The following is a kind of experiment. I’m not a professional writer. Actually, I’m not even an amateur writer. I just wrote these pages because I wanted to learn something about optical crystallography, specifically how to make use of a polarizing microscope to identify minerals without destroying them. My idea is based on the often repeated observation that the best way to learn something is to try to explain it to someone else. The..

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The following is a kind of experiment. I’m not a professional writer. Actually, I’m not even an amateur writer. I just wrote these pages because I wanted to learn something about optical crystallography, specifically how to make use of a polarizing microscope to identify minerals without destroying them. My idea is based on the often repeated observation that the best way to learn something is to try to explain it to someone else. The result is what you see here.

Each of us follows her (or his) own unique path in our respective journeys toward understanding. The following chapters are my effort to document my personal learning curve. The paragraphs were written while I was learning the ideas expressed. Consequently, the explanations tend to focus on the points that I personally found challenging, and ideas that came to me easily are developed in a more summary fashion. I’ve made no attempt to smooth out the lumps that this style leaves in the stew of ideas presented. Some ideas are over-explained and other explanations may seem to be incomplete. My theory is that this will work to the advantage of some readers, the ones who think the way I do. I hope it will not overly annoy the ones who don’t. It’s my hope that these chapters will be a fairly easy read for most people, and that the reader motivated to push through the awkward parts will be rewarded with a working understanding of the use of the polarizing microscope and will be reasonably comfortable with the graphical tools of the optical crystallographer. This is not a dumbed-down version so much as a summary with an emphasis on practice over theory. 

There are a number of excellent texts on this subject. This is not intended to replace them. Ultimately, the reader interested enough to read through this essay will want to read all the references in the bibliography. This essay is, in a sense, an abstract. It’s my effort to pull out and present just the good stuff. I’ve tried to make it into the short booklet that I wish I could have read when I began. 

It’s a little terrifying to offer an essay of this kind to an audience that includes many readers whose knowledge of the field dwarfs my own. I offer this up with the hope that they will be kind in their criticism, particularly at the inevitable points where I’ve simply gotten it wrong. I hope there aren’t many of those. I welcome constructive criticism and I promise to make corrections promptly. This document is available in the form of a pdf file to anyone who would like a copy. It's a lot easier to read and print out (it's 34 pages) in that form. Just email me at jesse@lava.net and include "CRYSTALS" in the title (so your message can get past my spam filters).

Jesse Crawford

CHAPTER 1: 
SYMMETRY AND COORDINATE SYSTEMS

Everyone learns in school that a straight line is the shortest distance between two points but straightness is more fundamental and more useful than a mere measuring rod. Straightness is what allows us to recognize symmetry in the world. Planes, lines and points may only exist in the human imagination, but together they are the simple tools that enable us to recognize symmetry, and symmetry is one of our most useful tricks for making sense of the world. It could be argued that its role is more fundamental even than its utility in the measuring of distances. The bilateral symmetry of the human face, for example, is probably what enables an infant to differentiate a mother’s face from the confusing chaos of his or her first hours. Symmetry is the tool that enables science to uncover and catalog a vast amount of information from the physical world. Understanding crystals is to a great extent about understanding symmetry. Cataloging the symmetry properties of a crystal allows it to be identified as a member of one of only six crystal types. That fact in itself is remarkable, that the thousands of crystal shapes can be reduced to so few species. It’s a measure of the power of symmetry. 

Recognizing symmetry, at least in the present context, involves just two kinds of simple thought experiments, rotation and reflection. A crystal is said to have an axis of symmetry if it is possible to imagine a line running through it around which it can be rotated into another position that looks identical. If there is only one other position in which this is true, then the line defines a two fold axis of symmetry. If there are two such positions, then it is a 3-fold axis, 3 positions implies a 4-fold axis, and so forth. . 

A geometrical solid is said to have a plane of symmetry if one can imagine a plane running through it that divides it into 2 pieces that are identical except that one is the reverse of the other, as if formed by reflection in a mirror located at the plane. An object has a center of symmetry when its center is located such that any line running through it that intersects a point elsewhere on the object also intersects a third point at the exact same distance from the center, but in the opposite direction. Identifying a center of symmetry involves reflection, not in a plane but in a point. There are actually several other kinds of symmetry that are used by scientists in making sense of the physical world, but these are the only ones we will be concerned with in these pages. 


THE SIX FAMILIES OF CRYSTALS

Mastering the six crystallographic types takes a little effort. There’s nothing about it that’s especially difficult, but like anything mathematical it takes a little practice to reach the point where it becomes easy. Learning these mathematical concepts may not seem that useful at first, but they form the ground upon which new knowledge begins to take root. 

One of the early scientists who studied mineral crystals was Nicholas Steno, a Danish scientist who lived in the 17th century. He is credited with discovering that the various angles observed on a crystal makes up a unique set of properties for crystals of that particular substance. It follows that measuring the angles on a crystal is one way to identify what it’s made of. As a practical matter, measuring the angles on a small crystal can be difficult, so much of crystallography is motivated by the need to find easier, more practical methods. They all follow however from Nicholas Steno’s original discovery, which is known as the law of the constancy of interfacial angles. 

If the various angles measured on a crystal of a certain type of substance are unique to that substance, then it follows that crystals of that material can only take on certain shapes. Of course, in accordance with another scientific principal, which states that nothing is ever absolutely one way or the other (which in these pages shall be referred to as “Law 2”), this rule has some exceptions. Some crystals are pseudomorphs (false forms). That’s when a large crystal of one substance becomes chemically altered into a lot of smaller crystals of a different substance. The overall shape of the large crystal remains, even though it’s comprised of a lot of smaller crystals of a different shape. Thus, the larger crystal can be said to violate Nicholas Steno’s law, but it really doesn’t, it just looks that way. This illustrates another scientific principle which says that you can’t always believe what you see (Law 3).

There are thousands of types of minerals in the world, and each of them has a unique shape, or more correctly a unique set of shapes, since for any set of angles it’s almost always possible to make more than one shape. This leads to what we might call families of shapes that are characteristic of certain sets of angles. Organizing and understanding these families of shapes was quite a challenge in Steno’s day. To be completely truthful, it still is, but thanks to the efforts of many talented people, it’s not as hard now as it once was. Today we can say with confidence that the total number of families of shapes is six. That includes every mineral ever found and all the ones yet to be discovered! That’s pretty amazing, when you think about it. Mathematicians, who have thought about it quite a lot consider it trivial, so I guess it all depends on one’s point of view. In the interest of full disclosure and in accordance with Law 2, there are a few exceptions that don’t fit into the six families, but for present purposes, they’re not that interesting. As we proceed with the study of crystals, we will encounter many new names and unfamiliar terms. We’ll get to the names of these families soon enough. For the moment let’s just think of them as families one through six.

CRYSTAL CHEMISTRY

Crystals are made from atoms, or more often, ions. Ions are just atoms that have acquired an electrical charge by either gaining an electron from or surrendering one to another atom. Ordinarily atoms have no electrical charge because for every positively charged proton in its nucleus it has one negatively charged electron. Some atoms hold on to their electrons very strongly. Other kinds of atoms bind their electrons loosely, so that they can easily be removed, The ease with which an electron can be removed from an atom is one of the primary factors that determines its chemical properties. When two atoms interact in such a way as to cause an electron to be transferred from one atom to another, they each take on an electrical charge. The one that acquires the extra electron becomes negative, and the one that loses an electron becomes positive. Their physical properties change and the resulting particles are referred to as ions. Because opposite charges attract, positive and negative ions attract each other. It’s this tendency to stick together that causes molecules and crystals to form and to hold their shape. The process by which crystals form and grow involves an interesting process of self assembly that has been rather thoroughly studied.

