An Introduction to Crystallography and Mineral Crystal Systems

If you're stuck in some other web site's frames click here to escape them.

Introduction to Crystallography and Mineral Crystal Systems

Written by Mike Howard - Illustrated by Darcy Howard

Part 1: Introduction

Part 2: Crystal Forms and Symmetry Classes

Part 3: The Cubic (Isometric) System

Part 4: The Tetragonal System

Part 5: The Orthorhombic System

Part 6: The Hexagonal System

Part 7: The Monoclinic System

Part 8: The Triclinic System

Part 9: Summary and Conclusion, Further Reading

About the Authors...

Mike [email protected]

Darcy Howard

Mike has been a mineral collector for over 30 years, thelast 22 of which he has been employeed by theArkansas Geological Commission as a geologist.Intrigued by crystal forms and how they relate tominerals and the stories they tell about mineralformation, he is most interested in crystallizedspecimens, particularly from Arkansas. He has workedon a variety of mineral-related projects in the state andespecially enjoys leading field trips for collegemineralogy classes to his favorite collecting spot --Magnet Cove.

Darcy is a scientific illustrator and commercialartist. Specializing in airbrush technique, shealso works in electronic and traditional media.Darcy studied art and medicine, and isassociated with geology by marriage. A formeremployee of both the medical profession andstate government, she is now a freelance work-at-home mom who designs and produces theGeology Collector's Series T Shirts under theFriend of the World (tm) tag line andphilosophy.

Mike and Darcy are residents of Arkansas, which is a world famous locality for quartz crystals and many other minerals. Evendiamonds can be collected in Arkansas, and many rockhounds enjoy collecting trips and vacations there. If you're interested inArkansas collecting and minerals, be sure and check out the best Arkansas rockhound resource on the WWW, Mike and Darcy'sRockhounding Arkansas.

Table of Contents

Bob Keller


by Mike and Darcy Howard

Part 1: Introduction

Crystallography is a fascinating division of the entire study of mineralogy. Even the non-collector may have an appreciation for largewell-developed beautifully symmetrical individual crystals, like those of pyrite from Spain, and groups of crystals, such as quartz fromArkansas or tourmaline from California, because they are esthetically pleasing. To think that such crystals come from the ground "asis" is surprising to many. The lay person simply hasn't had the opportunity to learn about crystals and why they are the way theyare; however, neither have many rockhounds and hobbyists.

We hope to bring you to a greater appreciation of natural mineral crystals and their forms by giving you some background andunderstanding into the world of crystallography. CRYSTALLOGRAPHY is simply a fancy word meaning "the study of crystals". Atone time the word crystal referred only to quartz crystal, but has taken on a broader definition which includes all minerals with wellexpressed crystal shapes.

Crystallography may be studied on many levels, but no matter how elementary or in-depth a discussion of the topic we have, weconfront some geometry. Oh no, a nasty 8-letter word! Solid geometry, no less. But stop and think about it, you use geometry everyday, 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 geometryinvolved with the study of crystal forms.

Crystallography is easily divided into 3 sections -- geometrical, physical, and chemical. The latter two involve the relationships of thecrystal form (geometrical) upon the physical and chemical properties of any given mineral. We will cover the most significantgeometric aspects of crystallography and leave the other topics for later. We do not intend for this series to be a replacement for amineralogy textbook, but instead an introduction to the study of crystallography. During and after reading these articles, you willprobably want to examine one or two textbooks for more detail about individual subjects. I recommend two: Klein and Hurlbut'sManual 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 aroundlearning 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 bya chemical compound, due to the action of its interatomic forces, when passing, under suitable conditions, from the state of a liquidor gas to that of a solid. WOW, what a mouthful! Let's dissect that statement. A polyhedral form simply means a solid bounded byflat planes (we call these flat planes CRYSTAL FACES). "A chemical compound" tells us that all minerals are chemicals, justformed by and found in nature. The last half of the definition tells us that a crystal normally forms during the change of matter fromliquid or gas to the solid state. In the liquid and gaseous state of any compound, the atomic forces that bind the mass together in thesolid state are not present. Therefore, we must first crystallize the compound before we can study it's geometry. Liquids and gasestake on the shape of their container, solids take on one of several regular geometric forms. These forms may be subdivided, usinggeometry, 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 todescribe 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 ofnumerous specimens of the same mineral, he found that, when measured at the same temperature, the angles between similarcrystal faces remain constant regardless of the size or the shape of the crystal. So whether the crystal grew under ideal conditionsor 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 knowthat this is so because studies of the atomic structure of any mineral proves that the structure remains within a close set of givenlimits or geometric relationships. If it doesn't, then by the modern definition of a mineral, we are not comparing the same twominerals. We might be comparing polymorphs, but certainly not the same mineral! (Polymorphs being minerals with the samechemistry, like diamond and graphite or sphalerite and wurtzite, but having differing atomic structure and, therefore, crystallizing indifferent crystal systems) Steno's law is called the CONSTANCY OF INTERFACIAL ANGLES and, like other laws of physics andchemistry, 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 mayhave is a distorted crystal form where some faces may be extremely subordinate or even missing. Distorted crystals are commonand result from less-than-ideal growth conditions or even breakage and recrystallization of the mineral. However, remember that wemust also be comparing the angles between similar faces. If the faces are not present, then you cannot compare them. With manycrystals we are dealing with a final shape determined by forces other than those of the interatomic bonding.

During the process of crystallization in the proper environment, crystals assume various geometric shapes dependent on the orderingof their atomic structure and the physical and chemical conditions under which they grow. If there is a predominant direction or planein which the mineral forms, different habits prevail. Thus, galena often forms equate shapes (cubes or octahedrons), quartz typicallyis 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 anddescribe what are called CRYSTALLOGRAPHIC AXES. Since we are dealing with 3 dimensions, we must have 3 axes and, for theinitial discussion, let's make them all equal and at right angles to each other. This is the simplest case to consider. The axespass 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 wastalking the same language. Many different systems arose in the early literature. Then, as certain systems were found to haveproblems, 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 indicateforms. But let's get back to our 3 axes and we'll discuss these two systems later.

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 isthe -. 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

Luis Paez Sinuco

c and b axes, because it goes from front to back it appears shorter. It is hard torepresent a 3-dimensional figure on the 2-dimensional surface of paper, but you can doit. 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 bein the background or behind the paper).

This all sounds complicated, but look at Figure 1 if you are having problems with drawingthe 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 lookedat these lines and tried to decide which one comes forward, and which one recedes, and then have the illusion flip-flop so that itlooks the other way. That's why they are labeled + and -. It can be confusing, so don't feel like the lone ranger.

We have now reached the point in our discussion that we can actually mention the six large groups or crystal systems that allcrystal forms may be placed in. We will use our crystallographic axes which we just discussed to subdivide all known minerals intothese systems. The systems are:

(1) CUBIC (aka ISOMETRIC) - The three crystallographic axes are all equal in lengthand intersect at right angles (90 degrees) to each other. This is exactly what you drewto obtain Figure 1. However, we now will rename the axes a1, a2, and a3 becausethey are the same length (a becomes a1, b becomes a2, and c becomes a3).

(2) TETRAGONAL - Three axes, all at right angles, two of which are equal in length (a and b) and one (c) which is different in length(shorter or longer). Note: If c was equal in length to a or b, then we would be in the cubic system! Discussed in part 4.

(3) ORTHORHOMBIC - Three axes, all at right angles, and all three of different lengths. Note: If any axis was of equal length to anyother, then we would be in the tetragonal system! Discussed in part 5.

(4) HEXAGONAL - Four axes! We must define this situation since it can not be derived from ourFigure 1. Three of the axes fall in the same plane and intersect at the axial cross at 120 degreesbetween the positive ends. These 3 axes, labeled a1, a2, and a3, are the same length. The fourthaxis, termed c, may be longer or shorter than the a axes set. The c axis also passes through theintersection of the a axes set at right angle to the plane formed by the a set. Look at Figure 2 to seethese relationships more clearly. Discussed in part 6.

(5) MONOCLINIC - Three axes, all unequal in length, two of which (a and c) intersect at an obliqueangle (not 90 degrees), the third axis (b) is perpendicular to the other two axes. Note: If a and ccrossed at 90 degrees, then we would be in the orthorhombic system! Discussed in part 7.

(6) TRICLINIC - The three axes are all unequal in length and intersect at three different angles (any angle but 90 degrees). Note: Ifany two axes crossed at 90 degrees, then we would be describing a monoclinic crystal! Discussed in part 8.

As was stated earlier, all known crystal forms fit into the above six crystal systems. But why don't all crystals in a given set look thesame? Or, stated differently, why can't I learn six crystal shapes and know all I need to know? Well, crystals, even of the samemineral, have differing CRYSTAL FORMS, depending upon their conditions of growth. Whether they grew rapidly or slowly, underconstant or fluctuating conditions of temperature and pressure, or from highly variable or remarkably uniform fluids or melts, all thesefactors have their influence on the resultant crystal shapes, even when not considering other controls.

We must touch on a type of notation often seen in mineral literature known as Miller Indices. Before William H. Miller (1801-1880)devised this mathematical system for describing any crystal face or group of similar faces (forms), there existed a considerableamount of confusion due to the many different descriptive systems. Some of these systems used letter symbols to denote crystalfaces and forms. Also different mineralogical "schools" existed as to how a given crystal should be viewed or oriented beforeassigning the crystallographic axes and then describing the various faces and forms present. If you were of the German school, youhad one view; from the English school another thought; from the French school, still another opinion.

So the problem was really one of how to bring order to the literature's chaos. To the problem, Miller (University of Cambridge) appliedrelatively simple mathematics - the Universal Language. To the lettering systems, Miller described the a,b,c (for hexagonal crystalshis notation is four numbers long) intercepts of each planar crystal form as numbers and also made note of the form letter. Hisnumbering system became widely accepted and is known as Miller indices. The numbers are presented as whole numbers (fractionsare not allowed) and are the reciprocal of the actual intercept number, all whole numbers being reduced by their lowest commondenominator. Here's a couple of simple examples from the cubic system.

