only 2 rare minerals, hilgardite and clinohedrite, crystallize in this class.

The third monoclinic symmetry class is 2 and represents a 2-fold axis of rotation on the b crystallographic axis. Figure 7.7 represents the general {hkl}form ñ a sphenoid or dihedron. Since we have no a-c symmetry plane and with the b axis being polar, in the 2 symmetry class, we have different forms present at the opposite ends of b. The {010} pinacoid of 2/m becomes 2 pedions, {0l0} and {0-10}. Likewise, the {0kl}, {hk0} and {hkl} prisms of 2/m degenerate into pairs of right- and left-hand (enantiomorphic) sphenoids.

The general form, the sphenoid, is enantiomorphic and has the Miller indices {hkl} and {h-kl}. Mineral representatives are scarce for this class, but include the halotrictite group with the mineral pickeringite as the most commonly occurring member. For comparisonís sake, take another look at Figures 7.6 and 7.7, just to keep straight what we are talking about.

Well, we have only one crystal system left to discuss. Prepare yourself to enter that land of variability where we break out of our need for square angles and equal length axes. The land where our symmetry is the lowest possible and our options are wide open. Are you ready for the Triclinic System?

Part 8: The Triclinic System

Index to Crystallography and Mineral Crystal Systems

Table of Contents

Bob Keller Copyright 1997-1998 J Michael Howard