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 be transverse 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.
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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 7600 Å.
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.
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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 33A). 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 the primary
pigment 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
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all wavelengths, 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 the
same.
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 electron density 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.
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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:
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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.
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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.
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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 defines ordinary light as 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 plane
of 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
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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
be crossed 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.
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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
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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 ray is 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. The ray velocity surface is the surface formed by
the ends of all those rays. An isotropic medium is defined as one in which light travels with the
same 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 rayvelocity 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
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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.
The “Strange” Refraction of Calcite:
Let us consider the behavior of light in a cleavage rhomb of calcite (CaCO3). If we lay the
rhomb over a dot on a sheet of paper, we see two images of the dot through the rhomb (Figure 311) which are separated both horizontally and vertically. We say that calcite shows double
refraction, the quantitative measure of which is called birefringence.* If we rotate the rhomb
about an axis normal to the paper, we note that one dot is stationary and the other rotates around
it. We may also note that the two images of the dot on the paper lie at different depths below the
upper surface of the rhomb. The law of Duc de Chaulnes (Figure 3-12) states that the depth of an
image (di) is inversely proportional to the index of refraction (n) of the transmitting medium and
directly proportional to the true depth (d).
di = d/n
Since the two images of the dot below the calcite rhomb are at different depths, they must be
formed by light waves which experience different indices of refraction (n) in the calcite. Moving
our head back and forth (cobra style) we note that the stationary dot is the more shallow image,
or is formed by waves experiencing the larger refractive index. The deeper image is the one
which rotates and is formed by waves of lower refractive index. Since the dot rotation is
abnormal, i.e., it is refracted in violation of Snell’s law, we say the rotating dot is formed by the
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extraordinary wave, and the stationary dot, as predicted by Snell’s law, is formed by the
ordinary wave. If we rotate a polaroid sheet over the rhomb, we see only one dot at a time first
one dot then the other, indicating that the two waves forming the images are plane polarized
waves, vibrating normal to one another. We see each dot twice in 360° rotation. The ccrystallographic axis of the calcite rhomb, bisects the 120° rhomb angle and lies in that vertical
plane at an angle to the paper below. Another rotation of the polaroid sheet, relating each dot to
the plane of the c-axis, reveals that the ordinary wave vibrates normal to the c-crystallographic
axis of the calcite rhomb and the extraordinary wave vibrates in the vertical plane containing the
c-crystallographic axis, normal to the vibration direction of the ordinary wave and normal to the
wave propagation direction of the light wave.
If we were to make a sphere of the calcite and view directly down the c-crystallographic
axis, the two dots would merge to a single dot at the depth of the ordinary dot. If we rotated the
sphere 90° to view normal to the c-axis, the two dots would again merge horizontally but show
maximum separation vertically, the shallow (ordinary) dot never changing depth.
*This “strange refraction” of calcite is certainly not confined to calcite and is a characteristic of all anisotropic
crystals, usually to a lesser degree. Calcite has unusually large double refraction (birefringence).
Our observations of the “strange refraction of calcite” lead to the following conclusions:
1. A wave of ordinary light upon entering a uniaxial crystal becomes two plane polarized
waves, ordinary and extraordinary, vibrating in mutually perpendicular directions.
2. An ordinary wave always vibrates normal to its ray and normal to the ccrystallographic direction.
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3. An extraordinary wave always vibrates normal to its ray, normal to the vibration
direction of the ordinary wave, and in the plane defined by its ray and the ccrystallographic axis.
4. Ordinary rays behave as though they were being transmitted by an isotropic medium,
obeying Snell’s law and moving with equal velocity in all directions to form a spherical
ray-velocity surface.
5. All waves moving parallel to the c-crystallographic axis move with the velocity of
ordinary waves and are not polarized.
6. Extraordinary waves do not behave as though they were being transmitted by an
isotropic medium; they defy Snell’s law of refraction and move with different velocities
in different directions.
7. Traveling along the c-crystallographic axis the extraordinary ray becomes ordinary, but
away from the c-axis, its velocity increases over that of the ordinary ray and reaches a
maximum when its direction of propagation is perpendicular to the c-axis.
Ray-velocity surfaces of Calcite (Figure 3-14B). To understand what we have just seen let us
look at the crystal structure of calcite as shown in Figure 3-13. It consists of equal numbers of
alternating calcium ions Ca2+ and complex carbonate ions (CO3)2─ and would have the same
crystal structure as halite (NaCl) if the carbonate ions were spherical. The carbonate ions,
however, are in the form of flat, equilateral triangles with an oxygen ion at each corner of the
triangle and a tiny carbon ion at the center. The carbon-oxygen bond in the carbonate ion is
highly covalent (See chapter 8) so that electrons shared in the covalent bond lie in the plane of
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the triangle, and all triangles lie parallel to the basal plane {0001},i.e., normal to the ccrystallographic axis.