Let us first describe a face of an octahedron and later a cube using Miller's indices. First,we should realize that an octahedron is an eight-sided crystal form that is the simplerepetition of an equilateral triangle about our 3 crystallographic axes. The triangle isoriented so that it crosses the a1, a2, and a3 axes all at the same distance from the axialcross. This unit distance is given as 1. Dividing 1 into the whole number 1 (it's areciprocal, remember?) yields a value of 1 for each Miller number. So the Miller indices is(111) for the face that intercepts the positive end of each of the 3 axes. See Figure 3 forall possible numbers for the 8 faces. Note: A bar over the number tells me that theintercept was across the negative end of the particular crystallographic axis. Theoctahedral form is given the letter designation of "o".

Now to the cube face. A cube face that intercepts the a3 (vertical) axis on the + end willnot intercept the a1 and a2 axes. If the face does not intercept an axis, then we assign a

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

mathematical value of infinity to it. So we start with Infinity, Infinity, 1 (a1, a2, a3). Infinitydivided into 0 = 0 (any number divided into zero equals zero). So the Miller indices of the+a3 intercept face equals (001). See the drawing for all possible Miller indices for the 6faces of a cube (Figure 4).

I think we should briefly mention cleavage at this point. CLEAVAGE is the preferred planardirection of breakage that many minerals possess. It is due to planes of weakness thatexist in some minerals because the bonding strength between the atoms or molecules isnot the same in every direction. Because crystals are composed of orderly arrangementsof atoms or molecules, we really should expect cleavage to be present in many crystals.The notation that denotes cleavage is derived in much the same manner as Miller indices, but is expressed in braces. So a cubiccrystal, say diamond, no matter whether it exhibits cubic {001} or octahedral {111} crystal form, has an octahedral cleavage form thatis given as {111}.

Note: The Miller indices when used as face symbols are enclosed in parentheses, as the (111) face for example. Form symbols areenclosed in braces, as the {111} form for example. Zone symbols are enclosed in brackets, [111] for example and denote a zoneaxis in the crystal. So in the discussion of cleavage (above), you must use braces to denote cleavage. Cleavage is analogous to formas cubic, octahedral, or pinacoidal cleavage and does not refer to just one face of a form.

Now we are ready to discuss ELEMENTS OF SYMMETRY. These include PLANES OF SYMMETRY, AXES OF SYMMETRY, andCENTER OF SYMMETRY. These symmetry elements may be or may not be combined in the same crystal. Indeed, we will find thatone crystal class or system has only one of these elements!

Huh? These parts, when put together, make the planes in figure 6.

Any two dimensional surface (we can call it flat) that, when passed through the center of the crystal, divides it into two symmetricalparts that are MIRROR IMAGES is a PLANE OF SYMMETRY. I repeat: any plane of symmetry divides the crystal form into twomirror images. Planes of symmetry are often referred to as mirror image planes. Let's discuss a cube again. A cube has 9 planes ofsymmetry, 3 of one set and 6 of another. We must use two figures to easily recognize all of them.

In Figure 5 the planes of symmetry are parallel to the faces of the cube form, in Figure 6 the planes of symmetry join the oppositecube edges. The second set corresponds to the octahedral crystal form. Planes of symmetry are always possible crystal forms.This means that, although not always present on many natural crystals, there exists the possibility that other crystal faces may beexpressed. So even though a cube form does not present an octahedral face, it is always possible that it could have formed underthe right conditions.

The typical human has two hands, right and left. Place them together palms facing away from you and thetips of the thumbs touching. Assuming that you have the same number of fingers on each hand, you willnote that your right hand is the mirror image of your left and vise versa. The average person issymmetrical, having binary symmetry vertically from the head to the feet when viewed from the front orback (bilateral symmetry).

You can have a lot of laughs with friends and a long mirror using this symmetry element. You need atleast one other person to do this so you both can view the results! Using Figures 5 and 6 as guides, takea wood or plastic cube and see if you can draw with a marker all the planes of symmetry that are present.Refer to the two figures for help.

It is sometimes convenient to designate planes of symmetry as axial, diagonal, principle, or intermediate. Figure 7 is an example ofthe 5 planes of symmetry of the tetragonal system and the proper abbreviated notation.

AXES OF SYMMETRY can be rather confusing at first, but let's have a go at them anyway. Any line through the center of the crystalaround which the crystal may be rotated so that after a definite angular revolution the crystal form appears the same as before istermed an axis of symmetry. Depending on the amount or degrees of rotation necessary, four types of axes of symmetry arepossible when you are considering crystallography (some textbooks list five). Given below are all possible rotational axes:

When rotation repeats form every 60 degrees, then we have sixfold or HEXAGONAL SYMMETRY. A filled hexagon symbol is notedon the rotational axis.

When rotation repeats form every 90 degrees, then we have fourfold or TETRAGONAL SYMMETRY. A filled square is noted on therotational axis.

When rotation repeats form every 120 degrees, then we have threefold or TRIGONAL SYMMETRY. A filled equilateral triangle isnoted on the rotational axis.

When rotation repeats form every 180 degrees, then we have twofold or BINARY SYMMETRY. A filled oval is noted on the rotationalaxis.

When rotation repeats form every 360 degrees, then we use a filled circle as notation. This one I consider optional to list as almostany object has this symmetry. If you really want to know the truth, this means NO SYMMETRY!!

Note that rotational axes may be on the plane of the face, on the edge of where two faces meet,or on the point of conjunction of three or more faces. On a complete crystal form, the axis mustpass through the center of the crystal and exist at the equivalent site on the opposite side of thecrystal as it entered.

Take a solid cube, made of wood or plastic (a clear plastic cube box works well for thisexercise). Mark, using the rotational notation, every four-, three-, and two-fold axis of rotationthat you can find. I think you will be surprised how many there are! Examine Figure 8 (the cubefrom hades!) to see how many symbols you can draw on your cube.

Now I'm sorry that this is not all there is to rotational axes, but there is another situation that we

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

Luis Paez Sinuco

must consider -- AXES OF ROTARY INVERSION. This is where the twisted mind has one up on the rest of us (there's a pun in theresomewhere!). We will consider a couple of simple examples.

First, let's examine a a crystal as drawn in Figure 9a at left. Use a piece of "2 by 2" board and make thiscrystal form by cutting off the ends so the wood block looks like the drawing. Hold the block in your lefthand with your thumb on the top and in the center of the 2-face edge join (long axis) and your index fingeron the same join on the underside. Your palm will be toward your body. Align your two fingers so that youare looking straight down on your thumb and can not see end of your index finger. The top of the block willappear as 2 equal-sized faces, sloping away from you. If you rotate the block 180 degrees, the faces willbe appear back in the same position (2-fold axis of rotation), but here's the tricky part -- rotate thespecimen 90 degrees and then turn your wrist where your index finger is on top (easiest done by turningyour wrist counterclockwise). You will see that the block's faces appear in the original position in theoriginal position. You have discovered an axis of rotary inversion!

Fig 9b: Wooden or plastic models areknown in a mineralogy class as 'idiotblocks'.

Fig 9c: Fig 9d: Block rotated 90 degreesaround the axis shown by the dot

Fig 9e: Block rotatedcounterclockwise 180degrees on the axis shownby the arrow.

See the series of photos (Figures 9 b-9 e) if you get confused. Some textbooks term these axes rotary reflection axes orrotoinversion axes. There may be 1-, 2-, 3-, 4-, and 6-fold rotary inversion axes present in natural crystal forms, depending upon thecrystal system we are discussing. I refer you to Klein and Hurlbut's Manual of Mineralogy (after J. S. Dana) if you want to sharpenyour axes of rotary inversion skills. With axes of rotation, there is a graphical notation used which looks like a very bold type-facecomma. For axes of rotary inversion, the same symbol is used, but appears dashed.

Both types of symmetrical rotational axes (discussed above) are commonly plotted on a circle (representing the complete cycle ofone 360 degree rotation). The simple axes of rotation symbol for a face is plotted at the center of the circle and the axes of rotationand rotary inversion are plotted on the circle's boundary at whatever rotational angle is appropriate. See Figure 10 for examples.

We have finally come to our last topic of geometric crystallography -- the CENTER OF SYMMETRY. Most crystals have a center ofsymmetry, even though they may not possess either planes of symmetry or axes of symmetry. Triclinic crystals usually only have acenter of symmetry. If you can pass an imaginary line from the surface of a crystal face through the center of the crystal (the axialcross) and it intersects a similar point on a face equidistance from the center, then the crystal has a center of symmetry. We maydiscuss this in a little more detail in the article about the triclinic system.

We now have to consider the relation of geometrical symmetry to CRYSTALLOGRAPHIC SYMMETRY. The crystal facearrangement symmetry of any given crystal is simply an expression of the internal atomic structure. This internal structure isgenerally alike in any parallel direction. But we must keep in mind that the relative size of a given face is of no importance, only theangular relationship or position to other given crystal faces. Refer back to Steno's law concerning the CONSTANCY OFINTERFACIAL ANGLES.

Let's consider a crystal in the cubic system with both cube {001} and octahedral {111} forms represented(Figure 11). In our figure, we have used the letter designation of -a- for the cube faces and -o- for theoctahedral faces. Despite the initial observation that both the various cube and octahedral faces areunequal in size, the example displays all the symmetry elements and relationships of a crystal from thecubic system. I hope you now begin to grasp the difficulty of learning crystallography using naturalcrystals. Due to a variety of factors, many natural crystals have some degree of distortion to their growth,causing the faces to vary in size and sometimes shape. In college mineralogy, this problem was resolvedby requiring the classroom use of a set of crystal forms, sometimes made of wood or plastic. These setswere not-so-fondly termed "idiot blocks" by exasperated students. Once you mastered the various forms

and understood symmetry planes, rotational axes, and form names, then you became recognized as a "complete idiot" and could goon to examine real minerals!

