IV.3 Interference figures of biaxial minerals perpendicular to acute bisectrix

Let's start the discussion of the conoscopic view of the optically biaxial minerals with the section perpendicular to the acute bisectrix. In Fig. 4.7 the sections with their schematic inteference schemes are given for both optically positive (A) and negative minerals (B). First, have a look at the optically positive crystal. The acute bisectrix γ is perpendicular to the section, the optic axial plane (with the optical axes A1 and A2) are oriented vertically E-W. On the top of the crystal is the elliptical section with the axes α and β indicated, i.e. the elliptical section with the perpendicular incoming front normal in the orthoscopic situation. On the side of the crystal we see the third main section through the indicatrix. Bringing in the Bertrand lens will result in an interference figure. Fig. 4.7 shows the interference scheme with the orientation of the various sections of the indicatrix.

Fig. 4.7 Conoscopic interference schemes for biaxial minerals sectioned perpendicular to the acute bisectrix. A = optically positive, B = optically negative.

The centre of the figure represents again light travelling perpendicular to the section, but in contrast to the previous section where we looked parallel to the optical axis of an uniaxial crystal, now the section through the indicatrix has the shape of an ellipse. When we vary the direction of the light in the E-W direction, then we cross the two optical axes on both sides of the acute bisectrix, where the sections have the shape of a circle. Further variation of the direction of the light will result in an increase in birefringence similar to what we have seen for the optical axis in the uniaxial crystal. When the birefringence is high enough we will observe concentric circles around each of the optical axes; but further out to the rim of the field of view they will form more or less oval shapes around both axes (Fig. 4.8a). This is sometimes known as the 0° position, where we see again a black cross very similar to that in an uniaxial interference figure but at the two melatopes the isogyres are normally showing a strong thinning. But what will happen when we rotate the microscope stage?

Fig. 4.8 Interference figure of a biaxial crystal; a) in 0° position, b) in 45° position.

In order to show what is happening, let's have a look at Fig. 4.9, which shows how the zones will look corresponding to the ellipse axes oriented N-S and E-W when the stage is rotated. These zones form the isogyres of the interference figure. When we rotate the stage from the 0° position to the 45° position we that the cross will open up to a maximum. The distance between the two melatopes, of course in relation to the aperture of the objective used, is directly related to the 2V angle between the two optical axes. Further rotation to the 90° position results again in a closure and the forming of the cross again. So, over a full 360° rotation we'll see the cross open and close four times; two time E-W and two times N-S.

Fig. 4.9 Interference scheme of a biaxial positive crystal similar to Fig. 4.8. The variation of the ellipse orientations is given by the solid lines for the longest axes, also compare to Fig. 4.7A.

Similar to the uniaxial crystals, the perpendicular to the section oriented indicatrix axis (γ for positive and α for negative crystals) will be projected radially by under an angle incoming light. Because of this, we can use the gypsum plate again to determine which indicatrix axis is oriented vertically. The NE and SW quadrants will show addition when the longest axis is vertical. If we also know that this axis is the acute bisectrix, then we have an optically positive crystal. We can also determine the optical sign in the 45° position, when we know which quadrant the isogyre is showing us. For crystals with large 2V angles, the isogyres can move out of our field of view making it very difficult to determine which bisectrix we're looking at.

Fig. 4.10 Determination of the optical sign of a biaxial mineral

IV.4 Determination of angle between the optical axes

We have already seen that in the interference figure the opening distance between the isogyres in the 45° position is determined by the numeric aperture of the objective, the refractive index β and the angle 2V. We can thus use the interference figure to estimate 2V. In the interference figure actually we don't have 2V but the angle 2E in air (Fig. 4.11). Since we're looking at the optical axes, we have the following relationships:

sin E/sin V = nmin

or more precisely: sin E/sin V = β

Fig. 4.11 relationship between the angles 2V and 2E, OA and OB are the optical axes.

When the focus of the objective, where the interference figure is formed, has the shape of a sphere, we can use the following relationship according to Mallard (Fig. 4.12):

Fig. 4.12 Mallard's relationship

b = k sin E, where b = half the distance between the isogyres, k = constant, depending on the lenses used, E = half the angle between the optical axes in air

Also:

r = k sin U = k NA, where r = half the diameter of the field of view, U = half the top angle of the light cone going through the objective and NA = numeric aperture of the objective.

