Optical Mineralogy - Summary of Nesse PDF

INTRODUCTION TO OPTICAL MINERALOGY by William Nesse CHAPTER 1: Light The Nature of Light Light is a form of energy transmitted from one place to another at finite velocity. It behaves as if it is composed of numerous tiny particles that travel bullet-like or behave a...

INTRODUCTION TO OPTICAL MINERALOGY by William Nesse CHAPTER 1: Light The Nature of Light Light is a form of energy transmitted from one place to another at finite velocity. It behaves as if it is composed of numerous tiny particles that travel bullet-like or behave as a wave phenomena. Thus, two theories were developed: a. Particle Theory - light is considered to be composed of subatomic particles called photons which are released when electrons revert back to their normal energy level after being excited. b. Wave Theory - light is considered as a form of radiant energy that travels wave-like from one point to another. These waves have electrical and magnetic properties and thus, are called electromagnetic radiation. Electromagnetic Radiation All electromagnetic radiation consists of electric and magnetic vectors that vibrate at right angles to the direction in which the radiation is moving. The interaction of the electric vector with the electrical character of the atoms and chemical bonds in minerals affects the behavior of light. The vibration direction of the electric vector is transverse as it vibrates perpendicular to the direction in which the light wave is propagating. Visible light has wavelengths of 400 −7 nm - 700 nm (1 nm = 10 cm) in a vacuum. A light wave can be described using velocity, frequency and wavelength: 𝑉 (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦) 𝑓= λ (𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ) Frequency - number of wave crests per second that pass a certain point. It is expressed in cycles per second or hertz (Hz). Wavelength - distance from one wave crest to another. The frequency remains constant regardless of the material that the light travels through. 1  The intensity or brightness of light is proportional to the square of the amplitude (A). Amplitude is the height of the wave. Wave front - surface that connects similar points on adjacent waves. Wave normal - a light constructed at right angles to the wave front which represents the direction that the wave is moving. Light ray - direction of propagation of light energy. In isotropic materials, light ray and wave normal coincide or parallel. While in anisotropic materials, wave normal and light ray directions are not parallel. Phase Retardation (∆) - the distance that one wave lags behind the other. a. IN-PHASE: When retardation equals an integral number (i.e. 1, 2, 3) of wavelengths. The two waves constructively interfere with each other producing a resultant wave which is the arithmetic sum of the two. b. OUT-OF-PHASE: When retardation equals 1/2,1 1/2,2 1/2, etc. wavelengths. The two waves destructively interfere and cancel each other. c. PARTIALLY IN-PHASE OR PARTIALLY OUT-OF-PHASE: When retardation is some intermediate value (i.e. 1/3, 1 1/3, etc.). The two waves are partially constructive or partially destructive relative to each other. If two waves vibrate at an angle to each other, they can be resolved into a resultant wave by means of vector addition. A component of a single wave may be resolved into any arbitrary vibration direction. The amplitude of the new vibration direction, Z, is obtained by Z = X cos θ. Perception of Color Light whose wavelength in a vacuum is ~660 nm is perceived as orange, and so forth. Monochromatic light - all one wavelength. It is perceived as whatever wavelength is present. Polychromatic light - consist of more than one wavelength and a combination of wavelengths is perceived as a single color. The sensation of all colors, except 420 nm (violet), 500 nm (green) and 660 nm (red), can be produced by suitable combinations of two or more different wavelengths. When all of the visible spectrum is present, it is perceived as white. Complementary light colors are combinations of two colors perceived as white. No complementary light color set that includes green exists. Sensation of purple is mixtures of red and violet light while brown is mixtures of red, blue and yellow light. Interaction of Light and Matter A. Transmitted Light a. Velocity - the velocity of light depends on the nature of the material that it travels through and the wavelength of the light. The maximum possible velocity is 10 17 3. 0 𝑥 10 𝑐𝑚/𝑠𝑒𝑐 (3 𝑥 10 𝑛𝑚/𝑠𝑒𝑐) in a vacuum. 2 b. Index of refraction (n) - light is bent when passing from one transparent material to another at any angle other than perpendicular. It is a measure of how effective a material is in bending light coming from a vacuum. It can be obtained by: 𝑉 (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡 𝑖𝑛 𝑎 𝑣𝑎𝑐𝑢𝑢𝑚) 𝑣 𝑛= 𝑉 (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙) The index of refraction of a vacuum is 1.0 and for all other materials, n > 1.0. Most minerals have n = 1.4 - 2.0. High index indicates low velocity and vice versa! Snell’s Law - calculates how much light will be bent on traveling from one material to another. 𝑠𝑖𝑛 θ1 𝑛2 𝑠𝑖𝑛 θ2 = 𝑛1 Where n1 and n2 are indices of materials 1 and 2, and θ1 and θ2 are angles between the wave normal and the normal boundary. In general, light is refracted towards the normal to the boundary upon entering a material with higher refractive index and is refracted away from the normal upon entering a material with lower refractive index. Snell’s law applies for both isotropic and anisotropic materials. In anisotropic materials, the angles must be measured from the wave normals, not the rays. c. Reflection - the amount of light reflected from the surface depends on the index of refraction of the two materials. When light is reflected from the boundary between two materials, the angle of incidence and the angle of reflection are identical. Percentage Reflection, R - used when both materials are transparent. 2 2 1 𝑠𝑖𝑛 (𝑖−𝑟) 𝑡𝑎𝑛 (𝑖−𝑟) Percentage Reflection (R) = 2 ( 2 + 2 ) 𝑥 100 𝑠𝑖𝑛 (𝑖+𝑟) 𝑡𝑎𝑛 (𝑖+𝑟) 𝑛2 − 𝑛1 2 Normal incidence (R) = (𝑛 + 𝑛1 ) 𝑥 100 2 Absorption coefficient (k) - a unitless number that varies from zero for transparent materials to over 5 for extremely opaque materials. d. Critical Angle and Total Internal Reflection Critical Angle - angle of incidence that yields an angle of refraction of 90 degrees. Small CA corresponds to a high index of refraction. Light with an angle of incidence greater than the critical angle (CA) cannot be refracted into the low-index material Because an angle greater than CA would prevent light from entering the low-index materials, it is internally reflected instead. 1 Total Internal Reflection = 𝑛1 = 𝑠𝑖𝑛 𝐶𝐴 3 e. Optical Class Optically Isotropic - includes minerals of an isometric system which have velocities that do not vary with direction, a single unit cell dimension, single index of refraction and the chemical bonds are the same in all directions. Optically Anisotropic - includes minerals of hexagonal, tetragonal, orthorhombic, monoclinic and triclinic systems. Light velocity is different in different directions. They have lower symmetry, show different strengths of chemical bonding in different directions and light passing in most directions is doubly refracted. This means that light is split into two plane polarized rays vibrating at right angles. Optic axes are directions along which light is not split into two rays. a. Optically Uniaxial - minerals under tetragonal and hexagonal systems. They have a single optic axis and two different indices of refraction. b. Optically Biaxial - minerals under orthorhombic, monoclinic and triclinic systems. They have two optic axes and three different indices of refraction. f. Dispersion - a consequence of the interaction of light with natural resonant frequencies of the electron clouds around each other. The electric vector causes the electron cloud to resonate at the frequency of light. Violet is more strongly refracted, has the shortest wavelength, higher energy while red is less refracted, has the longest wavelength and lower energy. The longer the wavelength, the lower the energy. The longer the wavelength, the lower the indices of refraction. Normal dispersion - refractive indices increase with increasing frequency and decreasing wavelength. Occurs if the frequency of light is significantly different from a resonant frequency of the electron clouds. Abnormal dispersion - refractive indices decrease with increasing frequency and decreasing wavelength. Occurs if the frequency of light is nearly the same as the natural resonant frequencies of the electron clouds. Thus, light is strongly absorbed. To describe the dispersion of the material, index of refraction is reported at several wavelengths: 486 nm, 589 nm and 656 nm. These are Fraunhofer lines or absorption lines. It is 𝑛𝐷, the index for light of 589-nm is meant when the index of a material is reported. Coefficient of Dispersion = 𝑛𝐹 − 𝑛𝐶 Where f and c corresponds to indices at 486 nm and 656 nm, respectively. A large coefficient of dispersion means the material shows a large change of index as a function of wavelength. 4  𝑛𝐹−𝑛𝐶 Dispersive Power = 𝑛𝐷−1 A large value for dispersive power means that the material shows a large change of index as a function of wavelength. g. Light Absorption and Color The color of a mineral or any other object is the color of light that is NOT absorbed on transmission or reflection. A white object reflects all visible light spectrum. A clear mineral transmits all visible light spectrum. A black object absorbs all wavelengths and if a mineral is colored, it selectively absorbs certain wavelengths and transmits or reflects the remaining. The colors of light depend on dispersion. If the frequency of light is significantly different from the natural resonance, then light is transmitted (normal dispersion). If the frequency of light is nearly the same as the natural frequency of the electron clouds, then light is absorbed (abnormal dispersion). Polarized Light Ordinary light - is unpolarized and vibrates in all directions at right angles to the direction of propagation. Polarized light - the vibration direction is constrained and it is not uniformly distributed around the direction of propagation. Three types of polarization: 1. Plane Polarization (also called linear polarization) - the electric vector vibrates in a single plane and light wave is a simple sine wave with vibration direction lying in the plane of polarization. 