Depending upon what elements of symmetry are present, all crystals may be divided into 32 distinct groups called CLASSES OFSYMMETRY. Remember, we concern ourselves with the symmetry elements we learned above, not the malformed crystal shapes ofmost minerals. Only forms which belong to the same class can occur in combination together in nature. We can not find a cube faceon a hexagonal crystal. Likewise, we will never discover the rhombic dipyramid termination of a hexagonal crystal on a tetragonalcrystal. So our laws, rules, and symmetry elements previously discussed prevent chaos in our beautifully symmetrical world ofcrystallography. Certainly, when dealing with real crystals, distortion problems can arise! Think of capillary pyrite. Here you have acubic crystal which, due to a growth phenomenon, has one axis nearing infinity in length in relation to the other two. But this iscaused by special conditions during growth, not the crystallography.

There are graphical methods of plotting all possible crystal faces and symmetry elements on a type of diagram called a stereo net.Stereo nets give a way to represent three-dimensional data on a two-dimensional surface (a flat sheet of paper). A discussion ofstereo nets is out of the scope of this paper because to present it adequately would require much graphing, mathematics, and that

each reader possess stereo-net paper. If you wish to attempt any exercises with stereo nets, I refer you to the previously mentionedtextbooks. You can purchase the graph paper at most major college and university bookstores.

Well, if your faces are all shining, your symmetry now in order, and your axes properly aligned, then stay tuned for the next articlewhen we consider crystal forms and the 32 symmetry classes. Then we will have the background necessary to discuss the sixcrystal systems!


Index to Crystallography and Mineral Crystal Systems

Table of Contents

Bob Keller




Let's discuss CRYSTAL FORMS and the 32 SYMMETRY CLASSES! Again we must begin with some definitions. Unfortunately,the term "form" is loosely used by many people to indicate outward appearance. However, we must "tighten up" our definition whendiscussing crystallography. HABIT is the correct term to indicate outward appearance. Habit, when applied to natural crystals andminerals, includes such descriptive terms as tabular, equidimensional, acicular, massive, reniform, drusy, and encrusting.

Drusy Quartz in Geode Tabular Orthoclase Feldspar Encrusting Smithsonite

As a crystallographer, I use "form" with a more restricted meaning. A FORM is a group of crystal faces, all having the samerelationship to the elements of symmetry of a given crystal system. These crystal faces display the same physical and chemicalproperties because the ATOMIC ARRANGEMENT (internal geometrical relationships) of the atoms composing them is the SAME.The relationship between form and the elements of symmetry is an important one to grasp, because no matter how distorted anatural crystal may be, certain key elements will be recognizable to help the student discern what form or forms are present. Theterm general form has specific meaning in crystallography. In each crystal class, there is a form in which the faces intersect eachcrytallographic axes at different lengths. This is the general form {hkl} and is the name for each of the 32 classes (hexoctahedralclass of the isometric system, for example). All other forms are called special forms.

Let's look at an octahedron as an example (fig. 2.1). All the crystal faces present are the expression ofthe repetition of a single form having the Miller indices of {111} about the three crystallographic axes(remember those from the first article?). Each face on a natural crystal (octahedral galena or fluorite areexamples), when rotated to the position of the (111) face in the drawing, would have the same shapeand orientation of striations, growth pits or stair steps, and etch pits, if present.

The presence of these features is true whether or not the crystal is well formed or distorted in its growth.Note that I did NOT state that the faces are necessarily the same size on the natural crystal!

In fact, due to variations in growth conditions, the faces are usually not. In the literature, you may see anotation, given as {hkl}. This is the notation, presented as Miller indices for general form. The octahedral form is given as {111}, thesame as the face that intersects all positive ends of the crystallographic axes. A single form may show closure, as with anoctahedron, or may not, as in a pinacoid (an open two-faced form). So every form has an {hkl} notation. In the case of generalnotation concerning the hexagonal system, it is {hk-il} and is read as "h, k, minus i, l".

Before leaving this discussion of form, here are a couple of examples of how knowledge of theinterrelationships of forms and crystal systems may be used. Someone gives you a quartz crystaland says, "Look at this crystal. It's not normal." Normal to this person, we assume means anelongate (prismatic), 6-sided crystal form with a 6-faced termination on the free-growth end. Whenyou examine the "abnormal" crystal, it is highly distorted, broken, and has regrowth faces. Prismfaces on quartz crystals almost always have striations at right angles to the c crystallographic axisand parallel to the plane of the a1, a2, and a3 axes. These striations are due to the variable growthrates of the terminal faces as the mineral crystallized. Knowing about the striations and theirorientation, you examine the surface of the specimen and, with reflected light, find the prism facesby their striations.

Terminal (or pyramidal) faces on quartz crystals often exhibit triangular pits or platforms. By findingthem, you can then determine if any other faces that would be really unusual are present. Not finding any unusual faces, you canreturn the specimen to the person with the comment, "Well, your crystal is certainly interesting, but it does not have any unusualforms. What it does display is a complex growth history reflected by its less than ideal crystal shape."

Most people and many collectors recognize unusual habits, but not unusual forms. They note thatthe shape of a crystal is odd looking, but don't have the background in crystallography to know ifthe crystal is truly unusual. A broken and regrown quartz crystal is not particularly special, but aquartz crystal with a c pinacoidal termination is worth noting, as it is a very uncommon form forquartz. I have only seen a few from one locality. Being the skeptic that I am, I purchased onecrystal which had another mineral coating the termination. I mechanically removed the coatingmineral with the edge of a pocket knife. There for my examination was the c pinacoid termination{0001}, satisfying me that it was a natural growth form!

A crystal's form may be completely described by use of the Miller's indices and the Hermann-Mauguin notation of its POINT GROUP SYMMETRY. The latter notation tells us how to orient thecrystal, in each specific crystal class, to recognize which axis (a, b, or c) is designated as havingthe highest symmetry. It also tells us what other symmetry elements may be present and where

they are in orientation to the other elements.

I hope I haven't LOST you here. When considering all the literature written about this subject, sometimes I even feel like I'm rotatingon one symmetry axis and not really getting anywhere! Point group symmetry is too complicated to get into in this discussion, so Irefer you to Klein and Hurlbut's Manual of Mineralogy for detailed information. However, I will discuss some pertinent portions ofpoint group symmetry under each crystallographic class and introduce its notation for selected crystal forms in each crystal-system article.

Types of Crystal Forms

Types of Crystal Forms

Note that there are TWO GENERAL TYPES OF FORMS: those that by repetition close on themselves creating a complete form(termed closed) and those that do not (termed open).

Now for the BAD NEWS. Every form has a name and there are many of them.

The GOOD NEWS is that you already know a few of them, particularly some of the closed forms because we have been usingthem in our previous discussions. The cube and octahedron are examples. There are 32 (some say 33) forms in the nonisometric(noncubic) crystal systems and another 15 forms in the isometric (cubic) system. Let's start to familiarize ourselves with them bymaking a tabulation and including the number of faces (below). I begin the listing with the isometric forms first, all of them beingclosed forms.

Isometric Crystal Forms

Name Number

of Faces Name Number

of Faces

(1) Cube 6 9)Tristetrahedron 12

(2) Octahedron 8 (10) Hextetrahedron 24

(3) Dodecahedron 12 (11) Deltoid dodecahedron 24

(4) Tetrahexahedron 24 (12) Gyroid 24

(5) Trapezohedron 24 (13) Pyritohedron 12

(6) Trisoctahedron 24 (14) Diploid 24

(7) Hexoctahedron 48 (15) Tetartoid 12

(8) Tetrahedron 4

Non-Isometric Crystal Forms

Name Number

of Faces Name Number

of Faces

(16) Pedion* 1 (32) Dihexagonal pyramid 12

(17) Pinacoid** 2 (33) Rhombic dipyramid 8

(18) Dome or Sphenoid 2 (34) Trigonal dipyramid 6

(19) Rhombic prism 4 (35) Ditrigonal dipyramid 12

(20) Trigonal prism 3 (36) Tetragonal dipyramid 8

(21) Ditrigonal prism 6 (37) Ditetragonal dipyramid 16

(22) Tetragonal prism 4 (38) Hexagonal dipyramid 12

(23) Ditetragonal prism 8 (39) Dihexagonal dipyramid 24

(24) Hexagonal prism 6 (40) Trigonal trapezohedron 6

(25) Dihexagonal prism 12 (41) Tetragonal trapezohedron 8

(26) Rhombic pyramid 4 (42) Hexagonal trapezohedron 12

(27) Trigonal pyramid 3 (43)Tetragonal scalenohedron 8

(28)Ditrigonal pyramid 6 (44) Hexagonal scalenohedron 12

(29) Tetragonal pyramid 4 (45) Rhombohedron 6

(30) Ditetragonal pyramid 8 (46) Rhombic disphenoid 4

(31) Hexagonal pyramid 6 (47) Tetragonal disphenoid 4

*Pedion may appear in several crystal systems

**Pinacoid drawing displays 3 pairs of pinacoid faces from the Orthorhombic system.Pinacoids appear in several crystal systems.

Now you know why mineralogy students hate idiot blocks! It is important to note that these are simply the possible INDIVIDUALFORMS, not the combinations of forms seen on a single natural crystal.

Fig. 2.4 Peruvian Pyrite, Various Crystal Forms

Pyrite is a common mineral which often exhibits several forms on a single crystal. One formis usually dominant, presenting the largest faces on the crystal. Peruvian pyrite commonlyhas cubic, octahedral, and dodecahedral forms on a single crystal; sometimes evenpyritohedral and diploid faces may be present. Any of these individual forms may be thedominant one. Crystals with the same forms present, but with different dominant forms willeach appear very different (fig. 2.4). As we explore each crystal system, there will beillustrations displaying most of the ideal forms and some drawings showing combinations offorms often exhibited by individual mineral crystals.



Table of Contents

Bob Keller




Now that you have read the two previous articles, you are ready to consider the first of our 6 crystal systems. So let's begin.

There are 15 forms, all closed, in the ISOMETRIC CRYSTAL SYSTEM-- more than in any other single system we will examine.You may wish to briefly refer back to the first article in this series, when we built an axial cross. In the isometric system, all 3crystallographic axes are at right angles to each other and are the same length. The axes are renamed a1, a2, and a3. We needto remember that a3 is vertical, a2 is horizontal, and a1 is front to back.

