محاضرة صخور ومعادن

Faculty of Science
Kafrelsheikh University
Introduction to Crystallography and Mineral Crystal Systems
Part I

Chapter1: Introduction
Crystallography is a fascinating division of the entire study of mineralogy. Even the non-collector may have an appreciation for large well-developed beautifully symmetrical individual crystals, like those of pyrite from Spain, and groups of crystals, such as quartz from Arkansas or tourmaline from California, because they are esthetically pleasing. To think that such crystals come from the ground "as is" is surprising to many. The lay person simply hasn't had the opportunity to learn about crystals and why they are the way they are; however, neither have many rock hounds and hobbyists.
We hope to bring you to a greater appreciation of natural mineral crystals and their forms by giving you some background and understanding into the world of crystallography. CRYSTALLOGRAPHY is simply a fancy word meaning "the study of crystals". At one time the word crystal referred only to quartz crystal, but has taken on a broader definition which includes all minerals with well expressed crystal shapes.
Crystallography may be studied on many levels, but no matter how elementary or in-depth a discussion of the topic we have, we confront some geometry. Oh no, a nasty 8-letter word! Solid geometry, no less. But stop and think about it, you use geometry every day, whether you hang sheetrock, pour concrete, deliver the mail, or work on a computer. You just don't think of it as geometry. Geometry simply deals with spatial relationships. Those relationships you are familiar with are not intimidating. The key word here is "familiar". We want this series of articles to help you become more familiar, and, therefore more comfortable, with the geometry involved with the study of crystal forms.
Crystallography is easily divided into 3 sections -- geometrical, physical, and chemical. The latter two involve the relationships of the crystal form (geometrical) upon the physical and chemical properties of any given mineral. We will cover the most significant geometric aspects of crystallography and leave the other topics for later. We do not intend for this series to be a replacement for a mineralogy textbook, but instead an introduction to the study of crystallography. During and after reading these articles, you will probably want to examine one or two textbooks for more detail about individual subjects. I recommend two: Klein and Hurlbut's Manual of Mineralogy (20th edition, 1985) and Ford's Textbook of Mineralogy (4th edition, 1932). Both of these are based on E. S. Dana's earlier classic publications.
In any type of study, there exists special words used to summarize entire concepts. This is the special language of the "expert", whether you speak of electrical engineering, computer science, accounting, or crystallography. There's no real way to get around learning some of these basic definitions and "laws" so we might as well jump right into them. Get Ready!
First, let's define what we're dealing with. A CRYSTAL is a regular polyhedral form, bounded by smooth faces, which is assumed by a chemical compound, due to the action of its inter-atomic forces, when passing, under suitable conditions, from the state of a liquid or gas to that of a solid. WOW, what a mouthful! Let's dissect that statement. A polyhedral form simply means a solid bounded by flat planes (we call these flat planes CRYSTAL FACES). "A chemical compound" tells us that all minerals are chemicals, just formed by and found in nature. The last half of the definition tells us that a crystal normally forms during the change of matter from liquid or gas to the solid state. In the liquid and gaseous state of any compound, the atomic forces that bind the mass together in the solid state are not present. Therefore, we must first crystallize the compound before we can study it's geometry. Liquids and gases take on the shape of their container, solids take on one of several regular geometric forms. These forms may be subdivided, using geometry, into six systems.
But before we can begin to discuss the individual systems and their variations, let's address several other topics which we will use to describe the crystal systems. There's also some laws and rules we must learn.
Way back in 1669, Nicholas Steno, a Danish physician and natural scientist, discovered one of these laws. By examination of numerous specimens of the same mineral, he found that, when measured at the same temperature, the angles between similar crystal faces remain constant regardless of the size or the shape of the crystal. So whether the crystal grew under ideal conditions or not, if you compare the angles between corresponding faces on various crystals of the same mineral, the angle remains the same.
Although he did not know why this was true (x-rays had not yet been discovered, much less x-ray diffraction invented), we now know that this is so because studies of the atomic structure of any mineral proves that the structure remains within a close set of given limits or geometric relationships. If it doesn't, then by the modern definition of a mineral, we are not comparing the same two minerals. We might be comparing polymorphs, but certainly not the same mineral! (Polymorphs being minerals with the same chemistry, like diamond and graphite or sphalerite and wurtzite, but having differing atomic structure and, therefore, crystallizing in different crystal systems) Steno's law is called the CONSTANCY OF INTERFACIAL ANGLES and, like other laws of physics and chemistry, we just can't get away from it.
Now, some of you may be thinking: I have a mineral crystal that does not match the pictures in the mineral books. What you may have is a distorted crystal form where some faces may be extremely subordinate or even missing. Distorted crystals are common and result from less-than-ideal growth conditions or even breakage and recrystallization of the mineral. However, remember that we must also be comparing the angles between similar faces. If the faces are not present, then you cannot compare them. With many crystals we are dealing with a final shape determined by forces other than those of the inter-atomic bonding.
During the process of crystallization in the proper environment, crystals assume various geometric shapes dependent on the ordering of their atomic structure and the physical and chemical conditions under which they grow. If there is a predominant direction or plane in which the mineral forms, different habits prevail. Thus, galena often forms equate shapes (cubes or octahedrons), quartz typically is prismatic, and barite tabular.
To discuss the six crystal systems, we have to establish some understanding of solid geometry. To do this, we will define and describe what are called CRYSTALLOGRAPHIC AXES. Since we are dealing with 3 dimensions, we must have 3 axes and, for the initial discussion, let's make them all equal and at right angles to each other. This is the simplest case to consider. The axes pass through the center of the crystal and, by using them, we can describe the intersection of any given face with these 3 axes.
Mineralogists had to decide what to call each of these axes and what their orientation in each crystal was so that everyone was talking the same language. Many different systems arose in the early literature. Then, as certain systems were found to have problems, some were abandoned until we arrived at the notational system used today. There exists two of these presently in use, complimentary to each other. One uses number notation to indicate forms or individual faces and the other uses letters to indicate forms. But let's get back to our 3 axes and we'll discuss these two systems later.
Fig. 1

We are going to draw each axis on a sheet of paper and describe its orientation. All you need for this exercise is a pencil and paper. Make the first axis vertical, and we'll call it the c axis. The top is the + end and the bottom is the - end. The second axis, the b axis, is horizontal and passes through the center of the c axis. It is the same length as the c axis. The right end is the +, and the left is the -. The third axis is the a axis and passes at a right angle through the join of the b and c axes.
It is somewhat tricky to draw because, even though the a axis is the same length as the c and b axes, because it goes from front to back it appears shorter. It is hard to represent a 3-dimensional figure on the 2-dimensional surface of paper, but you can do it. You have to use a sense of perspective, as an artist would say. The front end of a, which appears to come out of the paper, is the + and the back end the - (appears to be in the background or behind the paper).
This all sounds complicated, but look at Figure 1 if you are having problems with drawing the final axis. We always refer to the axes in the order - a, b, c in any type of notation. The point of intersection of the three axes is called the AXIAL CROSS.
Perspective is a key to drawing 3 dimensional objects on a flat 2D piece of paper. Perspective is what makes railroad tracks look like they come together in the distance. It is also what causes optical illusions when trying to draw axial crosses, or line drawings of crystal models. Perhaps you've looked at these lines and tried to decide which one comes forward, and which one recedes, and then have the illusion flip-flop so that it looks the other way. That's why they are labeled + and -. It can be confusing, so don't feel like the lone ranger.