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Chapter 6. Light Optics

6.1 Properties of Light

6.1.1. Snell's Law (Refraction)

When a beam of light strikes a polished surface of an isotropic material with a different index of refraction, it is split into two rays, a reflected and a refracted ray. The reflected ray always has the same angle to the normal as the incident beam. The refracted beam does not, but the angles which the incident and refracted ray make with the normal to the surface are related by the equation:

ni sin i = nr sin r

The reflected and refracted beams are partially polarized in a complimentary fashion and maximum polarization is achieved when the angles of incidence and refraction are complimentary (i.e. i + r = 90º) (sin r = cos i). In going from a high index to a low index medium, there is a critical angle above which the ray cannot escape and there is Total Internal Reflection. For our glass block (n = 1.50):


                 1.5 sin i = 1.0 sin 90º
                 sin i = 1/1.5 sin 90º = 0.67
                        i = 41.8 º

An ordinary beam of light has a radially symmetry distribution of electric and magnetic vectors in the plane normal to its path. It has a distribution of wavelengths across the visable spectrum and the electric and magnetic vectors of the various corpuscles are not in phase.

We can insert a color filter which only permits a certain range of wavelengths. If this range is narrow or if the light originates from a peculiar source, it can be all of one wavelength. Such a beam is said to be monochromatic. We can insert a different kind of filter which only permits light vibrating in one direction to pass. Such a filter is called a polarizer .

Light from a laser is monochromatic, coherent (all in phase) and can be polarized. In our microscopy we will generally use polarized, polychromatic light, although we sometimes use monochromatic light for specific applications. Light is an electromagnetic radiation having a wavelength (lambda) and a frequency (nu) such that the wavelength times the frequency is the velocity

c = lambda * nu

where c is the speed of light and is equal to 3x108 m/sec or 3x1018 Å/sec in a vacuum or in air; nu is the frequency in cycles per second (hertz), and lambda is the wavelength. In a vacuum, light travels at the same speed in all directions, but when it enters a non-opaque substance it slows down. The ratio of the speed of lighta in a substance to the speed of light in a vacuum is called the Refractive Index of the substance. More about this later, but first let's look at some properties of light.

Light may be treated as a wave, but it also is corpuscular in nature. It has an electric vector (E) and a magnetic vector (H) which are normal to each other and to the path vector (C).

Each corpuscle of light is called a photon and has energy equal to the frequency times a constant (Planck's constant = 6.625x10-34 joules sec) so that high frequency (short wavelength) photons are high energy, and low frequency (long wavelength) radiation is of low energy.

6.1.3. Dispersion

The separation of a beam of light into its component colors is known as dispersion. It occurs because the velocity (i.e. index of refraction) of the various wavelengths of light is different. This causes the dispersion of sunlight into a spectrum of colors by a glass prism.

These are concave-upward curves unless plotted on a specially scaled paper called Hartmann paper, on which they plot as straight lines. The relative dispersion curves of minerals and oils are important for precisely determining the index of refraction for a given mineral. The index of refraction may be used to identify a mineral.

6.1.4. Absorption

Light is attenuated on entering any dense medium by an amount which is proportional to the distance travel in the medium. The attenuation equation is known as Lambert's Law and applies to any radiation.

I / Io = exp(-kt)

where k is the absorption coefficient in m-! and t is the distance traveled in m. The absorption may be a function of wavelength or direction (in a non-isotropic medium). We may plot the transmission ratio (I/Io) as a function of wavelength.

6.1.5. Color

Selective absorption as a function of wave length gives rise to colored minerals (such as spinels). Color in minerals observed in plane-polarized light in the optical microscope (i.e. strong color) is an indication of the presence of transition metals (principally Ti, V, Cr, Mn, Fe) in the mineral. This is because elements with unfilled d-orbitals have electronic tran sitions at about the same wavelengths as optical photons and so cause strong absorptions at specific wavelengths.

Note: color observed in crossed polarizers is due to interference and is an entirely different phenomenon.

6.1.6. Pleochroism.

Selective absorption as a function of both wavelength and crystallographic direction gives rise to pleochroic minerals (such as biotite). Pleochroism may only be observed in plane polarized light. The trioctahedral micas are strongly pleochroic and change color when rotated in plane polarized light. This is not to be confused with interference or birefringence which causes pronounced color changes in cross-polarized light as noted above.

6.1.7 Summary of Properties of Light

6.2. The Petrographic (polarizing) Microscope

6.2.1. Grain Mounts

We will illustrate the principals of optical properties of minerals using grain mounts). A mineral is crushed (and seived, maybe) and placed on a standard petrographic slide. A cover slip is placed over the grains and a drop of oil of known index is placed beside the cover slip. This oil will be drawn under the cover slip by surface tension.