2. Circular Polarization - produced by two waves of polarized light with the same amplitude and whose vibration directions lie at right angles to each other. One wave is retarded ¼ relative to the other. The outline of the helix is a circle. 3. Elliptical Polarization - the same as circular polarization but the wave is retarded by a value different from ¼. a. Polarization by Double Refraction - when light enters an anisotropic material, it experiences double refraction where light is split into two rays vibrating at right angles to each other, have different velocities and may have different absorption. Selective Absorption - the basis for polarization with polarizing films. Pleochroism is observed through this concept where light vibrating at one direction is strongly absorbed while the other is transmitted. - Herapathite - material used for the original polarizing film developed in 1928 by Edwin Land. A second means of eliminating one of the two plane polarized rays produced by the double refraction utilizes the critical angle effect. A device known as Nicol prism is used, which is a clear calcite cut diagonally and glued back together with balsam cement (n = 1.537). 5  b. Polarization by Reflection - when unpolarized light strikes a smooth surface, the reflected light is polarized light so that the vibration direction is parallel to the reflecting surface. The reflecting light is not completely plane polarized unless the angle between the reflected and refracted rays is 90 degrees. Brewster’s Angle - angle of incidence needed to produce the 90 degrees angle between the reflected and refracted rays. 2 𝑛 𝑛1 = 𝑡𝑎𝑛 θ𝐵 Complete polarization of the reflected light is achieved if the angle of incidence is Brewster’s angle which places the reflected and refracted rays at 90 degrees to each other. c. Polarization by Scattering - when light passes through the air, an amount is scattered by the dust particles and the air molecules. Scattered light is polarized so that it vibrates in a plane at right angles to the original path of the light. The wavelengths that are most strongly scattered depend on the size of the scattering particles. In this case, blue light is the most strongly scattered. CHAPTER 2: Petrographic Microscope Direction Conventions The convention used is compass directions. The NE-SW and NW-SE are also referred to as 45 degrees. Samples Grain mounts, thin sections and the spindle stage are all used to examine transparent minerals with transmitted light. Polished sections are used to examine opaque minerals using reflected light. A. Grain Mounts - consists of small grains of a mineral covered with a piece of 0.17 mm thick glass. A dropper is used to introduce a liquid called an immersion oil between the slide and coverslip to surround and cover the grains. B. Thin Sections - thin slices of rock or mineral mounted on the microscope slide with a thickness of 0.03 mm. They are studied with transmitted light and used in the study of rock mineralogy and textures. C. Spindle Stage - consist of a wire spindle on which a single mineral grain is cemented. D. Polished Sections - samples of rock or mineral whose top surface has been polished to a mirror-like shine. Illuminator - an incandescent light mounted in the base to provide transmitted light. Rheostat control - adjusts light intensity. Filters are sometimes used. Field diaphragm - an iris diaphragm that controls the size of the area on the sample that is illuminated. 6 Substage Assembly - consists of lower polarizer, aperture diaphragm, condensing lens, auxiliary condensing lens and one or more filter holders. Lower polarizer - allows the vibration direction of the polarized light passing through the microscope to be set in any desired orientation (usually, E-W). Sometimes referred to as Nicols because Nicol prisms were used to provide polarized light on early petrographic microscopes. Aperture diaphragm - an iris diaphragm above or below the condensing lens. It adjusts the size of the cone of light that passes up through the microscope. Closing the aperture diaphragm will decrease the size of the cone of light and increase the contrast in the image. Condenser lenses - concentrates the light onto the area of the sample immediately beneath the objective lens. The light reaching the sample from the fixed condenser is only moderately converging (orthoscopic illumination). Auxiliary condensing lens - provides conoscopic illumination which consists of strongly converging light. It also allows production of optical phenomena called interference figures. Gray filters - used to adjust the intensity of the illumination. Colored filters - used to adjust the color balance of the light or to produce roughly monochromatic light. Microscope Stage - the circular stage of the petrographic microscope. Stage goniometer - allows the accurate measurement of angles of rotation. Objective Lenses - provides the primary magnification of the optical system. Numerical Aperture (NA) - measure of the size of the cone of light that it can accommodate. 𝐴𝐴 𝑁𝐴 = 𝑛 𝑠𝑖𝑛 2 Angular Aperture (AA) - the size of the cone of light that the lens can accept. Free-working distance (FWD) - distance between the end of the lens and the top of the sample. Resolution of a lens - measure the ability to reveal fine detail. - Limit of Resolution (d) - the smallest distance between two points that can still be distinguished. λ 𝑑= 2𝑁𝐴 - A small limit of resolution means the lens has a high resolving power. - As the numerical aperture increases, the resolving power of the lens increases but the limit of resolution decreases. - The numerical aperture and resolving power can be increased by placing something with a higher index of refraction between the lens and the objective. Oil immersion lenses are used. The oil used has n = 1.515, so the maximum NA is about 1.4. 7  Depth of the field - distance between the lower and the upper limits of reasonably sharp focus. Lenses with low magnification and low NA have a relatively large depth of field, and vice versa. P indicates that the lens is constructed of strain-free lens elements and is intended for use with polarized light. Vertical Illuminator - used to examine opaque minerals with reflected light Upper Polarizer - also known as analyzer. It is located above the objective lens. The vibration direction of the lower and upper polarizers are set so that they are at right angles to each other. When the upper polarizer is inserted, the polarizers are said to be crossed. If the upper polarizer is removed, the polarizers are said to be uncrossed and the view through the microscope is with plane light. Bertrand Lens - allow the observer to view optical phenomena called interference figures. Ocular - also known as eyepieces. These are lenses that slide into the upper end of the microscope tube. They magnify the image provided from the objectives and focus the light so that it can be accepted by the eye. The total magnification of a microscope = magnification of the objective lens x magnification of the ocular. A higher power ocular increases the magnification but not the resolution. Cross hairs or other markings are mounted in the ocular. Each ocular micrometer division is equal to 0.0062 mm. The ocular micrometer scale is 0.31 mm with respect to the stage micrometer. - The calibration is done by focusing on the stage micrometer and determining how many millimeters are presented by each unit on the ocular micrometer. Focusing Mechanism Focusing is accomplished by raising or lowering the stage, or by raising or lowering the microscope tube. Coarse focusing - allows rapid changes in the distance between the lens and the stage. Fine focusing - allows for very precise adjustment. Accessories Gypsum plate (∆ = 550 𝑛𝑚 𝑜𝑟 537 𝑛𝑚) - also known as one-wavelength or first-order red plate. It may be marked Gips, Gyps, rot I, Quartz-sensitive tint, 1λ, ∆ = 550 nm or 537 nm. Mica plate - also known as quarter-wavelength plate. It may be marked mica, glimmer, ¼ λ, or ∆ = 147 nm. Quartz wedge - a piece of quartz ground into a wedge shape. Additional Equipment 8  Mechanical stage - grasps the slide so that it can be moved in a systematic way. Universal stage - allows thin sections to be rotated about several axes. Adjustment of the Microscope Adjusting the Oculars: the cross hairs or other markings in the ocular is brought into focus by turning the upper part of the lens in or out of the tube. Focusing: the microscope is focused by turning the focusing knobs mounted on the microscope base. Adjusting the Illuminator: Ideally, the intensity of illumination should be adjusted by inserting or removing neutral gray filters to preserve the same color balance at all intensities. But a rheostat control is also used. Centering the Objectives: when the objectives are centered, the cross hairs are centered on the point about which the image rotates as the stage is turned. Adjusting the Substage: insert the auxiliary condenser and stop the aperture diaphragm down until it is almost closed. With the low power objective, only a small spot of light will be seen somewhere near the center of the field of view. Move the substage with its adjusting screws until the spot of light is centered on the cross hairs. Alignment of Polarizers: lower polarizers may pass light that vibrates either E-W or N-S. The simplest way to check this alignment is with a thin section containing biotite. - In a plane light (analyzer removed), biotite should be darkest when the cleavage is parallel to the vibration direction of the lower polarizer and lightest when it is at right angles. (darkest - E-W; lightest - N-S) - If the field of view is dark, but not entirely black, then the polarizers are not oriented at exactly 90 degrees from each other. CHAPTER 3: Refractometry Immersion Method - this method compares the index of refraction of a mineral to the known index of an immersion oil. The simplest and most convenient way of measuring refractive indices. Cases: 𝑛𝑚𝑖𝑛 ≠ 𝑛𝑜𝑖𝑙 : light is refracted into the mineral making it stand out. 