## 2 Crystal Orientation and Practical Determination of Principal Indices of Refraction
### Biaxial Crystals  
For biaxial symmetries, it is necessary to obtain values for all three principal refractive indices (Figure 1), in addition to a determination of their directions in the crystallographic frame, depending on the presence or absence of two-fold symmetry elements in their point group. Triclinic, monoclinic, and orthorhombic symmetries belong to the crystal classes that are biaxial, where \(n_{1}\
eq n_{2}\
eq n_{3}\). In triclinic crystals, the absence of symmetry elementscauses there to be no relation between the directions of the principal refractive indices and the crystallographic directions, other than by coincidence [7; 8; 13; 14]. In order to measure and obtain this relationship, it is necessary to combine birefringence measurements in the polarising microscope together with an indexing determination for the crystal morphology. Starting with the latter, measurements of the angles between crystal faces using the microscope and an appropriate three-axis rotation stage can result in unambiguous indexing for selected or all crystal faces. The unit cell dimensions must be determined by X-ray crystallographic indexing using rotation ranges. This approach will be facilitated if crystal faces include singular indices rather than higher order indices. This condition is met for many protein crystals [15]. If indexing by visual means fails, it will be necessary to perform a face-indexing using X-ray crystallography, which provides the orientation matrix for the correct indexing solution. Monoclinic crystals include one two-fold symmetry axis, which by convention is the \(b\)-axis. The two-fold symmetry ensures that one of the principle refractive index axes is in the direction of the crystallographic \(b\)-axis. The consequence is that, for a k-vector perpendicular to the \(b\)-axis, double refraction occurs such that one polarisation direction corresponds to the \(b\)-axis and is furthermore wavelength-independent. For any other k-vector, knowledge of the directions of \(n_{1}\) and \(n_{3}\)nsals, each axis is associated with a two-fold symmetry operation which fixes the directions of the principle refractive indices in the crystallographic directions. Indexing of orthorhombic crystals may present particular challenges. The unit cell is rectangular, and crystals may either appear rectangular or show more facets (e.g., they can have a dodecahedral shape). The visual measurement under the microscope of the interfacial angles, together with the unit cell dimension information, can usually allow indexing in this manner [15]. When crystal morphology is rectangular, or appears rectangular in at least four directions, the crystal faces can either reflect the orientation of the unit cell or they are combination directions. As an example, [PERSON] showed that orthorhombic crystals of carboxy-haemoglobin appear almost square in cross section, but end in bipyramidal prisms with {1 1 1} faces. The faces for the central part with a square cross section are {1 1 0} [16]. The direction of the principal refractive indices can therefore be determined; however, there can arise an ambiguity for assignment of index to those directions. Typically, X-ray crystallographic face indexing is needed. In general, for biaxial crystals it is necessary to make measurements of the magnitude of all three refractive indices \(n_{1}\), \(n_{2}\), and \(n_{3}\) and of the intrinsic birefringence values. In order to make measurements of the magnitude of birefringence, it is first necessary to orient crystals according to a known index. Crystals must also be embedded in mother liquor such that refraction at the surface is minimised. Moun  
Figure 1: Negative uniaxial (**left**), positive uniaxial (**middle**), and biaxial (**right**) indicatrix representations. The plane that is normal to the input k-vector bisects the indicatrix to create an ellipse, the semi-major axis of which are the decomposed electric field vectors. Due to symmetry one of the axes in the uniaxial case is always the same (\(n_{o}\)).