UNIT CELLS

Crystals are built up of identical repeating units that are referred to as unit cells. Each unit cell contains a number of atoms in a ratio determined by the chemical formula of the substance. They’re arranged so as to constitute the smallest possible unit that can be extended by simple translation to make up a crystal of that material. The number of atoms in a unit cell can be any number from one to several hundred. The criterion that determines whether a unit cell is to contain one molecule of a chemical compound or several dozen of them is the symmetry of the resulting cluster of ions. Each new unit cell grows on the surface of a crystal by a process of self organization until the symmetry of the new group of atoms matches the symmetry of the crystal upon which it is growing. The new unit cell then becomes bound to the crystal by virtue of the fact that the total energy of the bound system is lower than the total energy of the system comprised of the new cell existing separately from the crystal. That energy is what holds the crystal together. In this way, the crystal grows by assembling new unit cells onto it’s surface in accordance with the constraints formed by the nature of the chemical bonds involved, and the symmetry of the growing unit cell. We can imagine a cluster of atoms trying all kinds of combinations and configurations until suddenly the symmetry matches. Then, as a tiny quantum of energy is released, the new unit cell becomes bound to the body of the growing crystal, and the process continues at some other location on the crystal until all the available material has been attached to the crystal in the form of new unit cells. 

The crystal grows by adding these self assembled unit cells onto the surface of the crystal in accordance with the laws of symmetry. The shape of the unit cell places it (and the crystal) into one of the six families of crystals. It’s not surprising then that the family to which a crystal belongs is recognized by its observable symmetry properties.

THE SIX COORDINATE SYSTEMS

Mathematicians make extensive use of coordinate systems. The most common is the x, y, z coordinates intersecting at zero with all three axes mutually perpendicular, known as the cartesian coordinate system. It’s very useful, but for characterizing crystals, it has been found more convenient to modify the cartesian system for most cases.

The coordinates of the six families of crystals are as follows:
Family 1 (Isometric) uses the conventional cartesian coordinate system. Each axis makes a right angle with the other two, and the unit distances in each direction are equal. So, the unit cells of this family are shaped like cubes.
Family 2 (Tetragonal) is the same except that one of the axes uses a unit length that’s different from the other two. The other two are equal to one another.
Family 3 (Orthorhombic) is the same except that each axis has its own unit length different from either of the others.
Family 4 (Monoclinic) is like Family 3 except that two of the axes intersect at an angle other than 90 degrees. The third axis meets the other two at 90 degrees.
Family 5 (Triclinic) is like Family 3 except none of the axes make right angles with one another.
Family 6 (Trigonal/Hexagonal) is like Family 1 except that there are 4 axes, one at right angles to the other three. Consequently, the three remaining axes are in the same plane and divide it equally (120 degrees apart).

Any crystal shape can be mapped onto one of these six coordinate systems. The logic of the system follows from the way that crystals are structured at the scale of the unit cells that make them up. It might seem at first that the shapes of crystals should just be the shape of its unit cell, but in fact a given crystal can take on any number of shapes so long as the symmetry properties of the shape are the same as the symmetry properties of the unit cell, or in other words the symmetry properties of the coordinate system to which it belongs. This is the most important point. The shapes that a crystal may take on are the shapes that have the same symmetry properties as the coordinate system of the family the crystal belongs to. 

Family 1 (Isometric or cubic) has the highest amount of symmetry. Isometric crystals may contain many kinds of atoms, but frequently only one or two. Typically the faces of these crystals make cubes or octahedrons or dodecahedrons, all being shapes that have the same symmetry properties as the Family 1 coordinates. The crystals are usually blocky or equant, the term used to describe shapes in which the height, width, and length are all about the same. The three axes in the isometric system are designated as a1, a2, and a3, indicating that the unit cell takes on the same dimensions in each direction. 

Family 2 (Tetragonal) Unit cells in this family are typically elongated (or shortened) in one direction. Most often these crystals are compounds of only a few component ions. In the tetragonal system, the horizontal axes are designated a1 and a2. The vertical axis is c. Unit distance in the direction of the c axis is expressed as a decimal fraction of the unit distance in the directions of the a1 and a2 axes, which are equal and defined as unity.

Family 3 (Orthorhombic) Unequal development extends to all three directions in space. Minerals of this family are of intermediate complexity, typically consisting of several types of ions. Orthorhombic axes are designated a, b and c. The unit dimension for the b axis is defined to be unity. The units for the dimensions in the directions of the other two axes are expressed as a decimal fraction of the b unit distance.

Family 4 (Monoclinic) These crystals are typically more complicated minerals composed of numerous types of ions. Layers become shifted or offset as the unit cell is filled up. Asymmetric distribution of bonding forces causes one of the axes to tilt. In the monoclinic system, β is the angle between the a and c axes. The b axis is at right angles to the plane of the a and c axes. The unit distance in the direction of the b axis is defined as unity. The unit distances in the directions of the other axes are expressed as decimal fractions of the unit distance for the b axis.

Family 5 (Triclinic) Family 5 minerals are typically the ones with the greatest number of different types of ions and have the least symmetry of all the families. Each axis is inclined with respect to the others. The minerals in this family are typically made up of a variety of different ions and atoms bonded in complicated ways that cause the unit cell to twist or lean in all three dimensions. Crystals of this family are recognized by the absence of symmetry. Only a center of symmetry is possible for minerals of this family, planes or axes of symmetry are absent. In the triclinic system, as in the others, the c axis is vertical. The a and b axes are both inclined to c and to one another, defining three angles: α ( the angle between the b and c axes), β (the angle between the a and c axes), and γ (the angle between the a and b axes). Unit distances in the directions of the a and c axes are expressed as decimal fractions of the unit distance in the direction of the b axis, which is defined to be unity.

Family 6 (Trigonal/Hexagonal) These are crystals of often fairly simple composition in which the arrangement of the ions favors the hexagonal packing in one plane instead of the less efficient cubic centered packing. This is a family of fairly high symmetry. Axes and planes as well as centers of symmetry are common. Quartz is the best known member of this family. As usual, the c axis is the vertical axis, The other three axes are a1, a2, and a3. All three are in the horizontal plane at 120 degrees to one another. Unit distances in the directions of the a1, a2, and a3 axes are the same and are defined to be unity. The unit distance in the direction of the c axis is a decimal fraction of the unit distance so defined.

As described in each family, the length of one of the axes is defined to have a length of one. The unit lengths in the directions of the other two axes are reported as decimal fractions of that unit length. The ratio of the three (or 4) axes expressed in this way is called the axial ratio. The axial ratio together with the angles between the coordinates comprises the elements of crystallization, and are a unique property of any given mineral. 

CRYSTAL FACES

Assuming that the plane of some crystal face cuts the three axes of its coordinate system at points A, B, and C at distances a, b, and c from the origin of the coordinates, the ratio a:b:c is called the parametral ratio of the plane ABC. Obviously, there are a total of 8 such planes arranged around the axial center that have the same ratio. These 8 planes constitute a crystal form, represented by the general ratio a:b:c. Forms, such as this one that enclose a volume of space are referred to as closed forms. Other forms can be defined using planes that cut only one or two axes, running parallel to the axes not intersected. These are called open forms. Open forms are those which do not enclose space, but extend indefinitely in one or more directions. The form that cuts the three axes at unit lengths is designated the fundamental or unit bipyramid. 

Several forms may combine to make the faces of an actual crystal, Such combinations give rise to a great many possible crystal shapes. The intercept of any given crystal face with any axis is always an integral fraction of the basic axial units. This constitutes a strong constraint on the permissible shapes that may be assumed by a crystal, and is referred to as the law of rational coefficients. It’s a consequence of the fact that a crystal is constructed of discreet unit cells. 

MILLER INDICES

There are several conventions that are used to designate the forms and faces of crystals. The most common are called the Miller indices. The Miller indices are integer ratios of the points of interception with each axis. A Miller index of zero implies that the form does not intercept the axis but runs parallel to it. The Miller Indices are reciprocals. Some examples may serve to make the use of the Miller Indices clear. A crystal face designated 100 (read one zero zero) would be the plane intersecting the a1 axis at the unit distance from the origin, and running parallel to the other two axes. 110 would be a face that intersects the a1 and a2 axes at the unit distance and runs parallel to the third. A face designated 111 would be the face that cuts each axis at a distance of one unit for each axis, Another possible face might intersect the a axis at 1/3, the b axis at 1/2 and the c axis at 1/4th of each axes respective unit distances. Such a face would be designated 324. These notes on the Miller Indices are oversimplified, and should only serve to demystify them a little when they are encountered in the literature. An in depth discussion of the use of the Miller Indices is beyond the scope of this essay.


