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Biaxial Minerals
Francis 2013
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2. Biaxial Minerals
A triaxial ellisoid or indicatrix is required to represent the refractive indices of anisotropic minerals belonging to the orthorhombic, monoclinic, and/or triclinic crystal systems. Such an ellipsoid has 3 rather than 2 principle axes, each at right angles to each other:
ηγ or Z with the maximum refractive index
ηβ or Y with an intermediate refractive index
ηα or X with the smallest refractive index
Such an ellipsoid has two circular sections of radius ηβ. Each has an optic axis perpendicular to it, hence the name biaxial. The angle between the two optic axes is referred to as the 2V, and is a characteristic feature of the mineral of the mineral. Biaxial minerals whose 2V is acute about ηγ are said to be biaxial positive (+), while those whose 2V is acute about ηα are biaxial negative (-).
The principal of double refraction still applies:
Light entering a biaxial mineral is split into two rays that are each plane polarized at right angles to each other, parallel to the major and minor axis of the ellipse of section made by the plane of the microscope stage through the indicatrix.

3. Principle Sections:
There are 4 principle sections through a biaxial indicatrix, each having a characteristic interference figure:
Acute Bisectrix Section (BXA)
Contains ηβ and either ηγ or ηα, with the other principle vibration direction perpendicular to the microscope stage. Grains with this orientation are characterized by the intermediate to low interference colours.
Biaxial positive (+) minerals: (ηβ - ηα) < (ηγ - ηβ) and thus the angle
between the two optic axes is acute about ηγ, which is perpendicular to the microscope stage.
Biaxial negative (-) minerals: (ηβ - ηα) > (ηγ - ηβ) and thus the angle
between the two optic axes is acute about ηα, which is perpendicular to the microscope stage.
Obtuse Bisectrix Section (BXO)
Contains ηβ and either ηγ or ηα, with the other principle vibration direction perpendicular to the microscope stage. Grains with this orientation are characterized by the intermediate to high interference colours.
ηα is perpendicular to the microscope stage if the mineral is biaxial positive.
ηγ is perpendicular to the microscope stage if the mineral is biaxial negative.
Optic Normal Section (ON)
Contains both ηγ and ηα, with the ηβ principle vibration direction perpendicular to the microscope stage. Grains with this orientation are characterized by the highest interference colours, in an analogous way to uniaxial minerals whose C axis lie in the plane of the microscope stage. Such grains give centered flash interference figures that are identical to those exhibited by uniaxial minerals.
Optic Axis Section (OA)
Circular sections of radius ηβ, with an optic axis perpendicular to the stage. Note that there are two circular sections and 2 optic axes. Grains with this orientation are characterized by the lowest interference colours, in an analogous way to uniaxial minerals whose C axes are perpendicular to the microscope stage.

4. Acute Bisectrix Section
Optic Axis
Section
Optic Axis
Section
Obtuse Bisectrix
Section
Optic Normal
Section

5. Orientations: Orthorhombic minerals (2/m 2/m 2/m):
The three principle vibration directions of the indicatrix (ηγ, ηβ, and ηα ) must coincide with the three crystallographic axes (a, b, and c) of the mineral, which in turn coincide with the three axes of two fold rotational symmetry. Which principle vibration direction coincides with which crystallographic axis, however, varies from mineral to mineral and is characteristic of any given mineral.
Monoclinic minerals (2/m):
Only one of the three principle vibration directions of the indicatrix (ηγ, ηβ, or ηα ) must coincide with the b crystallographic axis of the mineral, which by convention coincides with the axis of two fold rotational symmetry. The other two principle vibration directions lie in the plane of the a-c crystallographic axes, at right angles to b, but do not necessarily coincide with either of these crystallographic axes. Which principle vibration direction (ηγ, ηβ, or ηα) coincides with b crystallographic axis varies from mineral to mineral and is characteristic of any given mineral.

6. Orientations: Triclinic minerals:
None of the three principle vibration directions of the indicatrix (ηγ, ηβ, or ηα ) coincides with any of the three crystallographic axes (a, b, or c) of the mineral. The angular relationship between the principle vibration directions and the crystallographic axes, however, may be characteristic of the mineral and sensitive to its chemical composition.

7. In order to understand biaxial interference figures we need a way of determining the vibration directions of light emerging in the field of view when looking through a Bertrand lens.
The Biot Fresnel rule:
The vibration directions of the two rays associated with any wave normal are given by the lines bisecting the angles formed by joining the wave front normal to the points of emergence of the two optic axes.
Proof:
Take any elliptical section of the indicatrix, whose major and minor axes give the vibration directions for the two rays associated with the wave front normal perpendicular to it. Somewhere in this section must be the traces of the two circular sections of the indicatrix, which must be symmetrically disposed about the major and minor axis of the ellipse of section. The optic axis for each of the circular sections must lie in the plane perpendicular to the trace of the circular section. The traces of these planes are also the projection of the lines joining the optic axes to the wave front normal, and must also be symmetrically disposed about the major and minor axis of the ellipse of section. Thus the two vibration directions bisect the angle formed by the lines joining the two optic axes to the wave front normal.

