IndicatrixIndicatrix Imaginary figure, but very usefulImaginary figure, but very useful The figures show and/or define:The figures show and/or define:

Location of optic axisLocation of optic axis Positive and negative mineralsPositive and negative minerals Relationship between optical & crystallographic axesRelationship between optical & crystallographic axes

Three type – each with characteristic shape:Three type – each with characteristic shape: IsotropicIsotropic Uniaxial (anisotropic)Uniaxial (anisotropic) Biaxial (anisotropic)Biaxial (anisotropic)

Primary use is to understand/visualize vibration Primary use is to understand/visualize vibration directions of slow and fast raysdirections of slow and fast rays

IndicatrixIndicatrix

Primary uses:Primary uses: Determine vibration directions within mineralDetermine vibration directions within mineral

Vibration direction determines index of refraction of Vibration direction determines index of refraction of slow and fast rays – and thus birefringence and slow and fast rays – and thus birefringence and interference colorsinterference colors

Determine wave front direction and ray paths if Determine wave front direction and ray paths if refractedrefracted

Show relationship between optics and Show relationship between optics and crystallographic axis/crystallographic featurescrystallographic axis/crystallographic features

IndicatrixIndicatrix

Possible shapes:Possible shapes: A sphere or oblate/prolate spheroidA sphere or oblate/prolate spheroid

Radii of the figures represent Radii of the figures represent vibration directionsvibration directions Length of radii represent the values of nLength of radii represent the values of n Plots of all possible values of n Plots of all possible values of n

generates figure generates figure Shows vibration directions and Shows vibration directions and

associated n for all ray pathsassociated n for all ray paths

Biaxial IndicatrixBiaxial Indicatrix

ConstructionConstruction Plot Plot primaryprimary indices of indices of

refraction along three refraction along three primary axes: X, Y, and primary axes: X, Y, and ZZ

Always 90º to each otherAlways 90º to each other nnqq & n & npp are two of the are two of the

principle vibration principle vibration directionsdirections

Fig. 7-22

Biaxial IndicatrixBiaxial Indicatrix Observe slice of figure Observe slice of figure

perpendicular to wave perpendicular to wave normal.normal.

Vibration directions Vibration directions perpendicular to wave perpendicular to wave normalnormal

Principle vibration Principle vibration directions and values of directions and values of index of refraction shown index of refraction shown by semi-major and semi-by semi-major and semi-minor axes of ellipseminor axes of ellipse

Fig. 7-22Fig. 7-22

Wave front – plane perpendicular to wave normal• Long axis = nslow

• short axis = nfast

Biaxial IndicatrixBiaxial Indicatrix

Ray paths Ray paths constructed by constructed by tangents to the tangents to the surface of the surface of the indicatrix that indicatrix that parallel vibration parallel vibration directionsdirections

Ray directions

Procedure to useProcedure to use

Imagine a section through the center Imagine a section through the center of the indicatrix and perpendicular to of the indicatrix and perpendicular to the the wave normalwave normal

Axes of sectionAxes of section are parallel to are parallel to fastfast (short axis) and (short axis) and slowslow (long axis) rays (long axis) rays

Ray paths of fast and slow rays are Ray paths of fast and slow rays are found by constructing tangents found by constructing tangents parallel to vibration directionsparallel to vibration directions

Generally used in a qualitative way:Generally used in a qualitative way: Understanding difference between Understanding difference between

isotropic, uniaxial, and biaxial mineralsisotropic, uniaxial, and biaxial minerals Understanding the relationship between Understanding the relationship between

optical properties, crystallographic axes, optical properties, crystallographic axes, and crystallographic propertiesand crystallographic properties

Isotropic IndicatrixIsotropic Indicatrix

Isometric minerals onlyIsometric minerals only: Unit cell has : Unit cell has only one dimensiononly one dimension Crystallographic axis = aCrystallographic axis = a

Minerals have only one index of Minerals have only one index of refractionrefraction Different for each mineralDifferent for each mineral

Shape of indicatrix is a sphereShape of indicatrix is a sphere All sections are circlesAll sections are circles

Light not split into two raysLight not split into two rays Birefringence is zeroBirefringence is zero