CHAPTER 2 
LIGHT AND MATTER

The perfect crystals described by theories are rare. The theory sounds great, but practically speaking, how useful is it when the crystals that we deal with in the real world are so rarely perfect? The theory wouldn’t be nearly as useful if it were not for light, and the things we’ve learned about the way that light interacts with matter. Luckily for us, light doesn’t seem to know that crystals are not perfect, and responds to them as if they were. Light has a lot to do with what makes crystals interesting in the first place, so perhaps it’s not so surprising that paying close attention to the nature of the interactions between light and crystals can yield a lot of information about them. 

Light is a phenomenon that has been something of a mystery for most of the history of civilization. It’s only in the past hundred years or so that we have begun to gain an understanding of what it is. For a long time a debate raged over whether light is a wave moving through some medium like sound moving through the air, or whether it’s a stream of particles like little pellets streaming through space. It was a long time until scientists realized that light is actually both of those things. Light is a form of energy, which has effects like waves and also acts like a stream of particles. Both models have contributed to our understanding of light and the things that it can do, and neither one by itself is sufficient to account for all the facts. Current theories of light are difficult to turn into an intuitive model. It doesn’t help very much to say that light is like a stream of wave packets. Such explanations seem to create more questions than they answer. Nevertheless, that’s the best we can do. Light is what it is, and as long as we can predict what it will do in any given situation, who cares if it makes sense? Light is a stream of discreet wave packets that move through space at a constant speed, and through material objects at a speed that depends on certain physical characteristics of the material. The actual speed of light through most transparent crystals has been measured and tabulated. Refractive index is the property of a transparent material defined as the ratio of the speed of light passing through it to the speed of light traveling through space. We’re interested in this figure for crystals because the refractive index of a given crystal is unique and because it’s fairly easy to measure. There are several methods available to us for determining refractive index.

The way that light behaves as it passes through is one of the best ways to determine the symmetry of the unit cells that make up a crystal. Light is energy, as has already been mentioned. By definition energy is a force acting through a distance. The bonds that hold molecules and crystals together are forces acting through a distance and are therefore a type of energy. The type of energy that holds crystals together is, like light, electromagnetic in nature. It is reasonable therefore to imagine that light, these waves of electromagnetic energy, might interact with the forces that hold the crystals together. Chemists know a lot about the details of these interactions, but most of those details need not concern us here. It’s sufficient, for our purposes to realize that whatever the nature of the interactions, as a photon of light passes through a crystal, if the interactions are not the same in every direction, then there will be preferred directions, directions in which the interactions are stronger or weaker than they are for others. This “sameness” of the environment of forces within the crystal is reflected in the symmetry of the crystal. That’s why observing the way in which a crystal affects polarized light enables us to assign a crystal to one of the families of crystals described earlier, families that are based on their symmetry. 

Tests based on the way that light interacts with crystals have the virtue of leaving the specimen undamaged. At least that’s usually the case (see law 2, above). The study of light and its interaction with crystals is the discipline known as optical crystallography. Using a polarizing microscope and the methods of the optical crystallographer one can quickly determine a whole list of exotic sounding properties like plechroism, optical dispersion, refractive index, and birefringence. Each of these is a useful item of information. Taken together, they comprise what amounts to a fingerprint of the chemical compound making up the mineral. The elegance of these concepts complements and enhances the beautiful forms and colors of the crystals themselves. 


USING POLARIZED LIGHT 

Before we can understand the use of polarized light, we need to have a pretty clear idea of what it is. Under current theories, ordinary light, as was mentioned earlier, consists of packets or bundles of waves of electromagnetic energy. We know that each bundle of light consists of electric and magnetic forces that oscillate at a very high rate back and forth at right angles to one another and at right angles to the direction in which the light is moving. The particular direction of the oscillations within a given beam of light can be any direction as long as they stay within the plane of the wavefront. The wavefront is usually the plane that is at right angles to the direction in which the light is moving (see Law 2). In unpolarized light, each photon, or light particle, oscillates in a different direction within the plane of the wavefront. Polarized light is light that oscillates in such a way that the electromagnetic forces of all photons are aligned not only in the same plane but are also oriented in the same direction. Creating polarized light was a difficult problem for those who conducted the early experiments with it, but modern polarizing filters make things much easier. Producing polarized light now is simply a matter of putting a filter made of polaroid plastic in series with the light source.

CHAPTER 3 
TOOLS AND METHODS

THE POLARIZING MICROSCOPE

Geologists and chemists make use of a special type of microscope called a petrographic microscope or polarizing microscope. It’s constructed much like a conventional microscope, but has some additional components. A polarizing microscope has a light source that includes a filter that causes all the light to be polarized in one direction, usually in a left to right direction as one looks through the eyepiece. In addition there is a second polarizing filter located inside the microscope tube above the objective lens. The polarization of this second polarizer is ordinarily oriented at right angles to the first one. The view with these crossed polarizers in the optical path is dark. Very little light is allowed through because of the orientation of the polarizers. Objects placed on the microscope stage, in order to be seen, must alter the direction of the polarization of the light as it passes through. In these microscopes, there is usually some provision for switching the upper polarizer (called the analyzer) out of the optical path so the microscope can also be used in the conventional mode. In these chapters, when reference is made to viewing a specimen using a polarizing microscope, it is assumed that both polarizers are in the optical path in the crossed orientation.

Polarizing microscopes are ordinarily used in one of two modes, called orthoscopic and conoscopic. Orthoscopic mode is the conventional way a microscope is used for producing magnified images of objects. In this mode the microscope condenser is adjusted so that light strikes the specimen as a beam of parallel rays. If instead the condenser is adjusted so that the light converges toward a point, and the apex of the resulting cone of light is directed at the specimen from below, then the microscope is said to be in conoscopic mode. Observations made in conoscopic mode are usually made with a special lens inserted between the eyepiece and the objective called a Bertrand lens. In the absence of a Bertrand lens, conoscopic observations can be made without the use of an eyepiece at all. Instead, the image is viewed by removing the eyepiece and looking down the microscope tube directly at the back of the microscope’s objective lens. 

REFRACTOMETER

A refractometer is an instrument used to measure the refractive index of a transparent material. Various kinds of refractometers have been invented for measuring the refractive index of solids and liquids. Measuring the refractive index of liquids is generally easier because sample preparation is less critical. For that reason, practical methods for measuring the refractive index of a crystal usually involve finding a liquid that matches the refractive index of the crystal and then measuring the refractive index of the liquid.

Light traveling through air moves at about the same speed as it does through empty space. It slows down, often to half or less of that speed when it strikes a transparent object and begins to move through it. It’s actually this slowing down of the light that makes it possible for us to see colorless transparent objects. The refractive index of a material is the ratio of the speed of light traveling through that material to the speed of light in a vacuum. The fact that the boundaries of objects are invisible when they’re surrounded by a fluid of the same refractive index is the basis for the most usual methods used for measuring the refractive index of crystals. A set of oils of known refractive index is used. The crystal is simply immersed in each oil in turn until one is found in which the crystal can’t be seen. Oils made especially for this purpose can be purchased. One small problem is that sets of oils for refractive index determination are somewhat expensive, so various methods have been developed that make it possible to make determinations without using more oil than necessary. One method that’s used employs just two oils. It’s not even necessary to know the refractive indices of the oils in advance other than that one must be higher and one lower than the crystal to be measured. The crystal is placed under the microscope in a small quantity of one of the oils, and the other oil is added and mixed in small increments until the crystal disappears. Then a small sample of the mixed oil is tested with a refractometer. Obviously, the two oils employed must be miscible.

