Chapter 3 - Optical Crystallography - the Behavior of Light in Crystals

CHAPTER 3 - OPTICAL CRYSTALLOGRAPHY-The Behavior of Light in Crystals

The Nature of Light:

Light is radiant energy which stimulates our visual sense. As with electricity and some other forms of energy, we know it by its effects, and can predict its behavior and use it for our purposes without fully understanding what it is. For some applications light is best seen as a stream of particles, called quanta which enter our eye and stimulate our visual sense. For our use here it is more useful to consider light to betransverse waves where electrons transmitting the wave vibrate normal to its propagation direction*. A transverse wave (Figure 3-3) is defined by its amplitude and its wavelength. The amplitude of a light wave represents its intensity, or brightness, and its wavelength determines its color. We seldom see light of a single wavelength, but when we do, we see a specific color. The light from a sodium vapor lamp like those that illuminate our highways is intense orange-yellow and about as close to pure monochromatic light as we encounter in our everyday experience.

*If we tie a rope to a tree, step back to stretch the rope, and shake the rope, a waves moves down the rope toward the tree. The wave is a series of crests and troughs which arrive successively at the tree, but any individual point on the rope only moves up and down normal to the direction of the wave propagation as the wave passes. This is a transverse wave.

Visible light is a very small range of wavelengths within the electromagnetic spectrum of radiant energy (Figure 3-2) which ranges from wavelengths of 1,000 km. to less than 10─10 cm. This huge electromagnetic spectrum is divided into rather arbitrary wavelength ranges according to their useful application and the instruments used to detect them. Visible light is that tiny part of the electromagnetic spectrum which stimulates our eyes. Visible wavelengths range from the longest wavelengths at about 7.6 X 10─5 cm. (red) to the shortest at about 4 X 10─5 cm. (violet). Wavelengths longer than red are said to be infrared, and wavelengths shorter than violet are ultraviolet.

Several dimensional units are commonly used to describe these very short wavelengths. A micron (μ) = 0.001 mm. or 10─4 cm., a milli-micron (mμ) or nanometer (nm) consequently = 10─7 cm., and an angstrom unit (Ǻ) = 10─8 cm. The shortest wavelength of visible light (violet) is therefore about 400 nm or 4000 Å and the longest wavelength of visible light (red) is about 760 nm or 7600Å.

The energy of light (electron volts) is inversely proportional to its wavelength λ.

E = hc/λ

Both h and c are fixed values (h = Plank’s constant and c = velocity of light in a vacuum).Violet

light is short wavelength and high energy, red light is long wavelength and low energy.

Color:

Color, like light itself, is simply our perception of a visual stimulation. Each wavelength of light produces a different color sensation, but we seldom see light of a single wavelength, i.e.,monochromatic light. Our eyes may not be able to distinguish between the color of a monochromatic light and the color of a combination of wavelengths which give the same visual sensation.

Light Addition:

Light from the sun is said to be white light and contains essentially all wavelengths of the visible spectrum. When all wavelengths enter our eye, our perception is white and when no visible wavelengths enter our eye our perception is black. Since our eyes are responsive only to red, green and blue-violet, these three colors alone produce the same perception of white as the complete visible spectrum and are called the primary light colors. These three colors, projected onto a “white” screen as overlapping circles yield white, where all three circles overlap (Figure 3-3A). The overlap of any two primary colors produces the complimentary color of the third primary color. Color photography, television and other modern technology employ the principle of light addition, which will also be central to our consideration of light in crystalline matter.

Light Subtraction. My primary school art teacher had me paint a “color wheel” of theprimarypigment colors: red, blue and yellow. Where the three colors overlapped, I should see black (I usually got brown.) and, by proper mixing of the three primary colors, I could get any color I wanted. My art teacher employed the principle of color subtraction (Figure 3-3B).

Light incident on an object is partially reflected, partially absorbed, and partially transmitted. The color of a transparent object is the color of the light wavelengths that pass through it, i.e., transmitted, without being absorbed, i.e., subtracted. The color of an opaque object is the color of the light wavelengths which are reflected from it without being absorbed. We see what the object does not subtract from the incident light. A black object absorbs allwavelengths, reflecting nothing, and a white object reflects all wavelengths, absorbing nothing. A “white” object in red light is red, since only red light is available to be reflected, and a “red” object in blue or green light is black, since the red object absorbs both blue and green light and there is no red light to be reflected. Color is therefore not an inherent property of any object, but the object is the color we see it to be.

Light Velocity:

The velocity of light in free space is considered to be a constant value and, for out purposes, is 3 X 1010 cm./sec. or 186,000 mi./sec.The velocity of light in air is essentially thesame.

The velocity of light in any other transparent medium is less than the value above as determined by the optical density, i.e., electron density,of the medium. We may view the transmission of light as the harmonic vibration of electrons within the medium,* and the more electrons that must be set into harmonic motion by the light energy, the slower the light wave advances. Great electrondensity means slow light velocity and, when the electron density is not the same in all directions, the light velocity is not the same in all directions.