A wave of common light entering the calcite normal to the c-crystallographic axis
becomes two, mutually perpendicular, polarized waves, one vibrating normal to the c-axis
(ordinary wave) and the other vibrating parallel to the c-axis (extraordinary wave). The ordinary
wave, vibrating in the plane of the carbonate ion, encounters all the electrons concentrated there
by the covalent bond and its progress is greatly retarded. The ordinary wave is the slower wave
and has the larger index of refraction. The extraordinary wave vibrates normal to the plane of the
carbonate ions and encounters far fewer electrons to slow its velocity. The extraordinary wave is
the faster wave and has the smaller index of refraction.
A wave of common light entering the calcite parallel to the c-crystallographic axis is not
polarized. It vibrates in all directions normal to its ray, or propagation direction, which is the caxis; and it encounters the same concentrated electron density as the slower wave, in the
paragraph above, which travels normal to the c-axis and vibrates normal to the c-axis, i.e., the
ordinary wave. All waves traveling down, i.e. parallel to, the c-crystallographic axis are not
polarized and move with the velocity of the ordinary waves. Any direction within a crystal where
all light waves travel with the same velocity and are not polarized is called an optic axis. In
calcite, the c-crystallographic axis is the only optic axis.
Any wave of common light entering the calcite at an angle to the c-crystallographic axis
becomes two mutually perpendicular polarized waves, one vibrating normal to the c-axis
(ordinary wave) and one vibrating normal to the ordinary wave, normal to its propagation
direction and in the plane formed by the propagation direction and the c-axis (extraordinary
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wave). The ordinary wave again vibrates in the plane of the carbonate ion, encountering the same
high electron density as the waves traveling parallel to the c-crystallographic axis. All ordinary
waves travel with the same velocity irrespective of propagation direction. The ray- velocity
surface for ordinary waves is a sphere. The extraordinary wave vibrates at an angle to the plane
of the carbonate ions, encountering fewer electrons than waves moving down the c-axis but more
than the extraordinary wave traveling normal to the c-axis. The velocity of extraordinary waves
in calcite is dependant on the direction of wave propagation, and ranges from the velocity of the
ordinary waves, for waves traveling parallel to the optic axis, to a maximum value, for
extraordinary waves traveling normal to the c-crystallographic axis. The ray-velocity surface for
extraordinary waves in calcite is an oblate spheroid of revolution tangent at two points to the rayvelocity sphere of the ordinary waves (Figue 3-14B). Those two points mark the optic axis (caxis).
Optic Axis and Optic Sign:
We have already defined an optic axis as any direction parallel to which all light waves
advance with the same velocity. Note that the optic axis is a direction, not a single line. Waves of
light moving along the optic axis direction are not polarized but are free to vibrate in any or all
directions normal to their light path.
All crystals which have only one optic axis are said to be uniaxial crystals and the single
optic axis is always parallel to the unique crystal axis, i.e., c-crystallographic axis. Crystals
belonging to the tetragonal, trigonal or hexagonal crystal systems have only one unique
crystallographic direction, the c-crystallographic axis, which may be a unique tetrad (tetragonal),
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a unique triad (trigonal) or a unique hexad (hexagonal).
A uniaxial crystal is said to be optically negative, if the ordinary waves are slower than
the extraordinary waves or the index of refraction for the ordinary waves (nω) is greater than the
index of refraction for the extraordinary waves (nε). Recall that index of refraction of an optical
medium is inversely proportional to velocity of light in that medium (n = v/V). Calcite is
optically negative, i.e., nω > nε. A uniaxial crystal is said to be optically positive, if the ordinary
waves are faster than the extraordinary waves or the index of refraction of the ordinary wave
(nω) is less than the index of refraction of the extraordinary wave (nε).
All crystals belonging to the orthorhombic, monoclinic, and triclinic crystal systems are
said to be biaxial crystals and, indeed, they have two directions where light travels with uniform
velocity, as we shall see. The spatial relationship between the optic axes and crystallographic
axes is more complex than the simple uniaxial relationship, that we observed in the calcite
example, and will also be delayed to later pages.
The Uniaxial Indicatrix (Figure 3-15):
Ray-velocity surfaces, for a uniaxial crystal, represent a real phenomenon which we must
picture in our imagination, however. A more useful concept is an indicatrix* which is a single,
purely imaginary geometric surface which we may construct for any given uniaxial crystal, if we
know its principle indices of refraction, i.e., nω and nε. It is used to “indicate” the vibration
directions and magnitude of the indices of refraction of the ordinary and extraordinary waves for
light traveling in any direction we may chose within the uniaxial crystal.
The Indicatrix for a given uniaxial crystal is a spheroid of revolution, i.e., an ellipse,
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rotated about the c-crystallographic axis i.e., optic axis. It is a prolate spheroid of revolution for
optically positive crystals (Figure 3-15A) and an oblate spheroid of revolution for optically
negative crystals (Figure 3-15B). One-half the length of the rotation axis is made equal to the
maximum refractive index of the extraordinary waves (nε), for positive crystals; or one-half the
minimum refractive index of the extraordinary waves (nω), for negative crystals. The single
circular section through the spheroid center and normal to the optic axis is constructed with a
radius equal to the index of refraction of the ordinary waves (nω).