THE BECKE LINE TEST

Another frequently used method is based on the fact that under certain conditions a bright halo of light can be seen surrounding a crystal viewed through a microscope. This halo is called the Becke line after the scientist who first developed this method. The Becke line is observed whenever a crystal, immersed in an oil of slightly different refractive index, is viewed with the microscope slightly out of focus. The halo can be seen to grow outward from the crystal as the microscope objective is raised above the position at which the crystal is in focus. It grows, that is, provided the refractive index of the oil surrounding the crystal is greater than that of the crystal. If instead the refractive index of the crystal is greater, then the halo shrinks as the objective is raised. To see the halo grow larger than the crystal, the objective is moved below the point where the crystal is in focus. The key point to remember is that the halo moves to the side with the higher refractive index as the objective is raised to a higher position. Higher goes with higher, lower goes with lower. It takes a little practice to do this test right, but it is quite sensitive, and can be used to determine refractive indices with great accuracy. Its greatest usefulness is to provide guidance when trying to find the right oil to match the crystal and to be sure of the point where the two are the same. When there’s no halo, then the oil and the crystal have the same refractive index. Certain other adjustments of the microscope can enhance the Becke line effect. Adjusting the condenser aperture to a very narrow beam, for example, makes the halo easier to see.


CHAPTER 4 
THE LIGHT SHOW

Isometric materials have no optic axis, which is a way of saying that no matter which way an isometric crystal is oriented in a beam of polarized light, there is no alteration of the direction of the polarization of the light as it passes through. Such materials are called isotropic, and include most fluids, glass, and all isometric crystals. These crystals have the highest degree of symmetry and therefore light finds no preferred direction. Crystals of all other families ( families 2 through 6) split polarized light into two parallel beams, each beam with its own wavefront plane and its own refractive index. The two beams are polarized at right angles to one another. Crystals of this type are called anisotropic. The actual orientation that the polarization takes in each case depends on the orientation of the crystal. It changes as the crystal is rotated in the beam of light.

Nobody knows who it was who first thought of looking at a crystal between crossed polarizers, but it must have been an exciting day. That was the experiment that would eventually unlock a veritable flood of information from the inside of a stone! The first thing that one notices is a host of pretty colors. Crystal grains that in unpolarized light are colorless are seen in shades of red, green, yellow, blue and violet, ranging from bright hues to subtle pastel shades. If the crystal is rotated, something else interesting happens. When the stage of the microscope on which the crystal is resting is rotated to a certain direction the light goes out! The crystal turns black, only to come on again as the stage is rotated further. As the stage is rotated, the crystals blink on and off like christmas lights. There’s a lot going on here, and sorting it all out takes a little time. 

One might suspect that if we can see a crystal through a microscope that has crossed polarizers, then the crystal must be rotating the light somehow as it passes through so that it’s no longer vibrating in the same direction when it reaches the second polarizer. That’s exactly what happens. When a polarized beam of light strikes a birefringent crystal, also called an anisotropic crystal, (families 2 through 6)), it splits apart into two beams. Each one vibrates now with a new orientation at right angles to the other one, and each beam moves through the crystal at a different speed, that is to say, each beam has its own refractive index, so the direction of the two rays is slightly different. The amount of light in the original beam is divided between the two beams equally if the vibration directions are at 45 degrees to the directions of the polarizers, but otherwise, the light is directed more toward one beam than the other. As the microscope stage supporting the crystal turns, two things happen: 1) the amount of the light energy going into the beam that is approaching a right angle to the upper polarizer increases while the other beam becomes weaker, and 2) the amount of the light that is allowed to pass through the upper polarizer decreases (since it’s nearly at right angles to the polarization of the beam). At 90 degrees, just as all the light is being routed to the beam polarized at 90 degrees to the upper polarizer, the point is reached where none of it is allowed through! The image of the crystal goes black! The same thing happens again, only with the other beam as the crystal is rotated by another 90 degrees, and again as the specimen is rotated through another 90 degrees, and so on. Every 90 degrees, the light goes out! This phenomenon is called extinction. The direction that is observed at which extinction occurs relative to the straight edges of the crystals contain some valuable information. Specifically, they tell us whether the crystal is uniaxial or biaxial, that is, whether it has one optical axis or two. IF IT HAS ONLY ONE, THEN THE CRYSTAL IS A MEMBER OF EITHER FAMILY 2 OR FAMILY 6, OTHERWISE, IT’S A MEMBER OF FAMILY 3, 4, OR 5. That’s a lot of information to get from what are basically a set of idiot lights. The new polarization directions for the two new beams are at right angles to one another, and they turn as the crystal is turned, so that if the crystal is rotated through a full circle there are 4 positions at which the vibration direction of one or the other of the new beams is blocked by the upper polarizer, each one is blocked just as it’s allocated 100 percent of the light coming through the microscope. Uniaxial crystals exhibit what is referred to as parallel extinction or in some cases symmetrical extinction. These are recognized by observing any straight edge on the crystal, a cleavage trace or the edge of a crystal face, as the crystal is rotated through extinction. If, at the point of maximum extinction, the edge is parallel to either the upper or the lower polarizer then the crystal is said to have parallel extinction. These are the directions marked by the crosshairs in the microscope eyepiece. If the extinction is not parallel, then further measurements must be made on other grains to determine whether the extinction is symmetrical or inclined. If other extinction angles can be found that are the same as the first, but in the other direction, then the mineral exhibits symmetrical extinction, and the significance is the same as if it were parallel. In those cases, the mineral is uniaxial. Biaxial crystals go extinct consistently at some oblique angle to the crosshairs (however, see Law 2). Often it’s not easy to determine, using these criteria, whether a mineral is uniaxial or biaxial. In these cases there are other methods that can be employed. This involves altering light so that it converges as it illuminates the specimen. This is an important method that can also yield a great deal of information about a crystal. Convergent or conoscopic methods will be discussed in a subsequent chapter.



CHAPTER 5 
INDICATRIX THEORY

THE ISOMETRIC INDICATRIX

The refractive index of an anisotropic crystal depends on the direction of the polarization of the light. For purposes of visualizing the relationship between the refractive index and the direction of polarization, crystallographers use a graphic device called an indicatrix. An indicatrix is like a piece of three dimensional polar graph paper. Since nobody has yet figured out how to make three dimensional paper, you have to imagine it. First, imagine the refractive index of a material for a specific light beam as a vector, the length of which is the value of the refractive index, and the direction of which is the direction of the polarization of that light beam (not the direction of the light beam, but the direction of its polarization). Then imagine all possible such vectors, representing light beams of all possible polarization directions, arranged with their tails joined at a common origin. The heads of these vectors generates a surface that completely characterizes the refractive index for that material. Obviously, an indicatrix doesn’t fit on a flat page, it must be imagined as a three dimensional object suspended in space, like a globe. Since the refractive index of an isometric material is the same regardless of the direction of the polarization of the light, the indicatrix for an isotropic material is a sphere. Since there is no preferred direction, there is no rotation of polarized light in passing through an isometric material. This means isometric crystals are very easy to recognize using a polarizing microscope. These are the ones that stay dark no matter how they’re oriented.



