Uniaxial Minerals

Anisotropic minerals differ from isotropic minerals because:1. the velocity of light varies depending on direction through the mineral; 2. They show double refraction.

When light enters an anisotropic mineral it is split into two rays of different velocity which vibrate at right angles to each other.In anisotropic minerals there are one or two directions, through the mineral, along which light behaves as though the mineral were isotropic. This direction or these directions are referred to as the optic axis.Hexagonal and tetragonal minerals have one optic axis and are optically UNIAXIAL.Orthorhombic, monoclinic and triclinic minerals have two optic axes and are optically BIAXIAL.Calcite rhomb displaying double refraction

Light travelling through the calcite rhomb is split into two rays which vibrate at right angles to each other. The two rays and the corresponding images produced by the two rays are apparent in the above image. The two rays are:

1. Ordinary Ray, labelled omega w, nw = 1.658 2. Extraordinary Ray, labelled epsilon e, ne = 1.486.

Vibration Directions of the Two RaysThe vibration directions for the ordinary and extraordinary rays, the two rays which exit the calcite rhomb, can be determined using a piece of polarized film. The polarized film has a single vibration direction and as such only allows light, which has the same vibration direction as the filter, to pass through the filter to be detected by your eye.

1. Preferred Vibration Direction NS

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With the polaroid filter in this orientation only one row of dots is visible within the area of the calcite rhomb covered by the filter. This row of dots corresponds to the light ray which has a vibration direction parallel to the filter's preferred or permitted vibration direction and as such it passes through the filter. The other light ray represented by the other row of dots, clearly visible on the left, in the calcite rhomb is completely absorbed by the filter.

2. Preferred Vibration Direction EW

With the polaroid filter in this orientation again only one row of dots is visible, within the area of the calcite coverd by the filter. This is the other row of dots than that observed in the previous image. The light corresponding to this row has a vibration direction parallel to the filter's preferred vibration direction.

It is possible to measure the index of refraction for the two rays using the immersion oils, and one index will be higher than the other.

1. The ray with the lower index is called the fast ray

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o recall that n = Vvac/Vmedium

If nFast Ray = 1.486, then VFast Ray = 2.02X1010 m/sec

2. The ray with the higher index is the slow ray o If nSlow Ray = 1.658, then VSlow Ray = 1.8 1x1010 m/sec

Remember the difference between: vibration direction - side to side oscillation of the electric vector of the plane light

and propagation direction - the direction light is travelling.

Electromagnetic theory can be used to explain why light velocity varies with the direction it travels through an anisotropic mineral.

1. Strength of chemical bonds and atom density are different in different directions for anisotropic minerals.

2. A light ray will "see" a different electronic arrangement depending on the direction it takes through the mineral.

3. The electron clouds around each atom vibrate with different resonant frequencies in different directions.

4. Velocity of light travelling through an anisotropic mineral is dependent on the interaction between the vibration direction of the electric vector of the light and the resonant frequency of the electron clouds. Resulting in the variation in velocity with direction.

Introduction to Uniaxial MineralsUniaxial minerals are a class of anisotropic minerals that include all minerals that crystallize in the tetragonal and hexagonal crystal systems.  They are called uniaxial because they have a single optic axis.Light traveling along the direction of this single optic axis exhibits the same properties as isotropic materials in the sense that the polarization direction of the light is not changed by passage through the crystal.  Similarly, if the optic axis is oriented perpendicular to the microscope stage with the analyzer inserted, the grain will remain extinct throughout a 360o rotation of the stage.  TThe single optic axis is coincident with the c-crystallographic axis in tetragonal and hexagonal minerals.Thus, light traveling parallel to the c-axis will behave as if it were traveling in an isotropic substance because, looking down the c-axis of tetragonal or hexagonal minerals one sees only equal length a-axes, just like in isometric minerals.

Like all anisotropic substances, the refractive indices of uniaxial crystals varies between two extreme values.  For uniaxial minerals these two extreme values of refractive index are defined as (or No) and (or Ne). Values between and are referred to as '.

Uniaxial minerals can be further divided into two classes.  If the mineral is said have a negative optic sign or is uniaxial negative.

