A NOVEL AUTOMATED OPTICAL TECHNIQUE TO DETERMINE

THE ORIENTATION OF BIAXIAL MINERALS IN A THIN SECTION

A Thesis

Presented to

The Honors Tutorial College

Ohio University

In Partial Fulfillment

of the Requirements for Graduation

from the Honors Tutorial College

with the degree of

Bachelor of Science in Engineering Physics

by

Shawn Patrick Hurley

June 2005

ii

This thesis has been approved by

The Honors Tutorial College and the Department of Physics and Astronomy

______________________________

Dr. Larry Wilen

Professor, Physics and Astronomy

______________________________

Dr. David Ingram

Professor, Physics and Astronomy

______________________________

Dr. Ann Fidler

Dean, Honors Tutorial College

iii

ACKNOWLEDGEMENTS

I would like to thank my advisor, Ohio University (OU) Physics and

Astronomy professor Dr. Larry Wilen, for presenting me with the initial idea for this

thesis. Without his guidance I wouldn’t have been able to develop it into what it is

today. Much of the conversation I have had with him has spurred my own creativity

and allowed me to think independently. Above all, Dr. Wilen enabled me to be

intimately and responsibly involved in research, which has been the most exciting and

irreplaceable portion of my education.

Funding for this project was provided by the National Science Foundation

Office of Polar Programs.

I would like to thank two of Dr. Wilen’s previous researchers, Nathan and Dirk

Hanson, both of whom created computer programs I examined to guide my own

programming. I also would like to thank Phil Skemer and Zhenting Jiang at Yale

University Department of Geology and Geophysics for providing the olivine sample

and scanning electron microscope data used in this thesis. Furthermore, I thank Roger

Smith and Randy Mulford at the machine shop at Clippinger Laboratory for all of their

assistance.

I thank Carlos Di Prinzio, a post doctoral researcher at Ohio University, for his

friendship and company in a windowless Clippinger lab. I have gotten to know Carlos

very well and I appreciate him sharing both his enthusiastic school spirit and his

coffee with me! I also thank him for allowing me to accompany him in traveling to

the 2003 American Geophysical Union Meeting in San Francisco, CA.

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I would like to thank all the faculty and administration at the Honors Tutorial

College (HTC). I also thank my director of studies in HTC, OU Physics and

Astronomy professor Dr. David Ingram, for his advice and support.

I would also like to thank all my friends for encouraging me to succeed. I

especially thank my dear friend Tracey Hanna for reviewing my work and showing

her love and understanding.

Finally, I thank my parents for the love and encouragement I received from

them since birth. They truly are my biggest fans.

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Table of Contents

Title..................................................................................................................................i Approval ....................................................................................................................... ii Acknowledgements ..................................................................................................... iii Table of Contents ..........................................................................................................v List of Tables and Figures...........................................................................................vi Abstract...................................................................................................................... viii

1.0 INTRODUCTION...................................................................................................1

2.0 BACKGROUND .....................................................................................................5 2.1 CRYSTALLOGRAPHY BACKGROUND.......................................................................5 2.2 OPTICS BACKGROUND..........................................................................................12 2.3 OPTICAL MINERALOGY........................................................................................18

3.0 EXPERIMENTAL METHODS ..........................................................................29 3.1 EXPERIMENTAL SYSTEM ......................................................................................29 3.2 DATA COLLECTION AND PRE-ANALYSIS..............................................................36 3.3 THEORETICAL ANALYSIS METHOD ......................................................................39 3.4 ANALYSIS PROCEDURE ........................................................................................49

4.0 RESULTS AND DISCUSSION ...........................................................................62

5.0 CONCLUSIONS AND APPLICATIONS TO FUTURE RESEARCH...........83

6.0 REFERENCES......................................................................................................85

APPENDIX A..............................................................................................................88

APPENDIX B ..............................................................................................................89

APPENDIX C..............................................................................................................92

vi

List of Tables and Figures Table 1. The crystal systems and their properties...........................................................7 Table 2. The nine sample orientations. .........................................................................33 Table 3. Memory usage statistics for floating point vs. two integer bytes. ..................60 Table 4. The Results .....................................................................................................67 Table 5. Index of refraction data for olivine and OPX. ................................................76 Table 6. Final optical parameters: a, b, and c used to analyze olivine and OPX..........81 Figure 1. Defining parameters of the unit cell. ...............................................................7 Figure 2. Illustration of the grain growth process...........................................................8 Figure 3. A photo of the olivine sample that was studied.............................................10 Figure 4. Color image of the olivine sample.................................................................11 Figure 5. The electromagnetic wave as depicted by Abramowitz. ...............................13 Figure 6. Photographic demonstration of Snell’s Law. ................................................14 Figure 7. Drawing that depicts refraction at the air-glass boundary.............................15 Figure 8. Drawing showing the electric field directions in polarized and unpolarized

light. ......................................................................................................................16 Figure 9. Crossed polarizers. ........................................................................................17 Figure 10. The isotropic indicatrix. Image by Nelson. .................................................20 Figure 11. The uniaxial indicatrix. Image by Nelson. ..................................................22 Figure 12. The uniaxial negative and positive indicatrix. Image by Nelson. ...............23 Figure 13. Different cross sections in a uniaxial indicatrix. Image by Nelson.............24 Figure 14. The biaxial indicatrix. Image adapted from Nelson. ...................................25 Figure 15. Different cross sections in a biaxial indicatrix. Image by Nelson...............26 Figure 16. The Euler angle rotations.............................................................................28 Figure 17. Drawing of experimental setup. ..................................................................30 Figure 18. Photo of experimental setup. .......................................................................30 Figure 19. Color images of the nine sample orientations. ............................................33 Figure 20. Image used to calibrate the center of sample rotation. ................................35 Figure 21. Extinction curve...........................................................................................38 Figure 22. Extinction angles and contrast.....................................................................39 Figure 23. Coordinate axes convention used in the experimental setup.......................40 Figure 24. Diagram explaining rotation convention used.............................................42 Figure 25. Cross sectional ellipse showing the theoretical extinction angle. ...............47 Figure 26. Graph of ( )xxf 2sin)( 2= . ..........................................................................51 Figure 27. Depiction of 3-D Euler angle space.............................................................54 Figure 28. File structure for the Simple Grid Method. .................................................58 Figure 29. Image showing each analyzed grain’s location. Pixel resolution = 640 x

480.........................................................................................................................65 Figure 30. Euler angle plot of data from the optical method. .......................................70

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Figure 31. Euler angle plot of data from both methods side-by-side. SEM data has been adjusted to compensate for a sample mounting difference. .........................71

Figure 32. R2 vs. all Euler angles for one grain. ...........................................................73 Figure 33. R2 vs. the Euler angles theta and psi for one grain......................................74 Figure 34. R2 vs. the Euler angles theta and psi for one grain......................................74 Figure 35. Relationship between the crystal axes and the indicatrix axes in olivine....77 Figure 36. Optic sign vs. composition for olivine. .......................................................79

viii

ABSTRACT

A new computer automated optical method to determine the orientation of

biaxial mineral grains in thin sections has been developed. The system takes images

of a thin section sample between rotating crossed polarizers. The intensity of light

transmitted through each grain is used to fit a curve yielding the angle of extinction.

The procedure is repeated for nine sample positions. To specify the orientation of a

grain by the Euler angles, phi, theta, and psi, the theoretical extinction angles

corresponding to a grid of possible grain orientations is calculated and matched with

the experimental extinction angles for the nine sample positions. For comparison, a

sample of the orthorhombic mineral Olivine was analyzed with a conventional

scanning electron microscope (SEM) method. The new optical method produces

results in good agreement with the SEM method for a fraction of the cost in equipment

and software.

1

1.0 INTRODUCTION

The study of the mineral composition of rocks is a common aspect of geology,

yielding insight into the structural evolution of the earth. An important component of

this research is the study of the individual crystal orientations of mineral crystals.

Geologists and geophysicists who study the transmission of seismic (earthquake)

waves are interested in determining how the orientation of minerals in the earth’s

upper mantle affects seismic wave transmission direction and rate of travel. These

minerals are anisotropic, meaning they exhibit physical properties that depend on

direction. Because of this fact, the orientation of the mineral crystals in the upper

mantle will have an effect on the propagation of earthquake waves. The collective

“preferred” orientation of these mineral crystals, called “texture” or “fabric,” is an

average orientation of all the crystal grains over a large area. Because of the

dependence of seismic propagation upon texture (“seismic anisotropy”),

measurements of the texture of rock samples is important for interpreting the

information derived from seismic wave analysis.

In addition to seismological studies, the texture of minerals provides important

clues into the deformation and formation of rocks and rock formation. An American

Geophysical Union VHS tape that discusses properties of the earth’s mantle goes into

detail about the significance of studying the texture of minerals, as quoted here from

their web site:

These findings are of significant interest for seismologists, geodynamicists and structural geologists

2

who recognize the importance of anisotropy. The link between the crystal properties and structural features of a geological formation is evident: anisotropy in the mechanical properties of rocks influences their deformations, and the deformations themselves induce the texture that is the source of anisotropy in the properties (Wenk).

The mineral olivine is the most abundant mineral in the earth’s upper mantle.

Understanding what causes texture to develop in rocks in general and specifically in

olivine is an important area of research. Karato describes how the process of

dislocation deformation reorients olivine crystals, “...the shape of a crystal can change

only in a certain manner. Thus, by rotating its orientation, a crystal tries to match the

boundary condition imposed by its surrounding materials” (109). He also adds,

“Simply put, crystals tend to align to match the microscopic deformation by

dislocation motion with the imposed macroscopic deformation geometry” (109-110).

Therefore, anisotropy is sometimes influenced by the deformation processes in the

mantle. Olivine has the strongest elastic anisotropy, so its texture has the largest effect

on the anisotropic structure of the upper mantle (Karato 110). This makes the study of

the texture of olivine important to researchers investigating seismic waves as well.

A prime example of one scientist’s study of seismic anisotropy is Dr. Furlong

at Penn State University who researched the mantle under Tibet. Dr. Furlong’s

research, as documented on a Penn State web site, aimed to study “the role of plate

tectonics in the evolution of continents.” His research looked at the recordings of

seismic events in one region of Tibet and he found a systematic pattern of mantle

deformation. Dr. Furlong further stated, “Theoretically, the olivine fabric should

3

relate to the forces that existed during collision and the actual deformation of the

mantle rock.” He says that better models of continental collisions can be made once

more is known about the deformations in the mantle (Furlong).

Because of the importance of studying the deformation processes in the

mantle, it becomes necessary to design experimental techniques for determining the

texture of olivine and other biaxial minerals. One tool already available for studying

texture is the scanning electron microscope (SEM). This proven method finds the

orientations of the grains in a polycrystalline olivine sample by using electron

backscattering. However, there is a need for a less expensive, yet still reliable

technique to find the orientation of biaxial minerals. Although manual optical

techniques exist for this type of analysis, they are typically laborious and time

consuming to perform (Wilen). Therefore, this thesis introduces an automated optical

method to satisfy the need. My research verified the validity of the optical method by

obtaining the orientations of grains in a sample of olivine. The accuracy of the optical

method was verified by a direct comparison of the data with results obtained using a

SEM.

The optical technique has its advantages over the SEM method. The optical

method can be setup for 1/100th the cost of a SEM. Also, the new optical method uses

images of the grains taken with cross polarized visible light. These images show sharp

contrast along grain boundaries due to differing orientations between grains. This

enables the researcher that uses the optical method to calculate statistics concerning

grain size, deformation, and nearest neighbor correlations using existing software.

4

The SEM method is not as well suited to see individual grain boundaries, which limits

the automation of such additional analyses (Wilen).

Over the next several chapters, I will describe the details of the optical

technique I developed. To the best of my knowledge, this is a unique method because

it uses polarized light to the find the orientation of biaxial crystal grains in a mineral

thin section.

5

2.0 BACKGROUND

2.1 Crystallography Background

“A crystal is a solid body bounded by natural planar surfaces, generally called

crystal faces, that are the external expression of a regular internal arrangement of its

constituent atoms or ions” (Berry 10). Designing experiments involving biaxial

crystals requires an understanding of the basics of crystallography, optics, and optical

mineralogy. I will use Berry’s definition of the crystal as a starting point in this

chapter, which is dedicated to introducing those principles necessary to understand my

research.

Crystallography is the study of the characteristics and evolution of rocks as

well as the internal structure of all crystalline substances. Crystalline substances are

those materials that have a regular three dimensional arrangement of constituent

atoms, where each atom is bonded to its nearest-neighbor, but may or may not be

bounded by crystal faces. Therefore, minerals are crystalline solids by definition

(Berry 10-12, Callister 32).

Crystallography is a practically important area of study. This is because

crystal structure, along with chemical composition, determines all the physical

properties of crystalline materials. These, in turn, determine the uses of crystalline

materials (Berry 14).

6

Crystal Systems

All minerals are classified into crystal systems according to their crystal

structure. Crystal structure is “the manner in which atoms, ions, or molecules are

spatially arranged” (Callister 32). Each crystal system is based upon the symmetry of

the smallest repeatable arrangement of atoms in the crystal structure, called the unit

cell. The unit cell is the basic building block defining the crystal structure through its

geometry and the position of the atoms within it. Each crystal system’s unit cell is

modeled by a unique crystal lattice, a “three-dimensional array of points coinciding

with atom positions” (Callister 33).

The specific geometry of each crystal system is defined using an x, y, and z

coordinate system (not necessarily orthonormal) such that the origin is situated in the

corner of the unit cell and each axis coincides with one of the edges of the unit cell

extending from this corner. Each crystal system is defined by 6 lattice parameters,

which are a, b, c, α, β, and γ. The parameters a, b and c are the lengths of the edges

along the x, y, and z axes respectively, where α, β, and γ are the interaxial angles as

shown in Figure 1 below adapted from Callister (39).

7

Figure 1. Defining parameters of the unit cell.

Table 1 lists the seven possible crystal systems and some characteristics

associated with each, which will be explained later.