𝑛𝑚𝑖𝑛 = 𝑛𝑜𝑖𝑙 : light is unrefracted thus, the mineral do not stand out. If both mineral and oil are colorless, they are almost indistinguishable. Relief - the degree to which mineral grains stand out from the mounting medium. Based on the difference of refractive indices of the mineral and the oil. If the difference between 𝑛𝑚𝑖𝑛 and 𝑛𝑜𝑖𝑙 : ≥ 0. 12 - high relief 0.12 - 0.04 - intermediate relief 0.04 - low relief Comparison of Indices 9  𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : positive relief 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : negative relief To determine whether the index of refraction of the oil is higher or lower than the index of refraction of the mineral, two convenient methods can be used: (1) Becke Line and (2) Oblique Illumination. 1. Becke Line - when mineral grains in grain mount are slightly out of focus, a band of light called the Becke Line should be visible along grain boundaries in plane light. The band of light may either be on the inside or the outside of the grain. Procedure for observing Becke lines: it is most distinct if the aperture diaphragm is stopped down and an intermediate power objective (e.g. 10x) is used. Focus is raised and the Becke line should appear moving into the material with higher index. The production of the Becke line involves the lens effect and the internal reflection effect. a. Lens Effect - most mineral fragments are thinner on the edges than in the middle, so they act as crude lenses. Cases: 𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : the grain acts as a converging lens concentrating the light toward the center of the grain. 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : the grain acts as a diverging lens concentrating the light into the oil. b. Internal Reflection Effect - the edges of the grains must be vertical at some point. Moderately converging light is either refracted or internally reflected depending on the angles of incidence and the indices of refraction. Cases: 𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : the Becke line is inside the grain boundary. 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : the Becke line is outside the grain boundary. Both the lens effect and the internal reflection effect concentrates light into the material with the higher index of refraction in the area above the mineral grain. Movement of Becke Line: 𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : the cone of light converges above the mineral. 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : the cone of light diverges. If the microscope is crisply focused on the grain, the Becke Line is coincident with the edge of the grain or it may disappear. Hence, as the stage is lowered, the Becke line moves toward the material with the higher index of refraction. The Becke Line is often paired with a dark band moving in the opposite direction. It exists because the light that might otherwise be present there is reflected or refracted to form the Becke line. 10  Dispersion Effects - The dispersion of immersion oil is greater than the dispersion of most minerals, so it is possible to produce a match for the indices at only one wavelength. - If the dispersion curves intersect in the visible spectrum, the oil will have the higher indices of refraction for wavelengths shorter than the match, and the mineral will have higher indices of refraction for the longer wavelengths. This results in the formation of two Becke lines - one with shorter wavelength moving into the oil and the other with longer wavelength moving into the mineral. - The color of the lines depends on the wavelength at which the dispersion curves cross. - The longer wavelength forms a yellowish-orange Becke line. - The shorter wavelength forms a bluish Becke line. - The color band corresponding to higher IR is much brighter than the other. - If the dispersion curves for mineral and immersion oil do not intersect, it forms a single white Becke line. - If a sodium vapor lamp or other monochromatic light source is used, only one Becke line is produced. - If the indices of the mineral and oil are matched for a particular wavelength, no Becke line is produced. 2. Oblique Illumination Method - involves examining the “shadows” cast by the grains when part of the light coming up through the microscope is blocked. Case: 𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : if the grains are bright on the side facing the darkened part of the field. 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : if the grains are dark on the side facing the darkened part of the field. BRACKETING METHOD: bracketing the index of the unknown mineral to determine a match. 1. Prepare the first grain mount using an immersion oil with IR that is in the middle of the available range. Determine the relief of the grains. The oblique illumination method may give better results if relief is very high. 𝑛𝑚𝑖𝑛 > 𝑛𝑜𝑖𝑙 : all indices less than the oil is eliminated. 𝑛𝑚𝑖𝑛 < 𝑛𝑜𝑖𝑙 : all higher indices are eliminated. 2. Prepare the second grain mount using an immersion oil with IR that is in the middle of the remaining range. Examine the relief. If the relief in the first mount was moderate or low, the second oil can be a little closer to the first. Otherwise, the second can be a little further away. 3. Prepare the third grain mount using an oil with IR that is in the middle of the remaining range. Use the relief seen in the first and second mounts as a guide in selecting the oil. If the index of the mineral did not fall between the first and second oil and the mineral has a very high relief in the second oil, oil at the extreme ends of the remaining 11  range may be used to determine whether the mineral falls within the range of available oil. 4. Prepare the fourth grain mount using an oil with IR that is in the middle of the remaining range of possible indices. With luck, the first three steps will have provided a bracket narrow enough to allow making a close match with the fourth oil. 5. Repeat until a match is obtained. Accuracy of the Immersion Method The accuracy depends on which the oils are calibrated, quality of the microscope optics and whether white or monochromatic light is used. Intermediate indices can be obtained by mixing two of the oils in various proportions. The index of the mixed oil should be measured with a refractometer. An important source of inaccuracy is contamination of the index oils through dirty slides and allowing the oils to evaporate by leaving the container open. The indices of the oils vary as a function of temperature. Most are calibrated for 20 degrees Celsius. The index of refraction decreases with increasing temperature. A correction factor of 0.0004/C is applied when the temperature varies from 20 C. Determining Indices of Refraction in Thin Section It is generally not possible to determine IR of minerals accurately in thin sections. However, the IR of an unknown mineral can be compared to the IR of the cement or to known minerals in the thin section. Common cements: Canada Balsam: n = 1.537 Epoxy: n = 1.540 The relief gives an indication of how close the index of the mineral is to the cement. CHAPTER 4: Optics of Isotropic Materials In isotropic materials, the velocity of light is the same in all directions. Isotropic materials include liquids, gasses, glasses and minerals in the isometric system. If the solid is strained, some chemical bonds will be stretched and others will be compressed and the normally isotropic materials will become anisotropic. Isotropic Indicatrix Indicatrix - a geometric figure that shows the index of refraction and vibration direction for light passing in any direction through a material. It is constructed so that the indices of refraction are plotted on lines from the origin that are parallel to the vibration directions of the light. For isotropic materials, the index of refraction for 𝑛𝑎 and 𝑛𝑏 are equal. The indicatrix of an isotropic material is a sphere. All radii are the same length and the index of refraction is the same for all propagation and vibration directions. To find the index of refraction, the wave normal is constructed through the center of the indicatrix and a section through the center of the indicatrix is constructed perpendicular 12  to the wave normal. This section is parallel to the wave front. The index of refraction for this light is the radius of the section that is parallel to the vibration direction of light. Distinguishing between Isotropic and Anisotropic Minerals Isotropic and anisotropic minerals can be quickly distinguished by crossing the polarizers. All isotropic minerals are dark between crossed polarizers while anisotropic minerals are light unless they are in certain orientations. Isotropic minerals are dark because they do not affect the polarization direction of the light coming up from the lower polar. The light that passes through the mineral on the stage is absorbed by the upper polar. However, bits of light along edges or cracks may be seen because the polarization of the light is modified somewhat by reflection. Anisotropic minerals do affect the polarization of light, so some light is generally able to pass through the upper polar. Anisotropic minerals appear dark or extinct once in every 90 degrees. If only a few grains of a mineral are present, methods below can be used to tell if it is isotropic: 1. Obtaining an interference figure: if a grain is actually anisotropic, but remains dark, an optic axis figure will be produced. 