8. Biaxial Acute Bisectrix Figure (BXA)
If the 2V of a biaxial mineral is  60o, both optic axes will appear in the field of view of the Bertrand lens in a centered BXA interference figure. The isochrome interference bands will form a double bulls eye pattern about the two optic axis, and the optic axes will not leave the field of view upon rotation of the stage. The black isogyres that form a cross at the extinction position separate into two parabolic isogyres when the stage is rotated from the extinction position, which flex about the optic axes and reach maximum separation at the 45o position. The separation and the degree of curvature of the isogyres in the 45o position are respectively direct and inverse measures of the 2V of biaxial minerals. 0o
45o

9. Biaxial Acute Bisectrix Figure (BXA)

10. If the 2V  60o, then the two optic axes lie outside the field of view in a BXA interference figure, and the isogyres will leave the field of view upon 15-30o rotation of the microscope stage from the extinction position. The optic plane can be recognized as the thinner isogyre arm in the extinction position, even if the optic axes are outside the field of view. Furthermore, the arm corresponding to the optic plane counter rotates to the direction of the microscope stage, while the other arm (which corresponds to one of the other two mirror planes of the biaxial optical indicatrix) same rotates.

11. Biaxial Acute Bisectrix Figure (BXA)0o
45o 0o
45o

12. Isochrome Rings:
The isochrome colour rings of the BXA interference figure contour points of emergence of light with equal retardation between the two plane polarized rays. They form a double bulls-eye about each of the optic axes, with retardation increasing in all directions away from the optic axes. The number of isochrome rings seen on an interference figure is a function of the birefringence of the mineral. Biaxial minerals of low birefringence may exhibit no isochrome rings, just broad diffuse isogyres and first order white outside them.

13. Phlogopite 2V = 10o
Aragonite 2V = 20o
Muscovite 2V = 45o
2V = 40o

14. Obtuse Bisectrix Interference Figure (BXO):
BXO interference figures are similar to BXA interference figures, except that the isogyres always leave the field of view. They can typically be distinguished from BXA’s in minerals with 2V > 60o, by the fact that less than 15o of rotation of the microscope’s stage from the extinction position. As the 2V of a biaxial mineral increases towards 90o, it becomes increasingly difficult to distinguish between BXO and BXA interference figures. At a 2V of 90o, BXA  BXO, and the mineral is neither biaxial positive nor negative.
Bxa
Bxo

15. Obtuse Bisectrix Interference Figure (BXO):
BXO interference figures are similar to BXA interference figures, except that the isogyres always leave the field of view. They can typically be distinguished from BXA’s in minerals with 2V > 60o, by the fact that less than 15o of rotation of the microscope’s stage from the extinction position. As the 2V of a biaxial mineral increases towards 90o, it becomes increasingly difficult to distinguish between BXO and BXA interference figures. At a 2V of 90o, BXA  BXO, and the mineral is neither biaxial positive nor negative.

16. Optic Axis Interference Figure (OA):
OA axis interference figures consist of a single isogyre that is straight and coincides with the optic plane and a crosshair in the extinction position, but which flexes about the optic axis and counter rotates as the microscope stage is turned from the extinction position. The isogyre exhibits maximum curvature in the 45o position, with the degree of curvature in this position being inversely proportional to the 2V of the mineral. The isogyre will be straight in the 45o position for biaxial minerals with 2V’s near 90o. The curvature of the optic axis isogyre is convex towards the BXA vibration direction.

17. Optic Axis Interference Figure (OA):
For minerals of low 2V, the other isogyre associated with the other optic axis will be visible at the edge of the field of view, and the optic axis interference figure is essentially just an off centered BXA interference figure.
In biaxial minerals of large 2V, the optic axes and thus the isogyres cannot be seen in the 45o position. In such cases, the optic sign is more commonly determined using an optic axis (OA) interference figure. By remembering the isogyre is convex towards the BXA vibration direction in the 45o position, one can use the accessory plate to compare the orientations Nbig or Nsmall on either side of the isogyre, and apply the same logic as with the BXA interference figure.

18. Optic Axis Interference Figure (OA):
#NB: Many grain orientations will yield a single isogyre that rotates and flexes upon rotation of the microscope stage. Such a single isogyre is only an optic axis figure if:
The isogyre flexes about a point (the optic axis) that remains in the field of view for the full rotation of the microscope stage.
The isogyre counter rotates to the direction that the microscope stage is turning.