An optical medium where the electron density, and hence the light velocity, is the same in all directions is said to be an isotropic medium (Gr.isos: equal, Gr tropos: direction). An optical medium where the electron density, and hence the light velocity, is not the same in all directions is said to be an anisotropic medium.

*This electron paradigm is useful for our purposes but may be inadequate for other applications, since light travels very well in a total vacuum where there are no electrons to vibrate.

Index of Refraction:

When light passes from one optical medium into another of different optical density (Figure 3-4), the velocity of the light changes and the light path, or light ray, is bent, i.e., refracted. The quantitative measure of this refraction is a pure number. It is called the index of refraction (n), and for each medium, is defined as the ratio of the velocity of light in air (v ) to the velocity of light in that medium (V).

n = v/V

The index of refraction of a vacuum (or air),* is therefore unity (1.000) and that of any other optical medium must be greater than unity. Note that the index of refraction of an optical medium and the velocity of light in that medium are inversely proportional. Large index of refraction means slow velocity

*The optical density of air is, of course, slightly greater than that of a vacuum and its true index of refraction is 1.000274, at 15° C and at atmospheric pressure.

Snell’s Law: The behavior of light rays passing from one optical medium of refractive index n1 into another optical medium of refractive index n2 is governed by Snell’s Law (Figure 3-6A).

n2/n1 = sin i/sin r*

If the first medium is air, n1 is unity, and the index of refraction of the second medium n2 is:

n2 = sin i/sin r

If n2 is greater than n1 the angle of incidence ( i ) is greater than the angle of refraction ( r), and if n2 is less than n1, angle i is less than angle r. A light ray passing from a medium of low refractive index into a medium of higher index of refraction is refracted toward the normal to the interface. Conversely, a light ray passing from a medium of high refractive index into a medium of lower refractive index is bent away from the normal and toward the interface between the two media.

* For those unfamiliar with the standard notation of trigonometry, let us construct any right triangle, i.e., a triangle where one angle is 90°. The “sine” (sin) of one of the other angles is equal to the length of the triangle side opposite the angle divided by the length of the hypotenuse, i.e., the long side. The “cosine” (cos) of the angle is the adjacent side over the hypotenuse, and the “tangent” (tan) of the angle is the opposite side over the adjacent side or the sine over the cosine.

Total Reflection and Critical Angle: If we reverse the direction of the light rays in Fig. 3-6A so that the rays pass from a low index of refraction into a higher index of refraction, the angle of refraction, in Figure 3-6A, becomes the angle of incidence, in Figure 3-6B, and the angle of refraction is larger than the angle of incidence. As we increase the angle of incidence, in Figure 3-6B, at some incident angle the angle of refraction reaches 90° and the light rays do not enter the medium of low refraction index. This value of the incident angle is its critical angle (CA), and at any larger angle of incidence, the light rays are totally reflected back into the high index medium.

n2/n1 = sin 90°/sin (CA) = 1/sin (CA)

If the low-index medium is air, n1 = 1:

n2 = 1/sin (CA)

The larger the index of refraction, the smaller the critical angle.

The brilliance of any faceted gemstone depends upon critical angle and cutting proportions, i.e. angles (Figure 3-5), and the critical angle for any gem mineral is inversely proportional to its index of refraction. The greater the refraction index of the gem mineral, the greater its potential for brilliance. Improper cutting angles can destroy its brilliance.

Dispersion:

The “fire”of a cut gemstone is determined by its dispersion, and may be enhanced or diminished by the skill of the cutter. Dispersion is familiar to anyone who has ever seen a rainbow or used a glass prism or diffraction grating to spread the sun’s light into a continuous color spectrum (Figure 3-7). The separation of polychromatic light (i.e. light consisting of many wavelengths) into its component wavelengths is called dispersion. A prism of lead glass produces a much broader spectrum than one produced by similar prism of common lime glass and the lead glass is said to show greater dispersion, i.e., greater light spreading. The quantitative dispersion of a transparent material is an inherent property of that medium. It may be expressed as a pure number, defined as the difference between the indices of refraction of that material for two specific light wavelengths near the two ends of the visible spectrum.

Dispersion = nF− nC

Where nF and nC are the indices of refraction for the F blue-green (λ = 486 nm) and C red-orange (λ = 656 nm) wavelengths (Fraunhofer lines) of the solar spectrum. If one had access to the appropriate filters to produce the F and C monochromatic wavelengths, he could use them in place of the yellow sodium filter on a common gem refractometer to measure the indices of refraction for these two wavelengths and determine the numerical dispersion of a gem.

Dispersion for common gem materials ranges from about 0.0045 (fluorite) to 0.330 (rutile).The dispersion of diamond is 0.044.