Crystal forms in the isometric system have the highest degree of SYMMETRY, when compared to all the othercrystal systems. Did you know that there is only ONE object in the geometrical universe with perfectsymmetry? Considerthe sphere (fig. 3.1). Infinite planes of symmetry pass through its center, infinite rotationalaxes are present, and no matter how little or much you rotate it on any of its infinite number of axes, itappears the same! A sphere is the HOLY GRAIL of symmetry!!

No crystal system even approaches a sphere's degree of symmetry, but the isometric system is often quicklyrecognizable because some of the forms and combinations of forms somewhat approach sphericity (or, at

least, roundness), especially when the faces begin to be curved, due to the high degree of symmetry in the isometric system.

Let's begin by looking at the Hermann-Mauguin notation for the first seven isometric forms and each form's notation:

Cube {001} Dodecahedron {011} Trapezohedron {hhl} Hexoctahedron {hkl}

Octahedron {111} Tetrahexahedron {0kl} Trisoctahedronv {hll}

For these forms, the 3 crystallographic axes are axes of 4-fold rotation. There are also 4 diagonal axes of 3-fold rotary inversionthat pass through the form at the point where the cube's 3 faces would join. Furthermore, there are 6 directions of 2-fold symmetry(at the center of the line formed by the intersection of 2 planes). There is also a center of symmetry. There are 9 mirror planes (seefigs. 1.5 and 1.6 in first article in this series). This combination of symmetry elements defines the highest possible symmetry ofcrystals. So the Hermann-Mauguin notation is 4/m-32/m.

In a textbook, my notation (-3) is presented as the number 3 with a negative sign above it, but due to computers and web browsers,I can't place this special notation properly in cyberspace, so don't get confused if you look this up in a mineralogy book! It shouldbe pronounced as "negative 3" or "bar3". In this instance, the -3 is the notation for the 3-fold axis of rotary inversion. I willconsistently use the negative sign before the number when it is necessary. The same stands true for my notation when dealingwith Miller indices or general form notation.

Crystallographers group forms by their symmetry notation, the first seven we will consider have the same symmetry - 4/m-32/m.

CUBE-- The cube is composed of 6 square faces at 90 degree angles to each other. Each face intersectsone of the crystallographic axes and is parallel to the other two (fig. 3.2). This form, {001}, is one of theeasiest to recognize and many minerals display it with little modification. Think of galena, pyrite, fluorite,perovskite, or halite cubes!

OCTAHEDRON-- The octahedron is a form composed of 8 equilateral triangles. These triangle-shaped facesintersect all 3 crystallographic axes at the same distance, thus the form notation of {111} (fig. 3.3). Mineralscommonly exhibiting the simple octahedral form are magnetite, chromite, franklinite, spinel, pyrochlore,cuprite, gold, and diamond. Sometimes fluorite, pyrite, and galena take this form.

DODECAHEDRON (AKA Rhombic Dodecahedron) -- This form is composed of 12 rhomb-shaped faces (fig.3.4). Each of these rhomb-shaped faces intersects two of the axes at equidistance and is parallel to the 3rdaxis, thus the notation {011}. The different mineral species of the garnet group often display this form.Magnetite and sodalite sometimes exhibit this form.

TETRAHEXAHEDRON-- This form has 24 isoceles triangular faces. The easiest way to understand its shape is to envision a cubethat on each face has 4 equal-sized triangular faces (fig. 3.5) that have been raised from the center of the cube face. Eachtriangular face has its base attached to the edge of the cube and the apex of the two equal-length sides rises to meet the 4-foldaxis. Because of the variation of inclination to this axis, there exists a number of possible tetrahexahedral forms, but all meet thegeneral notation of {0hl}.

The most common form is {012}. It is interesting to note that as the combination of each set of 4 faces rise along the axis, thisform approaches the dodecahedron. As they fall, the form approaches a cube. The tetrahexahedron is rarely the dominant form onnatural crystals, instead being subordinant to the cube, octahedron or dodecahedron (fig. 3.6). Cubic minerals on which you may

sometimes see this form exhibited include fluorite (cube and tetrahexahedron), magnetite or copper (octahedron andtetrahexahedron) and garnet (dodecahedron and tetrahexahedron).

TRAPEZOHEDRON (AKA Tetragon-trioctahedron) -- This form has 24 similar trapezium-shaped faces. If my Webster's is correct, atrapezium is a 4-sided plane that has no sides parallel. Each of these faces intersects a crystallographic axis at a unit distanceand the other two axes at equal distances, but those intersections must be greater than the unit distance. It sounds prettycomplicated, but see the drawing (fig. 3.7). Because there may be various intercepts distances on the two axes, the form symbol{hhl} in general is used. The most common mineral form is {112}. Two silicate minerals, analcime and leucite, usually crystallize assimple trapezohedrons. This form is not uncommon, varying from dominant to subordinate, on many varieties of garnet, where it isoften combined with the dodecahedron (fig. 3.8).

TRISOCTAHEDRON (AKA Trigonal Trisoctahedron ) -- This is another 24-faced form, but the faces areisoceles triangles. Each face intersects two crystallographic axes at unity, and the third axes at somemultiple of unity; hence the form notation in general of {hll}.

To more easily visualize what a trisoctahedron looks like, first think of an octahedron. Each octahedralface is divided into 3 isoceles triangles by drawing 3 lines, each originating at the center of the octahedralface and reaching the 3 corners of that face. Repeat this operation for the remaining 7 faces on anoctahedron and you have a trisoctahedron (fig. 3.9). As a dominant form, the trisoctahedron is scarce,most commonly being reported for diamond, usually as a subordinant form (fig. 3.10).

It has recently been demonstrated that trisoctahedral diamond is probably not a true crystal form (truecrystal forms are growth forms), but instead a solution form caused by the differential dissolving ofoctahedral diamond during its transport from the mantle to the crust, but that's another story altogether!As a subordinant form, it has been reported in combination with the octahedron for fluorite and magnetiteand in combination with cube and octahedron on complex crystals of galena.

HEXOCTAHEDRON-- This form has 48 triangular faces, 6 faces appearing to be raised from each face of asimple octahedron. These may be envisioned by drawing a line from the center of each of the 3 edges ofan octahedral face, through the face center to the opposite corner. Repeat this for the remaining 7 faces ofan octahedron and you have a hexoctahedron (fig. 3.11).

Just like the trisoctahedral form, this form is most often seen on diamond, where it is thought to representa solution form derived from an octahedron, not true crystallization. With both the tris- and hexoctahedron,the faces are often curved, resulting in a near spherical shape. The combination of dominant dodecahedronand subordinant hexoctahedron is not uncommon for garnet (fig. 3.12)

We have 8 remaining forms in the isometric system to consider. The next 4 have the Hermann-Mauguin notation of -43m. Theseare the tetrahedron, tristetrahedron, deltoid dodecahedron, and hextetrahedron.

TETRAHEDRON-- The tetrahedron includes both a positive and negative form with the notation {111} and{1-11}, respectively. These are simple mirror images of one another. A tetrahedron is a 4-faced form, eachface being an equilateral triangle.

Each face intersects all 3 crystallographic axes at the same distance. You may derive this form from anoctahedron by extending alternate faces until they meet (this also shrinks the opposing set of alternatefaces until they disappear).

Figure 3.13 displays the orientation of the tetrahedral form in relation to the cube. We aren't justspeculating that both the positive and negative forms exist because they are often seen together (fig. 3.14)on a single crystal!

If both positive and negative forms are equal sized on a single crystal, then the initial appearance of thecrystal form is INDISTINGUISHABLE from an octahedron. Here is where the differences in and orientationof surface features become exceedingly important in form study. One mineral so commonly has thiscrystal form that the mineral was named after the form itself - tetrahedrite. Other examples are diamond,helvite, and sphalerite.

TRISTETRAHEDRON-- By now, I think you might be able to tell me from my previous examples how toderive this form. Yep, take a tetrahedron and raise 3 isoceles triangle-shaped faces on each of the 4tetrahedral faces. So this form has 12 triangular faces (fig. 3.15).

Just like the tetrahedron, there are both positive and negative forms, designated as {hhl} and {h-hl},respectively.This is only a relatively common form on tetrahedrite, usually subordinant to the tetrahedron(fig. 3.16), but has also been reported on sphalerite and boracite. The possibility of it being present ondiamond can't be overlooked, but as mentioned, it may be the result of solution processes, rather thancrystallization.

HEXTETRAHEDRON-- Again, we take a tetrahedron and, in similar manner as the hexoctahedron, weraise 6 triangular faces having a common apex from the center of the equilateral triangular face of thetetrahedron. Repeating this on the entire tetrahedron results in 24 faces (fig. 3.17). There are both positiveand negative forms, designated as {hkl} and {h-kl}, respectively. This form has been reported ontetrahedrite, but rarely on sphalerite. Also a possible solution form on diamond.

DELTOID DODECAHEDRON-- This is a 12-faced form, derived by raising 3 4-sided faces on the each faceof a tetrahedron (fig. 3.18). The shape of the resultant faces are rhombic. There are both positive andnegative forms, designated as {hll} and {h-ll}, respectively. This form is sometimes seen as a subordinateone on tetrahedrite or sphalerite, where it would appear as a set of 3 rhombic faces modifying the cornersof the dominant tetrahedral shape.

Now we have only 4 remaining forms to discuss in the isometric system. The first to consider is the gyroid.

GYROID (Pentagon-trioctahedron)-- This form has no center of symmetry! TheHermann-Mauguin notation is 432. There are two forms, based on right- and left-handedsymmetry (fig. 3.19). Older mineral textbooks state that this is a rare form, sometimesreported on cuprite. But most recent textbooks indicate that a restudy of cuprite'scrystallography showed cuprite to probably be hexoctahedral. If this is so, then we haveno natural mineral that crystallizes with this form, although some laboratory-growncrystals with this form are known.

Two of the 3 remaining forms have 3 2-fold rotational axes, 4 3-fold rotary inversion axes, and 3 of the axial planes are mirror planesof symmetry. The Hermann-Mauguin notation is 2/m-3. These forms consist of the pyritohedron and the diploid.