Table 1. The crystal systems and their properties. Crystal System Axial

Relationships Interaxial Angles Effect on

Light Optical

Classification Indicatrix

Shape Isometric (Cubic) a = b = c o90=== γβα Isotropic Isotropic Sphere

Hexagonal a = b ≠ c

o

o

120

,90

=

==

γ

βα

Anisotropic Uniaxial Spheroid

Tetragonal a = b ≠ c o90=== γβα Anisotropic Uniaxial Spheroid

Rhombohedral a = b = c o90≠== γβα Anisotropic Uniaxial Spheroid

Orthorhomic a ≠ b ≠ c o90=== γβα Anisotropic Biaxial Ellipsoid

Monoclinic a ≠ b ≠ c βγα ≠== o90 Anisotropic Biaxial Ellipsoid

Triclinic a ≠ b ≠ c o90≠≠≠ γβα Anisotropic Biaxial Ellipsoid

Adapted from Callister (40). More than 50% of minerals exist in the orthorhombic and monoclinic systems. This is

because the most abundant elements in the earth’s crust: silicon (Si), oxygen (O),

magnesium (Mg), iron (Fe), aluminum (Al), and calcium (Ca) usually form

compounds in those crystal systems (Sen 39). For example, the orthorhombic mineral

z

a

b

c y

x

α β γ

8

olivine consists of two iron or magnesium atoms bonded to a silicate compound of a

silicon atom and four oxygen atoms.

Polycrystalline Thin Section The experimental setup for this research utilizes polished thin sections. The

thin sections that are typically studied are made from natural polycrystalline rock

samples. This means the crystalline solid is composed of many small crystal grains,

each with their own specific crystallographic orientation within the sample. The grain

growth process begins with the solidification of small crystals, which are typically

randomly oriented. As they grow to completion, irregular grain shapes form from the

atomic mismatch at grain boundaries. Sometimes there develops a preferred

crystallographic direction, meaning a tendency for crystals to align their orientations

in one direction. The case of preferred orientation among crystal grains is called

texture (Callister 54-56).

A diagram demonstrating the grain growth process as adapted from Callister

(55) is depicted in Figure 2, below.

Figure 2. Illustration of the grain growth process.

9

It is important to note that the grains themselves are three dimensional, unlike the two

dimensional depiction above. The thin section, however, gives the researcher a two

dimensional cross section of the grains in the sample.

I now will briefly describe how a thin section is typically prepared using five

main steps. First, the sample is cut using a diamond saw. Then, the sample is

mounted on the glass slide. Then, the surface is ground flat by using a carborundum

grit and water until the section is ~30 µm thick. The thickness is estimated by

comparing the observed colors of the grains to the colors expected from minerals in

the section. Then the surface is polished, usually with diamond grit plus oil as a

lubricant. Finally, the surface is buffed by using a gamma alumina powder and water

for lubrication (Gribble 32-33).

The two images shown in Figure 3 and Figure 4 are of the thin section of

olivine that was used in the experiment. The first is a photo of olivine mounted in the

sample stage of the system. Note how it looks in ordinary unpolarized light (the

principle of polarization will be described later). The second image is of olivine seen

between crossed polarizers through the camera of the experimental setup. Although

the camera is black and white, the color image was created by combining 3 images

taken under red, green and blue filtered light.

10

Olivine Sample

Figure 3. A photo of the olivine sample that was studied.

11

Figure 4. Color image of the olivine sample.

1 mm

12

2.2 Optics Background

The study of the behavior of light is called optics. I used optical procedures

and principles to guide the design of the experimental setup and analysis. In order to

explain the apparatus used in data collection and the theory behind the data analysis, it

is necessary to describe the most relevant concepts in optics and how they pertain to

this experiment.

Light as an Electromagnetic Wave In the context of the experiment, I will treat light classically as a form of

electromagnetic radiation which can be described by the propagation of a wave

carrying the energy of the radiation. This radiation is a wave that consists of two

perpendicular components, the electric field vector ( Er

) and the magnetic field vector

( Br

). The Er

and Br

vibrate in a sinusoidal manner perpendicular to the direction the

wave propagates (Figure 5Figure 5. The electromagnetic wave as depicted by

Abramowitz). This defines light as a transverse wave, because the fields vibrate

perpendicular to the direction of travel. Ultimately, it is the interaction of the electric

field component of light with the electrically sensitive atoms in the crystal lattice of a

mineral that affects the behavior of light (Nesse 1). It is this basic phenomenon that

makes the experiment possible.

13

Figure 5. The electromagnetic wave as depicted by Abramowitz.

Electromagnetic radiation is characterized by its wavelength as shown in

Figure 5. The wavelength is the distance between successive peaks or crests of the

wave and for visible light it is best measured in nanometers. The frequency of light

describes the number of sinusoidal cycles of the wave that pass a point in space per

second and is measured in Hertz (Hz). Wavelength and frequency are characteristics

that are directly connected by the velocity of the wave. The speed of light, c,

illustrates this relationship as:

fc λ= (1) where lambda is the wavelength and f is the frequency. This relationship describing

the velocity of waves was derived by Newton in the year 1687 (Hecht 16).

So far, I have described the characteristics of light using a picture of single

wave. In practice, light can be depicted as a multitude of waves traveling as a single

unit. The wave front is defined as the plane connecting similar portions of the waves,

such as the crests. The wave normal is a vector that is perpendicular to the wave front

and it points in the direction the wave front is traveling (Nesse 3).

14

Refraction, Index of Refraction, and Snell’s Law As light encounters a boundary between two media it will become reflected,

refracted, or absorbed in some combination: such as partially, selectively, or totally.

Light will always reflect at an angle equal to the angle of incidence, due to the law of

reflection. In addition, if light transmits into another medium, it will undergo

refraction. Refraction is a “bending” of the light ray that occurs because the light

changes its speed. The change in speed causes the ray to change direction at the

boundary between the media (Sen 70). A familiar example of refraction is the pencil

sitting in a glass of water, as seen in Figure 6.

Figure 6. Photographic demonstration of Snell’s Law.

The speed of light in a vacuum, c, is a constant value of approximately 3 x 108

m/s. The refractive index, n, is defined as the ratio of the speed of light in vacuum to

the speed in the medium:

vcn = (2)

15

The refractive index is also simply related to the change in direction that occurs at the

boundary in conjunction with the angle of incidence and the angle of transmittance.

This relationship is known as Snell’s Law, named after its discoverer:

ttniin θθ sinsin = (3)

The refractive index of air can be taken to be approximately 1.00 and many common

minerals have values that range from 1.43 to 3.22 (Shelley 21-22).

Figure 7 is a pictorial demonstration of reflection and refraction with all the

components of the equation for Snell’s Law labeled.

Figure 7. Drawing that depicts refraction at the air-glass boundary.

Polarization Using the model of the electromagnetic wave previously described, I will

explain the nature of polarized light. I am limiting the discussion to only include

linearly polarized light since that is the only type of polarized light relevant to the

experiment. As an example, sunlight is naturally unpolarized, meaning the electric

θt

air ni

glass nt

θi θi

16

field component is randomly oriented in any direction perpendicular to the direction of

propagation. Linearly polarized light has only a single direction in which the electric

field vibrates, (Figure 8).

Ordinary Unpolarized Light Plane Polarized Light

Figure 8. Drawing showing the electric field directions in polarized and unpolarized light. A polarizer is a substance used to create polarized light from unpolarized light.

My experiments use a very common type of polarizer composed of a stretched film of

polyvinyl alcohol impregnated with iodine. This type of polarizer is called Polaroid

and was invented by Edwin Herbert Land (Hecht 335). Polaroid works through

selective absorption, thereby only transmitting light whose electric field is vibrating in

the preferred direction (called the transmission axis).

Another common process to produce linearly polarized light is reflection. At a

special angle of incidence, Brewster’s angle, only the electric field component

polarized parallel to the surface will be reflected. This angle is related to refractive

indices by Brewster’s Law:

i

tp n

n=θtan (4)

Er

Propagation Direction

Er

Propagation Direction

17

named after Sir David Brewster (1781-1868) (Hecht 348). This principle creates a

polarized glare from reflecting surfaces, which can be preferentially reduced by

Polaroid sunglasses. I use Brewster’s Law to calibrate the transmission axis of the

polarizer for the experiment and this procedure is described in detail in the

experimental methods section.

Cross Polarized Light When two linear polarizers are superimposed on each other, as shown in

Figure 9, such that their transmission axes are perpendicular, they are considered to be

crossed. If no polarizing material exists between them, then extinction occurs and no

light will be able to pass through. Therefore, the two crossed polarizers in Figure 9

would appear extinct (black) to the observer in the overlapping region and show some

lowered intensity otherwise (Shelley 20).

Figure 9. Crossed polarizers.

When a mineral sample is placed between crossed polarizers it is said to be

viewed under cross polarized light. The appearance of the mineral depends upon its

crystal symmetry and orientation. A useful theoretical tool called the optical indicatrix

18

describes the way light acts within a given material. Understanding the indicatrix and

how it relates to the experiment was a critical step in my thesis.

2.3 Optical Mineralogy The Optical Indicatrix According to Nesse, the optical indicatrix is “a geometric figure that shows the

index of refraction and vibration direction for light passing in any direction though a

material” (34). It is constructed by drawing vectors whose lengths are proportional to

the index of refraction of light that is vibrating parallel to the vector (Nesse 35). In

general, this geometrical representation is an ellipsoid. The indicatrix is the

theoretical foundation upon which the experimental procedures and calculations of

this experiment are based.

Once the indicatrix is constructed and aligned with the material, it completely

determines how light will behave when incident upon any transparent material

according to the following rules:

1. A plane perpendicular to a ray of light traveling in an arbitrary direction will

intersect the indicatrix ellipsoid (through its center) in an ellipse. The light ray

then splits into 2 rays, one with a polarization aligned parallel to the semi-

major axis of the cross sectional ellipse and the other with a polarization

aligned parallel to the semi-minor axis of the cross sectional ellipse.

2. The ray polarized along the semi-major ellipse axis travels with a velocity of

1nc , where 1n is the length of the semi-major axis. The ray polarized along the

19

semi-minor ellipse axis travels with a velocity of 2n

c , where 2n is the length of

the semi-major axis.

3. Because the two light rays have different indices of refraction, if the light ray is

incident on the material with an angle ≠ 0, then the two rays will physically

split.

4. If the intersection of the indicatrix is a circle, then the polarization of the

incident light is not affected. The direction normal to the circular section in the

indicatrix is called the optic axis.

5. The difference between the maximum and minimum indices of refraction used

to construct the indicatrix define the birefringence of the material.

6. If light incident on the material is already polarized along the direction of the

semi-major or semi-minor axes of the ellipse, then its polarization is unaffected

by the material.

I compiled these rules from the Neese (35), Hecht (336-337), and Shelley (21-23).

According to Table 1, there are three types of optical classifications in minerals and

each one has a different indicatrix geometry. These are based upon the crystal

structure of the atoms, as defined and classified by the crystal system to which the

mineral belongs. The indicatrix makes a progression of increasing complexity from

isotropic to uniaxial to biaxial. Therefore, I will introduce each specific indicatrix in

this order to further aid the understanding of its role in optical mineralogy and the

theoretical framework of this experiment.

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Isotropic Minerals The light passing through an isotropic mineral of any random orientation will

have the same velocity. On an atomic scale, this is due to a highly symmetric atomic

arrangement, only possible in the isometric (cubic) crystal structure. Other non-

crystalline materials that are isotropic include gases, liquids, and glasses (Nesse 34).

Because the velocity remains constant for all directions, the index of refraction also

remains constant in all directions. Therefore, the isotropic indicatrix is the simplest

indicatrix, a sphere (Figure 10).

Figure 10. The isotropic indicatrix. Image by Nelson.

In the example in Figure 10, the index of refraction is 1.540 for all crystal orientations

because the intersection with the indicatrix is always a circle. Also, the polarization

state of light is not affected, so when any isotropic mineral is viewed between crossed

polarizers, it will appear extinct (Nesse 34-35).

Optic Axis

As previously defined, the optic axis is the direction through any crystal where

the intersection with the indicatrix is a circle. This means that rays entering parallel to

21

the optic axis will not incur double refraction (being split into two rays) or have their

polarization state altered. Therefore, a crystal viewed along its optic axis between

crossed polarizers will show extinction for all rotations of the polarizers, as long as the

polarizers remain crossed. For isotropic minerals, every direction in the crystal is an

optic axis (Nesse 53).

Uniaxial Minerals

The description of uniaxial minerals will begin with a historical introduction to

double refraction. Anisotropic minerals were observed empirically to refract light

with two different indices of refraction. One such mineral, calcite, is commonly used

to illustrate the phenomenon of double refraction. If a calcite rhomb is placed over a

black dot on a piece of paper, then two images of the dot are observed. When the

calcite is rotated, one dot stays fixed while the other dot rotates with the crystal. The

image of the stationary dot behaves as would be expected from an isotropic mineral,

which is according to Snell’s Law, so therefore it was termed the ordinary ray (ω).

Because the image of the rotating dot behaves so differently and does not appear to

obey Snell’s Law, it was called the extraordinary ray (ε) (Nesse 53).

Minerals in the hexagonal and tetragonal crystal systems have only one optic

axis and are called uniaxial minerals. In uniaxial minerals, the relationship between

the indicatrix and the crystallographic axes is such that the optic axis in the indicatrix

and the c crystallographic axis (c-axis) coincide. This makes the indicatrix very useful

in relating the optical and physical properties of uniaxial minerals. Figure 11 is a

representation of the uniaxial indicatrix.

22

Figure 11. The uniaxial indicatrix. Image by Nelson.

As previously defined, the birefringence, δ, is a measure of the difference

between the (higher) index of refraction of the slow ray and the (lower) index of

refraction of the fast ray. Because changing the orientation of an anisotropic crystal

usually changes the birefringence, it is commonly listed as the maximum value for a

given mineral (Nesse 40).

In uniaxial minerals, the ω ray has a fixed index of refraction, whereas the

ε ray’s index of refraction can vary according to the orientation of the mineral.

Because it must always be higher or lower than the index of the ω ray within the same

mineral, uniaxial minerals are classified as optically positive or negative. In positive

minerals, the ε ray is slower because its index of refraction is higher. In negative

minerals, the ε ray is faster because its index of refraction is lower (Nesse 53). Figure

12 shows both classifications of uniaxial minerals (remember that for uniaxial

minerals, the optic axis is the same as the c-axis).

23

Figure 12. The uniaxial negative and positive indicatrix. Image by Nelson.

In addition to having different indices of refraction, the ω ray and ε ray were

shown to be polarized at right angles to each other by Fresnel and Arago in the year

1811 consistent with rule number 1. The ω ray always vibrates in a plane at right

angles to the ω ray path and the c-axis of the mineral. The ε ray always vibrates

within this plane containing the c-axis (Bloss 71).

Three intersections of the uniaxial indicatrix are shown in Figure 13. A section

containing the optic axis is called a principal section and light transmitting through at

this orientation will show maximum birefringence. The permitted vibration directions

are aligned with the semi-major and semi-minor axes of the principal section ellipse.

Light propagating parallel to the c-axis will make a circular section in the indicatrix.

