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Crystal Systems. GLY 4200 Fall, 2012. William Hallowes Miller. 1801 -1880 British Mineralogist and Crystallographer Published Crystallography in 1838 In 1839, wrote a paper, “treatise on Crystallography” in which he introduced the concept now known as the Miller Indices. Notation.

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crystal systems

Crystal Systems

GLY 4200

Fall, 2012

william hallowes miller
William Hallowes Miller
  • 1801 -1880
  • British Mineralogist and Crystallographer
  • Published Crystallography in 1838
  • In 1839, wrote a paper, “treatise on Crystallography” in which he introduced the concept now known as the Miller Indices
notation
Notation
  • Lattice points are not enclosed – 100
  • Lines, such as axes directions, are shown in square brackets [100] is the a axis
  • Direction from the origin through 102 is [102]
miller index
Miller Index
  • The points of intersection of a plane with the lattice axes are located
  • The reciprocals of these values are taken to obtain the Miller indices
  • The planes are then written in the form (h k l) where h = 1/a, k = 1/b and l = 1/c
  • Miller Indices are always enclosed in ( )
isometric system
Isometric System
  • All intercepts are at distance a
  • Therefore (1/1, 1/1, 1/1,) = (1 1 1)
isometric 111
Isometric (111)
  • This plane represents a layer of close packing spheres in the conventional unit cell
faces of a hexahedron
Faces of a Hexahedron
  • Miller Indices of cube faces
faces of an octahedron
Faces of an Octahedron
  • Four of the eight faces of the octahedron
faces of a dodecahedron
Faces of a Dodecahedron
  • Six of the twelve dodecaheral faces
octahedron to cube to dodecahedron
Octahedron to Cube to Dodecahedron
  • Animation shows the conversion of one form to another
negative intercept
Negative Intercept
  • Intercepts may be along a negative axis
  • Symbol is a bar over the number, and is read “bar 1 0 2”
miller index from intercepts
Miller Index from Intercepts
  • Let a’, b’, and c’ be the intercepts of a plane in terms of the a, b, and c vector magnitudes
  • Take the inverse of each intercept, then clear any fractions, and place in (hkl) format
example
Example
  • a’ = 3, b’ = 2, c’ = 4
  • 1/3, 1/2, 1/4
  • Clear fractions by multiplication by twelve
  • 4, 6, 3
  • Convert to (hkl) – (463)
miller index from x ray data
Miller Index from X-ray Data
  • Given Halite, a = 0.5640 nm
  • Given axis intercepts from X-ray data
    • x’ = 0.2819 nm, y’ = 1.128 nm, z’ = 0.8463 nm
  • Calculate the intercepts in terms of the unit cell magnitude
unit cell magnitudes
Unit Cell Magnitudes
  • a’ = 0.2819/0.5640, b’ = 1.128/0.5640, c’ = 0.8463/0.5640
  • a’ = 0.4998, b’ = 2.000, c’ = 1.501
  • Invert: 1/0.4998, 1/2.000, 1/1.501 = 2,1/2, 2/3
clear fractions
Clear Fractions
  • Multiply by 6 to clear fractions
  • 2 x 6 =12, 0.5 x 6 = 3, 0.6667 x 6 = 4
  • (12, 3, 4)
  • Note that commas are used to separate double digit indices; otherwise, commas are not used
law of huay
Law of Huay
  • Crystal faces make simple rational intercepts on crystal axes
law of bravais
Law of Bravais
  • Common crystal faces are parallel to lattice planes that have high lattice node density
zone axis
Zone Axis
  • The intersection edge of any two non-parallel planes may be calculated from their respective Miller Indices
  • Crystallographic direction through the center of a crystal which is parallel to the intersection edges of the crystal faces defining the crystal zone
  • This is equivalent to a vector cross-product
  • Like vector cross-products, the order of the planes in the computation will change the result
  • However, since we are only interested in the direction of the line, this does not matter
generalized zone axis calculation
Generalized Zone Axis Calculation
  • Calculate zone axis of (hkl), (pqr)
zone axis calculation
Zone Axis Calculation
  • Given planes (120) , (201)
  • 1│2 0 1 2│0
  • 2│0 1 2 0│1
  • (2x1 - 0x0, 0x2-1x1, 1x0-2x2) = 2 -1 -4
  • The symbol for a zone axis is given as [uvw]
  • So,
common mistake
Common Mistake
  • Zero x Anything is zero, not “Anything’
  • Every year at least one student makes this mistake!
zone axis calculation 2
Zone Axis Calculation 2
  • Given planes (201) , (120)
  • 2│0 1 2 0│1
  • 1│2 0 1 2│0
  • (0x0-2x1, 1x1-0x2,2x2-1x0) = -2 1 4
  • Zone axis is
  • This is simply the same direction, in the opposite sense
zone axis diagram
Zone Axis Diagram
  • [001] is the zone axis (100), (110), (010) and related faces
slide28
Form
  • Classes of planes in a crystal which are symmetrically equivalent
  • Example the form {100} for a hexahedron is equivalent to the faces (100), (010), (001),