In the opposite case, where the mineral is said to have a positive optic sign or is uniaxial positive.

The absolute birefringence of a uniaxial minerals is defined as | | (the absolute value of the difference between the extreme refractive indices).

Double RefractionAll anisotropic minerals exhibit the phenomenon of double refraction.  Only when the birefringence is very

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high, however, is it apparent to the human eye.  Such a case exists for the hexagonal (and therefore uniaxial) mineral calcite.  Calcite has rhombohedral cleavage which means it breaks into blocks with parallelogram - shaped faces.  If a clear rhombic cleavage block is placed over a point and observed from the top, two images of the point are seen through the calcite crystal.  This is known as double refraction.What happens is that when unpolarized light enters the crystal from below, it is broken into two polarized rays that vibrate perpendicular to each other within the crystal.

One ray, labeled o in the figure shown here, follows Snell's Law, and is called the ordinary ray, or o-ray.  It has a vibration direction that is perpendicular to the plane containing the c-axis and the path of the ray.  The other ray, labeled e in the figure shown here, does not follow Snell's Law, and is therefore referred to as the extraordinary ray, or e-ray.  The e ray is polarized with light vibrating within the plane containing the c-axis and the propagation path of the ray.

Since the angle of incidence of the light is 0o, both rays should not be refracted when entering the crystal according to Snell's Law, but the e-ray violates this law because it's angle of refraction is not 0o, but is r, as shown in the figure. Note that the vibration directions of the e-ray and the o-ray are perpendicular to each other.  These directions are referred to as the privileged directions in the crystal.

If one separates out the e-ray and the o-ray as shown here, it can be seen that the o-ray has a vibration direction that is perpendicular to the propagation direction.  On the other hand, the vibration direction of the e-ray is not perpendicular to the propagation direction. A line drawn that is perpendicular to the vibration direction of the e-ray is called the wave normal direction.  It turns out the wave normal direction does obey Snell's Law, as can be seen by examining the diagram of the calcite crystal shown above.  In the case shown, the wave normal direction would be parallel to the o-ray propagation direction, which is following Snell's Law.

 we can use the electromagnetic theory for light to explain how a light ray is split into two rays (FAST and SLOW) which vibrate at right angles to each other.

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The above image shows a hypothetical anisotropic mineral in which the atoms of the mineral are: 1. closely packed along the X axis 2. moderately packed along Y axis 3. widely packed along Z axis

The strength of the electric field produced by the electrons around each atom must therefore be a maximum, intermediate and minimum value along X, Y and Z axes respectively, as shown in the following image.

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With a random wavefront the strength of the electric field, generated by the mineral, must have a minimum in one direction and a maximum at right angles to that. Result is that the electronic field strengths within the plane of the wavefront define an ellipse whose axes are;

1. at 90° to each other, 2. represent maximum and minimum field strengths, and 3. correspond to the vibration directions of the two resulting rays.

The two rays encounter different electric configurations therefore their velocities and indices of refraction must be different.There will always be one or two planes through any anisotropic material which show uniform electron configurations, resulting in the electric field strengths plotting as a circle rather than an ellipse.Lines at right angles to this plane or planes are the optic axis (axes) representing the direction through the mineral along which light propagates without being split, i.e., the anisotropic mineral behaves as if it were an isotropic mineral.The colours for an anisotropic mineral observed in thin section, between crossed polars are called interference colours and are produced as a consequence of splitting the light into two rays on passing through the mineral.Monochromatic ray, of plane polarized light, upon entering an anisotropic mineral is split into two rays, the FAST and SLOW rays, which vibrate at right angles to each other.