PYRITOHEDRON (Pentagonal dodecahedron)-- There are 12 pentagonal faces, each of which intersectsone crystallographic axis at unity, intersects a second axis at some multiple of unity, and is parallel to thethird axis. There are positive and negative forms, designated as {h0l} and {0kl}, respectively. There are anumber of pyritohedral forms, differing due to the degree of inclination of the faces. The most common formis the {102}, the positive form (fig. 3.20). Pyrite is the only common mineral that displays this form. It isoften subordinate, combining with the cube, diploid (below), or octahedron.]

DIPLOID (Didodecahedron)-- There are 24 faces (fig. 3.21), each face corresponding to one-half of thefaces of a hexoctahedron. This is a rare form. You should compare figures 3.20 and 3.21. The diploidlooks like a pyritohedron where two faces are made from each pentagonal face of the pyritohedron. Theresulting faces are trapezia. There are both positive and negative forms, designated as {hkl} and {khl},respectively. Pyrite is the only common mineral that exhibits the diploid form.

Believe it or not, we just reached the last form of the isometric minerals! If you are still with me at this point (mentally andphysically), then I submit that you either 1) had little to do today to have the time to read this entire article, 2) are doing this toavoid starting a major project (like a term paper), 3) have a mental health problem and need serious counseling, or 4) are seriousabout learning more concerning Crystallography! So, let's grab this last form, the tetartoid (say it 10 times - real fast), and finish itoff.

TETARTOID (pentagon-tritetrahedron)-- The Hermann-Mauguin notation is 23. It has 3 2-fold rotationcrystallographic axes and 4 diagonal axes that have 3-fold rotational symmetry. There are actually 4separate forms in this class: positive right {hkl}, positive left {khl}, negative right {k-hl}, and negative left {h-kl}. See figure 3.22 for the positive right form. Cobaltite, an uncommon mineral, often crystallizes in thisform. The tetartoid may be present as a subordinate form in combination with the cube, dodecahedron,pyritohedron, tetrahedron, and deltoid dodecahedron.

Do you remember the naturalist Steno from the 16th century?? We mentioned him in the introductory article. Well, here's someinformation that he and later crystallographer's discovered about isometric crystals and the interfacial angles of some of thedifferent forms. This information could come in handy when you are deciding which of the common forms you have present oncomplex isometric crystals.

The angle between two adjoining cube faces is 90 degrees.The angle between two adjoining octahedral faces is 70 degrees 32 minutes.The angle between two adjoining dodecahedral faces is 60 degrees.The angle between a cube (100) and a octahedron (111) is 54 degrees 44 minutes.The angle between a cube (100) and a dodecahedron (110) is 45 degrees.

The angle between an octahedron (111) and a dodecahedron (110) is 35 degrees 16 minutes.

ALL RIGHT!! Whether you realize it or not, you have just gotten through the mostlengthy discussion of crystal forms in this series. Therefore, I humbly bestowupon you the Order of the Golden Axial Cross! There will be other awards andaccolades for stamina, perseverance, tenacity, and dedication as you proceed inyour study of geometrical crystallography. Now if you are feeling a bit toospherical after digesting all of the isometric system, let's shed some of oursymmetry by exercising our mental powers on some lower symmetry forms in the

next article!!



Table of Contents

Bob Keller




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 thiscrystallography stuff is addictive and will warp your sense of priorities! You either love these articles by now or are totallymasochistic. The lack of geometrical reasoning necessary to understand crystallography and symmetry is what drove a lot ofcollege wannabe geologists into the College of Business! Let's begin...

Our discussion of the TETRAGONAL SYSTEM starts by examiningthe tetragonal axial cross and comparing it to the isometric axialcross (Article 3). Remember that in the isometric system all 3 axeswere the same length and at right angles to each other. In thetetragonal system, we retain the same angular relationships, butvary the length of the vertical axis, allowing it to be either longer orshorter than the other two. We then relabel the vertical axis as c,retaining the same positive and negative orientation of this axis (seefig. 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 symmetryelements, in that order. The tetragonal prism and pyramid forms have the symmetry notation4/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, andditetragonal 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 thesame 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) andthe form symbol is {110}. The second order prism is essentially identical to the first order prism, but rotated about the c axis towhere 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 secondorder 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 areoriented to the two a axes. When these forms are combined (fig. 4.2c), then you may readily see their relationships, one to theother. 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 orderprisms, especially if they are equally developed. But compare the orientation of the ditetragonal prism tothe a axes in relation to the combination form. What you should do is envision looking down the c axis ofthe ditetragonal prism and the combined 1st and 2nd order tetragonal prisms, then you will see thesimilarity.

The ditetragonal prism {210} would closely approximate the combined prism forms, and with naturalmalformations, 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 thecombined 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 canunderstand the perspective of this eight sided "stop sign" form. Another form in thetetragonal system is the dipyramid and -- yes, you guessed it -- there are 3 typesof dipyramids. They correspond to the three types of prisms just described. Thename 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 thelength 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 secondorder dipyramid has the basic shape as the form of the first order dipyramid, differing only in its orientationto the axial cross (fig. 4.5, the common {011} form). The second order dipyramid faces are designated bythe letter e.

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

Now to the 3rd dipyramid form, the ditetragonal dipyramid. Yes, it's a closed termination form having 16faces (fig. 4.7). Think of this form as a double 8-sided pyramid whose 16 similar faces meet the 3 axes atunequal distances. The general symbol is {hkl}. This form is rarely dominant, but is common enough as asubordinant form on zircon to be nicknamed a zirconoid. Anatase may also have this form expressed. Theditetragonal prism is often combined with the 1st order prism. In figure 4.7, although the prism is notpresent and therefore is simply at the junction of the two faces, we have marked its position if it had beenexpressed by an arrow and the letter m.

The next forms in this system to consider have the Hermann-Mauguin symmetry notation of -42m. Theseclosed 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-foldaxis of rotary inversion.

The tetragonal disphenoid exists as both a positive and negative form. Ithas 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 andp1 for the negative form. This form differs from the tetrahedron of the isometricsystem 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 thoughtto be in this class must have very accurate interfacial angle measurementsmade 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 onchalcopyrite and stannite. It may be derived from the disphenoid form of this system by drawing a linefrom one corner of each disphenoid face to the center of the line joining the two opposite corners, andraising two faces from the resulting division. Thus, from a 4-faced disphenoid form, we derive an 8-facedform. If you are still having trouble visualizing the form in figure 4.9, you might try thinking of it as thecombination of 4 classic diamond - shaped kites, every other one in an upside down orientation! Thisform 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 is4mm. 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 aMiller indices of {001}. The pedion will be a single face perpendicular to the c axis that "cuts off" thesharp termination of the ditetragonal pyramid. There are upper and lower forms for both the ditetragonalpyramid and the pedion, the upper being considered positive and the lower negative (just like theorientation of the c axis).

The ditetragonal pyramid looks like one half of the ditetragonal dipyramid, but on a well-formed exampleis present on only one end of the c axis! This form is rarely dominant, usually being subordinant toother common prism and dipyramidal forms. Diaboleite is the only mineral known to represent thiscrystal class. It is interesting to note that although the mineral diaboleite was first described in 1923, itwas not until 1941 that crystallographers had comprehensively investigated its forms, allowing the recognition of this form. Inliterature 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 8trapezohedral faces, which correspond to half the faces of the ditetragonal dipyramid. Its symmetrynotation is 422, having a 4-fold rotational axis parallel to the c axis and 2 2-fold axes at right angles tothe 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 than4/m, but when viewed as displayed on an actual crystal of scheelite(blue faces on fig. 4.12b), the true symmetry is revealed.Mineralspossibly expressing this closed crystal form, aside fromscheelite,include powellite, fergusonite, and members of thescapolite group.

Our next form is an interesting one in that it possesses only a 4-fold axis of rotary inversioncorresponding to the c axis. Its symmetry notation is -4. The closed form of this tetragonaldisphenoid (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 havetwo vertical symmetry planes present, giving it the symmetry of -42m (like the disphenoid wediscussed 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 lastform, has very low symmetry. The tetragonal pyramid (AKA hemihedral hemimorphic) is an openform with only a 4-fold axis of rotation corresponding to the c axis (fig. 4.14). The term hemimorphicsounds fancy, but is simply a short way of saying that it appears that only half a form is displayed! Nocenter 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 formnotation of {hhl} and {0kl}, depending on their form orientation to the axial cross. Wulfenite is the onlymineral species to represent this form, although its crystals do not always show the differencebetween 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 withcrystallographer's terminology. We hope you now understand that by simply stretching or compressing the vertical axis of the axialcross we had used previously in the isometric system, we defined the tetragonal system. Then, by examining the presence orabsence of the various symmetry elements (mirror planes, axes of rotation, and center of symmetry), we were able to describe allpossible crystal forms in the tetragonal system. Many crystallographers prefer to tackle the hexagonal system next because it hasits corollaries in the tetragonal system, but we would rather play around and vary the length of yet another axis of our axial crossand 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!



Table of Contents

Bob Keller




Now we are ready to consider the Orthorhombic system. We again will start by examining the axial cross for this system. If youremember in Article 4, the tetragonal system, we held the a and b axes the same length (a1 and a2) and varied the length of the caxis. Well, in the orthorhombic system, we will continue the 90 degree angular relationships between all 3 axes, but will vary thelength of each individual axis. Note that THE 3 AXES MUST BE UNEQUAL IN LENGTH. If any two are equal, then, by convention,we are discussing the tetragonal system.

In Figure 5.1, by current practice we orient any crystal in this system so that the length of c isgreater than the length of a, which, in turn, is greater than the length of b. You will commonly findthis in textbooks as "c<a<b".

There may also be 3 mirror symmetry planes, which must be at right angles to each other. Butguess what! In the past, mineralogists have not always observed the axial length practice given here,and presently, the consensus is to conform when possible to the existing literature. This reason iswhy we will encounter some special orientation situations when dealing with certain commonorthorhombic minerals.

When examining an orthorhombic crystal, we find that the highest obtainable symmetry is 2-fold. In asimple form, like the combination of the 3 pinacoids (open form), the crystal takes on an elongate, andoften tabular appearance. These are typical forms to see expressed on barite and celestine.