In this situation, the mineral will behave as though it were traveling in an isotropic

mineral because there is zero birefringence. For light propagating in a random

direction, the section will be elliptical and again the permitted vibration directions are

aligned with the semi-major and semi-minor axes of the cross sectional ellipse (Neese

56).

24

Figure 13. Different cross sections in a uniaxial indicatrix. Image by Nelson.

Biaxial Minerals

Biaxial minerals have two optic axes and are classified in the orthorhombic,

monoclinic, and triclinic crystal systems. Biaxial crystals have the least symmetry

since they vary in crystal structure and chemical bonding in all directions. Therefore,

the relationship between the indicatrix axes and the crystallographic axes varies from

mineral to mineral. However, if this relationship is known it makes the indicatrix very

useful in relating the optical and physical properties of biaxial minerals. Figure 14 is a

representation of the biaxial indicatrix.

25

Figure 14. The biaxial indicatrix. Image adapted from Nelson.

As shown in Figure 14, three principal indices of refraction are needed to

construct the indicatrix. These values are α, β, and γ, which are not related or to be

confused with the interaxial angles in crystallography. (From now on in this text,

α, β, and γ, will refer to the indices of refraction in biaxial minerals). These indices of

refraction are defined such that γ > β > α is always true (Nesse 76-77).

The defining features of the biaxial indicatrix are depicted in Figure 15. The

biaxial indicatrix has two circular sections as depicted in Figure 15. The two lines

perpendicular to one or the other circular sections are the optic axes. The angle that

separates the two optic axes is called 2Vα or 2Vγ according to which principal

refractive index bisects it. 2Vα + 2Vγ = 180°, so unless both angles are 90° they will

not be equal. It is interesting to note, in the limits that either 2V angle becomes 0° or

180°, the two circular sections and optic axes converge into one and β = α or β = γ.

Therefore, a uniaxial crystal can be thought of as a special case of the biaxial crystal

(Nesse 76-78).

26

Figure 15. Different cross sections in a biaxial indicatrix. Image by Nelson.

As in uniaxial minerals, the permitted vibration directions in a biaxial crystal

are aligned with the semi-major and semi-minor axes of the elliptical cross section

from the incident wavefront. Light propagating parallel to either optic axis will

“slice” through a circular section in the indicatrix and will behave as though it were

traveling in an isotropic mineral because there is zero birefringence. To avoid

confusion, light in a biaxial mineral is still split into two rays despite needing three

indices of refraction to construct the indicatrix. Of these two rays, one is the fast ray

and one is the slow ray; however, both of these rays are now extraordinary. Biaxial

minerals are also classified as either biaxial positive or biaxial negative. An optically

positive mineral means that 2Vγ < 90° and an optically negative mineral means that

2Vγ > 90°. If the 2Vγ = 180°, then the mineral is optically neutral (Nesse 76-78).

Extinction

Throughout the course of this paper, the term extinction is used to mean that no

light is transmitted when observing a mineral under cross polarized light. Consider a

27

mineral sample observed between crossed polarizers that are rotated through 360

degrees. When an isotropic mineral is viewed in this manner, it will always be extinct.

As discussed above, the polarization state of the light wave is unaltered by the

mineral, regardless of the mineral’s orientation. So, the light is polarized by the first

polarizer and extinguished by the second (perpendicular to first) polarizer. If the

mineral viewed is anisotropic, it will generally show extinction four times as the

polarizers are rotated 360 degrees. The angles at which this extinction occurs are

called the extinction angles and they are spaced ninety degrees apart. The extinction

occurs because each of these four positions correspond to an alignment of the

transmission axis of the polarizer and the characteristic polarization of one of the light

rays transmitted through the mineral (Shelley 23).

A geometric condition for extinction is established if one knows the orientation

of the indicatrix with respect to the light propagation direction. The angle of

extinction can be determined by constructing the ellipse of intersection between the

indicatrix and the plane perpendicular to the propagation direction. If the incident

light is polarized parallel to the semi-major and semi-minor axes of this ellipse, it will

propagate with no change in polarization. Therefore, if the crystal is viewed between

crossed polarizers it will appear extinct if either of the polarizers is aligned with either

ellipse axis (Wilen).

The Orientation of Biaxial Minerals The orientation of biaxial minerals is commonly specified using three angles of

rotation. In my experiment, the three angles used are the Euler orientation angles, phi

28

(φ), theta (θ), and psi (ψ). These angles of rotation are expressed explicitly as rotation

matrices in the section of this paper that outlines the theoretical method. For

simplicity, they are visually depicted in Figure 16.

Figure 16. The Euler angle rotations.

I adopted the Euler angle convention described in a classical dynamics

textbook (Marion 431-433) because it was convenient to express the orientation in the

same three Euler angles as the data obtained from other methods, such as the SEM at

Yale University. This later facilitated a direct comparison of the two methods of

finding the orientation of biaxial minerals, which is detailed in the results section of

this paper.

φ

y

θ

ψ x

z z”

x’

29

3.0 EXPERIMENTAL METHODS

3.1 Experimental System The experimental system for taking images of crystals under thin section was

designed by Dr. Larry Wilen. During data collection and analysis, the system was

kept at Clippinger Laboratory, room 169, where I used it to collect data for my thesis.

My work on the system included writing a software program to collect data and

performing a system calibration. In this section, I will describe the parts of the system

and their functions, define the sample orientation, and explain some of the calibration

procedures.

Components The system consists of the following components mounted on an optical

bench: (A) a black and white CCD (charge coupled device) video camera, (B) a

microscope lens assembly with adjustable zoom and focus, (C) two crossed linear

polarizers mounted on rotation stages, (D) a sample rotation stage, (E) the table

rotation stage, and (F) a diffuse white light source. The system also includes a stage

controller unit and a lab computer (DELL Optiplex GX240, Pentium 4 - 1.7Ghz,

1024MB of RAM) equipped with a video capture card and a computer program that

automates the data collection procedure. Figure 17 is a drawing showing each part’s

location on the optical bench and Figure 18 is a photo of the experimental setup.

30

Figure 17. Drawing of experimental setup.

Figure 18. Photo of experimental setup.

31

I will now explain the details of each part in the system. Light from a white

light source is used to image the crystals. It travels through the first polarizer, then

through the sample, then through the second polarizer, and finally into the camera.

The light source has an adjustable intensity to obtain maximum contrast in the image.

If the intensity is too high, the sample washes out and some light bleeds across grain

boundaries. If the intensity is too low, there will not be enough contrast. This in turn

may cause errors in the experimental extinction angle.

Each of the two polarizers is set in a rotation stage. The stages are set to rotate

in the same direction. The first polarizer, between the light source and the sample, is

set with its transmission axis horizontal and the second polarizer, between the sample

and the camera, is set with its transmission axis vertical. When the sample is not

present in the system, no light will reach the camera because the polarizers are

crossed.

The system is used to image crystal samples in thin sections. Because the

sample is so thin (~30 µm), it is possible to consider the crystal section as a 2-D

sample. There are two rotation stages that control the sample’s position. The table

stage is mounted horizontally so its rotation is akin to a record player. In other words

the rotation occurs in the x-z plane and its rotation angle is called ξ (xi). Mounted on

the table stage is the sample rotation stage. Its rotation is in the x-y plane and its angle

is called φinner (phi inner). The origin of the coordinate system is at the intersection of

the axes of rotation of the sample and table stages. In the home position, both the

table stage and the sample stage are set to zero degrees. This puts the sample stage in

32

a plane parallel to the two polarizers, meaning its axis of rotation is co-linear with the

axis of rotation of the two polarizer stages.

The black and white CCD camera is used to view the sample. It has an

adjustable zoom and focus. These must be adjusted by hand and are set according to

the needs of the experiment. The camera is secured to the optical bench with the axis

of the optical lens system aligned with the axes of rotation of the two polarization

stages. Also, the center of the camera image is aligned with the center of rotation of

the sample as best as possible. Although the camera feeds video to the computer, I use

single image samples of that feed, also known as snapping an image. The image

resolution is 640 by 480 pixels in 8-bit black and white. This gives 256 levels of

intensity. However, during sample installation and testing, it is helpful to watch the

live video feed to be able to make adjusting the image easier and faster.

There is a cover placed over the system to prevent ambient light from spoiling

the images and to protect against excessive dust buildup. A servo controller by

Newport, model MM3000, is used to move the stages with a one thousandth of a

degree precision. The controller can be used to move the stages manually via the front

panel controls, but it is interfaced to a computer via a serial connection to be run

autonomously.

Sample Orientation

There are a total of nine sample orientation sequences, indexed by n. A

sequence of images is taken at each sample orientation. Each sequence is denoted in

the form ξsφinner, or (xi) s (phi inner), where the letter “s” separates the two sample

33

orientation angles in degrees. Table 2 shows the nine sequences used. Figure 19

shows the color images of the 9 sample orientations.

Table 2. The nine sample orientations. n Sequence Xi Phi Inner 1 0s0 0 02 45s0 45 03 45s45 45 454 45s90 45 905 45s135 45 1356 45s180 45 1807 45s225 45 2258 45s270 45 2709 45s315 45 315

Figure 19. Color images of the nine sample orientations.

34

Calibration As with any experimental system, some calibration procedures must be made

to evaluate the consistency and correctness of the data. In this system, the first

polarizer has its transmission axis aligned horizontally by using the principle of

Brewster’s Law, which I described earlier. The procedure involves a laser, a glass

slide, an adjustable linear polarizer, a mirror, a photodetector, and a voltmeter. The

parts are setup on another optical bench in the lab. Brewster’s Law enables me to

polarizer the laser light horizontally using the glass slide. The polarized laser then

shines through the polarizer to be calibrated, is reflected off the mirror, and is detected

by the photodetector. This is hooked up to a voltmeter which displays a voltage

corresponding to the light intensity it detects. With the room lights off, I iterated small

adjustments of the polarizer and the incident angle on the glass slide. When

Brewster’s angle is reached at the glass slide, the light will be completely polarized

horizontally. The laser beam can be extinguished by setting the adjustable polarizer to

transmit vertically. (The mirror is used to keep the laser beam aimed directly into the

photodetector since it would become misaligned as the incident angle on the glass

slide changes).

The next step is to place this vertically calibrated polarizer into the system with

no sample present. Once it is installed into the system, a computer program rotates the

two polarizers of the experimental system independently until maximum transmission

intensity is achieved. Then they are all aligned vertically. Now polarizer #1 is rotated

until the light intensity is as close to zero as possible. This will set polarizer #1 to be

35

transmitting horizontally (parallel to the x-axis of the lab reference frame). Then I

remove the calibrated polarizer I just added and make sure the second (system)

polarizer crosses with the first, meaning it is aligned vertically.

The pre-analysis routine, to be described later, needs the location in the image

(pixel coordinates) of the center of rotation of the stage corresponding to φinner. After

camera and polarizer alignment adjustments are complete, but before taking data, a

calibration must be made to determine this center of rotation. First, an opaque disc

with a centered pin hole is placed in the sample stage and the polarizers are uncrossed

by rotating the second polarizer to horizontal. Then, φinner is rotated through 360

degrees while many images of the pin spot are taken. One image is created by

summing the individual images and the centroid of the spot of light in the image is

found. The pixel location of this centroid is used as the center of rotation for all

calculations done in the pre-analysis routine. Figure 20 is one such image taken on

July 3, 2003.

Figure 20. Image used to calibrate the center of sample rotation.

36

Image pixels are indexed with the origin (0, 0) located in the upper left hand corner.

The x pixel location increases left to right from 0 to 639 and the y pixel location

increases from top to bottom 0 to 479. The center of the image is therefore (319.5,

239.5). For the image above, the center of rotation is calculated to be at x = 340.65

and y = 250.88. The major difference between the center of the image and the actual

center of rotation demonstrates the importance of this calibration.

3.2 Data Collection and Pre-Analysis During experimental data collection, the images from all nine sample

orientations are compiled. I have written a computer program in Labview to automate

steps 4 through 8 in the data collection procedure. The steps are as follows:

1. Remove the protective cover and install a sample into the sample stage. 2. Adjust the focus, zoom, and light intensity if necessary while monitoring the

camera’s view. 3. Replace the protective cover. 4. Set the sample to the home position, 0s0. 5. Rotate the polarizers through 95 degrees in 5 degree increments. An image is

taken at each step, which is an average of 5 video frames. These are grouped and saved as one file consisting of all 20 images.

6. Rotate the sample to the next orientation. 7. Repeat steps 5 and 6 until all 9 sequences are finished. 8. Return the sample and the polarizers to the home position.

Once a sample is in place it takes about twenty minutes for an entire data set to be

taken. A data file is saved for each sequence in a folder named: “(time of data

collection) (name of sample)” which is saved in a folder named: “(date of data

collection)” within the main directory “C:\Geo Data”. In this manner, different runs of

37

the same sample are separated by the date and time. Here is an example of the

directory structure of one of the data folders for olivine:

“C:\Geo Data\2003-11-04-Tue\12.38 Olivine MG”.

Pre-Analysis

The pre-analysis routine is an existing algorithm that is used for the uniaxial

analysis of ice crystals. It was developed and programmed by Dirk Hansen and Larry

Wilen. I have kept this routine essentially intact, implementing only a few

modifications for faster processing. The pre-analysis routine calculates the nine

experimental extinction angles, nexpε , of each grain being tested in a sample. The

procedure to find the nexpε completes the following steps: The intensity of a three by

three region of pixels in the center of the grain is averaged for each image in a

sequence. This intensity is a number from 0 to 255, because the image has 8-bit

grayscale resolution. The 20 data points from a given sequence are used to plot a

curve of intensity vs. polarization angle (with respect to the x-axis). An analytic

function is fit to these data points. The polarizer angle corresponding to the minimum

of this function is the experimental extinction angle, nexpε . The process is repeated for

each of the sequences n = 1 to 9 (and for each grain specified). At the end of pre-

analysis, the results are saved in a file within a folder named after the date and time

the results were calculated. This folder is placed is a directory within “C:\Geo

Results” that includes folders for each set of data analyzed. This organizes the results

first by the data it analyzed and then by the time it was analyzed. Here is an example

38

of the directory structure of one of the results folders for olivine: “C:\Geo

Results\2003-11-04-Tue\12.38 Olivine MG\2005-04-06-Wed, 16.45”.

The pre-analysis routine must be able to determine each grain’s position in the

image after the sample changes orientation. The software uses a mapping algorithm to

ensure it continues to analyze the same three by three region of pixels in the specified

grain. One critical parameter for this mapping algorithm is the center of rotation of

the sample stage, which was determined during the calibration procedure previously

described.