*An indicatrix gives no more information than the corresponding ray velocity surfaces, however, it is much easier to
use.
Only the maximum or minimum value for nε is really useful and this is the value reported
in all tables of refractive indices.
Birefringence:
Birefringence is defined as the difference between the indices of refraction of the two
polarized waves passing through an anisotropic medium. This pure number changes with the
direction of light propagation within the crystal, but has a maximum value for any given mineral
species. For a uniaxial crystal the birefringence is the index of refraction of the extraordinary
wave minus the index of refraction of the ordinary wave and is a positive value for an optically
positive crystal and negative for an optically negative crystal. For light traveling normal to the ccrystallographic axis, the birefringence is maximum (nε − nω) and, for light traveling parallel to
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an optic axis, the birefringence is zero. For light traveling in any other direction within the
crystal, the birefringence is intermediate between zero and a maximum absolute value, owing to
the change in the refractive index of the extraordinary wave with direction of the light
propagation.
The Biaxial Indicatrix:
The biaxial ray-velocity surfaces are not simple spheres or ellipsoids but are complex
surfaces which pass through one another, coincident at four points which define two optic axes.
The biaxial indicatrix is, however, simple and is a very useful geometric shape (Figure 3-20).
The biaxial indicatrix is a triaxial ellipsoid constructed on three mutually perpendicular
optical directions which we will designate X, Y, and Z (Figure 3-17), and which may have only
limited or no fixed relationship to crystallographic directions. From the origin, i.e., intersection
point of X, Y and Z, we measure distances equal to nα, nβ and nγ along X, Y and Z respectively
and connect the points with an ellipse in each of the XY, XZ, and YZ planes to form a triaxial
ellipsoid (Figure 3-18A). The XZ plane is the optic plane which contains the two optic axes
(Figure 3-18B). At some angle to the X direction within the optic plane, the distance from the
center point to a point on the ellipse in the XZ plane is equal to nβ, since nβ is between nα, for
light vibrating parallel to X, and nγ for light vibrating parallel to Z. The indicatrix section
through that point on the ellipse and containing the Y direction is a circle of radius nβ. The
normal to that circle is an optic axis, since all light rays traveling normal to this circular section
have equal velocity with index of refraction nβ and may vibrate parallel to any radius of the circle
(i.e. not polarized). A second circular section exists at the same angle from X measured in the
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opposite direction. Normal to the second circular section is a second optic axis in the XZ plane.
The acute angle between the two optic axes is called the 2V angle (Figure 3-18B) and the optical
direction (X or Z) which bisects that angle is called the acute bisectrix. The optical direction (Z
or X) which bisects the obtuse angle between the optic axes is called the obtuse bisectrix. If Z is
the acute bisectrix, nβ is nearer to nα.than to nγ and the mineral is optically positive. If nβ = nα,
the 2VZ angle is 0° on Z, the triaxial ellipsoid becomes a prolate spheroid of revolution and the
mineral is uniaxial positive (Figure 3-19), and Z becomes the optic axis. If X is the acute
bisectrix, nβ is nearer nγ and the mineral is optically negative. If nβ = nγ, the 2VX angle is 0° on
X, the triaxial ellipsoid becomes an oblate spheroid of revolution and the mineral is uniaxial
negative with X as the optic axis.
The biaxial indicatrix is used in the same way as the uniaxial indicatrix, i.e. cut a section
through the center of the indicatrix normal to the direction of light propagation. The resulting
ellipse has a major axis and a minor axis indicating the vibration directions of the two polarized
waves (Figure 3-20) and the index of refraction as indicated by the lengths of the semi-axes. One
refractive index is always between nα and nβ and the other refractive index is always between nγ
and nβ.
Positioning the indicatrix within the crystal of a given mineral (Figure 3-21), however, is
much less simple than positioning the uniaxial indicatrix, where the single optic axis is always
parallel to the c-crystallographic axis. Biaxial minerals may be orthorhombic, monoclinic or
triclinic. If the mineral is orthorhombic (Figure 3-21A) the optical directions X, Y and Z are
always parallel to the crystallographic axes a, b and c, not necessarily in that order, i.e. X may be
parallel to a, b or c. If the mineral is monoclinic (Figure 3-21B), one of the optical directions X,
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Copyright 2010 Wm. Revell Phillips All Rights Reserved
Y or Z is always parallel to the b-crystallographic axis (i.e. the diad symmetry axis). To fix the
indicatrix in position, we must know the angle between the c-crystallographic axis and one of the
other optical directions in the ac-crystallographic plane. If the mineral is triclinic (Figure 3-21C),
there is no fixed relationship between optical directions and crystallographic directions. The
orientation of the indicatrix is fixed for a given triclinic mineral but is unique to that mineral
species.
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Copyright 2010 Wm. Revell Phillips All Rights Reserved