II

II.5 Determination of vibrational directions and magnitude of the refractive indices from the indicatrix

Let’s have another look at the interface between air and a crystal with a light ray hitting the surface perpendicular (again assuming a planar wave front). In the crystal we will observe two planar wave fronts parallel to another, so with the same wave normal. The vibrational directions (polarisation) are perpendicular to another and perpendicular to the shared wave normal. This plane intersects the indicatrix in the shape of an elliptical section. All the vibrational directions have to be located in this ellipse. There exists only one possibility: the vibrational directions we are looking for have to coincide with the axes of the ellipse. The length of each axis represents the corresponding refractive index. This construction is valid for both uniaxial and biaxial indicatrices.

II.5.2 The elliptical section

In every mineral grain or crystal we can visualize an imaginary indicatrix. Microscope samples (both thin sections and grain samples) section a mineral according to a certain orientation. The same is then true for our imaginary indicatrix, which is known as the elliptical section. Light is vibrating (polarised) in the directions of the axes of this ellipse and are perpendicularly polarised. When the ellipse contains one or two of the main axes of the indicatrix, then we can note down this as the ellipse axis. In a random section through an uniaxial crystal the ellipse axes are indicated as ε’ and ω’ (see II.4), while the largest and smallest axis of a random elliptical section are indicated with γ’ and α’.

II.6 Interference colours

II.6.1 Monochromatic light

Two light rays can interfere with another when:

a) their wavelengths are identical and

b) the polarisation directions are identical

In the optical microscope light passes in order:

1) the polariser, which lets through only N-S vibrating light (P-P in Fig. 2.8)

2) the mineral section, which will split the N-S polarised light into two rays with vibrational directions T₁ and T₂ according to the axes of the elliptical section

3) the analyser, which will allow only the E-W components of the two rays to pass (A-A in Fig. 2.8)

Fig. 2.8 propagation of light in an anisotropic medium

When the amplitude = √(light intensity) of the polarised light through the plane P-P is equal to 1, then the amplitude of the two rays that pass the analyser will be ½ a sin 2α. The function f(α) = ½ a sin 2α has a maximum for α = 45°, 135ö, 225° and 315° (this is known as the 45° position) and a minimum of 0 for α = 0°, 90°, 180° and 270° (0° position or extinction position). So, when we rotate the microscope stage with our mineral 360° we will see 4x extinction and 4x maximum brightness. In the extinction orientations the ellipse axes coincide with the vibrational directions of the polarisers.

Fig. 2.9

The vibrations of the two rays have a phase difference of 1/2λ. When both rays have the same velocity, the resulting amplitude would be zero and we would see no light at all. However, in an anisotropic medium in general both rays do have different velocities. Have a look at Fig. 2.9. When a light ray enters a mineral section with thickness d at time t₀, then the ray polarised according to T₂ (higher refractive index n₂, lower velocity v₂) will leave the mineral section at A at time t₂. The other ray polarised according to T₁ (lower refractive index n₁, higher velocity v₁) will leave the mineral section at t₁ and at t₂ it will have travelled the additional distance AB. This distance is known as the retardation (Δ). When we assume that outside the crystal we have a vacuum we can deduct that:

//T₂: t₂ – t₀ = d/v₂

//T₁: t₁ - t0 = d/v₁

Δ = AB = c(t₂ - t₁) = cd(1/v₂ - 1/v₁) = d(n₂ - n₁)

This result does not change significantly when we replace vacuum by air. So, for any random section through a biaxial crystal we can rewrite this formula to Δ = d(γ’ – α’).

Since one ray is slightly ahead of the other we will see a certain amount of light intensity (Fig. 2.10).

Fig. 2.10 A mineral in monochromatic light between polariser and analyser. For clarity of the figure both resulting rays have been displaced a bit from OQ to O₂Q₂ and from OR to O₁R₁. It is assumed that n₁/n₂ = v₂/v₁ = 15/16. In this example we will then get Δ = 1/8λ(air). Above the analyser both rays will give interference, the solid line is the resulting wave.

When the thickness of the mineral is such that one ray is exactly one λ ahead of the other, we will see extinction. The same is true for Δ = 2λ, 3λ, 4λ, etc. In the situation that the difference is 1/2λ, the components of the two rays will be in phase and the resulting light coming through the analyser will show maximum intensity.

Replacing a flat mineral section by a prism shaped section we will see a regular pattern of dark and light bands (Fig. 2.11)

Fig. 2.11 a prism shaped mineral in 45° position between polariser and analyser, using monochromatic light with wavelength λ. Over the prism the thickness and thus Δ will increase regularly. Dark bands are observed where Δ = nλ (with n = 0, 1, 2, etc.) when we choose light with a smaller λ, the dark bands will be observed closer to one another.