The following images show an example of the pre-analysis results from an

actual olivine grain that was analyzed. The first image, Figure 21, depicts the

extinction curve for the 0s0 sequence. The data points are graphed in white and the

fitted curve is graphed in green. It produces a reliable extinction angle of 30.8

degrees. Note that it has a high contrast, which is the maximum intensity minus the

minimum intensity of the grain.

Figure 21. Extinction curve.

The extinction angles and contrast for each sequence for one particular grain are

indicated in Figure 22.

39

Figure 22. Extinction angles and contrast.

Sequences with contrast lower than 45 may lead to some error in the extinction angle

that is calculated from the fit. Therefore, there is an option to exclude those extinction

angles from the full analysis.

3.3 Theoretical Analysis Method

After the images of a sample have been taken and each grain’s nine extinction

angles are determined experimentally during pre-analysis, the known optical

parameters found in the literature are used to determine the nine theoretical extinction

angles of each grain for many possible combinations of Euler angles. Then, these

values can be compared to the experimental values to find a close match for some

unique set of Euler angles which specify the orientation of the crystal. The details of

this procedure are discussed next.

First, I will describe how to get the nthε (theoretical extinction angle) from a

grain’s crystal orientation φ, θ, and ψ (Euler angles) and a given sample orientation,

(indexed by n). I use a coordinate system where the z-axis is positive towards the

40

camera, the x-axis is positive towards due east, and the y-axis is positive towards due

north, as shown in Figure 23.

Figure 23. Coordinate axes convention used in the experimental setup.

I begin by assuming that the sample is in its home position (0s0) and the

indicatrix that corresponds to the grain under examination is aligned with each of the

three coordinate axes. This is mathematically represented by the general equation for

an ellipsoid centered at the origin:

12

2

2

2

2

2

=++cz

by

ax (5)

Now I describe a procedure that will produce an equation for the ellipsoid

having an arbitrary orientation specified by the 3 Euler angles and also a sample

orientation specified by ξ and φinner. The left side of the equation for an ellipsoid can

be decomposed into the multiplication of the following matrices:

[ ]

=

==zyx

c

b

azyx srq ,

100

010

001

,

2

2

2

(6)

When multiplied together as follows, the left side of equation (5) results:

x-axis

y-axis

z-axis out of page

41

[ ] 2

2

2

2

2

2

2

2

2

100

010

001

cz

by

ax

zyx

c

b

azyx ++=

=qrs (7)

The next step is to rotate the indicatrix through three Euler angles φ, θ, and ψ.

This rotation scheme was first published in 1776 by the famous mathematician

Leonard Euler (1707-1783) where each rotation can be described by a rotation matrix

that transforms one coordinate system into another. The first rotation is

counterclockwise through an angle φ about the z-axis. The rotation matrix is:

−=

1000cossin0sincos

φφφφ

φλ (8)

The next rotation is counterclockwise through an angle θ about the newly created x-

axis. The rotation matrix is:

−=

θθθθθ

cossin0sincos0

001λ (9)

The final rotation is counterclockwise through an angle ψ about the newly created z-

axis. The rotation matrix is:

−=

1000cossin0sincos

ψψψψ

ψλ (10)

The complete transformation is given by the matrix λ:

42

−+−−−

+−==

θφθφθθψψφθφψψφθφψθψψφθφψψφθφψ

φθψ

coscossinsinsinsincoscoscoscossinsincossincoscossinsinsinsincoscossincossinsincoscoscos

λλλλ (11)

The Euler angle derivation and equations are from Marion (431-433).

“When discussing a rotation, there are two possible conventions: rotation of

the axes, and rotation of the object relative to fixed axes” (Weisstein). The Euler

method is a coordinate transformation of the axes. In this study, it is more useful to

leave the lab coordinate reference frame unchanged and instead rotate the ellipsoid in

space. This is a rotation of an object relative to fixed axes. All positive angles are

taken to be counterclockwise about the given axis of rotation. Next, I will describe

how the rotation matrices are altered to accommodate this convention.

Consider an object of rotation, vector x = [1 0 0]. Now consider (A) a rotation

of the coordinate axes equal to φ = 45 degrees. This has the same relative effect as

(B) a rotation of the object equal to φ = −45 degrees. This is easily demonstrated

pictorially as shown in Figure 24.

Figure 24. Diagram explaining rotation convention used.

axis & object object

axis

θ = 45 axis

θ = −45

object

Initially A B

43

A quick and easy solution to change the Euler angle matrices from a coordinate axis

rotation to an object rotation is to substitute the opposite value for each angle.

However, this might cause confusion. Instead, I converted each matrix to accept the

angle of actual rotation in space so that a rotation of a positive angle corresponds to a

positive angle entered into the calculations. This conversion used the following

property of the sine and cosine functions:

)(cos)(cos

)(sin)(sinxx

xx=−

−=− (12)

This happens to be equivalent to the inverse Euler matrices, 111 ,, −−−

ψθφ λλλ :

−=−

1000cossin0sincos

1 φφφφ

φλ (13)

−=−

θθθθθ

cossin0sincos0001

1λ (14)

−=−

1000cossin0sincos

1 ψψψψ

ψλ (15)

φθψ ρρρρ

ρλ==−

havewe,Setting 1

ii (16)

To find the theoretical extinction angle of a grain with a known orientation, I first

rotate its indicatrix through the three Euler angles describing its orientation. This is

done using the inverse Euler matrices, equations (13-15). Remember that rotating the

body in a positive angle is the same as the coordinates rotating a negative angle. Since

44

the second rotation is defined about the newly created x-axis, angle θ will rotate about

the newly created “x-axis” of the object. Therefore, I must rotate the θ rotation matrix

by φ. The same process must be repeated for the ψ rotation, as it must be adjusted by

the two previous rotations. This is equivalent to applying the rotation matrices in

reverse order so that they act upon themselves. This mathematical process is shown

below and a full derivation is completed in Appendix A.

ψθφφφθφφθφψφθφ ρρρρρρρρρρρρρρρ == −−−− )()()( 1111corrected (17)

If the applied rotations are now inserted into equation (7) as follows: srρqρ 1

correctedcorrected− (18)

then I can determine the equation for the ellipsoid at the home position: orientation

0s0. It is also necessary to determine the ellipsoid for each of the other sequences.

This is done by inserting two more rotation matrices for ξ (xi) and φinner (phi inner):

−=

ξξ

ξξ

ξ

cos0sin010

sin0cosλ (19)

−=

1000cossin0sincos

innerinner

innerinner

inner φφφφ

φλ (20)

These matrices are also adjusted to rotate the object so that the inverse rotation matrix

is used. So by using equation (16) I have:

−==−

ξξ

ξξ

ξξ

cos0sin010

sin0cos1 ρλ (21)

45

−==−

1000cossin0sincos

1innerinner

innerinner

innerinner φφφφ

φφ ρλ (22)

Then equation (17) is then modified so that all rotations are applied as shown in

equation (23):

ψθφφξ ρρρρρρ inner=corrected (23)

Remember that the angle ξ causes the incident light to enter the sample at an angle

different from normal incidence and the speed of light is slower in the sample than in

air. Therefore, refraction occurs inside the sample. This effectively reduces the angle

ξ in the sample from the rotation angle specified during data collection. The

adjustment is a simple correction made using Snell’s Law, described earlier by

equation (3). Because the index of refraction of air is approximately 1, the corrected

angle is:

= −

sample of refractionofindexaverage)(sin

sin duncorrecte1 ξξ (24)

As an example, when ξuncorrected = 45o and the average index of refraction is 1.67, then

ξ = 25.05o.

I now have the equation of the ellipsoid, after all necessary rotations, as

referenced in the fixed coordinate axes of the lab reference frame. This is given by the

following equation:

46

[ ] 1)(

100

010

001

1

2

2

2

1correctedcorrected

=

=

−

−

zyx

c

b

azyx innerinner ψθφφξψθφφξ ρρρρρρρρρρ

srρqρ

(25)

To summarize, I now have the equation of an ellipsoid, originally aligned in

the home position along the lab coordinate reference frame, that has been rotated

through 3 Euler orientation angles and 2 sample orientation angles.

Next, I find the extinction angle from the ellipsoid equation using the criterion

for extinction mentioned earlier. The criterion for extinction is that one of the

polarizers must be aligned with the semi-major (or semi-minor) axis of the ellipse that

results from the intersection of the indicatrix ellipsoid with the plane defining the

wavefront of incident light. According to the coordinate convention of the

experimental system, this plane is the z = 0 plane. The ellipse of intersection is found

by taking z = 0 in equation (25). In practice, I take z = 0 at the outset of the derivation

and then equation (20) gives the appropriate equation for the ellipse directly. The

following generalized equation for an ellipse centered at the origin results:

122 =++ CyBxyAx (26)

(where A,B and C are constants) This is the general equation for an ellipse that may have any orientation within the xy-

plane. The semi-major and semi-minor axes correspond to the index of refraction of

the slow ray and fast ray through the crystal lattice, respectively. (If one of the optic

47

axes of the indicatrix is normal to the xy-plane, then the cross section will be a circle

and the grain will be extinct at all angles).

The theoretical extinction angle, each nthε , is defined as the angle between the

semi-major or the semi-minor axis of the ellipse and the x-axis of the lab reference

frame, whichever ellipse axis makes a positive angle (Figure 25). It will then naturally

fall within the range 0-90 degrees, which I choose as my convention. (Recall that the

extinction angles have a periodicity of 90 degrees, so this procedure just picks out the

extinction angle that is in the first quadrant.)

Figure 25. Cross sectional ellipse showing the theoretical extinction angle.

The constants A, B, and C are used to calculate the theoretical extinction angle

according to:

−

= −

CAB1n

th tan21180

πε

o

(27)

48

This equation is derived for the interested reader in Appendix B. To avoid computer

errors, when A = C, I set nthε = 45 degrees. Also, the computer’s inverse tangent

function has a domain of -90 to 90 degrees. I map the result of the inverse tangent to

the domain 0 to 180 degrees so that the extinction angle will fall in the first quadrant,

according to my convention. Both of these corrections can be summarized by:

nth

nth

nth

else

45 then , if

εε

ε

=

== oCA (28)

nth

nth

nth

nth

nth

else

90 then ,0 if

εε

εεε

=

+=< o

(29)

Now I have developed the procedure to determine the theoretical extinction

angle for a given indicatrix orientation and sample orientation. In the next section, I

will describe how I use this information to find a unique Euler angle orientation for a

particular grain.

I used the Maple mathematics program to setup and test the rotation matrices

to make sure the definitions and sign (+,-) conventions were correct. It also allowed

me to graph the ellipsoid cross section to visually verify the results of the rotation

matrices. Also using Maple, I was able to perform all of the rotations symbolically and

arrive at explicit (albeit complicated) expressions for A, B, and C (the coefficients of

the cross sectional ellipse) in terms of the rotation angles φ, θ, ψ, ξ and φinner and the

values of a, b, and c. Therefore, the matrix multiplication need not be carried out

during the analysis. In order to save space, these equations will be deferred to

Appendix C. The rest of the analysis was carried out using Labview, which is the same

49

development software used in all other aspects of the experiment, such as data

collection.

3.4 Analysis Procedure Definition of a “Match” using R2

The analysis routine I have devised will determine the Euler angle orientation,

(φ, θ, ψ), for each grain being analyzed within a thin section sample using the known

optical parameters and the nine experimental extinction angles. The method for

determining the solution, (φ, θ, ψ), involves working backwards by finding the

theoretical extinction angle for many combinations of φ, θ, ψ, ξ, and φinner. Then, the

set of nine extinction angles determined experimentally is compared with all the sets

of nine extinction angles determined theoretically and the combination with the closest

match overall is going to reveal the Euler angles of the grain under evaluation.

The function describing the closeness of a match between a set of experimental

extinction angles and a set of theoretical extinction angles is a least squares fit called R

squared (R2). The R2 value is calculated according to the following formula:

∑=

−⋅=9

1expth

22 }]),,({2[sin),,(n

nnR εψθφεψθφ (30)

Note that R2 values are unitless. If there is a perfect match in all nine sequences, then

the summation will add to zero, because 00sin o2 = . If all of the angles are the worst

match possible, i.e. ± 45o apart, then because 190sin o2 = , the summation will be

equal to the number of sequences used in the analysis. A value for R2 of <.01 total

50

over nine sequences is what I consider a close match. This would correspond to a

difference < ± .955o per sequence’s extinction angle if the error were evenly distributed

among the sequences, which is comparable to the error in the experimental

measurement. If there were eight sequences with a perfect match and one with a bad

match, then an R2 of <.01 would allow a maximum difference of < ± 2.87o for the bad

sequence. This nonlinearity of the R2 function means one bad experimental extinction

angle can make finding a close match in absolute terms rather difficult.

The reason for using the sine squared of the difference, instead of just the

difference squared, is due to the 90 degree periodicity of the extinction angle. When

the difference between theory and experiment is close to 90 degrees, then the function

should report a close match. For example, suppose I am comparing the two extinction

angles 89° and 1°. If I used the difference squared, ( nthε - n

expε )2, it would consider

these two angles a bad match. In reality, they are only 2° apart. So I use the sine

squared function with a factor of 2 multiplied to the difference, ( nthε - n

expε ), to change

the periodicity from 180° to 90°. The sine function is squared to always return a

positive value. To demonstrate, observe the graph in Figure 28 below of f(x), which is

R2 for one sequence where the variable x represents the theoretical extinction angle

when the experimental extinction angle is zero degrees.

51

Figure 26. Graph of ( )xxf 2sin)( 2= .

Two Conceptually Similar Methods of Minimizing R2

Here I will explain the two conceptually similar methods I devised to find the

unique Euler angle orientation φ, θ, ψ that minimizes R2. Later I will describe some

of the programming tricks I employed to get the most performance out of the two

methods. The building block for either method is to divide the continuous 3-D space

of Euler angles into a discrete grid and then find the minimum R2 value over all the

points on the grid. The total number of sample points depends on the range and grid

spacing. For example, suppose the grid covers the full range of the three Euler angles

(0-180 degrees for each angle) and the grid spacing is 2 degrees. I represent this

minimization with the notation [ ] [ ] [ ]))2,1800 ,2,1800 ,2,1800(min( 2 →→→ ψθφR .

As a second example, suppose the grid focuses on a smaller region with high detail

such as, [ ] [ ] [ ]))01.0,100,01.0,8575,01.0,5040(min( 2 →→→ ψθφR . Note that the

building block used in both methods requires the theoretical extinction angle for each

sequence to be calculated for every point on the grid.

