Interference Figures

An interference figure is the conoscopic image seen through the petrographic microscope when strongly convergent polarized light passes through a mineral with crossed polarizers and the Bertrand lens in place. It provides, rapidly and reliably, three pieces of optical information that would otherwise take far longer to obtain: whether the mineral is uniaxial or biaxial (optical character), the optic sign, and - for biaxial minerals - an estimate of the 2V angle. Interference figures can also confirm the orientation of a mineral grain in a thin section. ^[1]

Quick Revision

• Conoscopic setup: auxiliary condensing lens + open aperture diaphragm + upper polarizer + Bertrand lens (or remove ocular and look down the tube). ^[1]
• Isochromes = rings of equal retardation; increase in order outward from the melatope. ^[1]
• Isogyres = dark extinction bands where vibration directions in the figure are N-S or E-W. ^[1]
• Melatope = the dark point at the center of an optic axis figure; marks where the optic axis (or bisectrix) emerges. ^[1]
• Uniaxial OA figure = black cross + circular isochromes; cross stays centered if optic axis is exactly vertical. ^[1]
• Biaxial Bxa figure = two melatopes + oval isochromes; isogyres form a cross (optic plane E-W/N-S) or hyperbolic arcs (45° position). ^[1]
• Optic sign (biaxial): insert accessory plate (slow NE-SW) in Bxa figure; if retardations subtract between melatopes → mineral is optically positive. ^[1]
• 2V estimation: isogyre curvature in optic axis figure at 45° position; straight = 90°; progressively curved = smaller 2V. ^[1]

Setting Up for Conoscopic Observation

To view an interference figure, focus on a single mineral grain with the high-power objective. Then flip in the auxiliary condensing lens and open the aperture diaphragm fully. Insert the upper polarizer to cross it with the lower polarizer. Finally, insert the Bertrand lens - the interference figure is formed just above the objective lens and the Bertrand lens brings it into focus. If the Bertrand lens is not available, the ocular can be removed and the figure viewed by looking directly down the microscope tube. ^[1]

Isochromes and Isogyres

The interference figure consists of two overlapping patterns. Isochromes are rings of equal retardation and equal interference color. They form because the auxiliary condensing lens sends strongly convergent light through the mineral at a wide range of angles to the optic axis. Light following the optic axis itself experiences zero retardation and forms the dark central point called the melatope. Light at progressively greater angles to the optic axis encounters increasing birefringence and must travel a longer path through the mineral, so retardation grows with increasing inclination. Rings of equal retardation - the isochromes - are therefore formed concentrically around the melatope. Thick minerals and high-birefringence minerals show many isochromes; thin or low-birefringence minerals show few. ^[1]

Isogyres are dark bands superimposed on the isochrome pattern. They form wherever the vibration directions of the convergent light in the interference figure happen to be aligned N-S and E-W - that is, parallel to the lower and upper polarizers. In those areas the mineral is in the extinction position for that beam direction, and no light passes through, producing a dark band. The isogyres define the overall shape of the figure and change dramatically as the stage is rotated, making them the primary diagnostic feature. ^[1]

Uniaxial Interference Figures

Optic Axis Figure

The uniaxial optic axis figure is obtained when the grain’s single optic axis is oriented perpendicular to the microscope stage (i.e., vertical). The petrographer identifies such a grain because it shows the lowest retardation - lowest interference color - of all grains in the sample, since light along the optic axis experiences no double refraction. The interference figure consists of a black cross (the isogyres) centered on a point called the melatope, superimposed on a circular pattern of isochromes that increase in order outward from the center. In the extraordinary ray pattern, ε′ rays vibrate along radial lines (parallel to lines of longitude as seen from above), while ω rays vibrate tangent to the circular isochromes (parallel to lines of latitude). The isogyres form where these vibration directions are N-S and E-W. If the optic axis is exactly vertical, the cross remains fixed as the stage is rotated. The presence of a single melatope confirms that the mineral is uniaxial. ^[1]

Off-Center Uniaxial Figure

If the optic axis is not exactly vertical but is inclined only about 25° to 30° from vertical, the melatope is displaced from the center but still visible within the field of view. This is an off-center optic axis figure. The isogyres still form a cross centered on the melatope, but the cross swings in an arc around the field of view as the stage is rotated. As long as the melatope remains visible, the figure can still be used to determine optical character and optic sign. If the optic axis is inclined more than about 25° to 30° from vertical, the melatope exits the field of view entirely and only one or two isogyre arms sweep sequentially across the field as the stage is rotated. This fully off-center figure is the most commonly encountered figure when grains are selected at random, but it is the least useful - both uniaxial and biaxial minerals can produce similar off-center figures, so optical character and optic sign cannot be reliably determined from it. ^[1]