CAUTION: Oils contain trichloroethylene or bromo- or iodo-ethylene which are nasty and unpleasant. Skin contact should be avoided.

6.2.2. Relief

If a mineral grain is immersed in an oil with identical index of refraction, absorption and dispersion, it will disappear. This will only be true for glasses or isometric (cubic) minerals because no liquid has anisotropic optical properties. However, although minerals and oils are both commonly colorless and isotropic, they will generally have slightly different dispersion curves. If the mineral and oil differ substantially in their index of refraction, the mineral is said to stand out in "relief" and a dark line will appear to surround the mineral grain. Relief may be positive or negative. That is, the mineral may be of higher or lower index than the oil. Positive relief occurs when the mineral's refractive index is higher than the oil's. Negative relief occurs when the mineral's refractive index is lower than that of the oil.

6.2.3. Becke Line Method

In lab we saw the Becke Line method of determining the index of refraction at a wavelength of 589 nm, as well as the Oblique Illumination method. Both of these methods utilize white light and make use of the relative dispersion of the immersion oil and the mineral grain to match indices for the grain and oil at lD.

Table 6.2. Optical properties of common index oils.


     Oiln(205C)  
-dn/dtDispersionDensity   nf-n
c

Ethyl Valerate1.393.0004.slight.88.012 Mineral Oil 1.478.0004.slight.95.018 Monochloro-1.626.0004.moderate1.20.025 naphthalene Methyl Iodide1.737.0007.strong3.33.040 (Di-iodomethane)

Note that the dispersion factor is generally higher for liquids than for solids. This means that when plotted on Hartmann paper, the curve for liquids is steeper than for solids. If the two curves cross dor the D wavelength (5890 Å), then the blue colors will be in negative relief and the red in positive relief (blue Becke line out; yellow line in). Note also that dispersion increases sharply with increasing index of refraction. This gives rise to the Becke Line chart (Figure 4-2 in Jones and Bloss) and means that the dispersion colors are much more readily seen at high index. Note that oils change index rapidly with increasing temperature.

6.3. Non-Isotropic Materials

6.3.1. Introduction

The speed of light in a substance is an inverse function of density, as we saw when we heated up the oil. Cubic and amorphous solids have the same density in all directions, as do liquids. These substances are said to be isotropic. However, if we deform a cubic solid or have a non-cubic solid, light may have different indices of refraction in different directions. Most minerals are non-opaque, non-isotropic substances. Hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, and triclinic minerals may have different indices of refraction in different directions. However, the indices are constrained by the overall symmetry of the lattice. The property of having different indices of refraction in different directions is called birefringence). This property is illustrated by the calcite rhomb. When non-polarized light enters the calcite rhomb, it is split into two polarized rays. This gives rise to two images. One of the rays obeys Snell's Law, the other does not.

6.3.2. The Optical Indicatrix

Fig. 6.4. The optical indicatrix of uniaxial negative (oblate) and positive (prolate) minerals.

The index of refraction of a crystal for light vibrating in a given direction may be represented by a vector. For isotropic ma terials the index is the same in all direc tions, and the figure described by these vectors is a sphere. All sections through a sphere are circular. For hexagonal, trigo nal and tetragonal minerals, the figure is not a sphere, but a spheroid. A spheroid is a sphere which has been squashed (ob late) or extended (prolate). Such figures have only one circular section, and one axis normal to the circular section. They are called uniaxial. Minerals that crystallize in the hexagonal, trigonal, and tetragonal crystal systems are optically uniaxial. The uniaxial indicatrix figure is constrained, so that the optic axis (pole to the circular section) is always parallel to the c crystallographic axis. Therefore, any random section through the center will be an ellipse, one of whose axis will be normal to the optic axis (w = Omega). The other axis is called e' (Epsilon). Cubic crystals, amorphous solids, liquids, and gases are optically isotropic. Non-cubic crystals are optically anisotropic.

6.3.3. Wave Normals, Ray Paths, and Vibration Directions

For anisotropic media, the wave normal is normal (perpendicular) to the vibration direction, but is not (necessarily) parallel to the ray path. When polarized light enters an anisotropic crystal, it is broken up into two rays whose vibration directions are parallel to the major and minor axes of the elliptical section of the indicatrix of the crystal, which is normal to the incident ray. Therefore, if a ray enters parallel to the c axis of the uniaxial crystal, it is not broken up at all because E' = w, and the section is circular (just as if it were isotropic). When the rays emerge from the crystal and go back into an isotropic medium, the light rays destructure by interference with each other, so that the net polarization is different for the incident light and the wavelength of the emergent light. When the analyser is inserted, you see only the difference between the two rays that have interacted in the crystal. This appears as a play of interference colors.