Due to differences in velocity the slow ray lags behind the fast ray, and the distance represented by this lagging after both rays have exited the crystal is the retardation - D.The magnitude of the retardation is dependant on the thickness (d) of the mineral and the differences in the velocity of the slow (Vs) and fast (Vf) rays.The time it takes the slow ray to pass through the mineral is given by:

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during this same interval of time the fast ray has already passed through the mineral and has travelled an additional distance = retardation.

substituting 1 in 2, yields

rearranging

The relationship (ns - nf) is called birefringence, given Greek symbol lower case d (delta), represents the difference in the indices of refraction for the slow and fast rays.In anisotropic minerals one path, along the optic axis, exhibits zero birefringence, others show maximum birefringence, but most show an intermediate value.The maximum birefringence is characteristic for each mineral.Birefringence may also vary depending on the wavelength of the incident light.INTERFERENCE AT THE UPPER POLARNow look at the interference of the fast and slow rays after they have exited the anisotropic mineral. Fast ray is ahead of the slow ray by some amount = DInterference phenomena are produced when the two rays are resolved into the vibration direction of the upper polar.Interference at the Upper Polar –Case I

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1. Light passing through lower polar, plane polarized, encounters sample and is split into fast and slow rays.

2. If the retardation of the slow ray = 1 whole wavelength, the two waves are IN PHASE. 3. When the light reaches the upper polar, a component of each ray is resolved into the vibration

direction of the upper polar. 4. Because the two rays are in phase, and at right angles to each other, the resolved

components are in opposite directions and destructively interfere and cancel each other. 5. Result is no light passes the upper polar and the grain appears black.

Interference at the Upper Polar - Case 2

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1. If retardation of the slow ray behind the fast ray = ½ a wavelength, the two rays are OUT OF

PHASE, and can be resolved into the vibration direction of the upper polar. 2. Both components are in the same direction, so the light constructively interferes and passes

the upper polar. MONOCHROMATIC LIGHT

If our sample is wedged shaped, as shown above, instead of flat, the thickness of the sample and the corresponding retardation will vary along the length of the wedge.Examination of the wedge under crossed polars, gives an image as shown below, and reveals:

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1. dark areas where retardation is a whole number of wavelengths. 2. light areas where the two rays are out of phase, 3. brightest illumination where the retardation of the two rays is such that they are exactly ½,

1½, 2½ wavelengths and are out of phase. The percentage of light transmitted through the upper polarizer is a function of the wavelength of the incident light and retardation.

If a mineral is placed at 45° to the vibration directions of the polarizers the mineral yields its brightest illumination and percent transmission (T).Polychromatic LightPolychromatic or White Light consists of light of a variety of wavelengths, with the corresponding retardation the same for all wavelengths.Due to different wavelengths, some reach the upper polar in phase and are cancelled, others are out of phase and are transmitted through the upper polar.The combination of wavelengths which pass the upper polar produces the interference colours, which are dependant on the retardation between the fast and slow rays.Examining the quartz wedge between crossed polars in polychromatic light produces a range of colours. This colour chart is referred to as the Michel Levy Chart.

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At the thin edge of the wedge the thickness and retardation are ~ 0, all of the wavelengths of light are cancelled at the upper polarizer resulting in a black colour.With increasing thickness, corresponding to increasing retardation, the interference colour changes from black to grey to white to yellow to red and then a repeating sequence of colours from blue to green to yellow to red. The colours get paler, more washed out with each repetition.In the above image, the repeating sequence of colours changes from red to blue at retardations of 550, 1100, and 1650 nm. These boundaries separate the colour sequence into first, second and third order colours.Above fourth order, retardation > 2200 nm, the colours are washed out and become creamy white.The interference colour produced is dependant on the wavelengths of light which pass the upper polar and the wavelengths which are cancelled.Retardation in a uniaxial mineral: Retardation is the distance between the slow and fast rays after they have left the crystal. = t(nslow-nfast) (1) therefore depends on (i) the thickness of the mineral (or thin section); (ii) the refractive indeces of that mineral (one of which will be , the other a value less than or greater than which we will call ’); and (iii) the wavelength of the incident light.

The birefringence for a mineral in a thin section can also be determined using the equation for retardation, which relates thickness and birefringence. Retardation can be determined by examining the interference colour for the mineral and recording the wavelength of the retardation corresponding to that colour by reading it directly off the bottom of Plate I. The thickness of the thin section is ~ 30 µm. With this the birefringence for the mineral can be determined, using the equation:

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See the example below.

