## 2 Crystal Orientation and Practical Determination of Principal Indices of Refraction
### Biaxial Crystals
In the biaxial case the polarisation of the incident ray (P) is shown to decompose into \(\mathrm{e}_{1}\) and \(\mathrm{e}_{2}\) along the axis as defined by the semi-axis of the ellipse (\(s_{max}\) & \(r_{max}\)). can aid in this part. When birefringent crystals are viewed under crossed polarisers, two situations can result in extinction. Either the view is along one of the two optic axes (in which case the crystal appears to be optically isotropic) or one of the principal axes of refractive index is aligned with a polariser direction. Consequently, under these conditions, the crystals will appear bright under parallel polarisers. When a crystal is rotated to the extinction point, all colours vanish and the view appears dark. To measure the difference in refractive index between principal axes, the crystal is rotated \(45^{\circ}\) away from the extinction point and a Berek compensator is inserted in the light path between the polarisers. The retardance \((R)\) between two waves is the product of the birefringence (\(\Delta n\)) and the optical path length through the crystal (\(d\)) and must also be measured:  
\[R=\Delta n\times d \tag{2}\]  
The retardance is also given as a function of the wavelength \(\lambda_{0}\) and the phase difference \(\phi\) between the waves:  
\[R=(\phi\times\lambda_{0})/2 \tag{3}\]  
The fast axis of the Berek compensator should be along the slow axis of the crystal in order to arrive at a measurement of compensation of retardance. The compensator is calibrated such that the retardance value can be read after rotation of the drum, once the extinction point is obtained. The \(\Delta n_{a,b,c}\) values measured in the a, b and c directions should be measured individually. In addition, these values are related according to:  
\[\Delta n_{a}=\Delta n_{b}-\Delta n_{c} \tag{4}\]  
where \(\Delta n_{a}=n_{c}-n_{b},\ \Delta n_{b}=n_{c}-n_{a},\ \Delta n_{c}=n_{b}-n_{a},\ \ \text{if}\ \Delta n_{a}\) is the smallest birefringence and \(\Delta n_{b}\) is the largest birefringence. This relationship allows the verification of the independent intrinsic birefringence measurements. The values of birefringence of crystals are typically in the range of \(10^{-2}\)-\(10^{-4}\)[15]. An alternative to the Berek compensator measurement uses the conventional Michel-Levy chart [17]. The method is less precise and only provides an estimate. In addition, since [PERSON]'s series appear in multiple orders, there may be uncertainty in the order of assignment. Using either visual colour determination or a spectrometer measurement, there may also be significant error in the estimate of birefringence determination using this method. In optical mineralogy, the phase contrast (or oblique illumination) method would additionally provide a means for estimating the refractive indices, but this requires immersion in liquids with a large refractive index [18]. An additional method that is used in optical mineralogy that may be useful for certain cases of resolving indexing ambiguity is conoscopy analysis. Conoscopy is the examination of the interference figures and colour patterns in the focal plane of the microscope objective lens, produced by an anisotropic crystalline section viewed under convergent light illumination and crossed polarisers. This method requires the use of a Bertrand lens or phase telescope, or alternatively, a conoscope lens that is used in the polarising microscope. The method may be less suitable for protein crystals, as single crystals need to fill the field of view or they must be in an aperture to view the interference image. Conoscopy images are instructive when viewing either in the direction of an optic axis or perpendicular to an optic axis. The optics allow a two-dimensional projection and view of the interference pattern and colours. Conoscopy can directly distinguish uniaxial and biaxial crystals. The use of waveplates and quarter waveplates allow the identification of negative and positive crystals, hence aiding in the assignment of fast and slow crystal axes [19].