THE UNIAXIAL INDICATRIX

For uniaxial crystals, that is, crystals in families 2 (tetragonal) or 6 (trigonal/hexagonal), the indicatrix is not a sphere. It’s still like a globe, but it’s longer or shorter in the vertical direction depending on whether the refractive index is more or less in the vertical direction than in the other directions. If you imagine making a flat slice through the center of the indicatrix, depending on the direction, the face of the slice might be elliptical, or if the slice is exactly horizontal, it would be a circle. That circle is just like the circle that would be obtained if you sliced through the indicatrix of an isometric material , and it means the same thing. For light passing through, vibrating in that plane, there is no preferred direction. Polarized light, vibrating in that plane is not changed in any way. If, on the other hand, the crystal is turned to the side, so that the horizontal section is no longer a circle but an ellipse, then the indicatrix is telling us that there is a preferred direction and that light passing through the crystal is split into two beams. This phenomenon is the basis for much of the subsequent discussion. A great deal of information can be derived from it

THE BIAXIAL INDICATRIX

For a biaxial crystal, the indicatrix is similarly derived, but imagining its shape is somewhat more involved. Fortunately, it’s easier to imagine it than it is to explain it. In a biaxial crystal, there are two vibration directions at right angles to one another that exhibit respectively the highest and the lowest values of refractive index. These refractive index values are referred to as gamma (γ), the highest value, and alpha (α), the lowest. A third direction, at right angles to the first two has a refractive index that lies between the first two and is called beta (β). Bear in mind that we’re still talking about the direction of the polarization of the light, not to be confused with the path that the light takes in moving through the material. These three vibration directions are designated X, Y, and Z and are called the principal vibration axes. Each associated refractive index (γ,β, and α) is called a principal refractive index. The value of the refractive index measured for light polarized in directions between the principal axes will lie between the values measured for the principal refractive indices. The shape of the indicatrix for biaxial crystals is called a triaxial ellipsoid. It’s just like the one for the uniaxial indicatrix (which is called a uniaxial ellipsoid), except that it has a second elliptical shape superimposed over the first. It’s like two ellipsoids fused together. This shape has some interesting properties that relate directly to the mineralogical properties observed in biaxial crystals. Most sections or slices running through the center of a biaxial indicatrix are ellipses. There are two however that are perfect circles with a radius equal to β, the intermediate refractive index. The particular position of these two planes is determined by the comparative values of the principle refractive indices, gamma (γ), beta (β), and alpha(α). These two circles define two planes that intersect at the center of the indicatrix. Two lines drawn through the center of the indicatrix and perpendicular to these planes are referred to as the primary optic axes. The plane defined by the two primary optic axes is the optic plane. The optic plane also contains the X and Z principal vibration axes. The acute angle between the two primary optic axes is the optic angle. The optic angle is sometimes abbreviated O.A. or sometimes 2V. The optic axes always lie within the XZ plane, that is the plane defined by the directions of the greatest and least refractive indices (γ and α). The Optic angle is also sometimes reported as the angle measured between the optic axes in either the X direction or the Z direction (2Vx or 2Vz). The sum of these two angles always equals 180 degrees. Using this convention, the reported optic angle may be greater than 90 degrees, otherwise it’s limited to 90 degrees by the definition of an acute angle. The principal vibration axis that bisects the acute optic axial angle is called the acute bisectrix. The one bisecting the obtuse axial angle is the obtuse bisectrix. The acute bisectrix will always be either the Z principal vibration axis corresponding to the highest refractive index γ or the X vibrational axis corresponding to the lowest refractive index α. Biaxial positive minerals are defined as those in which Z is the the acute bisectrix. Biaxial Negative minerals are those in which the X axis is the acute bisectrix. Said another way, in positive minerals the intermediate refractive index β is nearer in value to α than it is to γ (see Law 2).

If your head isn’t feeling a little full right now, you’re probably not paying attention. The preceeding paragraph should seem pretty dense the first time through. It introduces a lot of new terminology and there’s no particular reason at this point why the beginning student of mineralogy would care much about any of it. It will become evident as we proceed however, why these terms are introduced at this point in the discussion. It’s worth going over a few times to fix these relationships in the mind. These are the players in the crystallographer’s drama. They have a story to tell. 


CHAPTER 6 
THE STORY

REFRACTION and BIREFRINGENCE

When light moves from one transparent medium, such as air, to another transparent medium of a different refractive index, such as glass, provided it’s not moving at exactly 90 degrees to the surface, the direction that it takes through the new material changes. The direction that it takes in an isotropic material (Isometric crystals, gasses, liquids, glass, etc.) is related to the direction in the air by the ratio of the refractive index of the air, the refractive index of the glass, and the sine of the angle that the ray path makes with a line perpendicular to the surface of the glass. It sounds a little intimidating, but it’s really pretty simple. It’s known as Snell’s Law. In mathematical shorthand it’s: 

ni sin i = nr sin r

Where ni is the refractive index of the air, nr the refractive index in the isotropic material, and i and r are the angles made between a line that’s at right angles to the surface of the crystal and the incident (i) and refracted (r) rays respectively. 

Snell’s law holds true for all isotropic materials, but not necessarily for anisotropic ones. The minerals that crystallize in families 2 through 6 are the anisotropic minerals. These crystals split the light into into two beams and only one of them obeys Snell’s Law.

ELONGATION

When a crystal grows in such a way as to be longer in one direction than the others, it’s often useful to know how the orientation of this direction of elongation compares to the direction of the principal vibration axes. This is referred to as the elongation of the crystal and may be positive or negative. A crystal is referred to as possessing positive elongation or is said to be length slow if the direction of the highest refractive index is within 45 degrees of being parallel to the direction of elongation. Otherwise it possesses negative elongation or is length fast. Measuring the direction of the principal refractive indices will be covered later when we discuss the use of the quartz wedge and the compensation plates.



COLOR AND PLECHROISM

Most minerals display a variety of colors, whether viewed through a microscope or not. Colors arise because of the selective absorption of certain wavelengths of light, and generally are different depending on whether the light is passing through the crystal or is reflected from the surface. Colors may be characteristic of the mineral itself or may result from the presence of strongly colored impurities that are present in very small amounts. For this reason, color is often not a particularly trustworthy indicator of the composition of a mineral. Colors observed in thin sections of mineral crystals are generally more likely to be actual mineral properties instead of the result of trace components. When viewed in polarized light without crossed polarizers, often colors can be observed to change in hue and intensity as the orientation is changed. This phenomenon is called plechroism. When present, it furnishes an important clue to the identity of the mineral. Plechroism is always associated with a particular refractive index. Thus, uniaxial minerals have two refractive indices and are usually dichroic, meaning that they exhibit plechroism in two directions and in two colors. Biaxial crystals, with three refractive indices, are trichrioc. Isotropic minerals do not display plechroism. If the intensity of the absorption varies significantly as the orientation of polarization changes to correspond with the various principal vibration directions, that fact is noted in the absorption formula by citing the direction of least absorption first, followed by the direction of intermediate absorption followed by the direction of highest absorption, thus: X 



INTERFERENCE COLORS 

When crystals of an anisotropic mineral are viewed through a polarizing microscope with the polarizers crossed they seem to light up in a beautiful display of colors. Isometric crystals, that is crystals that have only one refractive index, do not display these colors. Instead, they remain dark no matter which direction the light passes through them. Light entering an anisotropic crystal divides into two beams polarized at 90 degrees to one another. The refractive index is different for light of one polarization than it is for the other. This difference is called birefringence. The colors observed under a polarizing microscope depend on how thick the crystal is, and the birefringence of the crystal. The polarization direction of the light is determined by properties of the crystal. Both beams rotate together when the crystal is rotated. One of the two beams is referred to as the ordinary ray, the other is the extraordinary ray. These terms derive from the fact that the path of the ordinary ray obeys Snell’s Law, while that of the extraordinary ray doesn’t. Because the refractive indices for these two rays are different, they pass through the crystal at different velocities. As they emerge from the other side of the crystal they recombine into a single ray again, but with a difference. Since the two rays travel at different speeds, when they recombine the waves have shifted with respect to one another. The two rays that start at the same time arrive at the opposite side at different times. The faster ray does not have the option of waiting for the slower one to catch up, it just combines with the other ray that it finds there. Certain wavelengths within the two beams of the light, those that have shifted relative to the other wave by an odd number of half wavelengths, are reduced in intensity. When the two beams recombine, the wavelengths that are shifted by an odd number of half wave lengths. those that are 1/2, 3/2, 5/2, etc are out of phase and the intensity of those wavelengths is reduced in the recombined wave, That’s another way of saying those colors are reduced in intensity in the recombined beam. Other colors, representing wavelengths that have shifted an even number of half wavelengths reinforce one another. Those wavelengths are said to recombine in phase, and the result is that the colors corresponding to those wavelengths are twice as intense in the recombined beam as they were in the original light. The result is that the light is no longer the same color. It contains a different distribution of colors than it did originally. The emerging light has acquired a color that is a function of two properties of the mineral crystal: the birefringence, or difference between the refractive indices of the two beams, and the thickness of the mineral crystal. This is very valuable information. If you know the thickness, which is fairly easy to measure with a microscope, then from the color you can determine the birefringence. This is a lot like taking a mineral’s fingerprint. It’s sufficiently unique that when a geologist looks at a thin section of a rock, he can usually name the minerals that it contains just by observing the colors of each grain! 