2. Using a gypsum plate: Weak anisotropism can be detected by inserting the gypsum plate. If the grain displays an interference color different from the normal magenta color of the gypsum plate, it is anisotropic. Otherwise, it is isotropic. Identification of Isotropic Minerals a. Grain Mounts - determining the IR is the primary means of identifying isotropic minerals in grain mounts. The index of many minerals varies depending on the chemical composition. If samples of the unknown mineral are large enough, physical properties can aid the identification. The color of isotropic minerals remains the same on rotation of the stage. The cleavage commonly found in the isometric crystal system include: cubic {001} (three at right angles), octahedral {111} (four cleavages that outline an octahedron), and dodecahedron {110} (six cleavages that outline a dodecahedron). Isometric minerals do not systematically fracture into elongate or splintery fragments. b. Thin Sections - estimates of IR can only be made when working with thin sections in determining IR’s. Lacking a value for refractive index, other properties of the mineral must be used. It is also possible to get some idea of the crystal shape from the thin section. The shape seen in the thin section is the shape of a random slice through the crystal. - Cubic crystals show three- or four-sided shapes. - Octahedrons show four- or six-sided shapes. - Dodecahedrons show six- or eight-sided shapes. 13  Cleavage in the thin section usually shows as straight lines or cracks in the grains. CHAPTER 5: Optics of Anisotropic Minerals Anisotropic minerals are distinguished by: (1) velocity of light varies depending on direction through the mineral and (2) showing double refraction. As light enters anisotropic minerals, light is split into two rays with different velocities. Optic axis - the direction in which light behaves as though the mineral were isotropic. a. Uniaxial minerals (those in hexagonal and tetragonal) have only one optic axis. b. Biaxial minerals (those in orthorhombic, monoclinic, triclinic) have two optic axes. The two rays are plane polarized and vibrate at right angles to each other. a. Fast ray - the ray with the lower index. b. Slow ray - the ray with the higher index. Propagation direction vs. Vibration direction: Propagation direction is the direction that the light is traveling while the vibration direction is the side-to-side oscillation of the electric vector of the plane polarized light. Electromagnetic theory explains why the velocity of light varies depending on direction through the anisotropic mineral. Light velocity depends on the interaction between the vibration of the electric vector and the resonant frequencies of the electron clouds and so, velocity will vary with direction. Electromagnetic theory also explains why the light entering an anisotropic mineral is split into two rays vibrating at right angles to each other. - Within the wave front, the strength of the electric field varies with direction. The plot of the electric field strength within that is an ellipse. The axes of the ellipse represent the minimum and maximum field strengths, and are at 90 degrees to each other which then corresponds to the vibration direction of the two rays. The two rays experience different electron cloud densities and field strengths with different associated resonant frequencies and thus, their velocities and indices must be different. - Along the optic axes, instead of an ellipse, the plot of the electric field strength is a circle. Interference Phenomena When an anisotropic mineral is placed between crossed polars, it is light and may show vivid colors. The colors seen under crossed polars are called interference colors. Interference colors are produced because light is split into two rays on passing through the mineral. Monochromatic Illumination a. Retardation (∆) - the distance that the slow ray is behind the fast ray after both have exited the crystal. The magnitude of retardation depends on the thickness (d) of the crystal plate and the difference in the velocity of the slow ray (𝑉𝑠) and fast ray (𝑉𝑓). 14  ∆ = 𝑑 (𝑛𝑠 − 𝑛𝑓) b. Birefringence (δ) - the difference between the indices of refraction of the slow and fast rays. It depends on the path followed by the light through the mineral. δ = 𝑛𝑠 − 𝑛𝑓 Birefringence also may vary depending on the wavelength of light. Unless stated otherwise, numerical values of birefringence are for light whose wavelength is 589 nm. c. Interference of the Two Rays Interference phenomena are produced when the two rays are resolved into the vibration direction of the upper polarizer. Cases (**should not be confused with in-phase and out-of-phase vibrating IN THE SAME PLANE): a. IN-PHASE but vibrating at right angles: the slow ray is retarded exactly one wavelength relative to the fast ray. When the two rays are in phase but vibrating at right angles, the resolved components are in opposite directions so they destructively interfere and cancel each other. Thus, no light passes the polarizer and the mineral appears dark. b. OUT-OF-PHASE but vibrating at right angles: the retardation is equal to one-half wavelength. The rays are resolved into the vibration direction of the upper polarizer. However, both components are in the same direction so the light constructively interferes and light passes the upper polarizer. The brightest illumination is where the two rays are one-half wavelength out of phase. For a colorless mineral and ideal optical conditions with no losses from reflection or absorption, the amount of light that passes through the upper polarizer is given by: 2 180 𝑑𝑒𝑔 𝑥 ∆ T (% transmission) = (− 𝑠𝑖𝑛 λ 𝑥 𝑠𝑖𝑛 2τ 𝑥 𝑠𝑖𝑛(τ − 90 𝑑𝑒𝑔)) 𝑥 100 If the angle (τ) between the lower polarizer vibration direction and the closest vibration direction in the mineral is 45 degrees, this yields the brightest illumination and the equation becomes: 2 180 𝑑𝑒𝑔 𝑥 ∆ T = (− 𝑠𝑖𝑛 λ ) 𝑥 100 If the upper polarizer is rotated so that it is parallel with the lower polarizer, the relations described (in-phase and out-of-phase) are reversed. - When the waves are in-phase, they constructively interfere when resolved into the vibration direction of the upper polarizer. And they destructively interfere when they are one-half wavelength out of phase. Polychromatic Illumination If white light is used, all of the different wavelengths are present. Interference colors are produced when a combination of different wavelengths passes through the upper polarizer. The color depends on the retardation between the fast and the slow rays as well as the thickness and birefringence of the mineral. 15  Orders of Interference colors: if a quartz wedge is used, at the thin edge of the wedge, the thickness and retardation is zero, so all the wavelengths cancel at the upper polarizer and the color is black. As the thickness increases, the color (order) increases. a. If the retardation for all wavelengths is 250 nm, then the slow ray and fast rays for all of the visible spectrum are out-of-phase. Thus, the light appears while. b. If the retardation is 500 nm, a portion of the blue and green section is canceled at the upper polarizer and the light appears red. c. If the retardation is 2500 nm, part of each section of the spectrum is allowed to pass and the light is perceived as a creamy white. Order of Interference Colors First-order colors are produced by retardations of less than 550 nm. Second-order colors are produced by retardation between 550 and 1100 nm, and so forth. First-order and second-order colors are the most vivid. The highest-order colors become progressively more and more washed out degenerating into a creamy white. Anomalous Interference Colors Anomalous interference colors are produced when birefringence and retardation are significantly different for different wavelengths of light. A different complement of wavelengths passes the upper polarizer and is perceived as a different interference color. Mineral color also influences the interference color because some wavelengths of light are selectively absorbed by the mineral. Determining Thickness of the Sample The equation ∆ = 𝑑 (𝑛 − 𝑛 ) provides the basis for determining thickness of a 𝑠 𝑓 sample. Retardation can be determined by examining the interference color. The retardation corresponding to that color is read from the bottom of the interference color chart. a. Thin Section - Quartz is commonly used to determine the thickness of a thin section. Quartz grains range from black to a maximum color because birefringence varies from 0 - 0.009 depending on grain orientation. - The retardation is the vertical line in the chart and is read first. Followed by the thickness which is represented by the horizontal lines and then, followed by the diagonal lines which provides the birefringence. b. Grain Mount - The thickness of grains in a grain mount can be easily estimated using the same technique employed for thin sections. - Measure the dimensions of a number of grains using a calibrated micrometer eyepiece. Then, determine the maximum and minimum dimensions based on the size of sieves used to separate the grains. 