Want to represent 3-D crystal on 2-D paperWant to represent 3-D crystal on 2-D paperUse a Use a ProjectionProjection

A cubic xl like our modelA cubic xl like our model

Note Note polespoles (normals to xl (normals to xl face planes)face planes)

Stereographic ProjectionStereographic Projection

Fig 6.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Spherical ProjectionSpherical Projection

Click to run animation Case Klein animation Click to run animation Case Klein animation for Mineral Science, © John Wiley & Sonsfor Mineral Science, © John Wiley & Sons

The outer sphere The outer sphere is a is a sphericalspherical projectionprojection

Plot points Plot points where poles where poles intersect sphere intersect sphere

Planes now = Planes now = pointspoints

But still 3-DBut still 3-D

Stereographic ProjectionStereographic Projection

Fig 6.3

Stereographic ProjectionStereographic ProjectionGray plane = Gray plane = Equatorial PlaneEquatorial Plane

Want to use it as Want to use it as our 2-D our 2-D representation representation and project our and project our spherical poles spherical poles back to itback to it

This is a 2-D This is a 2-D stereographic stereographic projectionprojection

Fig 6.5 of Klein (2002) Manual of Mineral Science,

John Wiley and Sons

Stereographic ProjectionStereographic ProjectionD and E are D and E are sphericalspherical

D' and E' are D' and E' are stereographicstereographic

Distance GD' = f(Distance GD' = f() )

as as 90 D’ 90 D’ G G

as as 0 D’ 0 D’ O O

Fig 6.6 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Stereographic ProjectionStereographic ProjectionWe can thus use We can thus use the angles and the angles and calculate the 2-D calculate the 2-D distances from distances from the center to find the center to find the stereographic the stereographic poles directlypoles directly

Or we can use Or we can use special graph special graph paper and avoid paper and avoid the calculationthe calculation Fig 6.5 of Klein (2002)

Manual of Mineral Science, John Wiley and Sons

Inclined Planes and Inclined Planes and Great CirclesGreat Circles

Great Circle as stereographic Great Circle as stereographic projection calculated from angle projection calculated from angle

Great circles on stereographic Great circles on stereographic projection = locus of all points projection = locus of all points projected from the intercept of an projected from the intercept of an inclined plane to the equatorial planeinclined plane to the equatorial plane

(bowl analogy)- (bowl analogy)- structural geologystructural geology

Use your hand for dip and a pencil for Use your hand for dip and a pencil for the the polepole of (011) at 45 of (011) at 45oo from vertical from vertical

This is the graph This is the graph paper for avoiding paper for avoiding calculating the calculating the distance from the distance from the center as a function of center as a function of each time each time

It is graduated in It is graduated in increments of 20increments of 20oo

(= (= zonezone))

Thus all poles in a Thus all poles in a zone are on the zone are on the same great circle!! same great circle!!

How do we find the How do we find the zone axis??zone axis??

Back to Fig. 2.42 Back to Fig. 2.42

(111) (100) (111) (111) (100) (111) (011) (100) all (011) (100) all coplanar coplanar

Fig 6.3 of Klein (2002) Manual of Mineral Science, John Wiley & Sons

Small circles

Gives angles between any two points on a great circle

= the angle between 2 coplanar lines!!

20o

The Wulff NetThe Wulff Net

Combines Combines great circles great circles and small and small circles in 2circles in 2oo incrementsincrements

Stereographic ProjectionHow to make a stereographic projection of our crystalHow to make a stereographic projection of our crystal

Use a Use a contact goniometercontact goniometer to measure the interfacial to measure the interfacial angles angles (also measures normals: poles)(also measures normals: poles)

Fig 6.2 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Plot Cardboard ModelPlot Cardboard ModelIsometric System (p. 93)Isometric System (p. 93)Crystallographic AxesCrystallographic Axes

““The crystal forms of classes of the isometric system The crystal forms of classes of the isometric system are referred to three axes of equal length that make are referred to three axes of equal length that make right angles with each other. Because the axes are right angles with each other. Because the axes are identical, they are interchangeable, and all are identical, they are interchangeable, and all are designated by the letter a. When properly oriented, designated by the letter a. When properly oriented, one axis, aone axis, a11, is horizontal and oriented front to , is horizontal and oriented front to back, aback, a22 is horizontal and right to left, and a is horizontal and right to left, and a33 is is vertical.”vertical.”