II.6.2 White light

Normally we don’t use monochromatic light but white light. White light contains all colours of the visible part of the electromagnetic spectrum. Every colour has its own wavelength. When white light travels through a prism shaped mineral, every colour will go extinct or show maximum intensity at certain thicknesses. Extinction of a certain colour will result in the observation of its complementary colour. The prism will now show an array of coloured bands. These colours are called interference colours. The order of these interference colours is reproduced in Newton’s colour scale in the Michel Lévy chart (Fig. 2.13). A similar pattern of interference colours can be seen as concentric rings in mineral grains with increasing thickness from rim to centre.

Fig. 2.13 Michel Lévy colour chart (click on image for larger version)

The interference colours can be divided in orders. The boundary between each order is generally at the typical violet colour of 550 nm and multiples of that. The colours of lowest order are more intense that the colours of higher order. At orders of 7 and higher we talk about higher order white or crème. This is caused by the fact that at those higher orders a number of intensity maxima occur which together result in white light. A good example of this effect can be seen in calcite grains.

The interference colours can easily be identified as belonging to a certain order:

1^st order: black-grey-white-yellow-orange-red I

2^nd order: blue-green-yellow-red II

3^rd order; blue-green-meat colour-red III

4^th order: pale green-pale red IV

So, each order is separated from the next by the colour red. At higher orders the colours become increasingly pale, until the colour does not vary anymore and we reach the higher order white. First order red I can easily be identified by bringing in the gypsum plate. Red I is also known with the French term “teinte sensible” because a small change in R (as caused by bringing in the gypsum plate) results in a colour change to either orange (R smaller) or blue (R larger).

II.7 Birefringence

The maximum difference in refractive index for a double refracting mineral is known as the birefringence of a mineral. For uniaxial minerals the birefringence = │ε - w│, while for biaxial minerals the birefringence is │γ - α│. Besides direct determination of the two refractive indices the birefringence can also be determined from the interference colour. From II.6 we know that Δ = d │ε - w│for uniaxial minerals and Δ = d │γ - α│for biaxial minerals, where Δ is the retardation (in nm) that determines the interference colour and d is the thickness of the mineral section. In formula:

This is the basis for the Michel Lévy chart, from which we can determine directly the birefringence when we know d and Δ. To find the correct section through a mineral one has to search for the grain with the highest birefringence in the thin section. Later on we will discuss how we can use conoscopic viewing to check that we have the correct section (see IV.6). Sometimes handbooks use relative terms to indicate the birefringence. Roughly one can say:

Anomalous interference colours can be observed when the birefringence at different wavelengths are strongly dissimilar, i.e. dispersion of the birefringence occurs. This will cause changes in Newton's colour scale, especially in the first order range. For example first order gray will show indigo-blue or violet when the crystal is almost isotropic in the red part of the spectrum (see e.g. minerals such as chlorite, vesuvianite and zoisite), yellow-brown or reddish when this is the case in the blue part of the spectrum (e.g. chlorite, vesuvianite). In the higher order ranges the colour changes are much harder to detect. One example with strong anomalous interference colours in the third order can be observed for epidote.

In the tube of the microscope we can insert a gypsum plate or first order red compensator. This plate is inserted under a 45° angle with the N-S direction of the microscope. Gypsum is a birefringent mineral. In the gypsum plate the thickness of the crystal is chosen so that for green light with a wavelength of 550 nm exactly a retardation Δ of exactly 1λ occurs. This means that green light will be extinct after passage through the analyser. We can check this by using white light and without any sample on the stage. When we insert the gypsum plate we will see the complementary colour: red. This colour red is the border between first and second order. The gypsum crystal is so oriented in the plate that the fast ray (smallest refractive index) is parallel to the axis of the plate and the slow ray (highest refractive index) is perpendicular to the axis of the plate.

Previously we have seen that when the elliptical axes of a certain anisotropic grain are oriented N-S and E-W, we will observe extinction. In order to find out which one is the shortest and which is the longest axis, we have to use the gypsum plate. The mineral grain is rotated till the maximum light is observed, i.e. in the 45° orientation. Now we have two possibilities for the orientation of the ellipse.

Fig. 2.14 Top: elliptical section with N parallel the N(gypsum) and bottom: elliptical section with n parallel to N(gypsum)

When we bring in the gypsum plate, the retardation of the gypsum plate will be superimposed on that of the grain. This means that in the case shown left in Fig. 2.14 the interference colour of the mineral will increase by 1 order. This is known as addition. In the other case the retardation will be diminished and the interference colour of the mineral will be 1 order lower. This is called subtraction. This allows us to determine the maximum and minimum refractive index for this orientation of the elliptical section. Furthermore, if a mineral is pleochroitic (see V.1), we can determine the colour in both directions when we remove the analyser. The gypsum plate can in a similar way also be used as an extra help in order to determine the order of an interference colour in the Michel Lévy chart.