16 Determining Birefringence from the Color Chart The equation ∆ = 𝑑 (𝑛 − 𝑛 ) provides the basis for determining the birefringence of 𝑠 𝑓 a sample. Thickness must be known and the retardation is estimated by recognizing the interference color. There is a room for error in determining the maximum birefringence of a mineral based on interference colors. a. Thin Section - in most cases the value of maximum birefringence is needed. To measure the maximum birefringence, scan the slide to find one or more grains of the unknown mineral showing the highest order interference color. b. Grain Mount - because the thickness is not generally known with certainty, it is difficult to make accurate estimates of birefringence based on interference colors in grain mounts. Recognizing the Different Orders of Interference Colors Colors in the first several orders of interference colors may appear similar and can be difficult to distinguish at a glance. Colored minerals tend to mask the interference colors. The order of an interference color up to fourth order or higher may be determined by looking at the edges of grains. Many are thinner at their edges than in the center, so the entire interference color sequence may be present at the edge of the grain. The order of the color in the center of the grain can be determined by “counting” the colors in. First-order white is usually a clear white grading to bluish gray or yellow. High-order white tends to be somewhat creamy colored and may show pale pastel highlights of color due to irregularities on grain surfaces. If in doubt, use the gypsum plate. If the color is first-order white, it will change to an upper first- or lower second-order color. If the color is a high-order white, little change in color will occur. Extinction - occurs when one vibration direction of the mineral is oriented to the lower polarizer that no component of incident light can be resolved into the mineral vibration direction oriented parallel to the upper polarizer. Unless an optic axis is vertical, anisotropic minerals go dark or extinct every 90 degrees of rotation of the microscope stage. If the stage is rotated and the vibration directions of the mineral are in 45 degree positions, a maximum component of both slow and fast ray is available to be resolved into the vibration direction of the upper polarizer. The phase relation between the slow and fast ray as well as the interference color is unaffected by stage rotation. The equation for %transmission can be used to predict extinction. If the angle τ between the mineral vibration direction and the lower polarizer direction is 0 degrees, %transmission for all wavelengths is zero. If the τ = 45 degrees, then a maximum amount of light is allowed to pass the upper polarizer. Extinction Angle - the angle between the length of the cleavage of a mineral and one of the mineral’s vibration direction. 17  Procedure: 1. Rotate the stage until the length or the cleavage of the mineral grains is aligned with the N-S crosshair. Record the reading as EA1. 2. Rotate the stage until the mineral grain goes extinct. The rotation direction doesn’t matter. Record the reading as EA2. 3. The extinction angle is the angle of rotation needed to make the mineral go extinct. It is the difference between EA1 and EA2. In many cases, grains oriented to produce maximum birefringence are also in the orientation to produce the extinction angle. Categories of Extinction a. Parallel Extinction - the mineral is extinct when the cleavage or length is aligned with one or the other of the cross hairs. The extinction angle is 0. b. Inclined Extinction - the mineral is extinct when the cleavage or length is at some angle to the cross hairs. c. Symmetrical extinction - the mineral displays either two cleavage directions or two distinct crystal faces to which two extinction angles can be measured, one from each cleavage or crystal face. If the two EA are the same, the mineral displays symmetrical extinction. d. No extinction angle - there is no cleavage or elongation available from which to measure an extinction angle, although it goes extinct every 90 degrees. Different parts of a single mineral grain may go extinct at different points of stage rotation. These are caused by strain and chemical zoning. a. Undulatory Extinction - different parts of a single strained grain are in slightly different orientations and go extinct at different times. This extinction in a grain follows an irregular or wavy pattern. b. Chemical Zoning - the center of the crystal may go extinct at a different time than the outer part due to zoning. Use of Accessory Plates To determine which of the two rays coming through the mineral is the slow ray and which is the fast ray, accessory plates are used. SLOW on SLOW (Retardations add) - when a grain’s vibration directions are oriented 45 degrees and its slow ray vibration coincides with the slow ray direction in the accessory plate. The result is a higher total retardation. When the two rays reach the upper polarizer, a higher-order interference color is produced. (∆𝑀 = retardation when the two rays leave the mineral; ∆𝐴 = retardation of the accessory plate) ∆𝑇 = ∆𝑀 + ∆𝐴 SLOW on FAST (Retardation cancel) - when a grain is rotated so that its fast ray vibration direction is parallel to the slow ray vibration direction of the accessory plate, the result is that the accessory plate cancels some of the retardation produced by the mineral. A lower order interference color is produced. ∆𝑇 = |∆𝑀 − ∆𝐴| 18  The gypsum plate produces around 550 nm of retardation giving a distinctive magenta color at the boundary of first- and second-order. The mica plate produces 147 nm of retardation yielding a first-order white interference color. The fast ray (NW-SE) direction of the accessory plate is parallel to its length and the slow ray is across its width (NE-SW). To determine which vibration direction in a mineral grain belong to the slow ray and which to the fast ray, the following procedure is performed: 1. Rotate the stage of the microscope until the grain is extinct. 2. Rotate the stage 45 degrees clockwise accurately using the stage goniometer. Note the interference color and record the retardation. 3. Insert the gypsum plate. If the retardation added, a retardation of ∆𝑀 + 550 nm is produced resulting in higher order interference color. If the retardation canceled, a retardation of ∆𝑀 - 550 nm is produced resulting in lower order interference color. 4. If the retardations are added, then the slow ray in the mineral is oriented NE-SW parallel to the slow ray of the accessory plate. If the retardations cancel, then the fast ray in the mineral is oriented NE-SW. With grain mounts, it is often useful to employ the quartz wedge instead of the mica or gypsum plate. Grains often display the lowest order interference colors along its thin edges and the highest order color in the center. As the quartz wedge is inserted (thin end first), bands of interference colors along the edge move either into or out of the grain. a. If the color bands move out, retardation adds. b. If the color band moves in, retardations are canceling. Sign of Elongation (Only applies to elongated minerals!) a. Length Slow (Positive Elongation) - the slow ray vibrates more or less parallel to length or trace of cleavage of an elongate mineral. b. Length Fast (Negative Elongation) - the fast ray vibrates more or less parallel to the length or trace of cleavage of an elongate mineral. Procedure to determine the sign of elongation: 1. Place the grain under the cross hairs, insert the upper polarizers and rotate the stage so the mineral is extinct and the long dimension or cleavage is nearest the N-S crosshair. 2. Rotate the stage 45 degrees clockwise. Note the interference color and retardation. 3. Insert the accessory plate and note the new interference color and determine the total retardation. a. Length Slow - retardations add, so equation, ∆𝑇 = ∆𝑀 + ∆𝐴 is used. b. Length Fast - retardation cancel, so equation, ∆𝑇 = |∆𝑀 − ∆𝐴| is used. A sign of elongation cannot be assigned if the vibration directions are ~45 degrees to the length of a crystal. 19 Relief The change of relief is a consequence of the fact that the two rays coming through the mineral have different indices of refraction. If the stage is rotated so that the fast ray vibration of the mineral is E-W, all light coming through is fast ray. If the mineral has the same index as the oil, the relief is low. If the grain is oriented so the slow ray vibration direction is E-W, all the light coming through is slow ray. If the mineral has a substantially different index than the oil, the relief is high. In intermediate orientations, the relief is also intermediate because two rays of light come through. One produces an image of high relief, the other produces a superimposed image of low relief. There are also two different Becke Lines produced, one for each ray. If both rays have indices either higher or lower than the oil, the two Becke lines are superimposed and indistinguishable. But if the oil selected has an index between the indices of the slow and fast ray, one Becke line moves into the grain and the other moves out. Birefringence depends on the direction that the light passes through a mineral. Grains that are oriented to display maximum birefringence show a maximum change of relief as well as the highest interference color. If birefringence is low, the change of relief is not noticed. Pleochroism (Only in Plane Light/Uncrossed Nicols!) Pleochroism (also dichroism) - is the change of color as the stage is rotated in plane light (upper polarizer removed). Two rays of light are absorbed differently as they pass through the colored mineral and therefore have different colors. Pleochroism is seen with the upper polarizer removed (plane light) and is not related to interference colors that are seen between crossed polarizers. CHAPTER 6: Uniaxial Optics When a cleavage rhomb of calcite is placed on a dot, two images appear, each composed of plane polarized light vibrating at right angles to each other. The light ray that produces an image that moves and behaves differently from anything found with isotropic materials is called the extraordinary ray (ε). The light ray that produces an image that is stationary and behaves as if it is an isotropic material is called the ordinary ray (ω). The amount that the two images are split depends on the birefringence. The extraordinary ray always vibrates perpendicular to the ordinary ray in a plane that contains the c-axis which coincides with the optic axis. (An optic axis is a direction along which light can propagate without splitting into two rays) Birefringence is zero for light propagating along the optic axis (c-axis) and is maximum for light propagating at right angles to the optic axis. Optic Sign 20  The extraordinary ray (ε) has lower index of refraction than the ordinary ray (ω). This provides basis for defining the optic sign: OPTICALLY POSITIVE: 𝑛 > 𝑛ω ε OPTICALLY NEGATIVE: 𝑛 < 𝑛ω ε If the extraordinary ray is the slow ray, it is optically positive. If the extraordinary ray is