52

Simple Grid Method

The first method is called the Simple Grid Method because it involves only one

iteration of the basic building block above. In fact, it is exactly described by the first

example above: [ ] [ ] [ ]))2,1800,2,1800,2,1800(min( 2 →→→ ψθφR . However, the

calculation is optimized as follows: The number of repetitions of the calculation of a

theoretical extinction angle is specified by equation (32) described later in this chapter.

To save calculation time, I take advantage of the redundancies in calculating all

possible values of nthε over the range of the grid for different grains in a thin section.

Even though a sample may contain as many as 500 to 1000 grains, as long as each one

is composed of the same mineral composition, I can use the same set of nthε for each

grain. Rather than recalculate them each time, it makes sense to save that work in a

file to be used during subsequent runs of the analysis program (for different grains of

the same mineral). Such a file is written for each sequence and it lists the nthε for each

Euler angle specified on the grid. The optimized simple grid method opens up the

entire list and finds the unique Euler angle orientation φ, θ, ψ that minimizes R2 for

that grain.

In practice, the optimized simple grid method by itself had some shortcomings.

First, it takes approximately 20 minutes to write a set of nine sequence files at a 2

degree resolution on the lab computer. Since this is done once per mineral

composition, it is a tolerable wait. However, since all Euler angles are checked when

calculating R2, it still takes about 2 minutes per grain at this 2 degree Euler angle

53

resolution. For a typical sample containing hundreds of grains, this results in many

hours of calculation. Plus, I am left with poor precision (approximately two degrees).

Later, I show that the number of calculations (and hence the calculation time) is a

nonlinear function of the precision desired. This means that simply increasing the grid

resolution by a factor of one half does not simply double the calculation time, but

rather it increases the calculation time by nearly a factor of 8. The dilemma of how to

improve the precision without further increasing the calculation time led me to create

another method, which I will describe in the next section.

Adapted Newton Method

The second method is called the Adapted Newton Method. It was first devised

as a method to find the precise Euler angle match to .01 degrees, given the best match

from the two degree resolution of the Simple Grid Method as a starting point. The

volume of the Euler angle space is like a 3D cube 180 degrees on a side (Figure 27).

54

Figure 27. Depiction of 3-D Euler angle space.

The Simple Grid Method divided that cube into smaller 2 x 2 cubes. Unless the

minimum of the Simple Grid Method is far away from the global minimum, the lowest

minimum of R2 lies within a volume of 8 of these smaller cubes surrounding the best

match from the Simple Grid Method. A naïve approach is to consider using the Grid

Method again on this smaller domain with a grid spacing of 0.01 degree. However,

100 samples per degree in the smaller domain still requires calculating R2 over

64,481,201 Euler angle combinations. This is more than 85 times the number of

samples used by the Simple Grid Method in the first place! This strategy was

successful, but too time-consuming.

Rethinking the problem, I returned to picturing how the function describing R2

varies. It can be likened to a function that describes the temperature of a room given

the point in space (x,y,z). If you were asked to find the lowest temperature in this

room, as quickly as possible, you might employ the hot-cold method. This entails

Phi

0

180 Theta

180

180

Psi

55

moving around the room towards the lowest temperature reading you find. You might

initially just check the temperature at a point on the left and right sides of the room,

and then move to the colder side and then check a few points around there, etc. This

led to the development of the Adapted Newton Method. The premise of the Adapted

Newton Method is to start with a very coarse grid and iterate the grid method

building block, continually shrinking the grid while centering it on the minimum of the

previous grid.

Detailed Methodology of the Adapted Newton Method

The Adapted Newton Method begins with a coarse grid over the entire region

of interest. The region of interest could be two degrees above and below each Euler

angle of the best match from the Simple Grid Method. The region of interest could

also be the entire span of Euler angles used in the Simple Grid Method, 0-180 degrees.

The building block grid of the Adapted Newton Method includes a central point, two

endpoints, and two midpoints for each Euler angle. Using the notation developed

earlier while covering all Euler angles, it would be expressed as:

[ ] [ ] [ ]))45,1800,45,1800,45,1800(min( 2 →→→ ψθφR . There are 5 sampling points

for each Euler angle giving 53 = 125 combinations. This number of combinations is

used in each stage (i.e. iteration of the building block) of the process. During each

successive stage, a new building block grid is created with a resolution that is one half

the resolution used in the previous iteration. In addition, the new building block grid

is centered about the Euler angle combination with the lowest R2 from the previous

iteration.

56

As an example, observe the sampling angles for the first iteration in this case.

Over the entire range there is 0-180 degrees. The center point is 90 degrees, the

endpoints are 0 and 180 degrees and the midpoints are 45 and 135 degrees. The

Adapted Newton Method takes all 125 combinations and minimizes R2. As an

example, if the method finds the minimum at (90,135,90) for (φ, θ, ψ), then the new

coarse grid is centered there. The endpoints are now set to be the nearest points of the

previous iterations grid, which were 45 and 135 degrees for φ and ψ and 90 and 180

degrees for θ. Then the new midpoints are set at 67.5 and 112.5 for φ and ψ and 112.5

and 157.5 degrees for θ. Once a minimum R2 is found, the process continues. After

13 iterations, the resolution in each Euler angle reaches:

oo

13

o

iterations 00549.819245

245

2resolutioninitialresolutionfinal ≈=== (31)

This method is called the Adapted Newton Method because of its similarity to

Sir Isaac Newton’s method of finding the x-intercept of a function.

Comparison of the Number of Repetitions

Even though it includes hundreds of sine and cosine functions, the calculation

of a single extinction angle takes less than one millisecond. However, when the

analysis is done, the calculation of an extinction angle must be made over and over.

When using the Simple Grid Method, in order to test all possible Euler angle

orientations, a range of each angle and a sampling resolution must be defined. The

repetitions of this program are calculated according to the following equation:

57

sequencesnresolution

ofrangeresolution

ofrangeresolution

ofrangesrepetitiongrid ⋅

+

+

+= 111

ψψ

θθ

φφ (32)

The range of 0-180 degrees covers all orientations and was chosen as the convention

for this experiment. Here is a sample calculation of the number of repetitions using 9

sequences and a 2 degree sampling resolution:

139,782,69571,753991912

18012

18012

180 3 =⋅=⋅=⋅

+

+

+=srepetitiongrid (33)

The program needs 19.5 seconds to iterate 100,000 calculations on the lab computer.

In the case of the example above, before any comparisons are made, (like R2), the

program would require 22 minutes per grain. To make things worse, that still only

reveals a solution within two degrees! It is important to note that the number of

calculations is proportional to the inverse cube of the grid spacing.

In the case of the Adapted Newton Method, the total number of repetitions of

the standard calculation is far fewer. To achieve a .00549 degree resolution, 13

iterations are made of the Adapted Newton Method yielding:

14625913125125 =⋅⋅=⋅⋅= sequencesniterationssrepetitionnewton (34)

Therefore, in order to reduce the amount of calculations necessary, the Adapted

Newton Method is the method of choice. However, one must also verify that this (or

any) method yields the true global minimum of the R2 function. Later in the results

section, I will discuss the success rate in practice of all the methods.

58

Detailed Methodology of the Simple Grid Method

Each file is classified in an easy to understand file management system

describing its contents. The data directory is located at C:\Geo Orientation Angle

Data. This contains folders organized by the resolution of each Euler angle. Each of

these folders contains sub-folders organized by the ellipsoid axes a, b, and c specified

to 4 digits of precision. It also contains a file that tells the analysis program how long

each data file is in bytes (more on that later). Finally, each of these folders contains a

file for each sequence, i.e. 0s0 Extinction Angles.dat. The default is a two degree

resolution from 0 to 180 degrees. Figure 28 is an example of the file structure for the

Simple Grid Method.

Figure 28. File structure for the Simple Grid Method.

The theoretical extinction angle is originally found in radians because the

computer makes all sine and cosine calculations in radian mode. However, it is easier

and more intuitive when using degrees. Therefore, I save the extinction angle data

files as degrees by shifting all the radian values by the constant multiplier 180/pi. As

an added bonus, I am able to maintain a higher resolution in terms of digits needed

59

after the decimal point. For example, a .01 degree change is equivalent to a change of

30001745.0 radians. What takes only 2 place values after the decimal in degrees will

consume many more place values in radians.

Next, I made some careful comparisons to determine the best method to save

the files. If each extinction angle were saved as a floating point number it would

consume 8 bytes of hard disk space. If each extinction angle were saved as an integer

number it would consume 1 byte of hard disk space, but I would lose all the

information after the decimal point as each value is rounded to a whole number. I

found this information from the “help section” included with Labview, as indicated

below.

• Double-precision (DBL) - Double-precision floating-point numbers have

64-bit IEEE double-precision format. Double-precision is the default format

for numeric objects.

• Byte (I8) - Byte integer numbers have 8 bits of storage.

To save space and retain all precision, I have employed a creative solution that

converts each extinction angle to two integers and writes them to disk in a continuous

fashion. The first integer byte is equal to the number before the decimal point and the

next integer byte is equal to the 2 significant figures after the decimal point. Since

each extinction angle is calculated between 0 and 90 degrees, the integer’s range from

-128 to 127 is more than enough. Even though this requires a conversion to and from

the integer format, hard drive space is saved and overall execution time is reduced

60

since it will write to disk faster. Table 3 shows the amount of hard drive space saved

by this method which (theoretically) writes to disk four times faster.

Table 3. Memory usage statistics for floating point vs. two integer bytes.

Resolution Combinations Total Memory Used

Memory Used

Memory Saved

Memory Saved

in Degrees Per Angle Combinations Floating Point

2 Integer Bytes Per Sequence

Over 9 Sequences

10 19 6859 0.05 MB 0.01 MB 0.04 MB 0.35 MB

5 37 50653 0.39 MB 0.10 MB 0.29 MB 2.61 MB

2 91 753571 5.75 MB 1.44 MB 4.31 MB 38.81 MB

1 181 5929741 45.24 MB 11.31 MB 33.93 MB 305.37 MB

0.5 361 47045881 358.93 MB 89.73 MB 269.20 MB 2422.79 MB

Each data file is written according to the same standardization, which is a systematic

method similar to how one would count all combinations of three items, given a set

number of items, if order matters. First, φ and θ are held constant while ψ varies

incrementally according to the sampling resolution. When ψ reaches a maximum, θ is

incremented once, and the process repeats. When θ reaches a maximum, φ is

incremented once, and the process repeats. Finally, φ, reaches a maximum when all

nthε have been calculated. Each n

thε is written to the file in the order in which it is

calculated. This effectively “numbers” them according to their position in the file.

The analysis routine employs no advanced selection criteria to eliminate bad

comparisons, rather it compares all extinction angles calculated. The procedure

includes: open all the files in the data directory, read the first entry in each one,

assemble it into a floating point number, convert it to radians, calculate the R2

(summed over all sequences), then do the same for each subsequent entry. This is why

61

the program needs to know how long the files are in bytes. After that task in

complete, I have an array of R2 values ordered according to their Euler angles. I find

the lowest R2 value and its place in the array indicates what Euler angles are the

solution!

62

4.0 RESULTS AND DISCUSSION

The olivine sample that I analyzed using the new optical technique was

acquired from a research group at Yale University. I wanted to compare the results of

my experiment with the known orientations of the minerals in order to see if the

optical method would yield the correct results. The research group at Yale, with the

help of my advisor, Dr. Larry Wilen, used a scanning electron microscope (SEM) to

find the orientation of 21 grains in the sample. The details of the SEM procedure and

the side-by-side comparison between both methods will be explained in the next few

paragraphs. I will also discuss the corrections I made to the analysis.

In the typical application of a scanning electron microscope (SEM), a beam of

electrons is focused onto the surface of a sample. The reflected electrons are used to

map out an image of the surface to be displayed on a monitor. The surface must be

electrically conductive for this procedure to work. In my case, the olivine sample was

covered with a carbon film for conduction. A SEM can display magnifications of

50,000 times or higher, which is much higher magnifications than an optical

microscope can obtain. However, the grains in the olivine sample are visible to the

naked eye, so a high level of magnification was not necessary (Callister 84-85).

Electron backscattered diffraction (EBSD) is the technique utilizing a SEM

that is used by mineralogists and other materials scientists to find the orientations of

mineral grains. The EBSD technique I am about to describe is the same system by

which the Yale University researchers found the orientations in the olivine sample.

63

Since its development in 1954, EBSD has proved to be a fast and user-friendly method

for this type of experimentation. The procedure examines a polished sample, such as a

thin section, that is tilted to a relatively high angle of ~70°. Below is an excerpt from

the HKL Technologies website detailing the EBSD procedure:

The electron beam is... directed at the point of interest on the sample surface: initial elastic scattering of the incident beam causes the electrons to diverge from a point just below the sample surface and to impinge upon crystal planes in all directions. Wherever the Bragg condition for diffraction is satisfied by a family of atomic lattice planes in the crystal, 2 cones of diffracted electrons are produced. These cones are produced for each family of lattice planes (HKL Technology: EBSD Basics). The website continues by stating how the cones can be detected by a phosphor

screen detector attached to a digital CCD camera. The cones produce bands on the

screen, which is close to the sample to collect as many bands as possible. Each band

corresponds to a crystal lattice plane family and the bands are called “Kikuchi bands”.

There are many Kikuchi bands which are then matched with theoretical data to

calculate the 3-D crystallographic orientation in less than 0.02 seconds (HKL

Technology: EBSD Basics). In the case of the Yale University SEM, this output is in

the form of the Euler angles.

Dr. Wilen and the researchers at Yale University analyzed a total of 23 grains

from the olivine sample. Of these, 14 were olivine and 9 were another phase called

orthopyroxene (OPX). However, when analyzing the grains using the optical setup,

one of each phase from the original 23 grains was too close to the edge of the camera

view to be seen in every sequence and could not be analyzed. The remaining 21

64

analyzable grains were numbered 0-20 and tested using the most recent version of the

Adapted Newton Method analysis routine. After some additional theoretical issues

were taken into account, the analysis routine found each grain’s Euler angles in good

agreement with the results from the SEM method.

It was important to make sure I analyzed the exact same grains using both

methods to be able to compare the data effectively. Dr. Wilen worked with the Yale

group on the olivine sample. He told me that when viewing the sample with the SEM,

it was difficult to discern the grain boundaries. Without the cross polarized light, the

only clue available to determine which grain was being analyzing was the cracks in

the sample. He used the cracks as a roadmap to hit as many unique grains as possible

from looking at a printout of an image of the sample taken with a microscope under

crossed polarizers. Since the SEM analyzed a specific region of each grain, he

indicated this region on the printout so I could use the optical technique to analyze the

same region of the grain. This ensured we were both hitting the same grain and it also

minimized any differences that might have arisen from a variation in grain orientation

according to the region of the grain that was analyzed (Wilen).