THE MICHEL-LEVY CHART OF INTERFERENCE COLORS

The colors typically observed in a birefringent crystal through a polarizing microscope have been compiled in the form of a chart, showing the color that is observed as a function of the linear distance of the shift between the two rays (referred to as retardation), It’s called an interference color chart or a Michel Levy Chart after the scientists credited with its invention. It has some interesting features. The first thing one notices is that it looks like the conventional spectrum of white light, except that the colors are in a different sequence. Also, whereas in a conventional spectrum each color appears only once, in the Michel-Levy chart the colors repeat. Starting with black at a retardation of zero, the color shifts to blue at 100 and then white as the retardation goes from 100 to 200 nanometers. At about 250 nanometers, the color begins to take on a yellow hue, which shifts to orange and then red beginning at about 500 nanometers. Then, rather abruptly, at about 580 nanometers the color shifts to purple and then blue again at around 650 nanometers. As the retardation reaches 720 nanometers, the color takes on a greenish tint, then bright green at 750 through about 870. Then another rather abrupt shift to yellow which grades through orange to red again starting at about 1000 nanometers. The color sequence shifts from red to a narrow band of purple again at about 1100 nanometers then blue, green, yellow, orange, and red again starting at about 1600 nanometers. Red turns into a broad band of green beginning around 1750 nanometers and extending to 2000 nanometers. Starting at the outer edge of the green band at about 2000 nanometers the color turns almost white again, then takes on a pink cast which gradually increases in intensity to almost red by 2200 nanometers. There is another narrow band of pale yellow starting at 2250 nm which becomes greenish by 2300 nanometers and the green intensifies then as the retardation increases. Beyond 2500 nanometers the colors are just varying shades of pink. Most charts end around 2200 nanometers since there’s usually not much more to be learned at higher values of retardation. Another feature of the colors in the interference color chart is the way in which the intensity of the color varies as a function of retardation. At the low values of retardation, the colors are crisp and bright. The appearance softens as the higher values of retardation are reached, so that starting around 1300 nanometers the colors have taken on a distinctively pastel character. In a microscope field filled with mineral grains of varying thickness, all of these colors may be present at the same time, and with practice it’s possible to recognize, for instance, whether a particular red or green crystal is displaying the color from the first band or from the second or third band on the chart. These bands running from blue through yellow to orange and then red, repeated 4 times in the retardation range from 0 to 2000 nanometers are called orders. Colors are identified as “first order gray” or “third order green”, or sometimes just “high order pastels.” There are several places in which a small change in retardation produces a dramatic shift in the color. These are referred to as sensitive tints, the most often exploited of these is the “first order red”, also referred to simply as the sensitive tint at about 580 nanometers (Rot 1 in German). 

Along the vertical border of the interference color chart the thickness of the mineral grain is plotted usually in 10 nanometer increments with the thickest grains represented at the top. In addition, lines are plotted that run across the chart diagonally beginning in the lower left corner, each one labeled with a birefringence value starting with the lowest value at the extreme left. Each line represents an increment of 0.001 in birefringence, or the difference between the highest and lowest refractive indices of the specimen. These lines are also often labeled with the names of minerals that have values of birefringence corresponding to those values, so that knowing the thickness of a mineral grain, and its color between crossed polarizers can often yield the minerals identity directly from the chart. On any given line representing some particular value of birefringence, the colors that it passes through are the colors that one would see in a collection of randomly sized grains of a mineral with that birefringence. These charts are very useful tools for the optical crystallographer. A number of them have been published. A quick search of the internet will yield several examples.

In accordance with law 2, real minerals sometimes display exceptions to the colors shown in the Michel-Levy chart. If the mineral is strongly colored, the interference colors will be altered. Even for minerals that don’t have a native color, optical dispersion can modify the interference colors that are seen. This can sometimes result in anomalous color sequences within one or another of the orders. These phenomena are rare, and usually specific to a certain type of mineral, so that when they are observed, for the person trying to identify the mineral, they’re usually good news.



CONVERGENT ILLUMINATION

Polarizing microscopes are usually equipped with a special kind of condenser that can be operated in either of two modes. So far this discussion has centered on observations made using collimated light, that is, light consisting of parallel rays that pass through the specimen as it rests on the stage. The condenser of a polarizing microscope can be adjusted to alter the rays of light so that they are no longer parallel, but converge to a point, forming a cone of light that is then used to illuminate the specimen. Observations made using this convergent mode are referred to as conoscopic observations. Observations made using the microscope in the conventional way, using parallel collimated light, are called orthoscopic observations.

Conoscopic observations are usually made using a special lens, called the bertrand lens, or they can be made using the microscope without the use of an eyepiece at all. If no eyepiece is used, it helps to use a small hand held magnifying glass, since the image is otherwise quite small. The image of interest in conoscopic observation appears in the focal plane at the back of the objective lens, looking down the microscope tube with the eyepiece removed. When properly adjusted (which can require some patience) an image can be seen that’s known as an Interference figure. It’s worth going into some detail as to how the interference figure is formed, because it can tell us a great deal about the nature of the crystal we’re viewing. 



CONOSCOPIC OBSERVATIONS

Usually we use a microscope as a device for making objects appear larger, so that the smaller details can be seen. That’s not the case when we use the microscope in conoscopic mode. Conscopic observations tell us how a crystal effects light as it passes through it from different directions. We have already learned that the direction and polarization of the light is changed depending on the orientation of the crystal in a polarized beam. Conoscopic light generates an image that tells us something about all the directions at once. Well, not all, but all the directions within the cone of light illuminating the crystal (which in the case of the usual 40x N.A 0.65 objective lens is a cone with a solid angle of about 80 degrees). Lenses with a higher numerical aperture take in a wider cone, but can be difficult to adjust since they focus very close to the specimen. When needed, a lens with a numerical aperture of 0.85 can be used to create a cone of light of 116 degrees. Directions that result in extinction (the optic axes) are evident within the interference figure as dark regions (first order black). Directions that in orthoscopic mode would show the crystal in a particular interference color show the same color in conoscopic mode. Thus the interference figure consists of a pattern of dark regions and lighter regions in various bands of color. It’s important to keep in mind that several orders of color from the interference chart can be represented in the interference figure. The higher orders are the bands at the outer regions of the figure and the lower orders are the bands near the center. It’s easy to see why this is so by realizing that the rays on the outside of the cone travel further through the crystal than those at the center. Since the pattern depends on the polarization of the light passing through each region and the orientation of the crystal, it shifts as the stage is rotated. 

OPTICAL DISPERSION

The refractive index reported for a mineral is usually the refractive index measured using a specific color of light. The refractive index is always at least slightly different for other colors. This change in refractive index as a function of the color of the light is called optical dispersion. It’s the reason why a prism cut from a piece of glass can be used to spread out white light into the rainbow of the visible spectrum. Certain kinds of minerals, such as diamond, have high optical dispersion and can spread the light wider than glass. Other minerals have optical dispersion so close to zero that a prism made from that mineral has very little ability to spread the light. Fluorite is often used in making microscope lenses because of its low optical dispersion. The optical dispersion of a mineral is characterized by measuring the refractive index at various wavelengths of light. Optical dispersion is also sometimes evident in interference figures and can provide clues that enable the crystallographer to distinguish between orthorhombic, monoclinic, and triclinic crystals. Optical dispersion is often reported in terms of its effect on the optic angle of biaxial crystals by using the abbreviations r (for red) and v (for violet). Thus, r > v would be interpreted to mean that the optic angle (2V) for red light is greater that the optic angle for violet light. 