In order to remove the unknown constant k, we define d = 2b/2r (Fig. 4.13)

Fig. 4.13

d = 2b/2r = k sin E/k sin U = sin E/NA, since sin E/sin V = β

we also find:

d = 2b/2r = β sin V/NA

With the help of the cross hairs we can now determine d, when we know β and NA, and we can calculate sin V and thus 2V. fig. 4.14 gives an example of a graph for NA = 0.85 from which 2V can be directly determined from d and β. On the right y-axis is indicated the maximum measurable 2V angle (b = r) as a function of β.

Fig. 4.14 Nomogram for the determination of 2V in the acute bisectrix for objective with NA = 0.85.

IV.5 Interference figures of biaxial minerals perpendicular to an optical axis

When the acute bisectrix makes an angle with the normal of the crystal section, it will move around in a circle when we rotate the stage, similar to what we have seen for uniaxial crystals (Fig. 4.15). However, during rotating the stage over 360° the cross will now open and close four times. Fig. 4.15 gives an example of a crystal of a strong birefringent mineral with an average 2V angle that is more or less sectioned perpendicular to the optical axis. We can recognise these sections easily in orthoscopic view because of the almost isotropic behaviour. Since the isogyre always has to go through the central melatope, it will during rotation of the stage, rotate around the melatope. In the 0° and 90° positions the isogyre will form a part of a cross, while in the 45° position the isogyre will show the maximum opening and curvature.

Fig. 4.15 section perpendicular to the optical axis for a mineral with a 2V angle of about 50°.

The strength of the curvature is again a function of both 2V and β. (Fig. 4.16). A value of 2V = 0° corresponds of course with an uniaxial interference figure. When 2V increases we will observe a change in the curvature of the isogyre. For relatively small values the second isogyre will remain visible in the 45° position and we can determine the 2V angle according to the method described before. When the 2V angle becomes larger (roughly between 35 and 65°) the second isogyre will move out of our field of view, but the acute bisectrix remains visible as the cross in the 0 and 90° positions. At 2V angle larger than 65° we will observe only one isogyre rotating in our field of view, without forming the cross. When we can observe a clear curvature in the 45° position, the convex side points towards the acute bisectrix and the optical sign can be determined analogue to what we have seen before. When the isogyre remains straight upon rotation we have a 2V angle of 90° and the crystal is called optically neutral.

Fig. 4.16 Determination of 2V based on the separation of the isogyres in the acute bisectrix (β = 1.60, NA = 0.85) on the left and based on teh curvature of the isogyre on the right (top both melatopes in field of view, bottome one melatope in field of view).

IV.6 Centric interference figures with out of view moving isogyres

Interference figures that show a sharp or fuzzy cross in the 0° or 90° positions are known as centric interference figures. 2V can easily be determined when the isogyres remain in the field of in the 45° position, as we have seen before. When the isogyres disappear after a rotation of the stage of about 30° or more, we can't determine 2V, but we still can recognise that we have a section perpendicular to the acute bisectrix. Here we can still determine the optical sign, when we know at which quadrant we're looking. However, when the isogyres leave our field of view after rotation of the stage between roughly 15 and 30° it is very difficult to tell bisectrix we're looking at. Do the isogyres leave our field of view after rotation of about 10-15°, then we are looking at a section almost perpendicular to the obtuse bisectrix. We can also use this interference figure to determine the optical sign but the colour addition and subtraction after bringing in the gypsum plate will be opposite to the situation where we are looking at the acute bisectrix. For rotation angles smaller than 10° we are looking at a section perpendicular to the optical normal (β) and this interference figure is known as a flash figure. For small 2V angles this can be confused with the obtuse bisectrix interference figure; the isogyres are not much broader and the optical sign can be determined following the normal methods. Increasing the 2V angle will result in a flash figure that becomes more and more fuzzy and broader until at 2V = 0° the cross fills the whole field of view and disappears completely after rotation of the stage over just a few degrees without splitting up in two separate isogyres.

Fig. 4.17 Flash figure (section perpendicular to β) for a mineral with a small 2V angle.