Figure 29 is an image showing the location of each analysis point for both

methods and its corresponding number.

65

Figure 29. Image showing each analyzed grain’s location. Pixel resolution = 640 x 480.

The SEM made two different runs on the sample to collect Euler angle data.

Therefore, the first 15 grains were analyzed before the final 8 were analyzed. The

overlap between the two runs resulted in 18 unique grains included in the 21 analysis

points. Analysis numbers 3 and 8 are the same grain, 7 and 15 are the same grain, and

12 and 20 are the same grain. It was initially discovered visually that the same grain

was being analyzed in each case. Then both methods verified this by producing

similar results for the Euler orientation angles for each pair of analysis points on the

same grain.

1 mm

66

It is interesting to note that analysis points 14 and 16 had a similar orientation.

These points might be on one grain that has a discontinuous orientation resulting from

stress or strain. Or these two points are on separate grains that split apart from one

grain, also giving them similar orientations. Another interesting pair of analysis points

are point 5 and point 18. They too have similar orientations. However, they are not

anywhere near each other in the sample meaning there are many grains in between

them. In addition, these two grains are not even the same phase, one is olivine and the

other is OPX. Therefore, I attribute the close alignment of these grains to be a

coincidence.

Table 4 shows a comparison of the results from the two methods. These

results were obtained using the final version of the analysis algorithm, which I will

discuss later.

67

Table 4. The Results

Grain Orientations Using Two Methods Grain Euler

Angles Yale Univ.

SEM Ohio Univ.

Optical Phi 62.79 69.83 Theta 94.12 92.43 Psi 68.00 67.67

Olivine 0

R2 0.00445 Phi 134.31 140.84 Theta 62.66 61.82 Psi 90.93 90.85

Olivine 1

R2 0.00871 Phi 63.01 72.19 Theta 52.00 53.88 Psi 122.35 119.24

Olivine 2

R2 0.00244 Phi 81.19 89.60 Theta 55.74 55.01 Psi 140.36 138.69

Olivine 3

R2 0.00160 Phi 43.07 45.61 Theta 19.89 20.04 Psi 0.73 4.59

Olivine 4

R2 0.00010 Phi 169.01 175.13 Theta 18.17 17.35 Psi 101.10 101.15

Olivine 5

R2 0.00053 Phi 34.77 42.38 Theta 102.48 101.09 Psi 149.96 149.56

Olivine 6

R2 0.00181 Phi 57.68 64.30 Theta 104.40 103.37 Psi 83.51 83.20

Olivine 7

R2 0.00156 Phi 80.26 88.38 Theta 55.17 54.81 Psi 141.63 139.94

Olivine 8*

R2 0.00172 Phi 102.16 107.89 Theta 112.01 112.03 Psi 63.11 61.64

Olivine 9

R2 0.01670

68

Grain Orientations Using Two Methods Grain Euler

Angles Yale Univ.

SEM Ohio Univ.

Optical

Phi 40.46 47.52Theta 43.91 45.02Psi 174.93 174.04

Olivine 10

R2 0.00148Phi 75.85 82.20Theta 90.25 90.04Psi 16.50 15.86

OPX 11

R2 0.00139Phi 69.55 75.85Theta 52.40 51.41Psi 32.54 31.86

OPX 12

R2 0.00007Phi 138.56 149.59Theta 57.26 55.66Psi 56.19 52.52

Olivine 13

R2 0.00343Phi 86.40 93.80Theta 87.25 86.33Psi 14.75 16.04

OPX 14

R2 0.00103Phi 58.19 64.36Theta 102.01 103.23Psi 82.85 83.16

Olivine 15*

R2 0.00150Phi 89.93 94.64Theta 86.39 86.36Psi 8.96 9.93

OPX 16

R2 0.00097Phi 72.23 77.09Theta 118.49 119.31Psi 178.71 178.48

OPX 17

R2 0.00175Phi 168.05 173.62Theta 18.44 16.46Psi 91.40 90.67

OPX 18

R2 0.00344Phi 30.85 36.85Theta 50.86 51.03Psi 24.55 23.64

OPX 19

R2 0.00035

69

Grain Orientations Using Two Methods

Grain Euler Angles

Yale Univ. SEM

Ohio Univ. Optical

Phi 71.04 75.80 Theta 51.23 51.76 Psi 32.47 31.06

OPX 20*

R2 0.00052 *Repeat Grain

70

Figure 30 is a graph depicting the orientation of all 21 grains. The x

coordinate is the angle phi, the y coordinate is the angle theta, and the angle that each

line makes with horizontal is psi. If the line is horizontal, then phi is 0° and vertical is

90°. There isn’t a need to make the line a vector pointing in all directions, because a

180° rotation in psi is a rotation equivalent to 0° because of symmetry in the

indicatrix.

Figure 30. Euler angle plot of data from the optical method.

71

Figure 31 is a graph depicting the orientation of all 21 grains alongside the

SEM results. The data from the SEM has been adjusted to account for the difference

in mounting between the SEM setup and the optical setup, which was apparent by a

constant shift in the angle phi.

Red = Optical; Blue = SEM

Figure 31. Euler angle plot of data from both methods side-by-side. SEM data has been adjusted to compensate for a sample mounting difference.

72

Characteristics of Olivine and the Solution to an Initial Error of Psi ± 90° This section will describe the development process by which I arrived at the

final version of the analysis routine. Initially, both the analysis routines derived for

this experiment (discussed in section 3.4) were used to find Euler angles for each of

the 21 analyzable grains in the sample. The first was the Simple Grid Method (with

two degree resolution) followed by the Adapted Newton Method (using 13 iterations

of a 2° to 2° starting grid) to achieve a 0.01 degree resolution. The second was the

Adapted Newton Method by itself (13 iterations of a 0° to 180° starting grid) to

achieve a 0.01 degree resolution. The Euler angles from the two analysis methods

were then compared with each other and with the SEM results.

Neither the Adapted Newton Method alone, nor the Grid Method + Adapted

Newton Method gave results in complete agreement with the SEM data. Two issues

with the results became apparent. The first problem was that sometimes one method

or the other found an answer with a lower value of R2. This means that neither

method found the global minimum every time. The second problem was that for the

best solutions (those with the lowest R2 among the two methods), there was good

agreement with the SEM results only in the angles phi and theta. For the olivine

grains, the best solution (independent of the analysis technique from which it was

derived) differed from the SEM results in the angle psi by 90°. For the OPX grains,

the angle psi was in agreement between the two methods.

The first step in diagnosing the inconsistency between the two analysis

methods involved a close examination of the data for R2 using the grid method. This

73

was the best way to get an overview of the entire 3-D Euler angle space. As a

reminder, the grid method calculates R2 in the following order: first, φ and θ are held

constant while ψ varies incrementally according to the sampling resolution. When

ψ reaches a maximum, θ is incremented once, and the process repeats. When

θ reaches a maximum, φ is incremented once, and the process repeats. Finally,

φ, reaches a maximum and all calculations are complete with the results “numbered”

according to the order in which they were calculated. The images below, (Figure 32),

(Figure 33), and (Figure 34), show three plots of R2 vs. Euler angle taken from the

results of the Grid Method for olivine.

The first plot was created by plotting the R2 value at a two degree resolution in

each Euler angle and shows 753571 data points for

[ ] [ ] [ ])2,1800,2,1800,2,1800(2 →→→ ψθφR .

Figure 32. R2 vs. all Euler angles for one grain. The second plot was created by plotting the R2 value at a two degree resolution in two

Euler angles and shows 8281 data points for

[ ] [ ] [ ])2,1800,2,1800,constant(2 →→ ψθφR .

74

Figure 33. R2 vs. the Euler angles theta and psi for one grain. The third plot was created by plotting the R2 value at a two degree resolution in one

Euler angle and shows 91 data points for [ ] [ ] [ ])2,1800,constant,constant(2 →ψθφR .

Figure 34. R2 vs. the Euler angles theta and psi for one grain.

Each successive plot has the effect of zooming in on the apparent minimum

from the plot before it. From examining these plots, I discovered that there was a near

degeneracy with respect to the angle psi. R2 appears to reach a local minimum at a

periodic 90° interval in psi. So, I concluded that the main reason that one or the other

analysis method was sometimes finding the wrong global minimum was because there

75

were many close local minima over the entire 3-D space. This was corroborated by the

fact that when the results from the two analysis methods were different, the difference

was usually just ± 90° in psi. To check this idea out, I re-minimized the solution from

the Simple Grid + Adapted Newton Method for each grain about the point (phi, theta,

psi + 90°) and chose the lowest R2 between them. This always gave the correct global

minimum (i.e. the one that agreed with the best choice among the two methods).

However, recall that the Simple Grid method is very slow; later, I discuss a more

efficient method to get the global minimum.

At this point, I was still left with the second problem, namely that all the

olivine results differ from the SEM results by 90° in psi and that the OPX grains were

in agreement. Dr. Wilen had earlier suggested an inconsistency between the angle

conventions as the cause for discrepancy. However, this bothered me because it didn’t

happen with all of the grains; the OPX grains seemed to be immune from the problem.

I spent about 6 months trying to discover what might be causing the

discrepancy. The quest for the answer revealed a physical property of olivine to be the

cause - a mineral characteristic called solid solution. More specifically, the psi ± 90o

near degeneracy (and hence the discrepancy) was due to how olivine’s solid solution

affected the values for the refractive indices: α, β, and γ. To better describe this

relationship, I will first define solid solution as a phenomenon whereby one or more

ions can be substituted for other ions in the mineral’s atomic structure. This must be

done without distorting the structure of the crystal or introducing a chemical inequality

that may cause instability. Therefore, the best substitutions are within 15% of the size

76

of the ion replaced and of equal electronic charge. There are different forms of olivine

that depend on the chemical composition, which is written as: (Mg,Fe)2SiO4. In

olivine, the cations (positive ions) Mg2+ and Fe2+ can substitute for each other. This

causes what is called a solid solution series, which is a range of olivine minerals

varied by percentage composition of the pure end members. These pure end members

are Mg2SiO4 (called forsterite and abbreviated as Fo) and Fe2SiO4 (called fayalite and

abbreviated as Fa). The in between compositions are denoted by the relative

percentages of each end member (Sen 53-54). Since olivine’s chemical composition varies from end member to end member,

the values for the refractive indices also vary. These values are found in a book that

compiles experimental and observational data on different minerals, such as An

Introduction to the Rock-Forming Minerals by Deer, Howie and Zussman. The

refractive indices listed in that book for the end members of olivine and OPX are in

Table 5.

Table 5. Index of refraction data for olivine and OPX.

Olivine OPX forsterite,

Mg2SiO4 fayalite, Fe2SiO4

enstatite, MgSiO3

ferrosilite, FeSiO3

α 1.635 1.827 y 1.650 1.768 y β 1.651 1.869 z 1.653 1.770 x γ 1.670 1.879 x 1.658 1.788 z 2Vγ 82o 134o 55-127-55o

(Deer 3, 155)

Besides listing the refractive indices, the book also shows how the

crystallographic axes align with the axes of the optical indicatrix. For olivine, the γ

refractive index is aligned along the x-axis, the α refractive index is aligned along the

77

y-axis and the β refractive index is aligned along the z-axis. This means if the grain

had an Euler angle orientation of zero degrees in each angle, then maximum

birefringence would be seen in my experimental setup (which views the grain parallel

to the z-axis). Figure 35 illustrates the relationship between the crystal axes and the

indicatrix axes (Deer 3).

Figure 35. Relationship between the crystal axes and the indicatrix axes in olivine.

The values I used for α, β, and γ were calculated assuming the most common

chemical composition for olivine: a magnesium rich 90% Mg and 10% Fe, indicated

as 10% fayalite: Fa10. The calculation was done by creating a linear variation between

the end members and interpolating each refractive index at 90% Mg and 10% Fe. This

is where I found the biggest clue to what was causing the near degeneracy in the Euler

angle psi. It is related to the angle 2Vγ, which is the angle between the two optic axes

as measured so that the γ axis in the indicatrix bisects the angle. (Incidentally 2Vα is

measured across the α axis, but I choose to stay consistent with 2Vγ). The relationship

z = β

y = α

x = γ

78

between the three indices of refraction and the angle 2Vγ is as shown below from

Neese (78).

)()(Vcos 222

222

γ2

αγββγα

−−

= (35)

(where the angle Vγ is half of 2Vγ)

The four values are linked by an equation, so this means 2Vγ can be calculated from

known indices of refraction as opposed to a direct measurement. (My experimental

setup is not equipped to measure 2Vγ or the indices of refraction). In fact, if any three

of the four variables in the equation are known, then the fourth variable can be

calculated or found using a plot called a nomogram (Bloss 159). Even though the

angle 2Vγ is not used in the calculation of the theoretical extinction angle or R2, it

proved to be an important indicator of the correct geometry of the indicatrix ellipsoid.

When the angle 2Vγ surpasses 90o, olivine changes optic sign from positive to neutral

(at exactly 90o) to negative. (Neese 78). Deer said about olivine, “The optic axial

angle likewise varies systematically from 2Vγ = 82o for Mg2SiO4 to 2Vγ = 134o for

Fe2SiO4, the sign changing at about Fa13” (10). The problem is, when using

α, β, and γ as linearly interpolated values, the change in optic sign occurs between Fa6

and Fa7. At Fa10, olivine should be biaxial positive. Therefore, I was using

interpolated values of α, β, and γ that made olivine appear biaxial negative when in

fact it was biaxial positive!

The 2V angle can be measured experimentally with a higher degree of

precision than it can be calculated from the measured refractive indices. In other

79

words, a small variation in the ratio among β and the other two refractive indices

causes a large variation in the angle 2V. As a matter of fact, I calculated 2Vγ from the

indices of refraction listed for the end points and discovered that instead of 82o and

134o, I got 86o and 128.9o respectively. So I devised a new method for calculating

α, β, and γ for use in the analysis routine taking optic sign vs. composition into

account. It still applies a linear interpolation for α and γ, but now it linearly

interpolates the measured data for 2Vγ using the end values in the literature (82o and

134o). This causes a change of sign around Fa13 and gives a value of 87.2o at Fa10.

Then, using the interpolated value of 2Vγ, Ι calculate β from the equation relating all

four variables. The graph in Figure 36 illustrates the difference between the two

methods.

0 10 20 30 40 50 60 70 80 90 100

Percent Fayalite

Beta Calculated

Beta Interpolated

Optic Sign vs. Composition For OlivineBiaxial Positive

Biaxial Negative

Figure 36. Optic sign vs. composition for olivine.