This same technique can be used by the thin section technician while making a thin section. By looking at the interference colour, the thickness of the thin section can be judged.The recognition of the order of the interference colour displayed by a mineral comes with practice and familiarity with various minerals. In the labs you should become familiar with recognizing interference colours.

ExtinctionAnisotropic minerals go extinct between crossed polars every 90° of rotation. Extinction occurs when one vibration direction of a mineral is parallel with the lower polarizer. As a result no component of the incident light can be resolved into the vibration direction of the upper polarizer, so all the light which passes through the mineral is absorbed at the upper polarizer, and the mineral is black.

Upon rotating the stage to the 45° position, a maximum component of both the slow and fast ray is available to be resolved into the vibration direction of the upper polarizer. Allowing a maximum amount of light to pass and the mineral appears brightest.The only change in the interference colours is that they get brighter or dimmer with rotation, the actual colours do not change.Many minerals generally form elongate grains and have an easily recognizable cleavage direction, e.g. biotite, hornblende, plagioclase.The extinction angle is the angle between the length or cleavage of a mineral and the minerals vibration directions.

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The extinction angles when measured on several grains of the same mineral, in the same thin section, will be variable. The angle varies because of the orientation of the grains. The maximum extinction angle recorded is diagnostic for the mineral.

Types of Extinction1. Parallel Extinction

The mineral grain is extinct when the cleavage or length is aligned with one of the crosshairs.The extinction angle (EA) = 0°

e.g. o orthopyroxene o biotite

2. Inclined ExtinctionThe mineral is extinct when the cleavage is at an angle to the crosshairs.EA > 0°

e.g. o clinopyroxene o hornblende

3. Symmetrical ExtinctionThe mineral grain displays two cleavages or two distinct crystal faces. It is possible to measure two extinction angles between each cleavage or face and the vibration directions. If the two angles are equal then Symmetrical extinction exists.EA1 = EA2

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e.g.o amphibole o calcite

4. No Cleavage

Minerals which are not elongated or do not exhibit a prominent cleavage will still go extinct every 90° of rotation, but there is no cleavage or elongation direction from which to measure the extinction angle.

e.g.o quartz o olivine

Exceptions to Normal Extinction PatternsDifferent portions of the same grain may go extinct at different times, i.e. they have different extinction angles. This may be caused by chemical zonation or strain.Chemical zonation

The optical properties of a mineral vary with the chemical composition resulting in varying extinction directions for a mineral. Such minerals are said to be zoned.e.g. plagioclase, olivine

Strain

During deformation some grains become bent, resulting in different portions of the same grain having different orientations, therefore they go extinct at different times.e.g. quartz, plagioclase

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Uniaxial IndicatrixThe uniaxial indicatrix is constructed by first orienting a crystal with its c-axis vertical.  Since the c-axis is also the optic axis in uniaxial crystals, light traveling along the c-axis will vibrate perpendicular to the c-axis and parallel to the refractive index direction.

Light vibrating perpendicular to the c-axis is associated with the o-ray. Thus, if vectors are drawn with lengths proportional to the refractive index for light vibrating in that direction, such vectors would define a circle with radius .  This circle is referred to as the circular section of the uniaxial indicatrix.  

Quartz, calcite, corundum, beryl, rutile, tourmaline and nepheline are important uniaxial minerals.

Uniaxial minerals belong to either the hexagonal or tetragonal crystal systems and possess two mutually perpendicular refractive indices, (or No) and (or Ne), which are called the principal refractive indices.

Intermediate values occur and are called ', a non-principal refractive index.

The uniaxial indicatrix is an ellipsoid, either

prolate (), termed positive,or

oblate (), termed negative.

In either case, coincides to the single optic axis of the crystal, yielding the name "uniaxial."

The optic axis also coincides with the axis of highest symmetry of the crystal, either the 4-fold for tetragonal minerals or the 3- or 6-fold of the hexagonal class.

Because of the symmetry imposed by the 3-, 4-, or 6-fold axis, the indicatrix contains a circle of radius perpendicular to (perpendicular to the optic axis).