From earlier discussion, it’s apparent that in any case in which we see two areas of an interference figure that are always dark (two melatopes), then we must be looking at a crystal that has two optic axes, in other words a biaxial crystal, one from families 3, 4, or 5. Uniaxial crystals show a cross that rotates as the stage rotates, but remains singular, corresponding to what we would expect of a uniaxial mineral. In a uniaxial figure, depending on the way the crystal is tilted, only one arm of the cross may be visible at the time. In that case, the arms of the cross can be seen to sweep across the field of view as the stage is rotated. In the case of a biaxial crystal, when the crystal is oriented so that the axis of the microscope corresponds to the acute bisectrix, with the optic plane at 45 degrees to the crosshairs of the microscope, the optic angle can be measured directly from the interference figure. The farther apart the dark regions marking the outcrop of the optic axes are, the greater the optic angle. Other optical characteristics can also be determined, using accessories that are part of the optical crystallographers bag of tricks. 

In uniaxial minerals the melatope (the areas corresponding to first order black) is always a dark cross. It may appear as a dark cross for biaxial minerals as well, at certain orientations, but the cross becomes two curves that separate along an imaginary line at 45 degrees to the crosshairs as the microscope stage is rotated. This line marks the position of the optic plane. The shape of these curves contains a lot of information. As mentioned earlier, the distance between them is proportional to the optic angle. In addition, in many cases there will be colored fringes evident on both sides of the melatopes. These colored fringes are due to the optical dispersion of the mineral along the optic axes. The distribution of the colors can help to identify the family the crystal belongs to. Crystals of the orthorhombic family possess more symmetry that do those of the monoclinic and triclinic systems. Crystals of the triclinic system possess the least symmetry of any family. The presence or absence of symmetry within the bonds comprising the crystal is reflected in the symmetry of the color fringes on either side of the melatopes of biaxial crystals. If two planes of symmetry can be imagined separating the colored fringes, the crystal is orthorhombic. If only one symmetry plane divides them, it’s monoclinic. If no symmetry is evident at all, it’s triclinic. These are always subtle clues and are sometimes impossible to observe. Fiddling with the microscope sometimes helps, and sometimes it doesn’t.

COMPENSATION 

It has been mentioned that the optical crystallographer has a variety of accessories that are used with the polarizing microscope. The quartz wedge is one of the most useful of these accessories. Others that are commonly used are the first order red plate (also called a full wave plate) and the quarter wave mica plate. All three of these accessories are designed to be used by inserting them into a slot cut into the side of the microscope tube oriented at 45 degrees to the direction of the polarizers. The full wave or sensitive plate is usually made from gypsum or quartz, and is ground to the precise thickness necessary to produce a retardation of 550 to 580 nanometers. This is the part of the Michel-Levy chart where the color changes most dramatically with a very small change in retardation. The quarter wave plate is usually made of mica and is ground so as to produce a retardation of about one fourth that of the full wave plate. The direction of one of the refractive indices is marked on the mount of the compensation plate, usually the direction of the highest refractive index, also called the slow ray. 

The quartz wedge is usually cut from the long direction of a quartz crystal to form a narrow tapering wedge about 2 inches long. The thick end of the wedge, viewed between crossed polarizers produces a shift between the fast and slow rays of anywhere from 1500 to 3000 nanometers, depending on the manufacturer. The thickness decreases gradually along the length of the wedge to zero at the other end, so that inserting the wedge into the optical path of the microscope with the thin edge first causes a continuously variable shift between the fast and slow rays beginning at zero to a maximum that depends on how far the wedge can be inserted. The direction of polarization of one of the rays, usually the slow ray, is marked on the quartz wedge holder as well as a scale that shows the retardation value of the wedge at that point. Recall that the polarization of the fast and slow rays are at right angles to one another so if the slow ray is in line with the length of the wedge, then the polarization of the fast ray will run from side to side. In use, the specimen is first rotated so that the retardation of the quartz wedge subtracts from the retardation of the specimen, which is the condition that exists when the fast ray of the wedge is aligned with the slow ray in the specimen. Then the quartz wedge is inserted until the retardation from the wedge exactly cancels the retardation from the specimen, that is, the point where the specimen goes extinct. The value of the retardation can then be read from the scale marked on the quartz wedge. Other kinds of compensators are manufactured that exploit the fact that a birefringent material can be cut and oriented so that the retardation changes as the material is rotated in the light path of the microscope. The angle of rotation is calibrated with the corresponding retardation value. Very accurate determinations of retardation can be made with such devices.

A compensation plate can be used in orthoscopic mode to determine which of the preferred directions is the fast ray (corresponding to the lower refractive index) and which is the slow ray. When the crystal is at extinction, the preferred directions are aligned with the cross hairs of the microscope. To determine which of the two directions is the fast ray, the stage is adjusted so that the extinction position is 45 degrees to the cross hairs, making one of the extinction directions parallel to the compensator slot in the microscope tube. This is the position of maximum interference, yielding the brightest colors. When the compensator is inserted into the slot, the mineral crystal will be observed to change to a different interference color. The color observed will depend on whether the retardation of the crystal adds to or subtracts from the retardation due to the compensator. If the retardation adds then the slow direction on the compensator is aligned with the slow direction in the mineral crystal. If they subtract, then the slow direction in the compensator is parallel to the fast direction in the crystal. It’s easier to do it than it is to say it. It should be obvious that if one of the other extinction positions at 90 degrees to the first had been chosen for alignment with the compensator slot, then the result would be different, that is, if the retardation added in the first case it would subtract in the second. In this way, the polarization directions in the crystal for the fast and slow rays can be identified. The full wave plate is usually used for minerals with strong birefringence. The quarter wave plate is better for minerals with weak birefringence. The quartz wedge also works well, and may be necessary for minerals with very high birefringence, where the addition of the smaller increments of the other two compensators may be difficult to recognize. Most quartz wedges are capable of adding or subtracting at least 4 orders of retardation to that of the mineral on the stage, which is pretty hard to miss even with thick grains of very high birefringence. 

The sign of elongation, mentioned earlier, is determined in this way. If the lengthwise dimension of a crystal is within 45 degrees of the preferred direction of the slow ray, then the crystal is length slow or of positive elongation. Otherwise it’s length fast (negative elongation).



MEASURING THE OPTICAL SIGN OF A MINERAL

The manner in which the colored bands change in the interference figure as the quartz wedge moves into the optical path of the microscope yields information about how the refractive indices are related to one another and to the axes of the specimen. For this reason, the quartz wedge is very valuable for the determination of the optical sign of an unknown crystal. In order to understand how we get at this information, it’s necessary to go into some detail concerning the nature of the interference figure. A uniaxial interference figure, as mentioned earlier, consists of a dark cross, called a melatope, against a background of bands of interference colors, called isochromes. Each band of colors represents one of the ordered bands of colors in the interference color chart. Colors of the lower order are in the innermost band, and those of higher orders are arranged toward the outside of the interference figure. When a quartz wedge is inserted into the path of the light moving through the microscope, The retardation due to the wedge will combine with the retardation due to the specimen in such a way as to increase it or decrease it. In fact it will do both at the same time. In one direction relative to the wedge it will add to the retardation of the crystal and in the direction perpendicular to that it will subtract. Which direction gains retardation and which one loses depends on the optical sign of the crystal. Ordinarily the wedge is marked with the direction of polarization of the slow ray. As the wedge is inserted into the microscope tube, the colored bands will be observed to move. Some will move toward the center of the figure. Others will move in the opposite direction, from the center outward. In the first case, the retardation of the wedge is adding to the retardation of the specimen. In the latter case, where the colored bands are moving outward, the retardation of the wedge is subtracting from the retardation of the specimen. It will be observed that the areas of addition are on opposite sides of the figure, likewise the areas of subtraction are on opposite sides and at right angles to the areas of addition. Each of these areas is called a quadrant. If an imaginary line is drawn between the two quadrants in which subtraction is occurring, as the wedge is inserted, and this line is perpendicular to the direction of the slow ray marked on the wedge, then the specimen is optically POSITIVE. If the line connecting the areas of subtraction is parallel to the slow direction in the wedge, then the specimen is optically NEGATIVE. 