This showed that the relationship among α, β, and γ as compared to each other

is most important in calculating the Euler orientation angles. So how did

misinterpreting the optic sign cause the angle psi to be off by almost exactly 90°?

This has to do with the alignment of the crystal axes to the indicatrix axes and how the

Euler angle rotations are defined (all stated previously). Both optic axes of olivine

80

initially lie in the x-y plane. So, when the 2V angle is very close to 90°, an extra

rotation of psi by 90° will produce theoretical extinction angles as if the mineral were

the opposite optical sign and had not undergone the extra rotation.

The OPX in the sample was not off by psi ± 90° because there was not a

problem with the interpolated values of the indices of refraction not matching the optic

sign of OPX. OPX is also biaxial positive at a 90% Mg and 10% Fe composition and

if 2Vγ is used to find the β index of refraction it agrees with the optic sign. Therefore,

I did not need adjust the indices for OPX. I just used a straight linear interpolation

like I had originally used for olivine. Incidentally, the 2Vγ angle in OPX does not vary

linearly with composition over the entire range.

As an interesting note, had olivine’s indices of refraction been aligned with the

crystallographic axes differently, then the consistency of the discrepancy might not

have been detected as readily. This is because misinterpreted optic sign would not

have exclusively affected the angle psi. In other words there would have still been a

90° rotational difference from the switch in optic sign, but that would have been in a

different plane from the psi rotation. This means it would have changed all three

Euler angles making the consistency of the discrepancy more difficult to detect.

Table 6 below shows the final parameters used to run the analysis routine. As

a reminder, the values of a, b, and c are the input variables into the equation for the

indicatrix ellipsoid.

81

Table 6. Final optical parameters: a, b, and c used to analyze olivine and OPX.

Final Parameters Used Olivine OPX a 1.6909 1.6647 b 1.6542 1.6609 c 1.6714 1.6701

With these parameters, the analysis method (using one of the versions that

finds the global minimum, as discussed above) finds a solution consistent with the

SEM results.

The next task was to create one optimized theoretical analysis technique that

finds this global minimum for each grain. Ironically, the near degeneracy of the angle

psi which alerted me to the inconsistency in the first place causes the two theoretical

methods (Newton and Grid) to have difficulty in finding the global minimum for each

grain (but at least this global minimum is now correct!).

The adaptation that fixed this problem most efficiently involved dropping the

Simple Grid Method completely (because it is slow) and increasing the number of

repetitions of the Adapted Newton Method by decreasing the starting grid. The

Adapted Newton Method is run eight times and the lowest R2 of these eight is checked

against psi + 90° to find the final lowest R2 solution. Each of the 8 runs of the

Adapted Newton Method is one equal sub-section of the entire 3-D Euler angle space

that was shown in Figure 27. Instead of one grid using points spaced 45° apart, each

of the 8 grids uses points spaced 22.5° apart. As an example, the first grid uses

endpoints of 0° and 90°, a center point of 45°, and midpoints of 22.5° and 67.5°. On

the lab computer, the new Adapted Newton Method (eight times with a check at psi +

90°) takes between 20 and 25 seconds of calculation time per grain.

82

After this programming adjustment was made, the new Adapted Newton

Method became the best minimization method. It found results consistent with the

SEM results for each and every grain as displayed in Table 4. In addition, there was

no longer a need to write memory consuming files and the calculation time was

reasonably short.

83

5.0 CONCLUSIONS AND APPLICATIONS TO FUTURE RESEARCH

In conclusion, the optical experimental technique I developed has been shown

to be able to accurately determine the orientation of the biaxial minerals olivine and

OPX in a thin section. The experimental system images a thin section between

rotating crossed polarizers and determines an experimental extinction angle for nine

sample orientations. By calculating the theoretical extinction angles for a coarse grid

of possible Euler angle orientations and continually narrowing the grid down to the

desired resolution, the best matching Euler angles can be found.

The optical method I developed costs much less than a comparably accurate

scanning electron microscope that uses electron backscattered diffraction. The cost

savings is augmented by the ability to run other analyses from the image data such as

grain size, elongation, and nearest neighbor correlations. Because of the relatively

small size of the setup, it is conceivable that the system could be designed to operate

using a laptop and battery powered rotation stages. This would enable the setup to be

taken into the field, which is currently not possible with the SEM.

Future considerations for this procedure would be an analysis routine that

would be able to decipher the mineral and/or composition of a grain in a sample, given

the known Euler angle orientation determined by some other method (such as the

SEM). The routine would use the known Euler angles to calculate all nine theoretical

extinction angles for many combinations of α, β, and γ. Then, by finding the best

84

matches between the theoretical extinction angles and the experimental extinction

angles, the ratios among the three indices can be determined. A database of known

indices of refraction for all possible minerals and/or compositions would then be

consulted to determine what satisfies the determined ratios of α, β, and γ.

The success of this new application depends on factors such as whether or not

nine sequences are enough to uniquely distinguish two minerals and/or compositions.

However, one way to increase the data is to analyze a polycrystalline sample that

consists of many grains of the same unknown mineral composition with many random

(but still known) orientations. By combining the data it might be possible to constrain

the indices of refraction enough to find unique numbers for α, β, and γ, instead of just

ratios, making a unique solution possible.

As the procedure stands now, it is immediately applicable to geologists and

geophysicists in search of a low cost experimental tool for finding the texture of

biaxial minerals. The only needed parameters outside of those measured by the

system are: the three indices of refraction α, β, and γ, their relationship to the crystal

axes, and a verification that β is consistent with the optical sign. If there is a problem

with β, then it can be adjusted according to the measured value of 2V.

85

6.0 REFERENCES Abramowitz, Mortimer, Matthew J. Parry-Hill, and Michael W. Davidson. Molecular

Expressions Microscopy Primer: Light and Color - Electromagnetic Radiation:

Interactive Java Tutorial. 2003. Olympus America Inc. and The Florida State

University.

30 Mar. 2005 <http://micro.magnet.fsu.edu/primer/java/electromagnetic/>.

Berry, L.G., Brian Mason and R.V. Dietrich. Mineralogy - Concepts, Descriptions,

Determinations. New York: W.H. Freeman & Co., 1983.

Bloss, Donald F. An Introduction to the Methods of Optical Crystallography. New

York: Holt, Rinehart and Winston, 1961.

Callister, William D. Jr. Materials Science and Engineering An Introduction. New

York: John Wiley & Sons, Inc., 2003.

Deer, W.A., R.A. Howie and J. Zussman. An Introduction to the Rock-Forming

Minerals. Harlow, England: Pearson Education Limited, 1992.

Furlong, Kevin. “Earthquakes Illuminate Mantle Under Tibet.” 1996. Earth &

Environmental Systems Institute, Penn State University. 17 May 2005

<http://www.psu.edu/ur/archives/SCIENCETECH/Tibet.html>.

Gribble, Colin D. and Allan J. Hall. Optical Mineralogy Principle and Practice. New

York: Chapman & Hall, 1992.

Hecht, Eugene. Optics. San Francisco: Addison Wesley, 2002.

86

“HKL Technology: EBSD Basics.” 2005 HKL Technology. 27 May 2005

<http://www.hkltechnology.com/?pid=4,18&articles>.

Karato, Shun-ichiro. The Dynamic Structure of the Deep Earth. New Jersey: Princeton

University Press, 2003.

Marion, Jerry B. and Stephen T. Thorton. Classical Dynamics of Particles and

Systems. Fort Worth: Harcourt College Publishers, 1995.

Neese, William D. Introduction to Optical Mineralogy. New York: Oxford University

Press, 2004.

Nelson, Stephen A. “Biaxial Minerals.” Mineralogy. Course Website. 2002.

Tulane University. 30 Mar. 2005 <http://www.tulane.edu/~sanelson/geol211/

biaxial.htm>.

---. “The Isotropic Indicatrix, Isotropic Minerals, and the Immersion Method.”

Mineralogy. Course Website. 2002. Tulane University. 30 Mar. 2005

<http://www.tulane.edu/~sanelson/geol211/isotropic_minerals.htm>.

---. “Uniaxial Minerals, Uniaxial Indicatrix, Optic Sign, & Ray Path.”

Mineralogy. Course Website. 2002. Tulane University. 30 Mar. 2005

<http://www.tulane.edu/~sanelson/geol211/uniaxial_minerals.htm >.

Sen, Gautam. Earth’s Materials: Minerals and Rocks. New Jersey: Prentice Hall, 2001.

Shelley, David. Manual of Optical Mineralogy. New York: Elsevier, 1975.

Weisstein, Eric W. “Rotation Matrix.” MathWorld - A Wolfram Web Resource. 1999.

Wolfram Research Inc. 30 Mar. 2005

<http://mathworld.wolfram.com/RotationMatrix.html>.

87

Wenk, H.-R., Dawson, P.R., et al. “AGU Mantle Video.” Texturing of Rocks in the

Earth’s Mantle. 1999. UC Berkeley Texture Laboratory. 17 May 2005

<http://eps.berkeley.edu/~wenk/TexturePage/AGU-Mantle-Video.htm>.

Wilen, Larry. Private communication. 2003 - 2005.

88

Appendix A A derivation of the corrected rotation matrix. Given the matrix identity relationships: IAA =−1 1111)( −−−− = ABCABC Since mathematically the rotations of phi and psi are in the same plane when θ = 0,

then:

ψ

φψφ

ψψψψ

φφφφ

ψφφψψφφψφψψφψφφψ

φφφφ

ψψψψ

φφφφ

ρ

ρρρ

=

−=

−

+−−+

=

−

−

−=

−

1000cossin0sincos

1000cossin0sincos

1000sinsincoscossincossincos0sincossincossinsincoscos

1000cossin0sincos

1000cossin0sincos

1000cossin0sincos

1

So therefore,

ψθφ

φψφθφ

φφθφφθφψφθφ

φφθφφθφψφθφ

ρρρρρρρρ

ρρρρρρρρρρρ

ρρρρρρρρρρρρ

=

=

=

=

−

−−−−

−−−−

1

1111

1111corrected )()()(

89

Appendix B Derivation of the formula for the angle between the x-axis and the semi-major axis or

semi-minor axis of an ellipse.

Starting with the equation for an ellipse centered at the origin:

12

2

2

2

=+by

ax

Then rotate the body of the ellipse by an angle θ:

[ ]

[ ]

22

2cos2

2sin2cossin2

2cossin22

2

2sin2

2cos

22

2cos2

2sin2cossin

2cossin

2cossin

2cossin2

2

2sin2

2cos

2

2cos2

2cossin

2

2sin2

2cossin

2cossin

2

2sin2

2cossin

2

2cos2

2

2cos2

cossin

2

2sin2cossin

2cossin

2

2sin2cossin

2

2cos

cossinsincos

2cos

2sin

2sin

2cos

cossinsincos

21

0

021

cossinsincos

cossinsincos

21

0

021

cossinsincos

yba

xyba

xba

yba

xybaba

xba

b

y

b

xy

a

y

a

xy

b

xy

b

x

a

xy

a

x

yx

b

y

b

x

a

y

a

x

b

y

b

x

a

y

a

x

yx

b

y

b

x

a

y

a

x

yx

b

ayxyx

yx

b

ayx

++−++

=++−+−++

=+−

+++−++

=

+−

++

−++

=−

+−

+

=−

+−+

=−

−

θθθθθθθθ

θθθθθθθθθθθθ

θθθ

θθθθθθθθθ

θθθ

θθθ

θθθθθθ

θθθθθθθθ

θθθθ

θθθθ

θθθθ

θθθθ

90

Define A = coefficient of x2, B = coefficient of xy, and C = coefficient of y2

+=

−=

+= 2

2

2

2

222

2

2

2 cossincossin2cossin2sincosba

Cba

Bba

A θθθθθθθθ

Let’s try to determine what θ is. The following identities from trigonometry are

employed extensively:

θθθθ

θθθθ

θθ

2cos1sin2cossin2cos1cos2sincos

1sincos

222

222

22

−=−=−

=−=−

=+

A good place to start is to simplify things later by combining terms:

( )

+=

++=

++

+=

++

+=+

22

2222

222

222

2

2

2

2

2

2

2

2

11

11sincos

11sin11cos

cossinsincos

ba

ba

baba

babaCA

θθ

θθ

θθθθ

Now let’s try subtraction:

91

( ) ( )

( )

( )

( )θ

θθ

θθθ

θθθ

θθθθ

θθ

θθθθ

2cos11

2cos2cos

1sincos2cos

1cossin2cos

1cossin1sincos

11sin11cos

cossinsincos

22

22

222

2

222

2

222

222

222

222

2

2

2

2

2

2

2

2

−=

−=

−−=

−+=

−+−=

−+

−=

+−

+=−

ba

ba

ba

ba

ba

abba

babaCA

After taking another look at B, it appears I have another simplification that will

eliminate all dependence of A, B, and C on “a” or “b”:

( )

θθθ

θθθ

θ

θθ

2tan2cos2sin

2coscossin2

2cos11

cossin211

22

22

=

=

=

−

−

=−

ba

baCA

B

Now solving for θ is trivial:

−

= −

CAB1tan

21θ

Converting to degrees will add an extra term:

−

= −

CAB1tan

21180

πθ

o

92

Appendix C The coefficients of the 2D ellipse centered at the origin in terms of the rotation angles

φ, θ, ψ, ξ and φinner and the values of a, b, and c.