19. Flash Interference Figure (optic normal section ON):
Grains whose ηγ and ηα vibration directions lie in the plane of the microscope stage (and thus has ηβ vertical) show maximum interference colours and exhibit broad diffuse isogyres that leave the field of view on less than 5o of rotation of the microscope stage from the extinction position. These biaxial “flash” figures are indistinguishable from those exhibited by uniaxial minerals. In the case of biaxial minerals, however, the diffuse isogyres of the flash figure leave the field of view in the quadrants into which the BXA vibration direction is moving as the microscope stage is rotated from the extinction position. It is thus possible to determine the sign of a biaxial mineral in an analogous way to that used for flash figures in uniaxial minerals, that is by determining whether the BXA vibration direction is Nbig (biaxial +) or Nsmall (biaxial -) with the accessory plate in the 45o position.

20. Summary of the Types of Biaxial Interference Figures:

21. Summary of the Types of Biaxial Interference Figures:

22. Determination of Optic Sign with a BXA Interference Figure:
The optic sign of a biaxial mineral is commonly determined by using the accessory plate to distinguish the relative orientations of Nbig or Nsmall in the region between the two parabolic isogyres of the BXA interference figure in the 45o position, as compared to the regions outside them. In the 45o position, ηβ is oriented between the two isogyres, while either ηα (+) or ηγ (-) is oriented along the optic plane joining the two optic axes and thus bisecting the two parabolic isogyres. The question then becomes; is ηβ equal to Nbig or Nsmall compared to the vibration direction along the optic plane.
If ηβ is Nbig, then the other is ηα, and you are looking down ηγ and thus the mineral is biaxial positive.
If ηβ is Nsmall, then the other is ηγ, and you are looking down ηα and thus the mineral is biaxial negative.

23. Determination of Optic Sign with a BXA Interference Figure:

24. Optic Axis Interference Figure (OA):
In biaxial minerals of large 2V, the optic axes and thus the isogyres cannot be seen in the 45o position. In such cases, the optic sign is more commonly determined using an optic axis (OA) interference figure. By remembering the isogyre is convex towards the BXA vibration direction in the 45o position, one can use the accessory plate to compare the orientations Nbig or Nsmall on either side of the isogyre, and apply the same logic as with the BXA interference figure.

25. Summary of the Types of Biaxial Interference Figures:

26. Summary of the Types of Biaxial Interference Figures:

27. Other Optical Properties of Biaxial Minerals
Sign of Elongation:
By convention, the c crystallographic axis is typically defined such that it corresponds to the longest
dimension of a mineral. In orthorhombic minerals, one of three possible situations will exist:
c ≡ ηγ in which case the mineral is length slow and has a (+) sign of elongation.
c ≡ ηα in which case the mineral is length fast, and has a (-) sign of elongation.
c ≡ ηβ in which case the mineral’s sign of elongation will depend on the orientation it is viewed.

28. Platy minerals can also be thought of as having an apparent sign of elongation that can be a useful identification criteria.
The c crystallographic axes of monoclinic and triclinic minerals do not normally coincide with any of the principle vibration direction, however, commonly one of the principle vibration directions is closest to the c axis and the sign of elongation of the mineral can still be a useful identification parameter.

29. Extinction Angle:
Orthorhombic minerals typically exhibit parallel or symmetric extinction.
Extinction angles in
orthorhombic pyroxene:

30. Monoclinic minerals will only exhibit parallel extinction in orientations in which the b crystallographic axis (and the 2 fold symmetry axis) lies in the plane of the microscope stage. In other orientations, monoclinic minerals will typically exhibit inclined extinction. The extinction angle will typically be greatest in orientations in which the b crystallographic axis is perpendicular to the microscope stage. By knowing which of the principle vibration directions coincides with the b crystallographic axis, one can use the corresponding interference figure to obtain grains of the right orientation, and then measure the exact angle of extinction, which will be characteristic of the mineral.
Extinction angles in
monoclinic pyroxene:

31. Triclinic minerals exhibit inclined extinction
Triclinic minerals exhibit inclined extinction. In some cases, most notably plagioclase (Lab. 8), a measurement of the extinction angle can be used to determine the composition of the mineral.
Whenever an extinction angle is noted, the following information is required to properly interpret it:
The vibration direction that the angle is measure to, ie; ηγ, ηβ, or ηα, or just ηbig versus ηsmall.
The linear feature being measured to, ie: crystal length, cleavage, twin plane, etc.
The orientation of the grain being measured; typically a centered interference figure, but sometimes the
trace of a vertical crystallographic plane corresponding to a cleavage or twin plane.