USING THE QUARTER WAVE PLATE AND THE FULL WAVE PLATE

Depending on the type of interference figure, sometimes it’s difficult to get unambiguous results with the quartz wedge. In those cases, one of the other accessory plates, the full wave plate or the quarter wave plate, can often be used to identify the quadrants in which subtraction is occurring. To understand this, recall that in the interference figure the dark area at and adjacent to the optic axis (the melatopes) are dark because they are areas of first order black (areas of very low retardation), and the innermost white areas are first order white. If a quarter wave plate is inserted, the retardation will be shifted by about 150 nanometers in the direction that depends on whether it adds or subtracts to the retardation of the specimen. If subtraction occurs, the regions of first order white are reduced to first order black. They become first order yellow if addition occurs. so the quadrants in which subtraction has taken place are marked by two dark dots on either side of the center of the figure. If a line drawn between them is at a right angle to the slow ray in the quarter wave plate, the mineral is optically positive, otherwise it’s optically negative. Following the same logic, using a full wave plate, the quadrant of subtraction is indicated by first order yellow. Second order blue indicates addition. 

UNIVERSAL STAGES

When one begins to use a microscope to study mineral crystals, it becomes obvious fairly quickly that it’s inconvenient to rotate crystals mounted on a conventional slide. It would be useful to have some way to mount a specimen so that it could be turned to whatever position desired. A number of special stages have been manufactured to meet this need. They’re referred to as universal stages. Universal stages are based on the use of gymbals that permit the specimen to be rotated about multiple axes at 90 degrees to one another and locked for observation under the microscope. The origin of the term universal for a gymbal is unclear, but probably originated with Henry Ford, who coined the term for the use of a gymbal in an automotive drive shaft. In any case, it’s universal because it permits, in principal, all possible orientations. In practice, of course, there are limits, but the use of a universal stage is a great convenience in many cases. The problem with the universal stage is that it’s rather fragile, and costly. Much work has been done using the so called 5 axis universal stage, but its use has decreased in recent years. There’s a lot of redundancy in the conventional universal stage that is unnecessary for most work. They have been replaced by a simpler and more elegant solution, the spindle stage.




THE SPINDLE STAGE

A device that allows a small crystal to be mounted and rotated around a single axis that is parallel to the microscope stage has been the topic of a lot of interest in recent years. F. Donald Bloss, a noted Professor and author, has worked extensively with spindle stages of various designs and in 1981 published an enlightening volume entitled “The Spindle Stage: Principles and Practice.” In it he describes the construction and application of this device, useful for determining crystal parameters with great efficiency. 

The spindle stage is used in conjunction with a computer program by the clever and poetic name of EXCALIBR. This program takes, as input, a data set consisting of the extinction angles read from the microscope stage for each of 18 orientations of the crystal. These orientations are set by advancing the rotation of the spindle in 10 degree increments over the range of a half circle. A data point for each spindle position is determined by observing one setting of the microscope stage at which extinction occurs. These 18 data points enable the program to map the optic axes, the optic plane, the acute and obtuse bisectrix and report the settings of the spindle and microscope stage that can be used to orient the grain so as to enable the user to measure all of the principal refractive indices using oil immersion methods. It does all this without requiring the use of a single interference figure! That’s an especially nice feature because, though it hasn’t been emphasized in previous discussions, the truth is that it’s often difficult to impossible to get a decent interference figure, especially if you only have a small micro mount to work with. The effort can be very exasperating.

If used with a monochromator, a data set taken at each of 4 wavelengths is enough to enable EXCALIBR to characterize the optical dispersion properties of the mineral as well. Professor Bloss’ book is a wonderful contribution to the literature of the field, and greatly enhances the usefulness of the polarizing microscope, even in the hands of a beginner. Spindle stages are easy to make. EXCALIBR for windows, at the time of this writing, is available free of charge on the internet, as well as plans for spindle stages of several designs. I built one in an afternoon, using a small plastic bottle for the main structural piece. It’s otherwise similar to other published designs. Small notches filed at 10 degree intervals around the base of the bottle serve as detents to give this design the advantages of the detent type stage described by Bloss. The detents save a considerable amount of time when collecting data and improve the accuracy and repeatability of the results. 



A GLOSSARY of TERMS

Acute Optic Angle: The acute angle made by the two optic axes of a biaxial mineral.

2V: The optic angle measured within the crystal.

2E: The optic angle measured in the air surrounding the crystal.

2H: The optic angle measured in some other medium such as oil.

Obtuse optic angle: The obtuse angle between the two optic axes of a biaxial mineral.

Acute bisectrix: BXa. The line that bisects the acute optic angle. The Z axis for an optically positive biaxial mineral. The X axis for an optically negative biaxial mineral.

Obtuse bisectrix: BXo. The line that bisects the obtuse optic angle. The Z axis for an optically negative biaxial mineral. The X axis for an optically positive biaxial mineral.

Principal vibration axis: Any of the preferred vibration directions in a biaxial indicatrix..

Principal plane: Any of the three planes defined by the principal vibration axes

Principal section: See Principal Plane

Principal ellipse: The elliptical section of the indicatrix made by intersection with a principal section.

Wave normal: The line normal to the plane of the wave front.

Normal: At right angles.

Plane Of Incidence: The plane that contains the incident ray and the line normal to the surface at the point of intersection

Optic Normal: The line normal to the optic plane. The Y axis in a biaxial indicatrix.

Optic Plane: The plane of the two optic axes in a biaxial indicatrix. The XZ plane

Fast Ray: The ray with the lower refractive index.

Slow Ray: The ray with the higher refractive index.

Refractive index: The ratio of the speed of light in a transparent material to the speed of light in a vacuum.

Biaxial Negative: Possessing two optic axes with β closer to γ than to α. X is the acute bisectrix in a biaxial negative crystal.

Biaxial Positive: Possessing two optic axes with β closer to α than it is to γ. Z is the acute bisectrix in a biaxial positive crystal.

Uniaxial Negative: Having one optic axis in which ω is greater than ε.

Uniaxial Positive: Having one optic axis, in which ε is greater than ω

Uniaxial: Having one optic axis.

Biaxial : Having two optic axes.

Vibration Direction: The direction of the electric vector in a light wave. Any of the preferred directions in a crystal. 

Polarization Direction: See Vibration direction.

Ordinary Ray: The ray for which Snells Law is valid.

Extraordinary Ray: The ray for which Snell’s law is not valid.

Snell’s Law: The law that defines the direction of a refracted ray in terms of the refractive index of the two media, the angle of incidence and the angle of refraction defined mathematically as ni Sin I = nr Sin R

Birefringence: The phenomenon of Double refraction. The difference between the refractive indices of the ordinary and the extraordinary ray.

Anisotropic: Not the same in all directions

Isotropic; The same in all directions

Indicatrix: A type of 3 dimensional graph depicting the refractive index as a function of direction using a spherical polar coordinate system.

α: The lowest refractive index in a biaxial indicatrix.

β: The intermediate refractive index in a biaxial indicatrix.

γ: The highest refractive index in a biaxial indicatrix.

ε: The refractive index of the extraordinary wave in a uniaxial indicatrix

ω: The refractive index of the ordinary wave in a uniaxial indicatrix.






READING LIST

There are lots of good books on minerals and crystals. These are the best I’ve found, not in any particular order. I especially like the two by Professor Bloss.

Mineralogy; An Introduction to the Study of Minerals and Crystals by Edward Henry Kraus, Walter Fred Hunt, and Lewis Stephen Ramsdell

Dana’s Handbook of Mineralogy 

Optical Mineralogy by Paul F. Kerr

Microscopic Identification of Minerals by E. Wm. Heinrich

An Introduction to the Methods of Optical Crystallography by F Donald Bloss

The Spindle Stage: Principles and Practice by F Donald Bloss

The Polarizing Microscope by AF Hallimond

Dana’s Minerals and How to Study Them by Edward Salisbury Dana

Ore Microscopy and Ore Petrography by James R Craig and David J Vaughn

Polarized Light Microscopy by Walter C McCrone, Lucy B. McCrone, and John Gustav Delly

The Petrographic Microscope by Daniel E. Kile Supplement to the Mineralogical Record Published November-December 2003

The US Department of the Interior has from time to time published books by various authors under the title of The Microscopic Determination of The Nonopaque Minerals. These are rich sources of information and historical perspective.