IV.7 Dispersion

IV.7.1 Introduction

Previously we have seen the relationships between the indicatrix and the crystal classes (Table 2) that are valid for all wavelengths. This means that there is a possibility for dispersion as a function of the symmetry of the crystal. It will be clear that in isometric or cubic crystals the only possible form of dispersion is the dispersion of refractive index (see I.4.3). In optically uniaxial minerals (trigonal, tetragonal and hexagonal crystal classes) we will never observe dispersion of the optical axis, since this optical axis has to coincide with the crystallographic c-axis. However, dispersion of the birefringence can be observed; changing the length of the indicatrix axes does not change the symmetry (see II.8). In orthorhombic crystals the indicatrix axes are oriented parallel to the crystallographic axes, so we can only find dispersion of the optical axes. In the monoclinic system only one of the indicatrix axes is parallel to the crystallographic b-axis, which means that the other two can show dispersion in addition to the dispersion of the optical axes. In the triclinic system there is no relationship between the orientation of the indicatrix axes and the crystallographic axes, which means that all indicatrix axes can show dispersion. In the next couple of paragraphs we will have a look at the various forms of axial dispersion. All of them lead to what we have called before the dispersion of extinction (III.4). Since the orthoscopic rays are represented in the conoscopic view in the centre of the interference figure, dispersion of extinction will be observed as coloured fringes around the isogyres.

IV.7.2 Dispersion of the optical axes, type I

The angle between the optical axes is a function of α, β and γ. When the refractive indices are independently a function of the wavelength, then the 2V angle will also change as a function of the wavelength λ (Fig. 4.18).

Fig. 4.18 Example of refractive index curves resulting in axial dispersion.

So, dispersion of the optical axes results in different shapes of the indicatrix for different wavelengths. In the interference figure every colour has now its own isogyre (see exaggerated example in Fig. 4.19). In Fig. 4.19  2V  for red light is larger than the 2V for blue light (shorthand r>v), resulting in a reddish fringe on the outside of the isogyre and a bluish fringeon the inside of the isogyre. For r<v the situation is the opposite. This can be seen for example in minerals such as titanite and zoisite.

Fig. 4.19 Dispersion of the optical axes in a section perpendicular to the optical axis, r>v.

IV.7.3 Dispersion of the indicatrix axes

Disperion of the indicatrix axes means that the orientation of the indicatrix relative to the crystallographic directions depends on the wavelength of the light. This implies that dispersion of one or more indicatrix axes, and one of the optical axes. Every wavelength has its own indicatrix orientation and therefore its own orientation of its optical axes. Analogue to the type I dispersion described above, we will observe the blue and red rims around the isogyres. Though symmetry rules can restrict the possibilities, we will still find more possibilities than as for type I. For example in monoclinic crystals we will find three forms of dispersion (see also Fig 4.20 in next paragraph).

β = b

The optical normal will be parallel to the b-axis. This results in inclined dispersion (Fig. 4.20). For all wavelengths the axial plane is perpendicular to the crystallographic b-axis and is thus parallel to the symmetry plane of the monoclinic class. In this plane we will also find α and γ and therefore the optical axes. For each wavelength the orientation of the optical axes will be slightly different, in other words for blue light the indicatrix has slightly rotated around β relative to the indicatrix for red light.

Fig. 4.20 Inclined dispersion.

The interference figure will give coloured bands along the isogyres. This is the purest form of inclined dispersion. Often we will find it combined with dispersion of the optical axes type I, so that not only the orientation of α and γ change with the wavelength but also the 2V angle and the orientation of the optical axes. This combination will result in a diminished dispersion along one axis and increased dispersion along the other. Minerals that will show this effect are for example sapphirine and most of the clinopyroxenes.

BXO = b

Here the obtuse bisectrix is parallel to the b-axis. This is known as parallel or horizontal dispersion. (Fig. 4.21). For every wavelength the acute bisectrix is oriented parallel to the b-axis.. The position of the acucte bisectrix though changes as a function of the wavelength. As a result the orientation of the axial plane will change. We can imagine that for blue light the axial plane will be rotated around the obtuse bisectrix relative to the axial plane for red light. The fourth column in Fig. 4.23 shows the interference figure for  the acute bisectrix with the resulting colour fringes along the isogyres. Notice that the colour distribution still follows the rules for the symmetry plane.

Fig. 4.21 Parallel or horizontal dispersion.

BXA = b

Here the acute bisectrix is oriented parallel to the b-axis. This is known as crossed dispersion (Fig. 4.22). Again every wavelength results in a slightly different orientation of the axial plane. The axial plane for blue light is slightly rotated around the commone acute bisectrix relative to that for red light. The interference figure of the acute bisectrix will show the various colours, as shown in column 5 of Fig. 4.23.

Fig. 4.22 Crossed dispersion.

For dispersion in triclinic crystals are no rules because there is no symmetry plane available and the orientation of the indicatrix is dependent on the wavelength of the light.

Fig. 4.23 Dispersion of  the indicatrix and optical axes in interference figures perpendicular to the acute bisectrix. Only the coloured fringes are indicated, not the dispersion of the optical axes.