If the ray travels down the optic axis, the cross section perpendicular to the ray is a circle, and the crystal appears isotropic. The further the ray path is from the optic axis, the more elliptical the cross section.

Light that does not vibrate parallel to one of these special directions within the uniaxial indicatrix would exhibit a refractive index intermediate to and and is termed '.

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The ray path and vibration direction for isotropic minerals are perpendicular.

This is not always the case in anisotropic minerals.

For the case when linearly polarized light is vibrating parallel to either of the principal refractive indices the ray path and vibration direction will be perpendicular.

The indicatrix provides the framework for understanding optical measurements on crystals.

Light propagating along directions perpendicular to the c-axis or optic axis is broken into two rays that vibrate perpendicular to each other.

One of these rays, the e-ray vibrates parallel to the c-axis or optic axis and thus vibrates parallel to the refractive index.

Thus, a vector with length proportional to the refractive index will be larger than or smaller than the vectors drawn perpendicular to the optic axis, and will define one axis of an ellipse.

Such an ellipse with the direction as one of its axes and the direction as its other axis is called the Principal Section of the uniaxial indicatrix.

Thus, the uniaxial indicatrix is seen to be an ellipsoid of revolution.  Such an ellipsoid of revolution would be swept out by rotating the ellipse of the principal section by 180o.  Note that there are an infinite number of principal sections that would cut the indicatrix vertically.Light propagating along one of the ' directions is broken into two rays, one vibrates parallel to an ' direction and the other vibrates parallel to the direction.An ellipse that has an ' direction and a direction as its axes is referred to as a random section of the indicatrix.To determine the principal refractive indices for uniaxial minerals, linearly polarized light must be forced to vibrate parallel to them.

A crystal can be oriented on the microscope stage so that, for instance, the optic axis is parallel to the microscope stage.Then a rotation of the microscope stage is made so the E-W lower polarizer is made parallel to the optic axis.

Optic SignRecall that uniaxial minerals can be divided into 2 classes based on the optic sign of the mineral.

If the optic sign is negative and the uniaxial indicatrix would take the form of an oblate spheroid.  Note that such an indicatrix is elongated in the direction of the stroke of a minus sign.

If the optic sign is positive and the uniaxial indicatrix would take the form of a prolate spheroid.  Note that such an indicatrix is elongated in the direction of the vertical stroke of a plus sign.

Application of the Uniaxial IndicatrixThe uniaxial indicatrix provides a useful tool for thinking about the vibration directions of light as it passes through a uniaxial crystal.  Just like crystallographic axes, we can move the indicatrix anywhere in a crystal so long it is moved parallel to itself.  

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This is shown here for an imaginary tetragonal crystal. In this case the optic sign of the mineral is positive, and the uniaxial indicatrix is shown at the center of the crystal.

If the crystal is mounted on the microscope stage such that the c-axis or optic axis is perpendicular to the stage, we can move the indicatrix up to the top face of the crystal (face a) and see that such light will be vibrating in the direction even if we rotate the stage.  Thus we will see the circular section of the indicatrix.

If the crystal is mounted on the stage such that the c-axis is parallel to the stage, we can move the indicatrix to one of the side faces of the crystal (such as face c) and see that light will be broken up into two rays, one vibrating parallel to the direction and one vibrating parallel to the direction.  Thus we will see one of the principal planes of the indicatrix.

If the crystal is mounted on the stage such that the c-axis or optic axis is neither parallel to or perpendicular to the stage, we can move the indicatrix to some random face that is not parallel to or perpendicular to the c-axis (such as face b) and see that the light will be broken into two perpendicular rays, one vibrating parallel to the direction and the other vibrating perpendicular to an '  direction.  Thus we will see one of the random sections of the indicatrix.

We will next look at what we could observe for crystals oriented on the microscope stage for each of the general orientations described above, beginning with the unique circular section.

Circular SectionIf a crystal is mounted on the microscope stage with its optic axis oriented exactly perpendicular to the stage, the circular section of the indicatrix can be imagined to be on the upper surface of the crystal such as for the crystal face labeled a in the diagram above.