Equation of an ellipse using primes: 122 =′+′+′ yCxyBxA

Please Note: 222

1,1,1c

Cb

Ba

A ===

** = “to the power of”

A′ =B*cos(xi)**2*cos(psi)**2*cos(phi_in)**2+C*cos(theta)**2-2*cos(phi_in)* cos(theta)*B*cos(psi)*sin(psi)*cos(xi)**2*sin(phi_in)-1*C*cos(xi)**2* cos(phi_in)**2*cos(theta)**2-1*A*cos(psi)**2*cos(xi)**2*cos(theta)**2 +2*cos(xi)*sin(phi_in)*cos(phi)*C*sin(theta)*sin(xi)*cos(theta)+B*cos(xi)**2+A-1*B*cos(psi)**2*cos(theta)**2-1*B*cos(xi)**2*cos(phi_in)**2+C*cos(phi)**2 *cos(xi)**2+C*cos(xi)**2*cos(phi_in)**2+2*A*cos(psi)**2*cos(xi)**2+A*cos(xi)**2*cos(theta)**2+A*cos(psi)**2*cos(theta)**2-2*B*cos(xi)**2*cos(psi)**2-1*B* cos(xi)**2*cos(phi)**2+2*cos(phi_in)*sin(phi)*C*cos(phi)*cos(xi)**2*sin(phi_in)-2*cos(phi_in)*sin(phi)*C*cos(phi)*cos(xi)**2*sin(phi_in)*cos(theta)**2+2*A*cos (psi)**2*cos(phi)**2*cos(xi)**2*cos(phi_in)**2-4*cos(phi)*A*cos(psi)*sin(psi) *cos(theta)*sin(phi)*cos(xi)**2*cos(phi_in)**2-2*cos(xi)*cos(phi_in)*cos(phi)* B*sin(psi)*cos(psi)*sin(xi)*sin(theta)+2*sin(phi)*A*cos(psi)*sin(psi)*cos(theta) *cos(phi)*cos(xi)**2-2*cos(phi_in)*sin(phi)*A*cos(theta)**2*cos(phi)*cos(xi)**2 *sin(phi_in)*cos(psi)**2+2*cos(phi_in)*cos(phi)*B*sin(phi)*cos(xi)**2*sin(phi_in)*cos(psi)**2-2*cos(phi_in)*cos(phi)*B*sin(phi)*cos(xi)**2*sin(phi_in)-2*cos(xi)*sin(phi_in)*cos(phi)*cos(theta)*A*sin(xi)*sin(theta)+2*cos(xi)*sin(phi_in)*cos(phi)*cos(theta)*A*sin(xi)*sin(theta)*cos(psi)**2+B*cos(psi)**2*cos(xi)**2* cos(theta)**2-2*cos(xi)*cos(phi_in)*sin(phi)*cos(theta)*A*sin(xi)*sin(theta)+ 2*cos(phi_in)*cos(theta)*A*sin(psi)*cos(psi)*cos(xi)**2*sin(phi_in)-1*A*cos(psi)**2*cos(xi)**2*cos(phi_in)**2-2*B*cos(psi)**2*cos(theta)**2 *cos(phi)**2*cos(xi)**2*cos(phi_in)**2-2*C*cos(phi)**2*cos(xi)**2 *cos(phi_in)**2-1*C*cos(phi)**2*cos(xi)**2*cos(theta)**2-1*A*cos(theta)**2+2*A*cos(theta)**2*cos(phi)**2*cos(xi)**2*cos(psi)**2

93

*cos(phi_in)**2-2*sin(phi)*B*sin(psi)*cos(psi)*cos(theta)*cos(phi)*cos(xi)**2 +2*cos(xi)*cos(phi_in)*sin(phi)*cos(theta)*A*sin(xi)*sin(theta)*cos(psi)**2+A *cos(theta)**2*cos(phi)**2*cos(xi)**2-2*B*cos(xi)**2*cos(psi)**2*cos(phi)**2 *cos(phi_in)**2+B*cos(psi)**2-4*cos(phi_in)*A*cos(psi)*sin(psi)* cos(theta)*cos(phi)**2*cos(xi)**2*sin(phi_in)-1*A*cos(psi)**2-2*cos(xi)*sin(phi_in)*cos(phi)*cos(theta)*B*cos(psi)**2*sin(xi)*sin(theta)-2*cos(phi_in)*cos(phi)*A*cos(psi)**2*sin(phi)*cos(xi)**2*sin(phi_in)+2*cos(phi_in)*sin(phi)*B*cos(psi)**2*cos(theta)**2*cos(phi)*cos(xi)**2*sin(phi_in)+2*B*cos(xi)**2*cos(phi)**2*cos(phi_in)**2-2*cos(xi)*cos(phi_in)*sin(phi)*cos(theta)*B *cos(psi)**2*sin(xi)*sin(theta)+4*cos(phi)*B*sin(psi)*cos(psi)*cos(theta)*sin(phi) *cos(xi)**2*cos(phi_in)**2+2*cos(xi)*sin(phi_in)*sin(phi)*B*sin(psi)*cos(psi)*sin(xi)*sin(theta)-2*cos(xi)*sin(phi_in)*sin(phi)*A*cos(psi)*sin(psi)*sin(xi)*sin(theta) +4*cos(phi_in)*B*sin(psi)*cos(psi)*cos(theta)*cos(phi)**2*cos(xi)**2 *sin(phi_in)-1*C*cos(theta)**2*cos(xi)**2+2*cos(xi)*cos(phi_in)*sin(phi)*C *sin(theta)*sin(xi)*cos(theta)+2*cos(xi)*cos(phi_in)*cos(phi)*A*cos(psi)*sin(psi)* sin(xi)*sin(theta)+B*cos(xi)**2*cos(psi)**2*cos(phi)**2+2*C*cos(phi)**2*cos(xi)**2*cos(theta)**2*cos(phi_in)**2+2*cos(phi_in)*sin(phi)*A*cos(theta)**2*cos(phi)*cos(xi)**2*sin(phi_in)+B*cos(psi)**2*cos(theta)**2*cos(xi)**2*cos(phi_in)**2+A*cos(theta)**2*cos(xi)**2*cos(phi_in)**2-1*A*cos(theta)**2*cos(xi)**2* cos(phi_in)**2*cos(psi)**2-2*A*cos(theta)**2*cos(phi)**2*cos(xi)**2* cos(phi_in)**2-1*A*cos(theta)**2*cos(phi)**2*cos(xi)**2*cos(psi)**2-1*A*cos(psi)**2*cos(xi)**2*cos(phi)**2+B*cos(psi)**2*cos(theta)**2*cos(phi)**2*cos(xi)**2-1*A*cos(xi)**2; B′ =-2*cos(xi)*cos(phi_in)*C*sin(phi_in)*cos(theta)**2-4*cos(xi)*sin(phi_in)*B* cos(psi)**2*cos(theta)**2*cos(phi)**2*cos(phi_in)+4*cos(xi)*cos(phi_in)*A *cos(psi)**2*cos(phi)**2*sin(phi_in)-8*cos(xi)*B*sin(psi)*cos(psi)*cos(theta)* cos(phi)**2*cos(phi_in)**2+4*cos(xi)*cos(phi)*A*cos(psi)**2*sin(phi) *cos(phi_in)**2+2*cos(xi)*cos(phi)*C*sin(phi)-2*cos(xi)*B*sin(psi)*cos(psi) *cos(theta)+4*cos(xi)*B*sin(psi)*cos(psi)*cos(theta)*cos(phi_in)**2-2*sin(xi)*sin(theta)*B*cos(psi)*sin(psi)*cos(phi)*sin(phi_in)+2*cos(xi)*cos(phi)*B *cos(psi)**2*cos(theta)**2*sin(phi)+8*cos(xi)*cos(phi_in)*cos(phi)*B*sin(psi)* cos(psi)*cos(theta)*sin(phi)*sin(phi_in)+4*cos(xi)*cos(phi)*C*sin(phi)*cos(theta)**2*cos(phi_in)**2-4*cos(xi)*sin(phi)*B*cos(phi)*cos(psi)**2*cos(phi_in)**2+ 4*cos(xi)*sin(phi_in)*A*cos(theta)**2*cos(phi)**2*cos(phi_in)*cos(psi)**2-4*cos(xi)*cos(phi)*C*sin(phi)*cos(phi_in)**2-2*sin(xi)*C*cos(theta)*sin(theta) *cos(phi)*cos(phi_in)+2*sin(xi)*sin(theta)*B*cos(psi)**2*cos(theta)*cos(phi)* cos(phi_in)+2*sin(xi)*C*cos(theta)*sin(theta)*sin(phi)*sin(phi_in)-2*sin(xi) *sin(theta)*B*cos(psi)*sin(psi)*sin(phi)*cos(phi_in)-2*sin(xi)*sin(theta)* B*cos(psi)**2*cos(theta)*sin(phi)*sin(phi_in)+2*sin(xi)*sin(theta)*A*cos(theta) *cos(phi)*cos(phi_in)-2*sin(xi)*sin(theta)*A*cos(theta)*cos(phi)*cos(phi_in) *cos(psi)**2-4*cos(xi)*sin(phi)*B*cos(psi)**2*cos(theta)**2*cos(phi) *cos(phi_in)**2+2*sin(xi)*sin(theta)*A*sin(psi)*cos(psi)*cos(phi)*sin(phi_in)

94

+2*sin(xi)*sin(theta)*A*sin(psi)*cos(psi)*sin(phi)*cos(phi_in)-4*cos(xi) *cos(phi_in)*B*cos(phi)**2*sin(phi_in)*cos(psi)**2+2*cos(xi)*sin(phi_in)*B*cos(phi_in)*cos(psi)**2+2*cos(xi)*cos(phi_in)*B*cos(psi)**2*cos(theta)**2*sin(phi_in)+2*cos(xi)*cos(phi)*A*cos(theta)**2*sin(phi)-4*cos(xi)*cos(phi)*A* cos(theta)**2*sin(phi)*cos(phi_in)**2-8*cos(xi)*cos(phi_in)* cos(phi)*A*cos(psi)*sin(psi)*cos(theta)*sin(phi)*sin(phi_in)-4*cos(xi)*sin(phi_in)*A*cos(theta)**2*cos(phi)**2*cos(phi_in)-2*cos(xi)*sin(phi)*B*cos(phi)+4*cos(xi)*sin(phi)*B*cos(phi)* cos(phi_in)**2+2*cos(xi)*sin(phi)*B*cos(phi)*cos(psi)**2+4*cos(xi)* cos(phi_in)*B*cos(phi)**2*sin(phi_in)+2*sin(xi)*sin(theta)*A*cos(theta)*sin(phi)*sin(phi_in)*cos(psi)**2-2*cos(xi)*cos(phi)*C*sin(phi)*cos(theta)**2+ 4*cos(xi)*cos(theta)*B*cos(psi)*sin(psi)*cos(phi)**2-4*cos(xi)* cos(theta)*A*sin(psi)*cos(psi)*cos(phi)**2-2*sin(xi)*sin(theta)* A*cos(theta)*sin(phi)*sin(phi_in)+2*cos(xi)*A*cos(psi)*sin(psi) *cos(theta)-4*cos(xi)*sin(phi_in)*C*cos(phi)**2*cos(phi_in)+4*cos(xi)*sin(phi_in) *C*cos(phi)**2*cos(phi_in)*cos(theta)**2-4*cos(xi)*A*cos(psi)*sin(psi)*cos(theta) *cos(phi_in)**2-2*cos(xi)*cos(phi_in)*A*cos(theta)**2*sin(phi_in)*cos(psi)**2-2*cos(xi)*cos(phi)*A*cos(theta)**2*sin(phi)*cos(psi)**2+4*cos(xi)*cos(phi)*A *cos(theta)**2*sin(phi)*cos(psi)**2*cos(phi_in)**2+2*cos(xi)*cos(phi_in)*A*cos(theta)**2*sin(phi_in)-2*cos(xi)*sin(phi_in)*B*cos(phi_in)+8 *cos(xi)*A*cos(psi)*sin(psi)*cos(theta)*cos(phi)**2*cos(phi_in)**2-2*cos(xi)*sin(phi_in)*A*cos(psi)**2*cos(phi_in)+2*cos(xi)*cos(phi_in)*C*sin(phi_in)-2*cos(xi)*sin(phi)*A*cos(psi)**2*cos(phi); C ′ =-1*B*cos(phi_in)**2*cos(psi)**2-1*C*cos(theta)**2-2*A*cos(theta)**2 *cos(psi)**2*cos(phi)**2*cos(phi_in)**2+A*cos(psi)**2*cos(phi_in)**2+2*C *cos(phi)**2*cos(phi_in)**2+C*cos(theta)**2*cos(phi)**2+C*cos(theta)**2*cos(phi_in)**2-1*B*cos(phi)**2*cos(psi)**2-1*A*cos(theta)**2*cos(phi_in)**2-2*B*cos(phi_in)**2*cos(phi)**2-1*C*cos(phi)**2+2*cos(phi)*B*sin(psi) *cos(psi)*cos(theta)*sin(phi)+B*cos(phi_in)**2+B*cos(phi)**2-1*C* cos(phi_in)**2+B*cos(psi)**2*cos(theta)**2-2*sin(phi)*A*cos(psi)*sin(psi) *cos(theta)*cos(phi)-1*A*cos(psi)**2*cos(theta)**2+A*cos(psi)**2 *cos(phi)**2-1*B*cos(psi)**2*cos(theta)**2*cos(phi)**2+A*cos(theta)**2-1*A* cos(theta)**2*cos(phi)**2-2*sin(phi_in)*cos(theta)*A*sin(psi)*cos(psi) *cos(phi_in)+4*sin(phi_in)*A*cos(psi)*sin(psi)*cos(theta)*cos(phi)**2*cos(phi_in)-2*sin(phi_in)*sin(phi)*A*cos(theta)**2*cos(phi)*cos(phi_in)+2* sin(phi_in)*cos(theta)*B*cos(psi)*sin(psi)*cos(phi_in)+2*sin(phi_in)*sin(phi)*C*cos(phi)*cos(phi_in)*cos(theta)**2-2*sin(phi_in)*cos(phi)*B*sin(phi) *cos(phi_in)*cos(psi)**2+4*sin(phi)*A*cos(psi)*sin(psi)*cos(theta) *cos(phi)*cos(phi_in)**2-2*sin(phi_in)*sin(phi)*B*cos(psi)**2 *cos(theta)**2*cos(phi)*cos(phi_in)-2*C*cos(phi)**2*cos(phi_in)**2 *cos(theta)**2+2*B*cos(phi_in)**2*cos(psi)**2*cos(phi)**2-2 *A*cos(psi)**2*cos(phi_in)**2*cos(phi)**2+A

95

*cos(theta)**2*cos(psi)**2*cos(phi)**2+A*cos(theta)**2*cos(psi)**2*cos(phi_in)**2+C-2*sin(phi_in)*sin(phi)*C*cos(phi)*cos(phi_in)+2*sin(phi_in)*cos(phi)*B *sin(phi)*cos(phi_in)+2*A*cos(theta)**2*cos(phi)**2*cos(phi_in)**2+2*sin(phi_in)*cos(phi)*A*cos(psi)**2*sin(phi)*cos(phi_in)-1*B*cos(psi)**2*cos(theta)**2* cos(phi_in)**2+2*B*cos(psi)**2*cos(theta)**2*cos(phi)**2*cos(phi_in)**2 -4*sin(phi_in)*B*sin(psi)*cos(psi)*cos(theta)*cos(phi)**2*cos(phi_in)+2 *sin(phi_in)*sin(phi)*A*cos(theta)**2*cos(phi)*cos(phi_in)* cos(psi)**2-4*sin(phi)*B*sin(psi)*cos(psi)*cos(theta)*cos(phi)*cos(phi_in)**2;