Section 1.6.4.6. Determination of linear birefringence

The numerical determination of linear birefringence gives less information than a full set of refractive-index measurements, but is nevertheless highly useful in crystal identification, particularly in mineralogical and petrological applications where the thin section is the norm. It is also a most sensitive indicator of changes in the crystal structure at a phase transformation or as a function of temperature, pressure etc. Refractive-index determination is tedious, but birefringence determination is quick and easy.

Double refraction generates polarization colours when crystals are viewed between crossed polars, except where the crystal is by chance in an extinction position, or cut normal to an optic axis. The colours result from the interference of the two transmitted rays when they are combined into one vibration direction in the analyser. Polarization colours are best observed with the substage diaphragm moderately closed down, so that the transmitted light corresponds to a roughly parallel bundle of rays (if the diaphragm is wide open, and the supplementary condenser is inserted, the resultant rays are far from parallel, and the polarization colours will immediately be seen to degrade in the direction of whitening).

Considering a section showing two refractive indices, and , the time difference required for a ray to traverse the section iswhere z is the thickness of the section. The time difference between the two rays is thenwhich is referred to as the retardation. Multiplication by c gives the relative retardation or optical path difference, R, where R is usually expressed in nm (formerly mµ).

The possibilities of interference clearly depend on R, but also on wavelength. For complete destructive interference, because of the way the transmitted rays are resolved into the vibration direction of the analyser (see Fig. 1.6.4.2), R must either be zero (as in cubic crystals, and sections normal to optic axes in anisotropic crystals) or a whole number of wavelengths. Thus as R changes, either with thickness, orientation of the crystals or with variation in birefringence in different substances, a variety of colours are produced, essentially formed from white light with various wavelengths subtracted. There is a good discussion of this point in Wahlstrom (1959).

Figure 1.6.4.2 | top | pdf | Successive sections across the optical path of the microscope. (a) Above the polarizer, beneath the crystal. Polarized light vibrates E–W. (b) Within the crystal. Transmitted light is resolved into the two vibration directions of the crystal ( and ). (c) Above the analyser. The two transmitted rays are resolved into the N–S direction. Amplitudes are shown, but phase difference is not.

Fig. 1.6.4.3 shows the effect of increasing R on a variety of visible-light wavelengths. When R is zero, no light is transmitted since all wavelengths show total destructive interference. As R increases a little, all wavelengths continue to show interference, and the polarization colours are essentially greys, which decrease in darkness until the middle of the first order where the grey is very pale, almost white. Most wavelengths at this stage are showing relatively strong transmission. With increasing R, the region is reached where the shortest wavelengths of the visible light spectrum (violet) are beginning to approach a phase difference of . The transmitted light then takes on first a yellow tinge and then bright orange, as violet light (at ) and then blue are completely removed. Next, the removal of green light () results in the transmitted light being red and at the top of the first-order colours is reached with the removal of yellow light. The resultant polarization colour is a distinctive magenta colour known as sensitive tint. The accessory plate known as the `sensitive-tint' or `' plate is made to have R = 560 nm. Its use will be explained below, but meanwhile note that the polarization colour is so-called because with only very slight changes in R it becomes obviously red (falling R) or blue (rising R).

Figure 1.6.4.3 | top | pdf | Plot of retardation, R, versus wavelength, showing how polarization colours are formed.

Between R = 560 nm and R = 1120 nm, the second-order colours are produced, and are similar in appearance to the colours of the rainbow (blue, green, yellow, orange and red in sequence), as orange, red, violet, blue, and green are successively cut out. The third-order colours (R = 1120–1680 nm) are essentially a repeat of the second-order, but there is a subtle change of quality about them, as they take on slightly garish hues compared with rainbow colours (the red at the top of the order has for example a distinct air of `shocking pink' or even lipstick about it). This effect is a consequence of the increasing chances that two wavelengths will be cut out simultaneously (see Fig. 1.6.4.3), e.g. while first-order red results from the removal of green, second-order red results from the removal of both violet and yellow.

With increasing R, the distinction of colours within each order becomes weaker as the number of wavelengths simultaneously removed increases. Colours are diluted towards grey or white, so that from the fifth order upwards the range is little more than an alternation of pale pinkish and pale greenish tints. Eventually, at higher orders the polarization colours become a more-or-less uniform dull white. Fig. 1.6.4.4 shows the colours produced by a quartz crystal cut into a wedge shape.

Figure 1.6.4.4 | top | pdf | A quartz wedge between crossed Nicols. The thin end is on the left. The colours shown vary between about the middle of the first order up to high fifth order.