In this orientation the crystal behaves just like an isotropic mineral.

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Light polarized in an E-W direction entering the crystal from below remains polarized in an E-W direction as it passes through the crystal. 

Since light is vibrating parallel to an direction for all orientations of the grain, no change in relief would be observed as we rotate the microscope stage.

A comparison of  the refractive index of the grain to that of the oil using the Becke line method would allow for the determination of the refractive index of the mineral.

With the analyzer inserted the grain would go extinct and would remain extinct throughout a 360o rotation of the microscope stage, because the light exiting the crystal will still be polarized in an E-W direction.

 Principal SectionIf the mineral grain is oriented such that the optic axis is oriented parallel to the microscope stage, then we can imagine the principal section of the indicatrix as being parallel to the top of the grain such as would be the case for a crystal lying on face c in the diagram above.In this case, the mineral will show birefringence for most orientations, unless one of the privileged directions in the crystal is lined up with the E-W polarizing direction of the incident light entering from below.

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If the direction in the crystal is parallel to the polarizing direction of the microscope, the light will continue to vibrate in the same direction (E-W) as it passes through the crystal.

In this position, one could use the Becke Line test to measure the refractive index.

If the direction in the crystal is parallel to the  polarizing direction, again, light will continue to vibrate parallel to the polarizing direction as it passes through the crystal.In this position one could use the Becke Line method to determine the refractive index. 

Since the refractive index will be different for the direction and the direction, there will be some change in relief of the grain as it is rotated 90o between the two positions.

How much change in relief  would also depend on the birefringence of the mineral (| - |).

If the analyzer is inserted when the direction or the direction is parallel to the polarizing direction of the microscope,the grain will be extinct, because the light will still be vibrating parallel to the polarizer as it emerges from the grain.

If the and privileged directions in the crystal are at any other angle besides 0o and 90o to the polarizing direction, some of the light will be vibrating at an angle to the polarizer on emergence from the crystal and some of this light will be transmitted through the analyzer.This will be seen as color, called the interference color.

Thus, as one rotates the stage with the analyzer inserted, the grain will go extinct every 90o,and will show an interference color between these extinction positions.

 Random SectionIf the mineral grain is oriented such that the optic axis is oriented at an angle to the microscope stage,then we can imagine a random section of the indicatrix as being parallel to the top of the grain such as would be the case for a crystal lying on face b in the crystal diagram above.In this case, the mineral will also show birefringence for most orientations, unless one of the privileged directions in the crystal is lined up with the E-W polarizing direction of the incident light entering from below.But this time, one of the privileged directions corresponds to the direction and the other to an ' direction in the crystal.

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Again, if the direction in the crystal is parallel to the polarizing direction of the microscope, the light will continue to vibrate in the same direction (E-W) as it passes through the crystal.

In this position, one could use the Becke Line test to measure the refractive index.

If the ' direction in the crystal is parallel to the  polarizing direction, again, light will continue to vibrate parallel to the polarizing direction as it passes through the crystal.In this position one could use the Becke Line method to determine the ' refractive index,but this would be of little value, since ' could have values anywhere between and . 

Since the refractive index will be will be different for the direction and the ' direction, there may or may not be a change in relief of the grain as it is rotated 90o between the two positions.If ' is close to , there will be little change in relief, and if ' is close to , then there could be a large change in relief.

Note that by using the Becke Line method and mounting grains of the same mineral in oils with a variety of refractive indices, we could determine the and refractive indices of the mineral.

Once these are known we could determine the optic sign (if > the mineral is uniaxial negative or if > , the mineral is uniaxial positive), and the birefringence of the mineral (| - |).  But, this would be a time consuming operation and would be difficult.It would be less difficult to determine the and refractive indices of the mineral in the case where the grain is either elongated in the direction of the c-axis or has a cleavage parallel to the c-axis (such as {110}, {100}, {010}, or {10 0}).In these cases one could more easily keep track of which direction is associated with (the direction perpendicular to the c-axis) and which direction is associated with (the direction parallel to the c-axis).Fortunately, there are other means to determine optic sign and birefringence that are less time consuming.

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