Section 1.6.4.12. Uniaxial figures

To understand the formation of an interference figure, consider a simple example, a specimen of calcite cut at right angles to the c crystallographic axis. Calcite is uniaxial negative, with the optic axis parallel to c. The rays that have passed most obliquely through the specimen are focused around the edge of the figure, while the centre is occupied by rays that have travelled parallel to the optic axis (see Fig. 1.6.4.8). The birefringence within the image clearly must increase from nil in the centre to some higher value at the edges, because the rays here have had longer path lengths through the crystal. Furthermore, the image must have radial symmetry, so that the first most obvious feature of the figure is a series of coloured rings, corresponding in outward sequence to the successive orders. The number of rings visible will of course depend on the thickness of the sample, and when birefringence is low enough no rings will be obvious because all colours lie well within the first order (Figs. 1.6.4.9a and b). Fig. 1.6.4.10(a) illustrates, by reference to the indicatrix, the way in which the vibration directions of the o and e rays are disposed. Fig. 1.6.4.10(b) shows the disposition of vibration directions in the figure. Note that o rays always vibrate tangentially and e rays radially. The o-ray vibration directions lie in the plane of the figure, but e-ray vibration directions become progressively more inclined to the plane of the figure towards the edge.

Figure 1.6.4.9 | top | pdf | (a) Centred uniaxial optic axis figure of a crystal with low birefringence. The isogyres are diffuse and the polarization colours in the quadrants are likely to be first-order grey. (b) As above, but for a crystal with high birefringence. The isogyres are narrower, and circular rings of polarization colours are seen, progressing outwards from first order to higher orders. (c) Off-centre uniaxial optic axis figure in the straight position (extinction position of crystal). (d) The same rotated to 45°. The curved arrow shows the rotation of the figure as the stage is turned clockwise. Arrows inside the field of view show directions of motions of isogyres. If the optic axis lies outside the field of view, such figures are difficult to interpret and may easily be confused with an off-centre biaxial optic axis figure when 2V is small to moderate.

Figure 1.6.4.10 | top | pdf | (a) Vibration directions of an oblique ray passing through calcite, viewed relative to the indicatrix. OA is the optic axis, normal to the circular cross section of indicatrix, is the ray direction, e and o are, respectively, the vibration directions of the extraordinary and ordinary rays in the plane of the elliptical cross section of the indicatrix normal to . (b) Vibration directions as seen in the figure. The tangential set represents the o rays, the radial set represents e rays. The isogyres are also shown.

The shaded cross on the figure illustrates the position of dark `brushes' known as isogyres (Fig. 1.6.4.10b). These develop wherever vibration directions lie N–S or E–W, hence corresponding to the vibration directions of the analyser and polarizer. As the stage is rotated, as long as the optic axis is truly parallel to the microscope axis, the figure will not change. This is an example of a centred uniaxial optic axis figure, and such a figure identifies the crystal as belonging to the tetragonal, trigonal or hexagonal systems (see Fig. 1.6.4.11a).

Figure 1.6.4.11 | top | pdf | (a) Centred optic axis figure of quartz (uniaxial). The birefringence is low, resulting in a diffuse cross and quadrant colours in the lower half of the first order. (b) The same figure with the sensitive-tint plate inserted (slow vibration direction NE–SW). The isogyres take on the sensitive-tint colour. Compensation takes place in the NE and SW quadrants (slow directions of crystal and plate parallel to each other) resulting in second-order blue, that is, one whole order above the original. In the other quadrants, the colours also apparently rise, but are restricted to yellow (high first-order). Thus the crystal is optically positive, that is, e rays (radial) are slow and o rays (tangential) are fast. (c) Centred optic axis figure of calcite with sensitive-tint plate (slow direction lying NE–SW) inserted. The birefringence is very high so there are many coloured rings. In the top right quadrant, the colour immediately adjacent to the cross is first-order yellow, whereas in the top left it is second-order blue. Calcite is hence optically negative. It can be observed that all the coloured rings in the top right and bottom left quadrants are an order lower than their counterparts in the top left and bottom right.

From the point of crystal identification, one can also determine whether the figure coincides with the uniaxial positive ) or uniaxial negative () cases. Inserting the sensitive-tint plate will move the coloured ring up or down the birefringence scale by a complete order. Fig. 1.6.4.11(c) shows the centred optic axis figure for calcite, which is optically negative. The insertion of a tint plate with its slow vibration direction lying NE–SW lowers the colours in the NE and SW quadrants of the figure, and raises those in the other quadrants (Fig. 1.6.4.11b). The simplest general rule is to look at the dark first-order grey in the original figure, lying immediately adjacent to the optic axis, i.e. the centre of the cross formed by the isogyres (Figs. 1.6.4.11b and c). If the crystal is optically negative, this colour changes to first-order yellow in the NE quadrant, if positive to blue. When the crystal has low birefringence, these colours may occupy the whole quadrant.

An off-centre uniaxial optic axis figure is obtained when the optic axis is inclined to the microscope axis by an amount which is small enough for it still to be visible within the figure (roughly within 25° of the microscope axis, using a normal high-power objective). Such figures show an isogyre cross with attendant rings, but the centre of the cross does not lie in the centre of the figure, and as the stage is rotated the centre of the cross moves round the figure in a circle (Figs. 1.6.4.9c and d). The isogyres remain NS and EW throughout. If the figure is so off-centre that the centre of the cross is not visible, the behaviour of the figure becomes difficult to interpret, and may easily be confused with some sorts of off-centre biaxial figures (see below). In the extreme case, when the optic axis lies in the plane of the slide, a quite different figure, known as a flash figure, is obtained. This is similar to many of the figures obtained from biaxial crystals, and will be considered further below.