The 3 pinacoids are at right angles to each other and usually the orientation of a given crystal to theaxes is accomplished by an examination of the habit and any apparent cleavage. In topaz, theprominent pinacoidal cleavage is in the plane of the 2 shortest axes and perpendicular to the longestaxis, so by convention, it is considered perpendicular to the c axis.

However, you will often encounter the situation where a given crystal displays a very prominent pinacoidand the crystal is tabular in form. In such a case, we then consider the c axis at right angle to the prominent pinacoid and thecrystal is oriented as in Figure 5.2. This is a much different appearance than the example of topaz, noted in the paragraph above.

The orthorhombic system has 3 general symmetry classes, each expressed by its own Hermann-Mauguin notation.

Let's look at the forms designated by the symmetry 2/m2/m2/m. There are 3 of these (have you noticed that almost everythingmentioned in this article is in 3's!): the pinacoid (also called the parallelohedron); the rhombic prism; and the rhombicdipyramid.

The pinacoid consists of 2 parallel faces, and can occur in the 3 different crystallographicorientations. These are the pair that intercept the c axis and are parallel to the a and baxes {001}; the pair that intercept the b axis and are parallel to the a and c axes {010};and the pair that intercept the a axis and are parallel to the b and c axes {100}. They arecalled the c pinacoid, the b pinacoid, and the a pinacoid, respectively (fig. 5.3).

The rhombic prism, an open form, consists of 4 faces which are parallel to 1 axis andintersect the other two. There are 3 of these rhombic prisms and they are given by thegeneral notational forms: {hk0}, which is parallel to the c axis; {h0l}, which is parallel tothe b axis; and {0kl}, which is parallel to the a axis. Figure 5.4 a,b,c present the 3 rhombicprisms, each in combination with a corresponding pinacoidal form. Only the positive face

of the rhombic prism is labeled in these examples.

5.4a Prism {110} andpinacoid {001}

5.4b Prism {101} andpinacoid {010}

5.4c Rhombic prism {011}and pinacoid {100}

However, we may discover after examining a large number of different orthorhombic minerals that we see a large number of prismforms expressed on a single crystal, and these forms cannot be expressed with unity in their numbers because their intersectsupon the horizontal axes are not proportionate to their unit lengths. This is where our general symbol notation comes in handy. Inthe old days of crystallography, these forms were designated as either macroprisms or bracyprisms, depending on whether h > kor k > h. A macroprism has the general symbol of {h0l} and a bracyprism has the general symbol of {hk0}.

5.5a Macro- Brachy- andBasal Pinacoids

5.5b Prism and Basal Pinacoid 5.5c

With Figure 5.5, we have 3 sets of prisms expressed by the letter designations of m, l, and n, and a pinacoid face letter-designatedas a.

The rhombic dipyramid is the last form to consider of this symmetry class. It is designated by the general form {hkl} andconsists of 8 triangular faces, each of which intersects all 3 crystallographic axes. This pyramid may have several differentappearances due to the variability of the axial lengths (figs. 5.6 a,b,c).

5.6a Rhombic dipyramid 5.6b 5.6c Sulfur crystal

A relatively large number of orthorhombic minerals are encountered with combinations of the various forms presented so far. Theseinclude andalusite, the members of the aragonite and barite group, brookite, chrysoberyl, the orthopyroxenes, goethite, marcasite,olivine, sillimanite, stibnite, sulfur, and topaz.

Next to consider are the few forms having the symmetry mm2 (termed the rhombic pyramidal). The two-fold rotational axiscorresponds to the c crystallographic axis and the 2 mirror planes (at right angles to each other) intersect this axis. Due to the factthat no horizontal mirror plane exists, forms at the top and bottom of the crystal are different. Look at Figure 5.7a. Also, due to thelack of the horizontal mirror plane, there exists no prisms, but instead we have 2 domes in place of each of the prisms (do youremember that a dome consists of 2 faces that intersect each other, but have no corresponding parallel faces on the other end ofthe crystal?). Think of the minerals hemimorphite (fig. 5.7b), struvite (fig. 5.7c) or bertrandite when you think of this symmetryclass.

5.7a Rhombic pyramid 5.7b Hemimorphite 5.7c Struvite

And now to the last (and the lowest) symmetry class of theorthorhombic system, the rhombic disphenoid.

The form has also been called the rhombic tetrahedron. Ithas the symmetry notation of 222, that is 3 axes of 2-foldrotation which correspond with the 3 crystallographic axes.

I'm sorry, but there is just no other symmetry here! Theforms are, however, enantiomorphic, that is to say presentas right and left images (fig. 5.8). These closed formsconsist of 2 upper triangular faces which alternate with 2 lower triangular faces, the pair of upper faces being offset by 90 degreesin relation to the pair of lower faces.

Figure 5.9

Pinacoids and prisms may also exist in this class. The most common mineral in this crystal class is epsomite(fig. 5.9). Note that in Figure 5.9 the rhombic disphenoid is designated by the letter z and the unit prism by m.

Well, well. Now we have completed the orthorhombic system and we have looked at 3 (again a 3) of the 6 crystalsystems! What do you think? I hope it has been interesting to read and consider the geometry of these crystalsystems. It has been interesting for me to write about them. I think the artist began to get a little bored drawing allthese figures, so started to use some color to liven things up a bit! I do like it.

This is the "hump" article as we have completed 5 of the 9 in the series. In fact, you might even consider it the 'center of symmetry'of the series and our journey. But so much for looking in the rearview mirror at where we've been, better to look and see where weare going - to that land where symmetry becomes less and less a factor! But before we go begin to lose more of our symmetry, Iwant to take a side road to the Hexagonal world, where we can look at all manner of items from either 3 or 6 directions in the nextinstallment, Article 6.



Table of Contents

Bob Keller




Now we will consider the only crystal system that has 4 crystallographic axes! You will find that the Miller indices should actuallybe termed Bravais indices, but most people, probably out of habit, still call them Miller indices. Because there are 4 axes, thereare 4 letters or numbers in the notation.

The forms of the hexagonal system are defined by the axial cross relationships. The hexagonal axes(fig. 6.1) consist of 4 axes, 3 of which are of equal length and in the same plane, as proposed byBravais. These 3 axes, labeled a1, a2, and a3 have an angular relationship to each other of 120 degrees(between the + ends). At right angle (geometrical mathematicians say "normal") to the plane of the aaxes is the c axis. Its length may vary from less than to greater than the length of any of the a axes. Itwill not equal the length of an a axis, however.

Note the orientation of the 4 axes and their + and - ends. If viewed vertically (down the c axis), the axesdivide a circle into 6 equal parts and the axial notation reads (starting with a +) as +,-,+,-,+,-. The

positive and negative ends alternating. In stating the indices of any face, four numbers (the Bravais symbol) must be given. In theHermann-Mauguin symmetry notation, the first number refers to the principal axis of symmetry, which is coincident with c in thiscase. The second and third symbols, if present, refer to the symmetry elements parallel with and normal to the a1, a2, a3crystallographic axes, respectively.

Now, surprise!! We find that the Hexagonal system has two divisions, based on symmetry. There are seven possible classes, allhaving 6-fold symmetry, in the Hexagonal division and five possible classes, all having 3-fold symmetry, in the Trigonaldivision. The general symbol for any form in the Hexagonal system is {hk-il}. The angular relation of the three horizontal axes (a1,a2, a3) shows that the algebraic sum of the indices h, k, i, is equal to 0.

The Hexagonal Division

Now, let's begin to consider the first class of the Hexagonal division. The Normal or Dihexagonal dipyramidalclass has 6-foldsymmetry around the c or vertical axis. It also has 6 horizontal axes of 2-fold symmetry, 3 of which correspond to the 3 horizontalcrystallographic axes and 3 which bisect the angles between the axes. It's Hermann- Mauguin notation is 6/m2/m2/m. Confused?Check out figure 6.2a and 6.2b which show the symmetry elements of this class, associated with axes and mirror planes.

Rotational symmetry elements Symmetry planes

There are 7 possible forms which may be present in the Dihexagonal Dipyramidal class:

Form Number of Faces Miller Indices Form Expression

1. Base or basal pinacoid 2 (0001) open

2. First order prism 6 (10-10) open

3. Second order prism 6 (11-20) open

4. Dihexagonal prism 12 (hk-i0) example: (21-30) open

5. First order pyramid 12 (h0-hl) example: (10-11), (20-21) closed

6. Second order pyramid 12 (hh2hl) example: (11-22) closed

7. Dihexagonal dipyramid 24 (hk-il) example: (21-31) closed

See figures 6.3 through 6.8 (below) for what these forms look like.

First order hexagonal prismand c pinacoid

Second order hexagonal prismand c pinacoid

Dihexagonal prism andc pinacoid

First order hexagonaldipyramid

Second order hexagonaldipyramid

Dihexagonaldipyramid

The two faces of the Base, or basal pinacoid, are normal to the c axis and parallel to eachother, and are generally denoted by the italic letter c. Their Miller indices are (0001) and (000-1).

The first and second order prisms cannot be distinguished from one another, as they each appearas a regular hexagonal prism with interfacial angles of 60 degrees, but when viewed down the caxis, as in figure 6.9, the relationships of the two forms to each other and to the a axes arereadily noted.

The dihexagonal prism is a 12-sided prism bounded by 12 faces, each parallel to the vertical(c) axis. If you had both first and second order prisms equally expressed on the same crystal,you could not easily tell them apart from the dihexagonal form. See figure 6.5.

Corresponding to the 3 types of prisms are 3 types of pyramids. Note in the figures 6.6 and 6.7 on the previous page the similarshape, but difference in angular relation to the horizontal axes. The dihexagonal dipyramid is a double 12-sided pyramid (figure6.8 ). The first order pyramid is labeled p. The second order pyramid is labeled s. The dihexagonal dipyramid is labeled v.

These forms look relatively simple until several of them are combined on a single crystal, then lookout! You can even have several of the same form at different angles, thus 2 first order pyramids maybe labeled p and u, respectively.

See figure 6.10 for a beryl crystal having all these forms displayed. Molybdenite and pyrrhotite alsocrystallize in this class.