Optic Normal (Flash) Figure

A uniaxial optic normal figure is produced when the optic axis is exactly horizontal - parallel to the microscope stage. Grains in this orientation show the highest interference color in the sample because they are in the maximum birefringence orientation. The figure is characterized by broad, fuzzy isogyres that occupy nearly the entire field of view when the trace of the optic axis is E-W or N-S. With only a few degrees of stage rotation the isogyres split apart and rapidly exit the field of view from the quadrants into which the optic axis is being rotated - this rapid disappearance is why the figure is also called a flash figure. An optic normal figure confirms that the optic axis is approximately horizontal. However, it is not routinely used to determine optical character or optic sign because biaxial minerals can produce interference figures with nearly identical appearances. ^[1]

Determining Optic Sign (Uniaxial)

The optic axis figure is used to determine optic sign. The vibration directions of ω and ε rays are known at every point in the figure: ordinary rays vibrate tangent to the isochromes (lines of latitude) and extraordinary rays vibrate along radial lines (lines of longitude). Note the interference color retardation Δ1 in the four quadrants from the interference color chart. Insert an accessory plate with retardation ΔA whose slow ray is NE-SW. Retardations will add in two quadrants and subtract in two. In the NW and SE quadrants, ordinary rays vibrate NE-SW, parallel to the accessory plate slow ray. If retardations subtract in those quadrants, the ordinary ray is the fast ray and the mineral is optically positive. If retardations add, the ordinary ray is the slow ray and the mineral is optically negative. Where retardations add, isochrome bands appear to shift inward (higher colors); where they subtract, bands shift outward (lower colors). ^[1]

Biaxial Interference Figures

Biaxial minerals produce interference figures with two melatopes, one for each optic axis. This is the most immediate way to distinguish them from uniaxial minerals. The optic sign and 2V angle can also be determined. ^[1]

Isochromes in Biaxial Figures

In a biaxial acute bisectrix figure the isochromes form an oval or figure-eight pattern centred on the two melatopes. Light travelling exactly along either optic axis experiences zero retardation (melatope). At progressively greater angles to both optic axes, retardation increases, forming the oval rings. As the stage is rotated the isochrome pattern stays fixed relative to the melatopes even as the isogyres change shape. ^[1]

Acute Bisectrix (Bxa) Figure

An acute bisectrix figure is obtained when the acute bisectrix (Bxa) - the X or Z indicatrix axis depending on optic sign - is perpendicular to the stage. Identifying the correct grain is mostly a matter of chance because the required grain displays intermediate retardation. A piece of muscovite is a good practice specimen. When 2V is less than about 45° to 55°, both melatopes are visible in the field of view. When the optic plane is oriented E-W or N-S, the isogyres form a cross; the arm parallel to the optic normal is wider than the arm parallel to the optic plane, and the melatopes are marked by a narrowing of the isogyres. Rotating the stage a few degrees causes the cross to split into two separate hyperbolic isogyre segments that pivot about the melatopes, with the vertices of the hyperbolas at the melatopes. ^[1]

When 2V is greater than about 50° to 60°, the melatopes are outside the field of view. With the optic plane oriented N-S or E-W, only a single straight isogyre arm is visible, parallel to the trace of the optic plane. As the stage is rotated clockwise, this isogyre pivots counterclockwise about the melatope - opposite to the stage rotation direction. At the 45° position, the isogyre shows maximum curvature, and the acute bisectrix lies on the convex side of the curve. ^[1]

Obtuse Bisectrix (Bxo) Figure

An obtuse bisectrix figure is produced when the Bxo is perpendicular to the stage. Identifying the correct grain is again mostly a matter of chance and intermediate retardation means many grains may have to be examined. Because the angle between the Bxo and each optic axis is greater than 45°, both melatopes lie outside the field of view. The isochrome and vibration-direction geometry is essentially the same as in the Bxa figure, but without the visible melatopes. When the optic plane is E-W or N-S, isogyres form a broad cross; only about 5° to 15° of stage rotation is needed to push the isogyres out of the field of view. Obtuse bisectrix figures are not routinely used to measure optical properties; they mainly confirm that the mineral is biaxial. When 2V approaches 90°, Bxa and Bxo figures become very similar to each other; when 2V is small, the Bxo figure resembles an optic normal figure. ^[1]

