biaxial interference figures i

1

ERSC 2P22 – Brock University Greg Finn

Interference Figures

Biaxial Minerals


Biaxial Interference Figures• Figures are obtained the same way as

uniaxial figures• The appearance of the figure is dependant on

the orientation of the mineral grain and its corresponding indicatrix

• Figures to be examined:– Acute Bisectrix (Bxa)– Biaxial Optic Axis (OA)– Obtuse Bisectricx (Bxo)– Biaxial Optic Normal or Flash Figure (ON)– Random Orientation


Acute Bisectrix Figure

M MOpticPlane

OpticNormal

Bxa

Isochromes

Isogyre

2


Acute Bisectrix FigureThe Acute Bisectrix (Bxa) figure results when the Bxa is perpendicular to the microscope stage. If the 2V angle is < 60°, both Melatopes (M), marking the point of emergence of the Optic Axes, lie within the FOV.

Bxa

M MAt Extinction the Isogyre cross forms. The arm parallel to the Optic Plane contains the Melatopes (M) and is the thinner arm.

The arm parallel to the Optic Normal is the fatter arm.

The two arms intersect at the Bxa.

OpticPlane

OpticNormal

Isogyre

Isochromes

The Isochromes form a tear drop or figure ‘8’shape about the melatopes.

The Isochromes are assymmetrically arranged about the M.


Acute Bisectrix FigureOn rotating the stage, the Isogyre cross splits into two identical hyperbolae. Each arm is centered on and rotates about the Melatopes (M), which in turn rotate around the Bxa. The Isogyre arms are curved, with the convex side pointing towards the Bxa. The Isochromesalso rotate with the figure, but maintain their symmetrical arrangment about the Melatopes.

BxaM M

OpticPlane

OpticNormal

At extinction 45° from extinction

M

M

Bxa

OpticPlane

OpticNormal45°

Rotation


Acute Bisectrix FigureFormation of Isochromes I

BxaOA OA

Z

YConvergent cone of light from Auxillary Condensor

OAOA Bxa

Retardation increases outwards from the OA. Towards the Bxa the retardation increases at a slower rate than in the opposite direction. This is a function of lower birefringence and the length of the path the light follows.

Light following paths 1 - 4experience ~600 nm of retardation and when this light exits the mineral grain defines the 600 nm isochrome

1

2

3

4

3


Acute Bisectrix FigureFormation of Isochromes II

BxaOA OA

Z

YConvergent cone of light from Auxillary Condensor

OAOA Bxa

Light traveling along any other path experiences varying degrees of retardation, depending on the distance through the mineral and the corresponding birefringence

1

2

3

4

The Isochromes are developed along lines of equal retardation

MM1

2

3

4

Bxa

300 nm

600 nm900 nm

Light traveling along each OAexperiences 0 retardation

∆ = d(ns-nf)


Acute Bisectrix FigureThe vibration directions on the biaxial indicatrix can be derived in a similar manner to that used for UniaxialMinerals

X

Y

Z

OpticNormal

Principal Sections through the indicatrix contain the indicatrix axes X, Y and Z

XZ plane =Optical Axial Plane

(OAP)

OpticAxis

OpticAxis

Vibration directionsVibration directions of light rays emerging from the biaxial indicatrix, projected onto the indicatrix surface

By taking a series of slices through the indicatrix, at right angles to the wave normals, the vibration directions for all paths of light emerging from the indicatrix can be determined.


BxaVibrationDirections

M MBxa

FOV

Vibration Directions of light, on the surface of

the indicatrix, exiting the mineral

Vibration directions for a number of wave paths through the mineral,

projected onto the top surface of the mineral

Vibration Directions of light, within the

interference figure

Isogyre cross forms where the vibration directions of

the light rays passing through the mineral are parallel to the vibration directions of the Polars

4


Isogyre RotationIf the 2V < 60°, both Melatopes will remain within the FOV on rotation

Bxa MM

ON

OAP

With the Optic Axial Plane (OAP) oriented EW, the isogyre forms a cross a that:

1) Narrows at the Melatopes (M), and

2) Widens along the trace of the Optic Normal (ON)

With a rotation of the stage the cross splits into two segments that pivot about the position of the Melatopes (M). Again the isogyre is narrowest at the Melatope.