6.3.4. The Conoscope and Uniaxial Interference Figures

Setup the conoscope on the microscope

The conoscope can be used to obtain optical information not readily available in the orthoscope, by making use of the third dimension. The condenser causes light to converge in the sample, and the transmitted light gives an interference pattern on the back focal plane of the objective. You can visualize the figure by either removing the ocular or by inserting the Bertrand lens.

The light at the center of the figure has passed through the center. The light at the edge of the image (figure) has passed through the sample at some angle from vertical (up to 40º). Suppose you put in a uniaxial crystal, such that you are looking down the unique axis (c axis = optic axis). Light that comes straight through the center will be blocked by the analyser. This gives rise to a uniaxial optic axis figure. The limbs of the curves are called isogyres, and the center is called the melatope. Lines of equal retardation circle the center and are called isochromes. The melatope corresponds to the optic axis of the sample. If the optic axis is not exactly vertical, the figure will appear off center, and the center will precess about the center of the field of view as the stage is rotated.

The optic sign (- or +) can be readily determined from an optic axis figure by inserting one of the accessory plates. If addition occurs in the quadrants parallel to the slow ray of the plate, the crystal is positive. If subtraction occurs in these quadrants, the crystal is negative.

6.3.5. Review of Uniaxial Optical Properties

Determination of the indices of refraction in uniaxial minerals

  • 1. Determine the sign.
  • a. Find grain of lowest birefringence.
  • b. Determine the sign from the interference figure.
  • 2. Compare grain to oil.
  • a. If negative (i.e. e' << w) and the oil is higher in relief than the grain, estimate the relief, go to a oil of lower refractive index, and repeat.
  • b. If positive (i.e. e' > w) and the oil is lower in relief than the grain, estimate the relief, go to an oil of lower refractive index, and repeat.
  • c. If negative and the oil is less than w, or positive and greater w, estimate the relief.
  • 3. Determination of e' (epsilon) and w (omega). a. Look for a grain showing the highest colors in crossed nicols, rotate to extinction, then clock wise 45º and insert the accessory plate. If colors increase, the slow ray of the grain is parallel to the slow ray of the plate. If the colors decrease, the slow ray of the grain is perpendicular to the slow ray of the plate. If positive, epsilon is slow, and if negative, omega is slow. Rotate epsilon parallel to the lower nicol and uncross the nicols. Compare the grain to the oil. Rotate 90º and compare to the oil. This should be omega.

    6.3.6. The Biaxial Indicatrix

    The biaxial indicatrix is a 3-dimensional ellipsoid. All central sections are ellipses except 2 which are circular. The major axes of the ellipse are X, Y, and Z, and correspond to the three indices of refraction alpha, beta, and gamma ( a << b << g ; X << Y << Z). The planes containing the 2 principal directions are principal planes. The normals (poles) to the 2 circular sections are the optic axes (O.A.). Because the light travelling along these axes has the same index of refraction for a full rotation, there is no birefringence observed when looking down one of these axes, and such a grain is extinct for a full rotation, just as when you are looking down the axis of a uniaxial crystal.

    The angle between the optic axes is called 2V), and the plane in which they lie i is called the optic plane). This plane is always normal to beta, the intermediate axis, which is called the optic normal). 2V is taken as the acute angle, either 2Vz) or 2Vx). If 2Vz) is acute (i.e. < 90º), then Z is the acute bisectrix) or Bxa), X is the obtuse bisectrix or Bxo, and the crystal is biaxial positive. If 2Vx is acute (i.e. < 90º), X is the acute bisectrix (Bxa), Z is the obtuse bisectrix, and the crystal is biaxial negative. If 2V = 90º, the crystal is neither. 2V can be calculated from alpha, beta and gamma as follows: <"eqn"> Cos Vz = ( a / b ) [ ( g + b )( g - b )/( g + a )( g - a ) ] 1/2

    6.3.7. Biaxial Interference Figures

    A. Centered Acute bisectrix Figure (Bxa). If 2V << 60º, the two melatopes re main in the field of view for a full rotation. What happens when 2V approaches 0º? B. Centered Obtuse bisectrix Figure (Bxo)

    An obtuse bisectrix figure is just like an acute bisectrix figure except that the me latopes are never in the field of view, and the isogyres leave completely. The cross is broad and diffuse at extinction. C. Centered Optic Normal Figure A broad diffuse cross fills the field of view at extinction and leaves the field in the quadrant into which the acute bisectrix has been turned.

    D. Centered Optic Axis Figure A straight bar is formed when the optic plane is parallel to the cross hairs. When at 45º , the bar is curved. The 2V may be estimated directly from the curve in the bar at 45º.