Optic Normal (Biaxial Flash) Figure

A biaxial optic normal figure is produced when the optic normal (Y axis) is vertical. Grains in this orientation show maximum retardation because both X and Z are horizontal. The vibration direction pattern is nearly rectilinear and very similar to the uniaxial flash figure. When X and Z are aligned N-S and E-W, a broad fuzzy cross fills the field of view; a few degrees of stage rotation causes the isogyres to split and rapidly exit the field from the quadrants into which the Bxa is being rotated. For minerals with 2V close to 90° the diffuse cross simply dissolves as the stage is rotated. Optic normal figures confirm that the optic plane is parallel to the stage but are not routinely used to extract optical information. ^[1]

Off-Center Biaxial Figure

Grains in random orientations produce off-center interference figures whose isogyre and isochrome pattern depends on the details of that grain’s orientation. As the stage is rotated, isogyres sweep sequentially across the field of view, and the wide end of each isogyre sweeps faster than the narrow end. Crucially, the isogyres in a biaxial off-center figure are not parallel to the N-S or E-W crosshairs as they sweep - this is what distinguishes a biaxial off-center figure from a uniaxial off-center figure, where the isogyre arms remain essentially N-S and E-W. Unless a melatope or the acute bisectrix happens to be visible, off-center biaxial figures cannot be used to determine 2V or optic sign, but they do confirm that the mineral is biaxial. ^[1]

Determining Optic Sign (Biaxial)

Using an Acute Bisectrix Figure

In the center of an acute bisectrix figure, two rays propagate along the Bxa. One vibrates parallel to Y (the optic normal) and has index nβ. The other vibrates parallel to the Bxo along the trace of the optic plane and has index nBxo. For an optically positive mineral, the Bxo is X and nBxo = nα (fast ray). For an optically negative mineral, the Bxo is Z and nBxo = nγ (slow ray). The task is simply to determine whether the ray vibrating along the optic plane trace in the center of the field is fast or slow. To do this, rotate the stage so the trace of the optic plane is NE-SW. Insert the accessory plate (slow NE-SW). Observe the interference color change in the center of the figure. If retardations subtract (interference color drops), the NE-SW ray is the fast ray (nα), so the Bxo is X, the Bxa is Z, and the mineral is optically positive. If retardations add, the NE-SW ray is the slow ray (nγ), so the Bxo is Z, the Bxa is X, and the mineral is optically negative. Colors outside the melatopes change in the opposite sense to those between them. ^[1]

Using an Optic Axis Figure

The biaxial optic axis figure is more commonly used in practice because grains oriented to display it - low or zero retardation - are easier to locate. If 2V is small enough that both melatopes are in the field of view, the figure is treated as a half of an acute bisectrix figure and interpreted the same way. If only one melatope is visible, orient the isogyre N-S and then rotate the stage 45° clockwise. The isogyre will be convex toward the NE or SW, and the Bxa is on the convex side. The accessory plate is then used in the same way as for the Bxa figure to determine sign. If 2V is nearly 90°, the isogyre is nearly straight and the Bxa cannot be located reliably; in this case the optic sign is recorded as neutral. ^[1]

Determining 2V

Apparent Optic Angle (2E)

Light following the optic axes within the mineral has index nβ and is refracted on exiting into air (n ≈ 1) above the mineral. The refracted angle between the two optic axes is the apparent optic angle, 2E, which is larger than 2V. The relationship is sin E = nβ sin V (Equation 7.18). The melatope spacing in the interference figure corresponds to 2E, not 2V directly. For the melatopes to be visible at the edge of the field of view, the angular aperture (numerical aperture, NA) of the objective lens must equal the refracted ray half-angle: NA = nβ sin V⊂max (Equation 7.19). Table 7.1 gives the largest 2V (2V⊂max) that places melatopes within the field of view for a range of nβ values and for objective lenses with NA = 0.65 and 0.85. ^[1]

Because most biaxial minerals have nβ in the range 1.50-1.70, these values show that a standard 0.65 NA objective can display melatopes for 2V up to roughly 48° to 51°. This range covers a large proportion of common minerals and makes 2V estimation feasible for routine work, with accuracy of about ±10° to 15°.

Isogyre Curvature Method

A more convenient 2V estimate uses the biaxial optic axis figure. The curvature of the isogyre when the trace of the optic plane is in the 45° position is directly related to 2V. If 2V = 90°, the isogyre is perfectly straight. Progressively smaller values of 2V produce progressively stronger curvature. If 2V is less than about 25° to 30°, both melatopes are in the field of view and the distance between them provides a direct estimate. For 2V less than 5°, the figure is nearly indistinguishable from a uniaxial figure except for a small gap between the isogyres. ^[1]

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