M

MBxa

OAP

ON

OAP

ON

With the OAP in the 45° position the isogyres form hyperbole centred on the Melatopes (M). Light vibrating along the OAPhas an RI=nBxo, light vibrating along the trace of the ON has an RI=nβ.

M

M

Bxa

At Extinction 45° from Extinction


Isogyre RotationIf the 2V > 60°, both Melatopes will remain outside the FOV on rotation

Bxa

ON

OAP

Bxa for a mineral with a 2V > 60°. Both Melatopeslie outside the FOV, along the thinner arm of the cross. The Isochromesare oriented about the Melatopes. The OAP is oriented parallel to the EW crosshair.

MM

OAPON

Bxa

M

M

OAPON

Bxa

M

M

On rotation (30-45°) the Isogyre cross splits and the arms leave the FOV in the quadrants into which the OAP is being rotated. The Larger the 2V, the lower the angle of rotation for the Isogyres to exit. Isochrome shape is preserved.

Following a rotation of 45°, the OAP is oriented NE-SW and the Isogyreslie entirely outside the FOV. The Isochromesoccupy the FOV



Bxa Figure• For minerals with a 2V < 60°, the

melatopes and Isogyres will remain in the FOV as the stage is rotated

• For minerals with a 2V > 60°, the melatopes will lie outside the FOV

• And the isogyres will leave the FOV and are not visible in the 45° position

5


Estimating the 2V Angle• A reliable estimate of the 2V angle,

based on the separation of the isogyres, can be obtained with the BxaFigure in the 45° position, with the OAPoriented NE-SW


Estimate of 2V based on the separation of the isogyres in the Bxa Figure

2V = 15° 2V = 30°

2V = 45°2V = 60°


Optic Axis FigureFor Biaxial Minerals with 2V < 30°

• An Optic Axis Figure results when one Optic Axis (OA) is vertical

• The figure may be centred or off-centred, depending on how close to vertical the OA is

• For a Centred Optic Axis Figure, the melatope for that OA, is positioned directly under the crosshairs

6


Centred Optic Axis FigureFor Biaxial Minerals with 2V < 30°ON

OAPBxa MM

OAP

ON

Bxa

M

M

One Melatope (M) lies at the intersection of the crosshairs. The thin arm of the Isogyre marks the position of the OAPand contains the melatopes and the Bxa.

With a low 2V the figure resembles an off-centred Bxa Figure

With a rotation of 45°, the Isogyre splits into two hyperbolae, centred on the Melatopes. The Isochromes are rotated, yet retain their tear-drop/Figure ‘8’ shape



OA

OA

Bxo

Bxa

BxaM

OAP

OAP

ON

ON

Centred Optic Axis FigureFor Biaxial Minerals with 2V > 45°

IndicatrixIndicatrix is oriented such that one OA is vertical

At extinction one arm of the Isogyre cross will be visible. This arm will narrow at the Melatopeand be parallel to the OAP. This arm will be oriented parallel to

one of the crosshairs.

Principal sections and Vibration directionsVibration directions of light are shown on the indicatrix

surface.