    6.3.8. Determination of Optic Sign

    Using the acute bisectrix figure, rotate 45º off the extinction position, so that the melatopes are parallel to the insertion direction of the plate. If addition occurs parallel to the slow ray of the plate, the crystal is biaxial positive. If subtraction occurs parallel to the slow direction of the plate, the crystal is biaxial negative. This also applies to the centered optic axis figure. The isogyre is always convex towards the acute bisectrix.

    6.3.9. Methods for Estimation of 2V

    1. From the curvature of the centered optic axis figure

    2. Kambs Method

    6.3.10. Pleochroism

    Pleochroism results from different absorption of light in different directions in a crystal. It does not occur in isotropic minerals. Uniaxial minerals only have 2 optic directions, so dichroism is a more appropriate word to use. Pleochroic minerals change color as they are rotated in plane polarized light (not crossed polarizers!!!).

    Pleochroism is a characteristic property of biotite. Trioctahedral micas (i.e biotite) are strongly pleochroic while dioctahedral micas (i.e muscovite) are not. Sometimes the colors do not change markedly but total absorption does

    6.4. Extinction Angles

    Many minerals show characteristic optical properties in relation to their form, cleavage, twin plane or exsolution planes. In particular the angle made between extinction and a prominant lineation is called the extinction angle, and may be used to identify or characterize certain minerals.

    Relative to a prominant lineation, extinction may be parallel, inclined or symmetrical. In general, hexagonal, trigonal, tetragonal, and orthorhombic crystals will show parallel extinction. Commonly, orthorhombic minerals also show symmetrical extinction. Monoclinic and triclinic minerals usually show inclined extinction. The extinction angle is usually taken as the acute angle measured from the cross hair parallel to the lower polarizer.

    6.5. Thin Sections

    6.5.1. Introduction

    Most of you will utilize the material obtained from this course in the study of rocks in thin section. Hence, much of the emphasis of the course has been on the tools needed to aid this.

    Thin sections come in various types. Normally, they are covered and mounted in Lakeside Balsam (Nlakeside = 1.54). Their normal thickness is 30 micrometers (maximum color of quartz is slightly yellow). Therefore, you should be able to estimate the thickness of the thin section by looking at the color of quartz. Special sections may be in apoxy (N = 1.54 to 1.58) because it does not melt or soften (used for probe mounts), or in Crystal Bond (used for TEM) because it is acetone-soluble and may be uncovered and polished on one side or both. Normal sections cost from $3.00 to $5.00, polished sections from $10.00 to $15.00, and double polished sections from $15.00 to $20.00. As I said before, most sections are in Lakeside Balsalm and covered.

    6.5.2. What to do when you look at a rock in thin section



    6.5.3. Reflectance

    Reflectance is the percent of incident light on a surface that is reflected. It is a function of the index of refraction which, in turn, is a function of density (i.e. packing and mean atomic number). Hence, metals like gold (density = 19 gm/cm3) and platinum (density = 20 gm/cm3) are of high reflectance. Fe metal also has high reflectance. Sulfides are somewhat less than metals, oxides less than sulfides, and non-opaque silicates less than oxides. Any mineral that is a good electrical conductor is opaque.

    6.5.4. Reflected Light

    You saw in Lab 16 that you could not see through some of the minerals in your thin section. Most petrographers would simply pass these off as opaques, probably oxides, and let it go at that. With the advent of the microprobe in recent years, many petrographers routinely prepare polished sections, and a great deal of information is now available from a cursory optical examination in reflected light. Oxides give temperature and oxygen fugacity (fO2) estimates. Sulfides can give temperature estimates.

    Minerals with high density and mean atomic number will tend to be opaque. This includes most oxide minerals, all of the sulfide ores, and all of the native metal ores. A full treatment of ore microscopy is beyond the scope of this course, but I think that you should be exposed to some of the basics. What you need is a polished surface and a reflected light microscope. Many of the optical phenomena observed in transmitted light are equally well observed in reflected light. The standard reference work is The Ore Minerals and Their Intergrowths by Paul Ramdohr.

    Further Reading

    Bloss, F. D. (1961) An Introduction to the Methods of Optical Crystallography. Holt Rinehart Winston, New York 294 pp.

    Kerr, P. F. (1971) Optical Mineralogy. McGraw-Hill, New York. 492pp.

    Nesse, W. D. (1986) Introduction to Optical Mineralogy. Oxford, New York, 326 pp.


    GEOL 3010 Syllabus

    Chapter 5 Mineralogy Notes

    Chapter 7 Mineralogy Notes

    Mineral Structures and Properties Data Base

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