the fast ray, it is optically negative. The index of the extraordinary ray is variable, thus 𝑛ε refers to the maximum or minimum index of the extraordinary ray. The term 𝑛ε’ refers to an index for an extraordinary ray that lies between 𝑛ε and 𝑛ω. Crystallographic Considerations Uniaxial minerals are all either HEXAGONAL or TETRAGONAL. Their common characteristic is a high degree of symmetry about the c crystallographic axis. There is uniform chemical bonding in all directions within the (001) plane, which is at right angles to the c-axis, and different strength bonding between these planes. The ordinary ray has the same velocity regardless of the path because it always vibrates in the same electronic environment while the velocity of the extraordinary ray varies depending on direction. Whether the extraordinary ray has a higher or lower index than the ordinary ray depends on the chemical bonding and crystal structure of the mineral. Uniaxial Indicatrix If the indices for light traveling in all directions are plotted, the result is an ellipsoid of revolution whose axis is the optic axis. A. Uniaxial Positive (𝑛ε > 𝑛ω) Indicatrix: Prolate Spheroid If the mineral is optically positive, the ellipsoid is prolate and is stretched out along the optic axis. B. Uniaxial Negative (𝑛ε < 𝑛ω) Indicatrix: Oblate Spheroid If the mineral is optically negative, the ellipsoid is oblate and flattened along the optic axis. The radius of the indicatrix along the optic axis is always 𝑛ε. Principal Section - a section through the indicatrix that includes the optic axis. It is an ellipse whose axes are 𝑛ε and 𝑛ω. It always includes 𝑛ε. Circular Section - a section perpendicular to the optic axis whose radius is equal to 𝑛ω (thus, it always contains the 𝑛ω). Use of Indicatrix 21  Light is considered to pass directly through the center of the indicatrix. Determining the indices of refraction and vibration direction of wave normals passing in a random direction through the mineral is performed by the following: 1. Construct the wave normal direction through the center of the indicatrix. 2. Construct a slice through the center of the indicatrix perpendicular to the wave normal. Unless it is perpendicular to the optic axis, the section is an ellipse whose axes are 𝑛ε’ and 𝑛ω. 3. The vibration directions are parallel to the axes of the elliptical section and the indices of refraction are the lengths of the axes. If the angle θ between the extraordinary wave normal and the optic axis is known, the value of 𝑛ε’ can be calculated through: 𝑛ω 𝑛ε' = 2 1 2 𝑛ω 2 [1+( 2 −1) 𝑠𝑖𝑛 θ] 𝑛ε The extraordinary ray direction can be determined by constructing a tangent to the indicatrix from the wave normal and parallel to 𝑛ε'. The angle between the optic axis and the extraordinary ray is given by: 2 𝑛ω 𝑡𝑎𝑛 Ψ = 2 𝑡𝑎𝑛θ 𝑛ε If the ray direction for an extraordinary ray is specified instead of the wave normal direction, follow the steps below to determine the wave normal direction, vibration directions and IR. 1. Construct the ray direction through the center of the indicatrix at angle Ψ from the optic axis. 2. Construct a surface tangent to the indicatrix at the point where the rays pierce it. The angle between the tangent surface and the optic axis is given by: 2 𝑛ε 𝑡𝑎𝑛 (90 − θ) = 2 𝑡𝑎𝑛 Ψ 𝑛ω 3. Construct a section through the center of the indicatrix parallel to the tangent surface. 4. The extraordinary ray vibrates parallel to the ellipse axis for 𝑛ε’. The wave front for the extraordinary ray is parallel to the section through the indicatrix and the wave normal is perpendicular to the wave front. Birefringence and Interference Colors The birefringence and interference color of uniaxial minerals depends on the direction that light passes through the mineral. Case 1 to 3 involves light that is normally incident to the surface of a mineral cut in different orientations and Case 4 involves inclined incidence. Case 1-3 is applied during orthoscopic illumination while Case 4 is applied during conoscopic illumination. 22  CASE 1 (for orthoscopic illumination): Normal incidence on a sample cut so that the top and bottom surfaces are parallel to the optic axis. The birefringence is 𝑛ε - 𝑛ω so, there is a maximum birefringence and high interference colors. In this case, the extraordinary ray and its wave normal are parallel. The section through the indicatrix is a principal section. CASE 2 (for orthoscopic illumination): Normal incidence on a sample cut perpendicular to the optic axis. The mineral will appear dark on rotation like an isotropic mineral and have low interference colors. The section through the indicatrix is the circular section, so all light passes as ordinary ray. The extraordinary ray experiences double refraction and develops retardation. If the mineral has high birefringence, it may not appear entirely black. CASE 3 (for orthoscopic illumination): Normal incidence on a sample cut in a random direction. The birefringence is intermediate and will show an intermediate interference color. The extraordinary ray path diverges from the ordinary ray path. CASE 4 (for conoscopic illumination): Inclined Incidence. The mineral is cut in a random direction and the incident and refracted light lie in a principal section of the indicatrix. The birefringence is intermediate since 𝑛ε’ is intermediate. The interference color will be higher than if the same birefringence were experienced with normal incidence because the inclined path through the mineral is longer. Extinction In thin sections, cleavages typically appear as thin parallel cracks in a mineral grain. If the relief is high, cleavages can be recognized, but if relief is low, they may be difficult to see. A. Tetragonal Minerals - typically prismatic and either elongate or stubby parallel to c-axis. The usual cleavages are prismatic and pinacoidal. 1. CASE 1: The crystal is cut perpendicular to the optic axis, so the light traveling through the crystal follows the optic axis. No extinction angle can be measured. The section is uniformly dark on rotation. The indicatrix is circular displaying the ω color. An optic axis figure is produced. 2. CASE 2: The crystal is cut parallel to the optic axis. Shows maximum birefringence and parallel extinction. Light travels through the crystal perpendicular to the optic axis. The indicatrix is a principal section. In plane light, the ω is displayed; if rotated 90 degrees, the ε will be displayed. A flash figure is obtained. Optically Positive (𝑛ε > 𝑛ω) : length slow Optically Negative (𝑛ε < 𝑛ω) : length fast 3. CASE 3: The crystal is cut on an angle. Shows intermediate birefringence. The index for the extraordinary ray is intermediate and the ordinary ray vibrates parallel to the trace of the pinacoidal cleavage so extinction is always parallel to that cleavage. If rotated 90 degrees, will displayed a color intermediate of ω and ε. An off-center interference figure is obtained. Miter cut: extinction is parallel. 23  Diamond-shaped section: produced by a diagonal cut. Extinction is parallel to the pinacoidal cleavage and symmetrical to the prismatic cleavage. Parallelogram-shaped section: produced by a random cut. Extinction is parallel to the trace of the pinacoidal cleavage and asymmetric with respect to the prismatic cleavage. B. Hexagonal Minerals - forms include prisms, pinacoids, prisms and rhombohedrons. Common cleavages are prismatic, pinacoidal and rhombohedral. a. Rhombohedral Cleavage - three cleavage planes intersecting at angles other than 90 degrees. 1. CASE 1: The mineral is cut perpendicular to the optic axis. The crystal displays uniform dark color on rotation between crossed polars. The indicatrix section is circular and all three cleavages are visible. The ω is displayed in plane light. An optic axis figure is obtained. 2. CASE 2: The mineral is cut parallel to the optic axis. Extinction is symmetrical to two cleavage traces and parallel to the third. If the cut is in any other orientation parallel to the c-axis, then the extinction will typically not be symmetrical or parallel to any cleavages. Birefringence is maximum. In plane light, it displays ω color and if rotated 90 degrees will display the ε color. A flash figure is obtained. 3. CASE 3: The mineral is cut in a random direction. All three cleavages are visible but extinction is not parallel nor symmetrical to any of them. Birefringence is intermediate. In plane light, it displays ω color and if rotated 90 degrees will display an intermediate color between ω and ε. An off-center interference figure is observed. b. Prismatic and Pinacoidal Cleavage - three prismatic cleavages intersecting at 60 degrees and 120 degrees, and a pinacoidal cleavage at right angles to the c-axis. 1. CASE 1: The mineral is cut perpendicular to the optical axis. The prismatic cleavages are visible while the pinacoidal cleavages are not. The angles between the cleavages are 60 or 120 deg. The ω is displayed in plane light. The grain is uniformly dark and the indicatrix is a circular section. An optic axis figure is produced. 2. CASE 2: The mineral is cut parallel to the optic axis. Extinction is parallel to both prismatic and pinacoidal cleavages. The indicatrix section is a principal section. In plane light, it displays ω color and if rotated 90 degrees will display the ε color. Extinction is parallel and birefringence is a maximum. A flash figure is produced. 3. CASE 3: The mineral is cut randomly. All cleavages may be visible. The vibration direction of the ordinary ray is always parallel to the trace of the pinacoidal cleavage. Birefringence is intermediate. In plane light, it displays ω color and if rotated 90 degrees will display an intermediate color between ω and ε. An off-center interference figure is produced. 