BxaMOAP

ON

Centred Optic Axis FigureFor Biaxial Minerals with 2V > 30°

With a counterclockwise rotation the isogyre arm rotates

clockwise, pivoting around the M

Bxa

OAP

M

Bxa

OAP

M

With the optic plane in the 45° position the Isogyre will show its maximum curvature

and the position of the Bxa lies on the convex side of the Isogyre

7


Estimate of 2V based on the curvature of the Isogyre in the Optic Axis Figure

2V = 5°

2V = 75°2V = 60°2V = 45°

2V = 30°2V = 15°

OAP


Optic Axis Figure 2V=90°Isogyre is straight, no curvatureOAP lies at 45°, passing through the MelatopeCannot determine the position of the Bxa

OAP

M


Obtuse Bisectrix (Bxo) Figure• Results when Obtuse Bisectrix (Bxo) is

perpendicular to microscope stage• As the angle between Bxo and Optic Axes >

45°, Melatopes will always lie outside FOV• The pattern of the Isochromes and vibration

directions are similar to those of Bxa figure• The isogyre cross is generally fuzzier than

Bxa figure, but Optic Plane will still parallel EW or NS crosshair

8


Obtuse Bisectrix (Bxo) Figure• On rotating the stage the Isogyre cross will

split and leave the field of view in the quadrants into which the Optic Plane is being rotated, as with Bxa figure

• Isogyres split and leave FOV, usually with a rotation of 5° to 15°

• For a Bxo figure the Isogyres, when they split, will not be in the field of view

• If 2V = 90°, Bxa and Bxo Figures are identical

• If 2V is small, Bxo figure resembles an Optic Normal (Biaxial Flash) Figure


Bxo

OAOA

Bxa

ON

BxoM M

Obtuse Bisectrix (Bxo) Figure

IndicatrixIndicatrix is oriented such that the Bxo is vertical

The OAP is vertical, containing the Bxo, Bxa and OAs

In the interference figure the two Melatopes (M) lie outside the

FOV.

The Isogyre cross has a broad fuzzy appearance, with the

thinner arm lying along the OAP

OAP


BxoM M

Obtuse Bisectrix (Bxo) Figure

With a 45° rotation the arms of the Isogyre lie well outside the FOV and

the pattern of the Isochromes, if present will be visible

With a rotation of 5° to 15° the Isogyre cross splits and leaves the FOV in the quadrants into

which the OAP is being rotated

OAPOAP

Bxo

Bxo

OAP

9


Optic Normal Figure

• AKA Biaxial Flash Figure• Similar to Uniaxial Flash Figure• Results when the Optic Normal is vertical

and the Optic Plane is horizontal• A grain that will produce an Optic Normal

Figure will display the maximum interference colours

• The vibration directions in figure are similar to those for a uniaxial flash figure


Optic Normal Figure

• When X & Z indicatrix axes parallel the polarization directions, figure is a broad fuzzy cross with only the outer edges of each quadrant allowing any light to pass.

• Very small degree of rotation <5° causes the isogyres to split and leave the field of view from the quadrants into which the Bxa is being rotated

• If 2V = 90° the cross - shaped isogyre does not split as the stage is rotated, it simply dissolves away with a 5 - 10° rotation


Optic Normal Figure

ON

Bxo

BxaOA

OA

The Indicatrix is oriented such that the Optic Normal (ON) is vertical. The Optic Plane, containing the Bxa, Bxo and OAs, is horizontal

and lies in the plane of the section

With the Bxa and Bxo parallel to the polarization directions the Isogyresform a broad fuzzy cross, with only the outer edges of each quadrant

allowing any light to pass

ON

FOV

Bxa or

Bxo

Bxa or Bxo

10


Optic Normal Figure

The Indicatrix is oriented such that the Optic Normal (ON) is vertical. The Optic Plane, containing the Bxa, Bxo and OAs, is horizontal

and lies in the plane of the section

With the Bxa and Bxo parallel to the polarization directions the Isogyres

form a broad fuzzy cross

ON Bxa or

Bxo

Bxa or Bxo

At Extinction

5° Rotation

45° from Extinction

Bxo

Bxa

Bxo

Bxa

ON

ON


Off-Centred Figures• Most interference figures examined

during routine microscope work are off-centred figures.

• In these instances none of the indicatrixor optic axes is vertical.

• Any combination of orientations are possible for off-centred figures

biaxial interference figures i

Documents