++aa33

++aa11

++aa22

9090

90909090

Plot (100) (001) (010) (110) (101) (011): = top half o = bottom half

How plot (111) ?a) Plot (110) & then plot (111) between (110) and (001)

(110) (111) = 36.5o

- go in from primitive

b) No measure technique:

(111) must lie between (110) & (001) (zone add rule)

also between (100) & (011)

thus intersection of great circles (111)

The finished productThe finished product

face poles and principal zonesface poles and principal zones

symmetry elementssymmetry elements Fig 6.8 of Klein (2002)

Manual of Mineral Science, John Wiley and Sons

Once finished can determine the angles between any 2 faces w/o measuring.

What is (100) (111) ?

(54.5o)

(111) (111) ?

(70o)

Model #75-

How can you use the position of the (111) face on a stereonet to determine:

a/b?

b/c?

a/c?

TwinningTwinning Rational symmetrically-related intergrowthRational symmetrically-related intergrowth Lattices of each orientation have definite Lattices of each orientation have definite

crystallographic relation to each othercrystallographic relation to each other

TwinningTwinning

Aragonite twinAragonite twin

Note zone at twin Note zone at twin plane which is plane which is common to each common to each partpart

Redrawn from Fig 2-69 of Berry, Mason and Dietrich, Mineralogy, Freeman & Co.

Although aragonite is orthorhombic, the twin looks hexagonal due to the 120o O-C-O angle in the CO3 group

TwinningTwinning Twin Twin OperationOperation is the symmetry operation which relates the is the symmetry operation which relates the

two (or more) parts (twin mirror, rot. axis)two (or more) parts (twin mirror, rot. axis)1) Reflection1) Reflection (twin plane) (twin plane)

Examples: gypsum “fish-tail”, models 102, 108Examples: gypsum “fish-tail”, models 102, 1082) Rotation2) Rotation (usually 180 (usually 180oo) about an axis common to ) about an axis common to

both (twin axis): normal and parallel twins.both (twin axis): normal and parallel twins.Examples: carlsbad twin, model 103Examples: carlsbad twin, model 103

3) Inversion3) Inversion (twin center) (twin center) The twin element cannot be a symmetry element of the The twin element cannot be a symmetry element of the

individuals. Twin plane can't be a mirror plane of the crystalindividuals. Twin plane can't be a mirror plane of the crystal Twin Twin LawLaw is a more exact description for a given type is a more exact description for a given type

(including operation, plane/axis, mineral…)(including operation, plane/axis, mineral…)

ContactContact & & PenetrationPenetration twins twins Both are Both are simple twinssimple twins only two parts only two parts

Multiple Multiple twins (> 2 segments repeated by same law)twins (> 2 segments repeated by same law) Cyclic twinsCyclic twins - successive planes not parallel - successive planes not parallelPolysynthetic twins Polysynthetic twins Albite LawAlbite Law

in plagioclasein plagioclase

TwinningTwinningMechanisms:Mechanisms:1) Growth1) GrowthGrowth increment cluster adds w/ twin Growth increment cluster adds w/ twin

orientationorientationEpitaxialEpitaxial more stable than random more stable than random

Not all epitaxis Not all epitaxis twins twins

Usually simple & penetrationUsually simple & penetrationsynneusissynneusis a special case a special case

TwinningTwinningMechanisms:Mechanisms:1) Growth1) Growth

Feldspars: Feldspars: Plagioclase: Triclinic Albite-law-striationsPlagioclase: Triclinic Albite-law-striations

bb

a-ca-c

bb

a-ca-c

TwinningTwinningMechanisms:Mechanisms:1) Growth1) Growth

Feldspars: Feldspars: Plagioclase: Triclinic Albite-law-striationsPlagioclase: Triclinic Albite-law-striations

TwinningTwinningMechanisms:Mechanisms:2) Transformation2) Transformation (secondary) (secondary)