Polarized Light:

The transverse wave theory definesordinary lightas waves vibrating in all possible planes through the line of light propagation (Figure 3-8A). Plane polarized light waves vibrate in only one plane (Figure 3-8B), and are thus not symmetrical about the light ray, i.e., the line of propagation. Since all light waves passing through anisotropic crystals are polarized, an understanding of plane polarized light is basic to our understanding of light behavior in gem minerals. Plane polarized light is our greatest reason for using the transverse wave model as it is difficult to imagine a stream of quanta not symmetrical about its line of propagation. The planeof vibration of a plate of polarizing material (polar) is commonly called the polarization plane. “Vibration plane” and “polarization plane” will be used here interchangeably. When two polars are superimposed one in front of the other, with their polarization planes parallel, the second polar transmits the polarized light produced by the first polar and we see polarized light passing through both polars (Figure 3-9A). These are said to be parallel polars. Two polars superimposed one in front of the other, with their polarization planes at right angles, are said to becrossed polars and the second polar cuts out the polarized light produced by the first, and we see no light coming through the second polar (Figure 3-9B).

Polarization by Reflection: In 1809, Etienne Louis Malus (1775-1812) discovered that light reflected at certain angles from a smooth surface of a transparent, non-conducting material is plane polarized with its polarization plane parallel to the reflecting surface (Figure 3-10). Sir David Brewster (1781-1868) demonstrated that complete polarization of the reflected light occurs when the angle of incidence and the angle of refraction add to 90° (Brewster’s Law). The angle of incidence at which total polarization takes place is the polarizing angle (p). Applying Snell’s law and assuming n1 is the refractive index of air, then:

n2 = sin i/sin r = sin p/sin (90°─ p) = sin p/cos p = tan p

The index of refraction of the material making up the reflecting surface is the tangent of the polarizing angle. Viewing the light reflected from a polished table top through a polar sheet, we can find the polarization plane of the polar sheet by rotating it until the reflection is maximum brightness. The polarization plane of the polar is then parallel to the table top. By moving forward and back we change the angle of incidence and we may estimate the polarizing angle.

Brewster’s law suggests a method of measuring the refractive index of a gemstone by measuring the angle of incidence at which the light reflected from a polished surface has maximum polarization, and a Brewster-angle Meter is produced by the Gemmological Instruments Ltd., London. It reportedly measures Brewster angles 55°-73°, which translates, with some calculations, to index of refraction 1.43 - 3.30. It is most accurate for large indices of refraction, above the limits of the standard critical angle refractometer, for which it can be a useful supplement.

Polarization by absorption: Some crystals transmit a range of light wavelengths vibrating parallel to one crystallographic direction while absorbing those light waves vibrating normal to it. This property, called pleochroism, is only possible for an anisotropic crystal. Tourmaline, for example, transmits most of the light waves vibrating parallel to the c-crystallographic axis and absorbs most of the light waves vibrating normal to it. A plate of tourmaline thus creates and transmits polarized light vibrating parallel to its c-crystallographic axis. Two superimposed plates of tourmaline transmit polarized light when their c-crystallographic axis are parallel (Figure 3-9A) and transmit little or no light when their c-crystallographic axes are perpendicular (Figure 3-9B).

In 1852 William Herapath (1796-1868) described crystals of an organic compound which he called iodoquinine sulfate, which, like tourmaline, polarizes light by absorption. Myriads of these tiny slender crystals oriented in a uniform direction within a plastic binder are the basis of the original commercial“Polaroid”, which was developed in the 1930's by Edwin H. Land. Today the name “Polaroid” refers to several synthetic filters produced by the Polaroid Corporation of America. These polarizing filters have essentially replaced all other methods for producing polarized light.

Light in Anisotropic Crystals:

The crystal structure (i.e. arrangement of atoms) of an anisotropic crystal differs with direction, and hence its physical properties and electron density differ with direction. A light wave passing through such a crystal may encounter few or many electrons depending on the direction the light wave travels and the direction it vibrates. The more electrons the light wave encounters and must set in vibration, the slower it travels; and the light waves travel with different velocities in different crystallographic directions.

The Ray-Velocity Surface:

A light rayis the direction of light propagation and may be considered as a line segment beginning at the point of origin of the light wave and increasing in length as the wave advances. Let us consider the light rays emitted from a point light source. The rays move outward in all directions from the point source and, since light travels with finite velocity, all rays will have traveled a certain distance in a specified time. Theray velocity surface is the surface formed by the ends of all those rays. An isotropic medium is defined as one in which lighttravels with thesame velocity in all directions, and hence the ray velocity surface in any isotropic medium is a sphere which grows larger as the rays advance outward from the point light source. A ray-velocity surface, or surfaces, may be considered a real physical phenomenon which could be photographed, if our techniques were sufficiently keen. Every-day experience makes us familiar with air, water, glass and other transparent media where the atoms or molecules of the medium have somewhat random distribution and random orientation, making the medium statistically isotropic. Where ray velocity changes with direction, i.e., anisotropic media, ray velocity surfaces are not spherical.