32.  d  Birefrigence:  =  / d
The birefringence of a biaxial mineral must be estimated in grains that have both ηγ and ηα in the plane of the microscope stage, and thus ηβ vertical (optic normal section ON). Such grains will show the maximum the interference colours and should give a centered biaxial flash figure when viewed with the Bertrand lens. As in the case of uniaxial minerals, the exact birefringence () of the mineral can be obtained if the thickness (d) of the thin section is known:
 =  / d
Conversely, the exact thickness of a thin section can be determined by identifying the maximum interference colour () of a known mineral.  d 

33. Pleochroism and anomalous interference colours:
Many minerals are coloured in the plane polarized light of the microscope because of the selective absorption of certain wavelengths of light passing through them. In biaxial minerals, each of the different principle vibration directions may exhibit differential absorption, such that the colour of a mineral grain changes upon rotation of the microscope stage in plane polarized light (analyzer out). This property is called pleochroism and is described by determining the true colours of the ηγ, ηβ, and ηα vibration directions by successively rotating a grain having the desired vibration direction in the plane of the microscope stage to the extinction the position in which the vibration direction in question is parallel to the microscope’s polarizer and then identifying the grain’s colour with the analyzer out (ie. proceeding as if you were actually measuring the refractive indices). In some cases, pleochroism is a reflection of different degrees of total absorption and is not associated with any particular colour.
90o
hornblende

34. Pleochroism:
Most people determine the vibration direction of their microscope’s polarizer by taking advantage of the marked pleochroism exhibited by biotite. The ηα vibration direction of biotite is light yellow in colour, while ηγ, and ηβ have a dark reddish brown colour. The ηα vibration direction is approximately perpendicular to biotite’s perfect cleavage, thus when a biotite crystal is rotated to the extinction position in which the trace of the cleavage plane parallels the microscope’s polarizer, the grain will have a dark brown colour in plane polarized light.
90o Nα
polarizor Nα

35. anomalous interference colours
As in the case of uniaxial minerals, strong pleochroism can also be responsible for anomalous interference colours that are often characteristic of a given mineral. For example, epidote exhibits an anomalous first order yellowish green that is diagnostic once recognized. Similarly, some varieties of chlorite and serpentine exhibit an anomalous first order blue (Berlin blue) and brown colours that are diagnostic.
Anomalous green-yellow of epidote
Anomalous blue of chlorite
epidote
chlorite

36. Effects of Dispersion
The refractive index of materials varies with the wave light under consideration. This property known as dispersion is responsible for the ability of a prism to divide white light up into its constituent colours and the orange and blue “Becke” lines observed when the refractive index of the mounting medium is ~ that of the mineral (see notes on measuring refractive index of isotropic minerals with oils.
Dispersion in isotropic materials can be defined in terms of a single curve of R.I. versus λ, but in uniaxial and biaxial minerals, distinct dispersion curves must be defined for each of the principle vibration directions. This leads to the development of colour fringes on the isogyres of biaxial interference figures.

37. Sillimanite: Rhombic dispersion:
Rhombic dispersion develops in response to independent variations in the magnitudes of Nα, Nβ, & Nγ, but not in their vibration direction. This is the only type of dispersion observed in orthorhombic minerals, but it is also observed in monoclinic and triclinic minerals. Because the 2V of biaxial minerals is dependent of the relative magnitudes of the three principle vibration direction, a differential variation in their magnitude will cause a corresponding variation in 2V with wavelength.

38. Rhombic Dispersion

39. In monoclinic and triclinic minerals there may also be dispersion of the direction of vibration of the principle vibration directions with the wavelength of light. In monoclinic minerals, one of the principle vibration directions will coincide with the b crystallographic axis, which by convention coincides with the 2 fold axis of rotational symmetry, and the other two will not:
Parallel Dispersion (b axis coincides with BXO principle vibration direction):
Note: Parallel, inclined, and crossed dispersion can only be seen if rhombic dispersion is also present.

40. Inclined Dispersion (b axis coincides with Nβ principle vibration direction (Optic Normal ON)):

41. Crossed Dispersion (b axis coincides with BXA principle vibration direction):
In practice, it is rare that the symmetry of colour fringes can be confidently determined in monoclinic minerals, and in triclinic minerals the maximum symmetry is a centre of symmetry. The presence of rhombic dispersion, however, can frequently be detected, and most identification tables will indicate the character of rhombic dispersion (if present) by the notation r > v or r < v, where r and v stand for the 2V of red and violet light respectively.

42. An unknown biaxial mineral occurs as elongated prismatic crystals with a positive sign of elongation and a maximum extinction angle of 10o. BXA interference figures obtained on elongated grains of low to moderate birefringence exhibit inclined dispersion. Elongated grains with highest interference colours yield optic normal 'Flash' interference figures. Identify the crystal system to which the mineral belongs and describe the spatial relationships between the mineral's 3 principal vibration directions, its three crystallographic axes, and its crystal habit. What is the optic sign of the mineral? Explicitly explain your reasoning.