The ditrigonal dipyramid{hk-il}has a 6-fold rotoinversion axis,which is chosen as c. We should note that -6 is equivalent to a 3-fold axis of rotation with a mirror plane normal to it. Three mirrorplanes intersect the vertical axis and are perpendicular to the 3horizontal crystallographic axes. There are also 3 horizontal 2-foldaxes of symmetry in the vertical mirror planes. The Herman-Mauguin notation is -6m2.

This class is a 12-faced form with six faces above and 6 facesbelow the mirror plane that lies in the a1-a2-a3 axial plane. Figure6.11a is the ditrigonal dipyramid form and figure 6.11b displays adrawing of benitoite, the only mineral described in this class.

The Hemimorphic (dihexagonal pyramid) class. This class differs from the above discussedclasses in that it has no horizontal plane of symmetry and no horizontal axes of symmetry. There isno center of symmetry. Therefore, the Hermann-Mauguin notation is 6mm. The geometry of theprisms is the same. The basal plane is a pedion (remember a pedion differs from a pinacoid in that itis a single face) and the positive and negative pyramids of the 3 types. The difference may be readilynoted on a form drawing of this class (fig. 6.12 ) when compared to figure 6.8 (two pages back).

Several minerals including zincite, wurtzite, and greenockite fall in this class (figs. 6.13a, b, & c).

In the Hexagonal Trapezohedral class, the symmetry axes are the same as the Normal(dihexagonal dipyramidal class discussed initially in this section), but mirror planes andthe center of symmetry are not present. The Hermann-Mauguin notation is 622. Twoenantiomorphic (mirror image) forms are present, each having 12 trapezium-shaped faces(figure 6.14).

Other forms, including pinacoid, hexagonal prisms, dipyramids, and dihexagonal prisms,may be present. Only 2 minerals are known to represent this crystal class: high (beta)quartz and kalsilite.

The Hexagonal Dipyramid class (figure 6.15) has only thevertical 6-fold axis of rotation and a symmetry planeperpendicular to it. The Hermann-Mauguin notation is 6/m.

When this form is by itself, it appears to possess higher symmetry. However, in combination with otherforms it reveals its low symmetry content.

The general forms of this class are positive and negative hexagonal dipyramids. These forms have 12faces, 6 above and 6 below, and correspond in position to one-half of the faces of a dihexagonaldipyramid.

Other forms present may include pinacoid and prisms. The chief minerals crystallizing in this class arethose of the apatite group.

The Trigonal Dipyramid possesses a 6-fold axis of rotary inversion, thus the Hermann- Mauguin notationof -6. This is equivalent of having a 3-fold axis of rotation and a symmetry plane normal to it (3/m). Seefigure 6.16.

Mathematically, this class may exist, but to date no mineral is known to crystallize with this form.

In the Hexagonal pyramid class, the vertical axis is one of 6-fold rotation. No other symmetry ispresent. Figure 6.17 is the hexagonal pyramid. The forms of this class are similar to those of theHexagonal Dipyramid (discussed above), but because there is no horizontal mirror plane, differentforms are present at the top and bottom of the crystal. The hexagonal pyramid has four 6-faced forms:upper positive, upper negative, lower positive, lower negative. Pedions, hexagonal pyramids and prismsmay be present. Only rarely is the form development sufficient to place a crystal in this class.Nepheline is the most common representative of this class.

The Trigonal Division

Now we have worked through the first 7 classes in the Hexagonal System, all having some degree of 6-fold symmetry. Time toshed that 6-fold symmetry and look at the Trigonal Division of the Hexagonal System. Here, we will see that 3-fold symmetryrules.

Remember that prisms are open forms. In the trigonal division there are two distinctive sets of prisms to be concerned with. Thefirst is called the trigonal prism. It consists of 3 equal-sized faces which are parallel to the c crystallographic axis and which form a3-sided prism. You may think of it as one-half the faces of the first-order hexagonal prism.

In fact, the normal light-refracting 60 degree glass prism used in many physics labworkshops is this form, bounded on the end by the c pinacoid. There exists a secondorder prism, which on general appearance looks the same as the first order, but whenother trigonal forms are present on the termination other than the c pinacoid, the twoprisms may be readily distinguished, one from the other. The second order prism isrotated 60 degrees about the c axis when compared with the first order prism.

The second prism is the ditrigonal prism, which is a 6-sided open form. This form consists of 6 verticalfaces arranged in sets of 2 faces.

Therefore the alternating edges are of differing character; especially noticable when viewed by lookingdown the c axis.

The differing angles between the 3 sets of faces are what distingish this form from the first orderhexagonal prism.

The striations on the figure to the left are typical for natural trigonal crystals, like tourmaline. In thedrawing, c is the pinacoid face and m the prism faces.

I think these forms are simple enough that we don't need any drawings to explain them, but look forthem on figure 6.23 (below) - the tourmaline forms. They are given the normal prism notation of m and a.

Hexagonal-scalenohedral class. The first to consider are those forms with thesymmetry - 3 2/m (Hermann-Mauguin notation). There are two principal forms inthis class: the rhombohedron and the hexagonal scalenohedron.

In this class, the 3-fold rotoinversion axis is the vertical axis (c) and the three 2-fold rotation axes correspond to the three horizontal axes (a1, a2, a3).

There are 3 mirror planes bisecting the angles between the horizontal axes. Seefigure 6.18 to observe the axes and mirror planes for the rhombohedron.

The general form {hk-il} is a hexagonal scalenohedron(figure 6.19). The primary difference in therhombohedron and this form is that with a rhombohedral form, you have 3 rhombohedral faces above and 3rhombohedral faces below the center of the crystal.

In a scalenohedron, each of the rhombohedral faces becomes 2 scalene triangles by dividing therhombohedron from upper to lower corners with a line. Therefore, you have 6 faces on top and 6 faces below,the scalenohedron being a 12-faced form. These forms are illustrated in figure 6.20.

With this form, you can have both positive {h0-hl} and negative{0h-hl} forms for the rhombohedron...

and positive {hk-il} and negative {kh-il} forms for thescalenohedron.

To further complicate matters, the rhombohedron andscalenohedron, as forms, often combine with forms present inhigher hexagonal symmetry classes. Thus, you may find them incombination with hexagonal prisms, hexagonal dipyramid, andpinacoid forms.

Calcite is the most common, well crystallized, and collectiblemineral with these forms. See figure 6.21 for some crystallizationforms of calcite. Several other minerals, such as chabazite andcorundum, commonly show form combinations.

On the last 3 drawings in figure 6.21, see if YOU can name the faces present. I have already given the notation in the first 5 figures.Email me with your answer, and I'll tell you if you are right!

The next crystal class to consider is the Ditrigonal pyramid. The vertical axis is a 3-fold rotation axis and 3 mirror planesintersect in this axis. The Hermann-Mauguin notation is 3m, 3 referring to the vertical axis and m referring to three planes normal tothe three horizontal axes (a1,a2,a3). These 3 mirror planes intersect in the vertical 3-fold axis.

The general form {hk-il} is a ditrigonal pyramidform. There are 4 possible ditrigonal pyramids, withthe indices {hk-il}, {kh-il}, {hk-i-l}, and {kh-i-l}.

The forms are similar to the hexagonal-scalenohedral form discussed previously, but contain onlyhalf the number of faces owing to the missing 2-fold rotation axes. So crystals in this class havedifferent forms on the top of the crystal than on the bottom. Figure 6.22 shows the ditrigonal pyramid.

Figure 6.23 shows 2 tourmaline crystals, the mostcommon mineral crystallizing in this class, which display3m symmetry.

This form may be combined with pedions, hexagonal prisms and pyramids, trigonalpyramids, trigonal prisms, and ditrigonal prisms to yield some complicated, thoughinteresting, forms.

We now have come to the Trigonal-trapezohedral class. The 4 axial directions are occupiedby the rotation axes. The vertical axis is an axis of 3-fold rotation and the 3 horizontal axeshave 2-fold symmetry.

This is similar to those in class -32/m (hexagonal-scalenohedron), but the planes of symmetryare missing. There are 4 trigonal trapezohedrons, each composed of 6 trapezium-shapedfaces. Their Miller indices are: {hk-il}, {i-k-hl}, {kh-il}, and {-ki-hl}. These forms correspond to 2enantiomorphic pairs, each with a right and left form (one pair illustrated in figure 6.24).

Other forms which may be present include pinacoid, trigonal prisms,hexagonal prism, ditrigonal prisms, and rhombohedrons.

Quartz is the common mineral which crystallizes in this class, but onlyrarely is the trapezohedral face (s) displayed. When it is, it is a simplematter to determine if the crystal is right- or left-handed in form (figure 6.25).

Cinnabar also crystallizes in this class.

Cinnabar also crystallizes in this class.

The Rhombohedral class has a 3-fold axis of rotoinversion, which is equivalent to a 3-fold axis ofrotation and a center of symmetry. The general form is {hk-il} and the Hermann- Mauguin notation is -3.

This form is tricky because unless other forms are present, its true symmetry will not be apparent. Thepinacoid {0001} and the hexagonal prisms may be present.

Dolomite and ilmenite are the two most common minerals crystallizing in this class. See figure 6.26.

Now we arrive at the final class in the Hexagonal system. The Trigonal pyramid has one 3-fold axisof rotation as its sole element of symmetry. See figure 6.27. There are, however, 8 trigonal pyramidsof the general form {hk-il}, four above and four below. Each of these correspond to 3 faces of thedihexagonal dipyramid (discussed above). In addition to this, it is possible that there may be trigonalpyramids above and equivalent, but independent, pyramids below. Only when several trigonalpyramids are in combination with one another is the true symmetry revealed.

It appears that only one mineral, a rare species called gratonite, belongs to this class and it has notbeen studied sufficiently to remove all doubt in some crystallographer's minds.

All crystals in the Hexagonal system are oriented so that the negative end of the a3 axis (see againfigure 6.1) is considered to be 0 degrees for plotting purposes. This becomes important when looking atthe distribution of rhombohedral forms and determining if they are + or -.

I suggest you read page 88 of the Manual of Mineralogy - after J. D. Dana by Klein and Hurlbut (20thedition) if you wish further detail.

WOW! We have now wrapped up the Hexagonal system. I hope you are not feeling too hexed by all this discussion. If so, let's justlose that old feeling and prepare yourself to become even less symmetric as we move to the next Crystal System -- Monoclinic.