Isotropic indicatrixIsotropic indicatrix

Length of radii of sphere represent value for n

Ray path and Wave normal coincide

Light does not split into two rays, polarization direction unchanged

Circular Section

Uniaxial IndicatrixUniaxial Indicatrix

Tetragonal and hexagonal minerals onlyTetragonal and hexagonal minerals only: : two dimensions of unit cell (a and c)two dimensions of unit cell (a and c) High symmetry around c axisHigh symmetry around c axis

Two values of n’s required to Two values of n’s required to definedefine indicatrixindicatrix One is One is epsilonepsilon ,, the other is the other is omega omega

Remember – infinite values of nRemember – infinite values of n Range between nRange between n and n and n

Uniaxial IndicatrixUniaxial Indicatrix

Ellipsoid of revolution (spheroid) with axis of Ellipsoid of revolution (spheroid) with axis of rotation parallel the c crystallographic axisrotation parallel the c crystallographic axis

One semi-axis of ellipsoid One semi-axis of ellipsoid parallelsparallels c c nn

Other semi-axis of ellipsoid Other semi-axis of ellipsoid perpendicularperpendicular to to cc nn

Maximum birefringence is positive difference Maximum birefringence is positive difference of nof n and n and n Note Note nn < or > n < or > n, just as c > or < a, just as c > or < a

Fig. 7-23Fig. 7-23

Uniaxial Indicatrix

Note:(1)Axes designated X, Y, Z(2)Z axis always long axis for uniaxial indicatrix(3)May be c axis or a axis(4)Axis perpendicular to circular section is optic axis(5)Optic axis always c crystallographic axis

n>n

n<n

Y=Z

X=Y

Optic SignOptic Sign

Defined by nDefined by n and n and n

Optically positive (+)Optically positive (+) – n – n > n > n, Z = c = , Z = c = nn

Optically negative (-)Optically negative (-) - n - n < n < n, Z = a = , Z = a = nn

Ordinary and extraordinary Ordinary and extraordinary raysrays

In uniaxial minerals, one ray always In uniaxial minerals, one ray always vibrates perpendicular to optic axisvibrates perpendicular to optic axis Called Called ordinaryordinary or or ray ray Always same index = nAlways same index = n

Vibration always within the (001) planeVibration always within the (001) plane The other ray may be refractedThe other ray may be refracted

Called Called extraordinaryextraordinary or or ray ray Index of refraction is between nIndex of refraction is between n and n and n

Note that nNote that n < or > n < or > n

Fig. 7-24Fig. 7-24

Ordinary ray vibrates in (001) plane: index = n

C crystallographic

axis

Ordinary RayOrdinary Ray

Extraordinary RayExtraordinary Ray

Refracted extraordinary ray – vibrates in plane of ray path and c axisIndex = n’

How the mineral is cut is critical for what N the light experiences and it’s value of and

Sections of indicatrixSections of indicatrix Cross section perpendicular to the wave normal Cross section perpendicular to the wave normal

– usually an ellipse– usually an ellipse It is important:It is important:

Vibration directions of two rays Vibration directions of two rays mustmust parallel axes of parallel axes of ellipseellipse

Lengths of axes tells you magnitudes of the indices of Lengths of axes tells you magnitudes of the indices of refractionrefraction

Indices of refraction tell you the Indices of refraction tell you the birefringencebirefringence expected for any direction a grain may be cutexpected for any direction a grain may be cut

Indices of refraction tell you the angle that light is Indices of refraction tell you the angle that light is refractedrefracted

3 types of sections to 3 types of sections to indicatrixindicatrix

Principle sectionsPrinciple sections include c include c crystallographic axiscrystallographic axis

Circular sectionsCircular sections cut perpendicular to cut perpendicular to c crystallographic axis (and optic c crystallographic axis (and optic axis)axis)

Random sectionsRandom sections don’t include c axis don’t include c axis

Principle SectionPrinciple Section

Orientation of grainOrientation of grain Optic axis is horizontal (parallel stage)Optic axis is horizontal (parallel stage) Ordinary ray = nOrdinary ray = n ; extraordinary ray = ; extraordinary ray =

nn We’ll see that the wave normal and ray We’ll see that the wave normal and ray

paths coincide (no double refraction)paths coincide (no double refraction)

Fig. 7-25Fig. 7-25

Semi major axis

Semi-minor axis

Emergent point – at tangentsIndicates wave normal and ray path are the same, no double refractions

Principle Section

What is birefringence of this section?How many times does it go extinct with 360 rotation?