SiOSiO22: High T is higher symmetry: High T is higher symmetry

High Quartz P6High Quartz P6222222 Low Quartz P3Low Quartz P3222121

cyclic twinning in cyclic twinning in inverted low quartzinverted low quartz

TwinningTwinningMechanisms:Mechanisms:2) Transformation2) Transformation (secondary twins) (secondary twins)

Feldspars: Feldspars: Orthoclase (monoclinic) Orthoclase (monoclinic) microcline (triclinic) microcline (triclinic)

MonoclinicMonoclinic(high-T)(high-T)

bb

a-ca-c TriclinicTriclinic(low-T)(low-T)

bb

a-ca-c

TwinningTwinningMechanisms:Mechanisms:2) Transformation2) Transformation (secondary) (secondary)

Feldspars: Feldspars: K-feldspar: large K K-feldspar: large K lower T of transformation lower T of transformation

““tartan twins”tartan twins”

Interpretation wrt petrology!Interpretation wrt petrology!

TwinningTwinningMechanisms:Mechanisms:3) Deformation (secondary)3) Deformation (secondary) Results from shear stressResults from shear stress

greater stress greater stress gliding, and finally rupture gliding, and finally rupture Also in feldspars. Also in feldspars. Looks like transformation, but the difference in Looks like transformation, but the difference in interpretationinterpretation is tremendous is tremendous

Mechanisms:Mechanisms:3) Deformation (secondary)3) Deformation (secondary) Results from shear stress. Results from shear stress. PlagioclasePlagioclase

Mechanisms:Mechanisms:3) Deformation (secondary)3) Deformation (secondary) Results from shear stress. Results from shear stress. CalciteCalcite

X-ray CrystallographyX-ray CrystallographyX-ray wavelengths are on the same order of X-ray wavelengths are on the same order of

magnitude as atomic spacings. magnitude as atomic spacings.

Crystals thus makes excellent diffraction gratingsCrystals thus makes excellent diffraction gratings

Can use the geometry of the x-ray spots to Can use the geometry of the x-ray spots to determine geometry of grating (ie the crystal)determine geometry of grating (ie the crystal)

X-ray CrystallographyX-ray CrystallographyX-ray generationX-ray generation

W Cathode Cu Anode(-) (+)

X-rays

electronselectrons

X-ray CrystallographyX-ray CrystallographyX-ray generationX-ray generationContinuous & characteristic spectrum (Fig. 7.2)Continuous & characteristic spectrum (Fig. 7.2)

Continuous from E loss of collisionsContinuous from E loss of collisionsCharacteristic is quantizedCharacteristic is quantized

I

X-ray CrystallographyX-ray CrystallographyDestructive and constructive interference of wavesDestructive and constructive interference of wavesBragg Equation:Bragg Equation:

Y

x

d

in phasein phase in phasein phase

X-ray CrystallographyX-ray Crystallographynn=2dsin=2dsin n is the “order” n is the “order” As soon as the crystal is rotated, the beam ceasesAs soon as the crystal is rotated, the beam ceases

(This is (This is diffractiondiffraction, not reflection), not reflection)Only get diffraction at certain angles!Only get diffraction at certain angles!Relation between Relation between and d and and d and

Y

x

d

X-ray CrystallographyX-ray CrystallographyMethods:Methods:1) Single-Crystal: Laue Method1) Single-Crystal: Laue Method

Several directions simultaneously fulfill Bragg equationsSeveral directions simultaneously fulfill Bragg equationsGood for symmetry, but poor for analysis because distortedGood for symmetry, but poor for analysis because distorted

Fig 7.39 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray CrystallographyX-ray CrystallographyMethods:Methods:1) Single-Crystal: Precession1) Single-Crystal: Precession

Use motors to move crystal & film to satisfy Bragg Use motors to move crystal & film to satisfy Bragg equations for different planes without distortionsequations for different planes without distortions

Fig 7.40 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray CrystallographyX-ray CrystallographyMethods:Methods:2) Powder- 2) Powder-

EasiestEasiestInfinite orientations at once, so only need to vary Infinite orientations at once, so only need to vary

Cameras and diffractometersCameras and diffractometers