Table of Contents




Having dispensed with the hexagonal system in article 6, we are ready to resume our task of the removal of symmetry from 3-axissystems. 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 axialangles at 90 degrees to each other. But in the Monoclinic System, we will look at what happens when we have 3 axes of unequallength 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 papertowards the viewer, the vertical axis is c, and the remaining axis which is at right angleto the plane of the a and c axes is b. When properly oriented, the inclined axis aslopes toward the observer, b is horizontal and c is vertical. Both b and c axes are inthe plane of the paper.

In Figure 7.1, the angle between c and b remains 90 degrees and the angle ( )̂ betweenc and a is the one we will vary. Its called beta and is represented by the Greek letter inthe 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 themorphology. The 2-fold rotation axis (the direction perpendicular to the mirror plane) isusually 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) arerelatively 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 yourstandard mineralogy texts these days give the formulae to do these calculations. Aside from the axial constants necessary todescribe minerals in the monoclinic system, the ̂beta must also be given. Given this situation, you might wish to look up thisinformation for orthoclase in a standard mineralogy textbook, like Klein and Hurlbutís Manual of Mineralogy after E. S. Dana. Youwill find that for orthoclase a:b:c = 0.663:1: 0.559. b̂eta = 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 mineralorthoclase), then it is usually called the basal cleavage. In the monoclinic pyroxenes and amphiboles, where there are 2 equivalentcleavage 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 calledthe side pinacoid (used to be called the clinopinacoid), and the c is termed the basalpinacoid.

There are 2 additional pinacoids with the general form notations of {h0l} and {-h0l}. Thepresence 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 thecube in the isometric system. To further confuse the issue, most newer textbooks call thepinacoid 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 planeis 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. Thegeneral 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 of2-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 convertedinto each other by simply rotating their orientation about the b axis! Corollary to this situation, theprisms 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 letternotation where a, b, and c are the pinacoid forms (the diametrical prism, remember?); m is theunit prism and z is a prism; o, u, v, and s are pyramids; p, x, and y are orthodomes; and n is aclinodome.

Figures 7.5a, b, and c are common forms for the mineral orthoclase and 7.5d is a common form for selenite (gypsum). Manycommon 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) thatincludes 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 symmetricalacross 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 bcrystallographic axis. Figure 7.7 represents the general {hkl}form ñ a sphenoid or dihedron. Since wehave no a-c symmetry plane and with the b axis being polar, in the 2 symmetry class, we havedifferent 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}. Mineralrepresentatives are scarce for this class, but include the halotrictite group with the mineralpickeringite as the most commonly occurring member. For comparisonís sake, take another look atFigures 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 ourneed 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?



Table of Contents

Bob Keller




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 isrelatively 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 theTriclinic System (fig. 8.1). In this figure, we see that all 3 axes (a, b, and c) are unequal inlength to each other and that there are no axial angles of 90 degrees. In the MonoclinicSystem, 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 additionalangles to define, neither of which are equal to 90 degrees. One angle is termed alpha andis defined as the angle between the c and b axes and the second is gamma which isdefined as the angle between a and b. Now, we must have some accepted conventions orrules to follow to orient a triclinic crystal, or we will always be in a state of confusion withother folks over just the orientation.

Remember, in the orientation of any crystal, you also are determining the position of the 3crystallographic axes. So, the rules are: 1) the most pronounced zone should be verticaland 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 isdetermined by the intersection of {100} and {001}. Once this is done, the a axis should be shorter than the b axis so that theconvention becomes c < a < b. The axial distances and the 3 angles, alpha, beta, and gamma, can be calculated only withconsiderable difficulty. As in the Monoclinic system, the b axis length is defined as unity (1). The crystallography informationconcerning 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 willconsider 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 istermed the pinacoidal class after its general form {hkl}. So all the formspresent are pinacoids and therefore consist of two identical and parallelfaces.

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

Remember pinacoids intersect one axis and are parallel to the other 2 (in 3axes 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 acenter 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 thosethat intersect all 3 axes. The first type are the pinacoids {100}, {010}, and {001}. The {100} is the front pinacoid and intersects the aaxis, 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 caxis. 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 axisand 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 andintersects the a and c axes. It may be positive {h0l} or negative {-h0l}. Finally, the {hk0} pinacoid is parallel to the c axis andintersects 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. Anumber of minerals crystallize in the -1 class including plagioclase feldspar pectolite, microcline, and wollastonite. The secondsymmetry class of the triclinic system is the 1, which is equivalent to no symmetry! It is a single face termed a pedion and theclass called the pedial class after its {hkl} form. Because the form consists of a single face, each pedion or monohedron stands byitself. 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 hardlybelieve it! If you feel like pursuing the subject of symmetry further, go to Article 9 for my summary remarks and some suggestedadditional references and articles.

Part 9: Conclusion and Further Reading


Table of Contents

Bob Keller



Part 9: Conclusion and Further Reading

Well, here we are. We've finished the 6 crystal systems and we're feeling kind of smug with what we have learned. But listen herereaders, although we had some chalkboard discussions about the crystal systems and their symmetry elements and classes, nowwe must leave the ideal world of symmetrical crystal forms and enter the real world. The world where the growth mechanisms andthe environment of formation of a crystal have as much to do with the expression of the crystal forms we see as the moleculararrangement of atoms composing the mineral we are interested in.

Almost all minerals are crystalline (part of the definition of a mineral), but only under special conditions do you form minerals withwell expressed crystal form(s). The mineral may grow from a melt as in igneous rocks, from solids under elevated temperature andpressure, or in open crevices or pockets from hot, warm, or cold mineral-laden fluids.

So, when you examine euhedral crystals of any mineral, please realize that they are rather remarkable objects. Probably 99 % ofall minerals present in the earth's crust display no external crystalline form. A euhedral crystal is a flower of the mineral world!Treat crystals with care and respect because they probably survived millions of years before someone discovered them. Onecareless moment and they may be destroyed forever.

But enough of our soapbox lecture on the need to understand the scarcity of well-formed crystals! Throughout the series of articles,we have attempted to show you the world of symmetry present in the crystalline world around you. But symmetry is presenteverywhere. Symmetry is present in varying degrees in the biological world. From simple bilateral symmetry (mirror symmetry likeleft-right) to 5-fold symmetry not present in the crystalline world. Members of the echinoderm family are beautiful examples of thistype of symmetry. Next time you look at a starfish, a fossil blastoid, or a sea urchin spend some time examining its symmetry.You will recognize elements of several principles you learned from studying minerals.

Symmetry or the intentional lack of symmetry governs many artistic works. One of the most remarkable persons to use masterfullya variety of symmetry elements was M. C. Escher. His works not only display many symmetry elements, but he includes much ofhis personal theology hidden in his designs. An excellent recent book, available by ordering from your local bookstore is M. C.Escher: His Life and Complete Graphic Work, by M. C. Escher, L. J. Locher (ed.), and F. Bool, published by Harry N. Abrams,November, 1992, ISBN: 0810981130. You could also order this book from Amazon.com on the internet. There is a CD-ROM of M.C. Escher's Works also available for less than $50 through Amazon.com.

There are many articles written in journals concerned with various aspects of symmetry including anarticle in Scientific American on the symmetry of the patterns of dimples on golf balls!

One of the most interesting books we ever came across is Snow Crystals by W. A. Bentley and W. J.Humphries.

This remarkable book contains 2,453 photographs of snow crystals (the mineral ice).

W. A. Bentley's hobby was the photography of snow crystals. After his death, his collection of severalthousand snow crystal and frost crystal photographs were donated to the American MeteorologicalSociety. The AMS was charged with designating someone to put the collection of photographs into asensible order and overseeing their publication. W. J. Humphries oversaw the work which involved severalnoted mineralogists of the day, including S. B. Hendricks, H. E. Merwin, C. S. Ross, and W. T. Schaller.These men classified the photographs into basic types of formation and growth forms and worked up thedescriptive portion of the book's text on the crystallography of the Snow Crystal.

The presentation of so many illustrations of a single mineral (ice) all from the same crystal system makesthe book unique and gives the viewer the opportunity to examine how the various methods of growth of amineral may affect a crystal's appearance, even though the symmetry elements remain constant. Youhave probably heard the comment that no two snow flakes are identical. You will believe it after examiningthe photographs in this book! Every instructor teaching mineralogy should make this book available totheir class just to enlighten the students as to nature's infinite variety on the same theme!

We do not know if it is still in print, but it was first published by McGraw-Hill in 1931 and republished in1962 by Dover Publications, New York. The standard book number is 486-20287-9 and the Library ofCongress Card Catalog Number is 63-422 for the Dover edition.

I could rave on for a few additional pages, but it would do little good. By now, you are either hooked on symmetry or sick of mydiscussion of it. But never again will you look in a mirror and see just yourself. Instead you will see an organism with bilateralsymmetry. Symmetry is a way of looking at the world around you and seeing order in the apparent randomness of nature. It is trulya way to define order out of apparent chaos.

Randomness and chaos are popular topics these days, in mathematics and in philosophical discussionsconcerning the general nature of life, but I assure you that if you look around and are thinking of symmetryyou will find it. It is everywhere in the plant and animal world as well as the mineral kingdom. Just stop,bend down, and look at common clover, a plant whose individual leaves exhibit mirror plane symmetry, butwhose 3-leaf arrangement displays trigonal symmetry elements. Think about the symmetry of a 4-leafedclover. Now you are not interested in finding one for good luck, but so you can more thoroughly examineits symmetry!

Speaking of good luck, we wish all of you the best of it in your continuing quest for knowledge. We hopewe have brought you to a better understanding and appreciation of the symmetry and beauty of the world around you! For the effortinvolved in reading all this series, we salute you and you are hereby deserving of the Mineralogical Order of the Crystal Sphere,

that one object of infinite symmetry. May the sphere ever remind you of the infinite patience it takes to learn about Crystallographyand Geometry.

Mike & Darcy Howard


Table of Contents

Bob Keller

An Introduction to Crystallography and Mineral Crystal Systems

Documents

Transcript of An Introduction to Crystallography and Mineral Crystal Systems