24 Pleochroism - pleochroism, or change of color on rotation in plane light, occurs when the extraordinary and ordinary rays are absorbed differently on passing through a mineral. Colored uniaxial minerals are usually pleochroic. If the change of color is substantial, it is described as strong pleochroism and if there is relatively little color change, it is weak pleochroism. The following procedure can be used to determine the color of each ray for uniaxial minerals: 1. Cross the polarizers and search the sample for a grain that shows the lowest-order interference color. The optic axis is vertical and light passes as ordinary ray. (CASES 1) 2. Uncross the polarizers and note the color of the grain. This color is the color of the ordinary ray. The grain should remain the same on rotation as there is no extraordinary ray that can pass. 3. Cross the polarizers and search the sample for a grain that shows the highest-order interference color. The optic axis is horizontal and the optic plane is parallel to the stage. Both ordinary and extraordinary rays are present. (CASES 2) 4. Uncross the polarizers. Maximum change of color is observed as the stage is rotated. The extraordinary ray color differs from the ordinary ray color and is seen when the grain is in one of its extinction positions (at 90 degrees). [CASES 3] Grains cut in a random direction (CASE 4) display the color of the ordinary ray in plane light and if the stage is rotated 90 degrees, the ε’ is able to pass through and an intermediate color between the extraordinary and ordinary ray is displayed. Interference Figure - provides the basis for determining whether an anisotropic mineral is uniaxial or biaxial and also for determining the optic sign. Obtaining an Interference Figure 1. Focus on a mineral grain with a high-power objective. 2. Flip the auxiliary condenser and open the aperture diaphragm. 3. Cross the polarizers. 4. Insert the bertrand lens or remove the ocular and look down the microscope tube. **The interference figure consisting of a pattern of interference colors and dark bands appears near the top surface of the objective lens. The nature of the pattern depends on the orientation of the mineral grain. A. Optic Axis Figure - produced optic axis is perpendicular to the stage and thus, vertical. This does not move or change as the stage is rotated. Parts: 1. Isogyres - black bars forming a cross. Formation of Isogyres: formed where the vibration directions in the interference figure correspond to the vibration directions of the lower and upper polarizers. They are areas of extinction. Ordinary rays vibrate parallel to lines of latitude while extraordinary rays vibrate parallel to lines of longitude. If the polarizers are crossed, isogyres form where the vibration directions in the figure are parallel to the vibration directions of the polarizers. 2. Melatope - point in the center where the isogyres cross. It marks the emergence of the optic axis. Interference colors increase in order outward from the melatope. 25  Those nearest the melatope are low first order. (should not be confused with interference colors in grain mounts!) 3. Isochrome/s - each band of color. Formation of Isochromes: the auxiliary condenser provides strongly converging light that is collected by the objective lens. This converging light follows paths in different retardations as both birefringence and distance through the sample increase as the inclination of the light path to the optic axis increases. Because the optic axis is vertical and the optical properties are symmetric about the optic axis, rings of equal retardation are produced about the melatope. The number of isochromes depends on the birefringence and thickness of the sample. Samples that are thick or have high birefringence produce more isochromes because retardation is a function of both birefringence and thickness. The number of isochromes also depends on the numerical aperture of the objective lens. High numerical aperture accepts a larger cone of light and thus produces more of the higher-order isochromes. Determining the Optic Sign using an Optic Axis Figure - it can be determined whether the ordinary ray is the fast ray or the slow ray. OPTICALLY POSITIVE: if the ordinary ray is the fast ray. OPTICALLY NEGATIVE: if the ordinary ray is the slow ray. One of the accessory plates can be used to determine which is fast and which is slow. This performed by the following procedures: 1. Obtain an optic axis interference figure. Note the retardation of the interference colors. 2. Insert the accessory plate. 3. Observe the interference colors. In two quadrants, the retardations add while in the other two, retardations cancel. (Quadrants are labeled I-IV from upper right (NE) then clockwise) 4. Consider the SE quadrant (II quadrant). In this configuration, the ordinary ray vibrates NE-SW and the extraordinary ray vibrates NW-SE. The slow vibration direction of the accessory plate is NE-SW, parallel to the ordinary ray vibration direction. OPTICALLY POSITIVE: Retardations cancel in the SE quadrant (it turns yellow). OPTICALLY NEGATIVE: Retardation adds in the SE quadrant (it turns blue). Either the gypsum or mica plate may be used to determine the optic sign. If the interference figure has numerous isochromes, it may be desirable to use the quartz wedge to determine the optic sign. As the wedge is inserted at the thin end first, the isochromes move. - Consider the SE quadrant, once again. If the isochromes move outward (away from the melatope; low interference colors near the center displaces the high interference colors on the edge), it is optically positive. If the isochromes move inward, it is optically negative. With the accessory plate inserted, the isogyres adopt the interference color corresponding to the retardation of the accessory plate. 26 B. Off-Center Optic Axis Figure - produced when the optic axis is inclined somewhat from the vertical. The result is that the interference figure will no longer be centered in the field of view. By noting the direction of the isogyre movement as the stage is rotated clockwise, it is possible to identify the quadrant present in the field. Usually, it sweeps from quadrant I to IV. If the quadrant is identified, the procedure for optic sign determination in optic axis figures can be performed. A hazard in working with interference figures that do not contain a melatope is that you cannot be entirely certain whether the mineral is uniaxial or biaxial. C. Flash Figure - produced when the grain is oriented with the optic axis horizontal or parallel to the stage. The isogyre is a broad, fuzzy cross because vibration directions in all but the other parts of the four quadrants are parallel to the polarizers. If the stage is rotated a small amount, the isogyres quickly split and move out of the field of view in opposite quadrants corresponding to the quadrants into which the optic axis is being moved. When the optic axis is placed at 45 degrees (NW-SE), the entire field of view is occupied with interference colors and the isochromes concave outward. Interference colors decrease outward in the quadrants containing the optic axis and increase outward in the other quadrants. The number of isochromes depends on the thickness and birefringence. If the central part of the figure is first-order white, the quadrant containing the optic axis will be first-order gray and the other quadrants will be pale first-order yellow. Determining the Optic Sign using a Flash Figure: 1. Rotate the stage until the isogyre fills the field of view. The optic axis is now parallel to N-S or E-W crosshair. 2. Slowly rotate the stage and note the quadrants from which isogyres leave the field of view. The optic axis will enter in the quadrants from which the isogyres exited. 3. If the isogyres left the field NW-SE, continue rotating the stage to make a total of 45 degrees placing the optic axis NW-SE. If the isogyres left the field NE-SW, reverse the direction of rotation so that they leave from NW-SE and continue rotating until 45 degrees. Near the center of the field, the extraordinary ray vibrates NW-SE parallel to the optic axis while the ordinary ray vibrates NE-SW. 4. Insert the accessory plate (slow NE-SW) and note the change in interference colors. a. OPTICALLY POSITIVE: Retardation cancels and the ordinary ray is the fast ray. b. OPTICALLY NEGATIVE: Retardation adds and the ordinary ray is the slow ray. The flash figure does have some value because it confirms that the optic axis is nearly parallel to the microscope stage. 27 Selecting Grains to Give Interference Figures A. Optic Axis Figure - used to determine whether a mineral is uniaxial, and if so, its optic sign. The melatope should be in the field of view to be certain that the mineral is uniaxial. The optic axis must be nearly vertical so the grains display zero or low birefringence. Procedure: 1. Arrange the microscope for orthoscopic illumination by removing the auxiliary condenser. Use the low- or medium-power objective and cross the polarizers. 2. Scan the slide for a grain of the mineral with the lowest interference color or for one, which remains dark on rotation. 3. Obtain an interference figure by using high-power objective, inserting the auxiliary condenser and bertrand lens. If the grain is properly oriented the melatope is in the field of view, if not, look for other grains. B. Flash Figure - produced by a grain with its optic axis parallel to the microscope stage. It experiences maximum birefringence and displays the highest-order interference color. Determining Indices of Refraction A. Grain Mount - the indices of oil and mineral are compared using relief, Becke line, and oblique illumination methods. However, because two indices are needed for uniaxial minerals, find grains that are oriented to allow only one or the other of the two rays. Two ways in determining the index of the ordinary ray, 𝑛 : ω 1. Cross the polarizers and search for a grain whose optic axis is vertical so that all light that passes through is the ordinary ray, ω. To confirm that the optic axis is vertical, obtain an interference figure and determine the optic sign. If the optic axis is vertical, the melatope will be in the center of the field of view. Uncross the polarizers and return to orthoscopic illumination. Compare the index for the ordinary ray to the index of the oil by using the bracketing method until a match is obtained. 