want to represent 3-d crystal on 2-d paper use a projection a cubic xl like our model note poles...
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Want to represent 3-D crystal on 2-D paperWant to represent 3-D crystal on 2-D paper
Use 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 Projection
Gray 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 Projection
D 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 Projection
We 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 Projection
How 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 Model
Isometric 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, 108
2) 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 1033) 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 parallel
Polysynthetic twins Polysynthetic twins
Albite LawAlbite Law
in plagioclasein plagioclase
TwinningTwinningMechanisms:Mechanisms:1) Growth1) Growth
Growth increment cluster adds w/ twin Growth increment cluster adds w/ twin orientationorientation
EpitaxialEpitaxial more stable than random more stable than randomNot all epitaxis Not all epitaxis twins twins
Usually simple & penetrationUsually simple & penetration
synneusissynneusis 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 Crystallography
X-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 collisions
Characteristic is quantizedCharacteristic is quantized
I
X-ray CrystallographyX-ray CrystallographyDestructive and constructive interference of wavesDestructive and constructive interference of waves
Bragg 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 MethodSeveral directions simultaneously fulfill Bragg equationsSeveral directions simultaneously fulfill Bragg equations
Good 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: PrecessionUse 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