II.10 Biaxial minerals

II.10.1 the wave-surface of biaxial crystals

Using the same construction as was used before for the uniaxial crystals we can determine the wave surface from the indicatrix (Fig. 2.15).

Fig. 2.15 Biaxial indicatrix

In Fig. 2.16 one can observe the following things: when the front normal is parallel to the z-axis, then we have two wave fronts in the crystal with refractive indices α and β. The velocities along the z-axis are thus c/α and c/β, with c the velocity of light in air. Similarly, when the front normal is parallel to the y-axis, there are two velocities along the y-axis equal to c/α and c/γ. Finally, when the front normal is parallel to the x-axis then the velocities along the x-axis are equal to c/β and c/γ.

Fig. 2.16

When the front normal is oriented in the ZY plane, then there exist one wave fronts with refractive index α and one with refractive index γ' with a value between β and γ. In the ZY plane a circle is drawn corresponding with c/α. In Fig. 2.17 the ZY plane of the indicatrix has been drawn separately. Here ON is the front normal and OE the refractive index γ'. It turns out that the wave surface has to take the shape of an ellipse with as the two axes c/β and c/γ (Fig. 2.18). This ellipse is exactly the same as the ellipse shown in figure 2.15; but all distances have been multiplied by a factor c/βγ. So, OP = ON x c/βγ. Mathematics says that for an ellips in this situation the product ON x OE  must be constant and the same as the product of the two axes, βγ. This leads to ON = βγ/γ', so OP = c/γ', which is indeed the velocity of the wave front along ON. Following the same reasoning we find: when the front normal is oriented in the XY plane, then there exist one wave front with a circular section corresponding to a velocity of c/γ and with ellipse axes c/α and c/β, and when the front normal is oriented in the ZX plane, then there exist one wave front with a circular section corresponding to a velocity of c/β and with ellipse axes c/α and c/γ.

Fig. 2.17

Fig. 2.18

Fig. 2.16 gives the complete wave surface; it is a 4th order surface that sections the symmetry planes  according to 2nd order curves. In the ZY plane the circle lies outside the ellipse. In the XY plane the circle lies inside the ellipse and in the ZX plane the circle and ellipse cut one another. In these directions the radial velocities are the same (secondary optical axes). In the ZX plane the circle and the ellipse have a tangent line (Fig. 2.19).

Fig. 2.19 XY plane of biaxial wave surface

 Perpendicular to this line we'll find the front normal of the coinciding wave fronts and is called the primary optical axis. This optical axis is thus oriented perpendicular to the circular section through the indicatrix. The refractive index in this case is thus β. In all other directions which are not situated in the symmetry planes, we'll find two extraordinary rays. The refractive indices of the corresponding wave fronts are α' (between α and β) and γ' (between β and γ). We can observe this in Fig. 2.20, when we remember that the elliptical section cuts both circular sections. This gives the lines OB and OB' with a length equal to β, so α' (OA) has to be smaller than β and γ' (OC) has to be larger than β. The wave surface of uniaxial crystals can be deduced from this biaxial system by saying that α = β or that β = γ.

Fig. 2.20 Random elliptical section through biaxial indicatrix

II.10.2 The biaxial indicatrix

The biaxial indicatris has the shape of a three-axial ellipsoide. The refractive indixes belonging to the vibrational directions follow the axes α, β and γ of this ellipsoide (Fig. 2.21). In this ellipsoide the following  is always true: α < β < γ. An indicatrix with this shape has two circular sections and is therefore called optically biaxial. The two circular sections cut one another along β, which is therefore also known as the optical normal. This indicatrix axis is also the radius of the circular section.

Fig. 2.21a

The two normals to the circular sections, the optical axes A₁ and A₂. are located in a plane through α and γ, the plane between the optical axes. The angle between these two optical axes is known as the 2V angle.

Fig.2.21b

Be aware of the nomenclature; we speak of a three-axial ellipsoide and at the same time of a biaxial indicatrix. In the first instance the three indicatrix axes α, β and γ are meant, while in the second the two axes are the two optical axes A₁ and A₂.

The shape of the ellipsoide is determined by the angle between the optical axes, which in themselves are a function of the dimension of β relative to α and γ. Now there are two possibilities:

1) γ is the acute bisectrix of the optical axes = optically positive

2) α is the acute bisectrix of the optical axes = optically negative

For very small 2V angles the difference between the optical normal and the obtuse bisectrix is small and the indicatrix shape looks roughly like that of an uniaxial indicatrix.

Fig. 2.21c. Effect of 2V on the orientation of the optical axes (click on image for larger image)