2. Rotate the stage so that all light coming is ordinary ray. The index can be compared with the oil. Using the three methods below, place the ordinary ray vibration direction parallel to the lower polarizer vibration direction. a. Identify which of the two rays is the ω ray with one of the accessory plates. If the mineral is positive, the ordinary ray is the fast ray. If the mineral is negative, the ordinary ray is the slow ray. Place the ray corresponding to the ordinary ray parallel to the lower polarizer vibration direction. b. Scan the mount for a grain with low interference colors so that the optic axis is reasonably close to vertical. Obtain an interference figure. In most cases, it will be an off-center optic axis figure. Rotate the stage to place one isogyre N-S so it is bisected by the N-S crosshair. The ordinary ray vibrates E-W in the center of the field of view. Return to orthoscopic illumination and remove the upper polarizer. Now, all light passes as ordinary rays. 28  c. If the mineral is pleochroic, the ω ray vibration direction can be identified by observing the color. Rotate the stage until the mineral has the color of the ω ray. The ω ray can be placed exactly parallel to the lower polarizer by crossing the polarizers and rotating slightly to get complete extinction. Determining the index of the extraordinary ray, 𝑛 : Accurate measurement of the index of ε refraction of the extraordinary ray requires the grains oriented so that the optic axis is horizontal. If grain are in any other orientation, only a value of 𝑛 ’ can be measured. ε 1. Scan the grain mount for grains that show the highest interference colors with polarizers crossed. To confirm that the optic axis is horizontal, obtain an interference figure. If the optic axis is horizontal, a symmetrical flash figure is obtained. 2. Rotate the stage so that the ε ray vibration direction is parallel to the lower polarizer. There are two ways to do this: a. Use the accessory plate. If the mineral is optically positive, the extraordinary ray is the slow ray. If the mineral is optically negative, the extraordinary ray is the fast ray. From the extinction position, rotate 45 degrees and insert an accessory plate and determine which are the fast and slow rays. Then rotate the slow (for positive) or fast (for negative) vibration direction so it is parallel to the lower polarizer. b. Use the flash figure. Starting with the fuzzy isogyre, rotate the stage slightly clockwise. If the isogyres leave the field of view NE-SW quadrants, the ε ray vibration direction was oriented N-S before rotation. If the isogyres leave from the NW-SE quadrants, then the ε ray vibration direction was oriented E-W. Rotate the stage to place the ε ray vibration direction parallel to the lower polarizer, return to orthoscopic illumination, and compare the index to the index of the oil. B. Thin Section - estimates can only be made and it is through the use of the Becke line to compare the indices of an unknown mineral with the index of the cement or other known minerals in the thin section and by examining the relief. It is generally more successful to compare the indices of the unknown mineral with the cement, rather than with another anisotropic mineral whose indices vary depending on orientation. C. Spindle Stage - allows a single grain of the unknown mineral to be rotated about a horizontal axis. This makes it possible to place the optic axis horizontal to accurately measure the indices. The procedure is as follows: 1. Using orthoscopic illumination, rotate the spindle until the grain shows maximum birefringence and the highest interference color. 2. Change to conoscopic illumination and obtain an interference figure. When properly oriented, a symmetrical flash figure is obtained. Rotate the microscope stage placing the optic axis at right angles to the lower polarizer as mentioned in the previous procedures. 3. Return to orthoscopic illumination and the grain should be in an extinct position. If not, rotate for a few degrees needed to make it extinct. Uncross the polarizers 29  and compare the index of the ordinary ray to the index of the oil using the Becke line. 4. Rotate the stage 90 degrees so that the optic axis is parallel to the lower polarizer which allows all extraordinary rays to pass. Compare the index of the extraordinary ray to the index of the immersion oil. 5. Remove the index oil from the spindle stage and repeat the process using new index oils until a match is obtained. Use the bracketing method. **Note: This procedure presumes that it is already known that the mineral is uniaxial. If not known, obtain an interference figure to check whether it is biaxial or uniaxial. If in doubt, proceed as if the mineral is biaxial. If two or the three indices of refraction measured using the biaxial technique is the same, then the mineral is uniaxial and the two same indices are the index for ordinary ray. CHAPTER 7: Biaxial Optics Biaxial minerals include the orthorhombic, monoclinic and triclinic crystal systems. To describe their crystallographic properties, it is necessary to specify the lengths of the unit cell along all three crystallographic axes as well as the three indices of refraction. The three indices are identified as 𝑛α < 𝑛β < 𝑛γ. And the maximum birefringence is always 𝑛γ − 𝑛α. The 𝑛α has the smallest index while 𝑛γ has the largest index. Two points to avoid confusion, (1) although there are three indices, light is still broken only into two rays, the fast and the slow ray and (2) both rays behave as the extraordinary ray did in uniaxial minerals. Index of the Slow Ray (𝑛γ'): 𝑛β ≤ 𝑛γ' ≤ 𝑛γ Index of the Fast Ray (𝑛α'): 𝑛α ≤ 𝑛α' ≤ 𝑛β Biaxial Indicatrix - it is constructed to allow the indices of refraction and vibration direction of the light to be determined. The indicatrix for biaxial minerals is a triaxial ellipsoid, elongate along Z and flattened along X. 𝑛α is plotted along the X-axis. 𝑛β is plotted along the Y-axis. 𝑛γ is plotted along the Z-axis. The indicatrix has three principal sections: X-Y section (axes 𝑛α and 𝑛β), X-Z section (axes 𝑛α and 𝑛γ), and Y-Z section (axes 𝑛β and 𝑛γ). The indicatrix has two circular sections with radius 𝑛β that intersects the Y-axis. Directions perpendicular to the two circular sections are the optic axes and both lie in the X-Z plane. The X-Z plane that contains the optic axes is called the optic plane. The acute angle between the optic axes is the optic angle or 2V angle. The axis (either X or Z) that bisects the 2V angle is the acute bisectrix or Bxa. 30  The axis (either X or Z) that bisects the obtuse angle between the optic axis is the obtuse bisectrix or Bxo. The Y-axis, which is perpendicular to the optic plane, is called the optic normal. Determining the Optic Sign: based on whether X or Z-axis is the acute bisectrix. OPTICALLY POSITIVE: Bxa = Z-axis. OPTICALLY NEGATIVE: Bxa = X-axis. In cases where 2V = 90 degrees, the mineral is OPTICALLY NEUTRAL. The angle between the optic axes bisected by the X-axis is called the 2Vx. While the angle between the optic axes bisected by the Z-axis is called the 2Vz. The angles can vary between 0 to 180 with the restriction that: 2Vx + 2Vz = 180 degrees. Determining the Optic Sign: based on the size of 2Vx and 2Vz. OPTICALLY POSITIVE: 2Vz < 90, 2Vx > 90. OPTICALLY NEGATIVE: 2Vz > 90, 2Vx < 90. Optical data for minerals that change from optically positive to negative as the chemical composition changes. Uniaxial indicatrix as special cases of Biaxial Indicatrix: a. If 𝑛α = 𝑛β, the Z-axis is the optic axis, the X-Y plane is the circular section and the optic sign is uniaxial positive. b. If 𝑛β = 𝑛γ, the X-axis is the optic axis, the Y-Z plane is the circular section and the optic sign is uniaxial negative. Mathematical Relationships 2 2 2 𝑥 𝑦 𝑧 1. The equation for the indicatrix is conveniently expressed as: 1 = 2 + 2 + 2 𝑛α 𝑛β 𝑛γ 2. The length of any radius (n’) is the index of refraction of light vibrating parallel to the radius whose wave normal is perpendicular to the radius. It is given by: 1 𝑛' = 2 2 2 2 2 1/2 𝑠𝑖𝑛 𝑝 𝑐𝑜𝑠 θ 𝑠𝑖𝑛 𝑝 𝑠𝑖𝑛 θ 𝑐𝑜𝑠 𝑝 [ 2 + 2 + 2 ] 𝑛α 𝑛β 𝑛γ where p is the angle between Z and n’, and θ is the angle between X and Y’. 3. The relationship between the optic angle and the principal indices of refraction is given 2 2 2 2 𝑛α (𝑛γ −𝑛β) by: 𝑐𝑜𝑠 𝑉 = 2 2 2 𝑧 𝑛β (𝑛γ −𝑛α) Use of the Indicatrix To find the indices of refraction, vibration directions for the slow and fast waves, and the ray paths, the following procedure is used. 31  1. Construct a section through the indicatrix perpendicular to the wavelength. This section is parallel to the wave front and is an ellipse, unless the wave normal is parallel to an optic axis. 2. To find the point of emergence of the slow ray, construct a line through the wave normal that is parallel to the slow vibration direction and tangent to the indicatrix. The point of emergence is the point where the tangent line for the slow ray or fast ray touches the indicatrix. 3. Biot-Fresnel Rule - used to determine the vibration directions associated with a wave normal. Two planes are constructed through the indicatrix: one plane contains the wave normal and one of the optic axes and the other contains the wave normal and the second optic axis. The vibration directions for light waves traveling along the wave normal bisect the angles between the two planes. Behavior of light passing through a mineral plate: with normal incidence, the wave normals are not refracted, so the wave normals for slow and fast rays are parallel. With inclined incidence, the wave normals for the two rays are refracted because the two rays have different indices of refraction. 1. CASE 1: Normal Incidence Parallel to an Indicatrix Axis (wave normal not refracted): The indicatrix is situated so that it is cut in half by the bottom surface of the mineral. The incident light passes parallel to the Y indicatrix axis and the elliptical section is perpendicular to the wave normal. 2. CASE 2: Normal Incidence Parallel to an Optic Axis (wave normal not refracted): The