III.  ORIENTATION OF THE INDICATRIX

 

III.1 Relation between the crystal structure and the orientation of the indicatrix

 

The indicatrix has qua shape and orientation to follow the symmetry of the crystallographic structure, which means that there is a direct relationship between the crystallographic class and the indicatrix (see Table 2 below).

- isometric crystals have, because of the fact that the three crystallographic axes are equal in length and perpendicular a ball-shaped indicatrix and are therefore optically isotropic.

- Tetragonal, trigonal and hexagonal crystals have one clearly different crystallographic axis, the c-axis. In relation with this the indicatrix has an optical axis parallel to the c-axis. These crystals are in general optically uniaxial.

- Orthorhombic crystals have three different but perpendicular crystallographic axes. As a result the indicatrix is a three-axes ellipsoide, where the three indicatrix axes are parallel to the three crystallographic axes. These crystals are thus in general optically biaxial (Fig. 3.1).

- Monoclinic crystals have three different crystallograpic axes from which only, the b-axis, is oriented perpendicular to the other two axes. As a result the indicatrix is optically biaxial. With relation to the orientation, the only restriction is that one indicatrix axis has to be parallel to the b-axis (does not matter which one of the two axes).

- Triclinic crystals have no symmetry axes or planes and therefore there is no restriction with respect to the orientation of the indicatrix. Again, in general triclinic minerals are optically biaxial

 

Fig. 3.1 Example of an orthorhombic, optically biaxial crystal with the 3 indicatrix axes parallel to the crystallographic axes.

 

III.2 Extinction angle: parallel and inclined extinction

 

As we have seen before a birefringent mineral grain or crystal shows extinction at 90° intervals on rotation when placed between crossed nicols. When in a specific mineral section one can identify a prominent crystallographic direction, such as the face of a idiomorphic crystal or a cleavage plane, one can determine the angle between this plane and the orientation at which extinction occurs. This angle is known as the extinction angle of that specific section. It is enough to determine one angle; if we assume that the crystallographic direction is A and the ellipse axes α’ and γ’ then we can say that A Λ α’ + A Λ γ’ = 90°. It is important thought to identify relative to which ellipse axis the angle has been determined.

 

The extinction is called to be parallel when the extinction angle is 0°, in the other cases it is called inclined extinction. A special case of inclined extinction occurs when two equal crystal faces or cleavages are visible and the ellipse axes cut the angles between those two planes exactly in half. This special situation is known as symmetrical extinction (Fig. 3.2).

 

 

Fig. 3.2 Relative positions of greatest and least illumination in parallel, inclined and symmetrical extinction

 

 

Which one of these three possibilities is observed depends on the orientation of the indicatrix in the crystal and of the chosen crystallographic orientation. Prismatic sections of tetragonal or trigonal crystals will show parallel extinction, while a pyramid plane will show symmetrical extinction. The cleavage rhombohedrals of calcite will also show symmetrical extinction, with the ε’ as the bisectrix of the  largest angle (>90°) between the two cleavages. Orthorhombic crystals will show parallel (or symmetrical) extinction, when at least one of the crystal axes (indicatrix axes) lies in the plane of the section. In other random sections the extinction angle will seldom be larger than 10°. Monoclinic crystals exhibit parallel (or symmetrical) extinction, when the b-axis (and the corresponding indicatrix axis) are in the plane of the section. In all other sections the extinction will be inclined. When extinction angles are reported for monoclinic minerals, one normally means the angle between the c-axis and the nearest indicatrix axis in the plane (010). Well known examples are the angles c Λ γ for amphiboles and pyroxenes. Triclinic crystals have an undefined indicatrix orientation and therefore will generally show inclined extinction. If one wants to report an extinction angle in this situation, one have to know exactly what orientation one is looking at. For example: α’ Λ (010) in the section perpendicular to a, as is used for the determination of plagioclase.

 

Table 2 Relationships beween crystallographic classes and indicatrix.

 

 

A very useful application for the measurement of extinction angle is in the determination of the plagioclase feldspar compositions. Feldspars are most ubiquitous rock forming minerals and occur in many igneous, metamorphic and sedimentary rocks. Their range of compositions has led to their use as a means to classify igneous rocks. Compositions of feldspars are normally expressed in terms of mole percentages of the three essential components anorthite (An), albite (Aab) and orthoclase (Or) and are written as An_xAb_yOr_z, where x, y and z indicate the mole percentages of each. Because the amount of orthoclase in plagioclase is normally small, especially in calcic plagioclases, composition is generally expressed only in terms of anorthite-albite (CaAl₂Si₂O₈-NaAlSi₃O₈) content (recalculated to 100%) and is simply reported as An_%.

 

Fig. 3.3 Polysynthetic twinning of the albite type (a) and a cleavage fragment showing albite twinning, cleavage parallel to (001).

 

The composition of a plagioclase can be determined by measuring the symmetrical extinction angles of albite twins (Fig. 3.3) on sections at right angles to the a-axis (sections normal to {010}). In this orientation twin lamellae appear as very sharp, dark and light stripes or bands under crossed polarisers and display maximum ectinction angles. When the twin lamellae are parallel to the vertical cross hair of the microscope they merge into a single grey tone across the entire mineral. In the correct orientation, rotating the microscope stage to the left and right of this vertical position should produce similar extinction angles of the albite twin lamellae (Fig. 3.4). If the extinction angles are within 5° they can be averaged and the composition of the plagioclase from the Michel-Levy chart (Fig. 3.5).

 

 

Fig. 3.4 diagram showing the method of determining the extinction angles in albite twins in plagioclase in sections normal to (010)

 

Fig. 3.5 Curve showing the maximum extinction angle of albite twins in plagioclase sections normal to (010) (Michel-Levy method) (click on image for larger image)

 

In order to identify plagioclase in the An₀-An₂₁ range and in the An₂₁-An₃₈ range (extinction angles 0-20°), the refractive indices or optical sign may be used. For the portion of the curve less than An₂₀ the optical sign is positive and the refractive index is less than the mounting medium or quartz (1.544). For the portion of the curve greater than An₂₀, the optical sign is negative and the refractive index is higher than that of the mounting medium or quartz.