Circular SectionCircular Section

Optic axis is perpendicular to microscope Optic axis is perpendicular to microscope stagestage

Circular section, with radius nCircular section, with radius n Light retains its polarized directionLight retains its polarized direction Blocked by analyzer and remains extinctBlocked by analyzer and remains extinct

Fig. 7-25Fig. 7-25

Light not constrained to vibrate in any one direction

Ray path and wave normal coincide – no double refraction

Optic Axis

Circular Section

What is birefringence of this section?Extinction?

Random SectionRandom Section

Section now an ellipse with axes nSection now an ellipse with axes n and n and n’’ Find path of extraordinary ray by Find path of extraordinary ray by

constructing tangent parallel to vibration constructing tangent parallel to vibration directiondirection

Most common of all the sectionsMost common of all the sections

Fig. 7-25cFig. 7-25c

Random Section

Point of emergence for ray vibrating parallel to index ’

What is birefringence of this section?Extinction?

Line tangent to surface of indicatrix = point of emergence

Biaxial IndicatrixBiaxial Indicatrix

Crystal systems: Crystal systems: Orthorhombic, Orthorhombic, Monoclinic, TriclinicMonoclinic, Triclinic

Three dimensions to unit cellThree dimensions to unit cell a ≠ b ≠ ca ≠ b ≠ c

Three indices of refraction for Three indices of refraction for indicatrixindicatrix nn < n < n < n < n alwaysalways Maximum birefringence = nMaximum birefringence = n - n - n alwaysalways

Indicatrix axesIndicatrix axes

Plotted on a X-Y-Z systemPlotted on a X-Y-Z system Convention: nConvention: n = X, n = X, n = Y, n = Y, n = Z = Z

Z always longest axis (same as uniaxial Z always longest axis (same as uniaxial indicatrix)indicatrix)

X always shortest axisX always shortest axis Requires different definition of positive Requires different definition of positive

and negative mineralsand negative minerals Sometimes axes referred to as X, Y, Z Sometimes axes referred to as X, Y, Z

or nor nxx, n, nyy, n, nzz etc. etc.

Biaxial Biaxial IndicatrixIndicatrix

Note – differs from uniaxial because n ≠ n

Fig. 7-27Fig. 7-27

Biaxial indicatrix has two Biaxial indicatrix has two circular sectionscircular sections Radius is nRadius is n

The circular section ALWAYS contains the Y The circular section ALWAYS contains the Y axisaxis

Optic axis:Optic axis: perpendicular to the circular sectionsperpendicular to the circular sections Two circular sections = two optic axesTwo circular sections = two optic axes Neither optic axis is parallel to X, Y, or ZNeither optic axis is parallel to X, Y, or Z

Circular sections

Fig. 7-27Fig. 7-27

Both optic axes occur in the X-Z Both optic axes occur in the X-Z planeplane Must be because nMust be because n = Y = Y Called the Called the optic planeoptic plane Angle between optic axis is called Angle between optic axis is called 2V2V Can be either Can be either 2V2Vxx or or 2V2Vzz depending depending

which axis bisects the 2V anglewhich axis bisects the 2V angle

Optic signOptic sign

Acute angle between optic axes is Acute angle between optic axes is 2V angle2V angle Axis that bisects the 2V angle is Axis that bisects the 2V angle is acute acute

bisectrixbisectrix or or BBxaxa

Axis that bisects the obtuse angle is Axis that bisects the obtuse angle is obtuse obtuse bisectrixbisectrix or or BBxoxo

The bisecting axis determines optic sign:The bisecting axis determines optic sign: If If BBxaxa = X = X, then , then optically negativeoptically negative If If BBxaxa = Z = Z, then , then optically positiveoptically positive If If 2V = 90º2V = 90º, then , then optically neutraloptically neutral

Fig. 7-27Fig. 7-27

X-Z plane of Biaxial Indicatrix

+ -

Optically negativeOptically positive

Uniaxial indicatrixes are special Uniaxial indicatrixes are special cases of biaxial indicatrix:cases of biaxial indicatrix: If nIf n = n = n

Mineral is uniaxial positiveMineral is uniaxial positive nn = n = n and n and n = n = n, note – there is no n, note – there is no n

If nIf n = n = n

Mineral is uniaxial negativeMineral is uniaxial negative nn = n = n and n and n = n = n

Like the uniaxial indicatrix – there Like the uniaxial indicatrix – there are three primary sections:are three primary sections: Optic normal section – Y axis vertical so Optic normal section – Y axis vertical so

X and Z in plane of thin sectionX and Z in plane of thin section Optic axis vertical Optic axis vertical Random section Random section

Fig. 7-29Fig. 7-29

Optic normal – Maximum interference colors: contains n and n

Optic axis vertical = Circular section – Extinct: contains n only

Random section –Intermediate interference colors: contains n’ and n’

Crystallographic orientation of Crystallographic orientation of indicatrixindicatrix

Optic orientationOptic orientation Angular relationship between Angular relationship between

crystallographic and indicatrix axescrystallographic and indicatrix axes Three systems (biaxial) orthorhombic, Three systems (biaxial) orthorhombic,

monoclinic, & triclinicmonoclinic, & triclinic

Orthorhombic mineralsOrthorhombic minerals

Three crystallographic axes (a, b, c) Three crystallographic axes (a, b, c) coincide with X,Y, Z indicatrix axes – all 90ºcoincide with X,Y, Z indicatrix axes – all 90º

Symmetry planes coincide with principal Symmetry planes coincide with principal sectionssections

No consistency between which axis No consistency between which axis coincides with which onecoincides with which one

Optic orientationOptic orientation determined by which axes determined by which axes coincide, e.g.coincide, e.g. Aragonite: X = c, Y = a, Z = bAragonite: X = c, Y = a, Z = b Anthophyllite: X = a, Y = b, Z = cAnthophyllite: X = a, Y = b, Z = c

Fig. 7-28Fig. 7-28

OrthorhombOrthorhombic Mineralsic Minerals

Here optic Here optic orientation is:orientation is:

Z = cZ = cY = aY = aX = bX = b

MonoclinicMonoclinic

One indicatrix axis always parallels b One indicatrix axis always parallels b axisaxis 2-fold rotation or perpendicular to mirror 2-fold rotation or perpendicular to mirror

planeplane Could be X, Y, or Z indicatrix axisCould be X, Y, or Z indicatrix axis Other two axes lie in [010] plane (i.e. a-c Other two axes lie in [010] plane (i.e. a-c

crystallographic plane)crystallographic plane) One additional indicatrix axis may (but One additional indicatrix axis may (but

usually not) parallel crystallographic axisusually not) parallel crystallographic axis

Optic orientationOptic orientation defined by defined by 1.1. Which indicatrix axis parallels bWhich indicatrix axis parallels b2.2. Angles between other indicatrix axes and a Angles between other indicatrix axes and a

and c crystallographic axesand c crystallographic axes Angle is positive for the indicatrix axis Angle is positive for the indicatrix axis

within obtuse angle of crystallographic within obtuse angle of crystallographic axesaxes

Angle is negative for indicatrix axis Angle is negative for indicatrix axis within acute angle of crystallographic within acute angle of crystallographic axesaxes

Fig. 7-28Fig. 7-28

Monoclinic mineralsMonoclinic minerals

Symmetry – rotation Symmetry – rotation axis or perpendicular axis or perpendicular to mirror planeto mirror plane

Positive angle Positive angle because in obtuse because in obtuse angleangle

Negative angle Negative angle because in acute because in acute angleangle

> 90º

Triclinic mineralsTriclinic minerals

Indicatrix axes not constrained to Indicatrix axes not constrained to follow crystallographic axesfollow crystallographic axes

One indicatrix axis may (but usually One indicatrix axis may (but usually not) parallel crystallographic axisnot) parallel crystallographic axis

Fig. 7-28Fig. 7-28

Triclinic minerals

P. 306 – olivine informationP. 306 – olivine information

All optical properties

Optic Axes

Optical orientation