,

,

isometric 1111
Isometric [111]
  • {111} is equivalent to (111),

,

,

,

,

,

,

closed form isometric 100
Closed Form – Isometric {100}
  • Isometric form {100} encloses space, so it is a closed form
closed form isometric 111
Closed Form – Isometric {111}
  • Isometric form {111} encloses space, so it is a closed form
open forms tetragonal 100 and 001
Open Forms –Tetragonal {100} and {001}
  • Showing the open forms {100} and {001}
pedion
Pedion
  • Open form consisting of a single face
pinacoid
Pinacoid
  • Open form consisting of two parallel planes
  • Platy specimen of wulfenite – the faces of the plates are a pinacoid
benitoite
Benitoite
  • The mineral benitoite has a set of two triangular faces which form a basal pinacoid
dihedron
Dihedron
  • Pair of intersecting faces related by mirror plane or twofold symmetry axis
    • Sphenoids - Pair of intersecting faces related by two-fold symmetry axis
    • Dome - Pair of intersecting faces related by mirror plane
slide37
Dome
  • Open form consisting of two intersecting planes, related by mirror symmetry
  • Very large gem golden topaz crystal is from Brazil and measures about 45 cm in height
  • Large face on right is part of a dome
sphenoid
Sphenoid
  • Open form consisting of two intersecting planes, related by a two-fold rotation axis
  • (Lower) Dark shaded triangular faces on the model shown here belong  to a sphenoid
  • Pairs of similar vertical faces that cut the edges of the drawing are pinacoids
  • Top and bottom faces are two different pedions
pyramids
Pyramids
  • A group of faces intersecting at a symmetry axis
  • All pyramidal forms are open
apophyllite pyramid
Apophyllite Pyramid
  • Pyramid measures 4.45 centimeters tall by 5.1 centimeters wide at its base
uvite
Uvite
  • Three-sided pyramid of the mineral uvite, a type of tourmaline
prisms
Prisms
  • A prism is a set of faces that run parallel to an axes in the crystal
  • There can be three, four, six, eight or even twelve faces
  • All prismatic forms are open
diprismatic forms
Diprismatic Forms
  • Upper – Trigonal prism
  • Lower – Ditrigonal prism – note that the vertical axis is an A3, not an A6
citrine quartz
Citrine Quartz
  • The six vertical planes are a prismatic form
  • This is a rare doubly terminated crystal of citrine, a variety of quartz
vanadinite
Vanadinite
  • Forms hexagonal prismatic crystals
galena
Galena
  • Galena is isometric, and often forms cubic to rectangular crystals
  • Since all faces of the form {100} are equivalent, this is a closed form
fluorite
Fluorite
  • Image shows the isometric {111} form combined with isometric {100}
  • Either of these would be closed forms if uncombined
dipyramids
Dipyramids
  • Two pyramids joined base to base along a mirror plane
  • All are closed forms
hanksite
Hanksite
  • Tetragonal dipyramid
disphenoid
Disphenoid
  • A solid with four congruent triangle faces, like a distorted tetrahedron
  • Midpoints of edges are twofold symmetry axes
  • In the tetragonal disphenoid, the faces are isosceles triangles and a fourfold inversion axis joins the midpoints of the bases of the isosceles triangles.
dodecahedrons
Dodecahedrons
  • A closed 12-faced form
  • Dodecahedrons can be formed by cutting off the edges of a cube
  • Form symbol for a dodecahedron is isometric{110} 
  • Garnets often display this form
tetrahedron
Tetrahedron
  • The tetrahedron occurs in the class bar4 3m and has the form symbol {111}(the form shown in the drawing) or {1 bar11}
  • It is a four faced form that results form three bar4 axes and four 3-fold axes
  • Tetrahedrite, a copper sulfide mineral
forms related to the octahedron
Forms Related to the Octahedron
  • Trapezohderon - An isometric trapezohedron is a 12-faced closed form with the general form symbol {hhl}
  • The diploid is the general form {hkl} for the diploidal class (2/m bar3)
forms related to the octahedron1
Forms Related to the Octahedron
  • Hexoctahedron
  • Trigonal trisoctahedron
pyritohedron
Pyritohedron
  • The pyritohedron is a 12-faced form that occurs in the crystal class 2/m bar3
  • The possible forms are {h0l} or {0kl} and each of the faces that make up the form have 5 sides
tetrahexahedron
Tetrahexahedron
  • A 24-faced closed form with a general form symbol of {0hl}
  • It is clearly related to the cube
scalenohedron
Scalenohedron
  • A scalenohedron is a closed form with 8 or 12 faces
  • In ideally developed faces each of the faces is a scalene triangle
  • In the model, note the presence of the 3-fold rotoinversion axis perpendicular to the 3 2-fold axes
trapezohedron
Trapezohedron
  • Trapezohedron are closed 6, 8, or 12 faced forms, with 3, 4, or 6 upper faces offset from 3, 4, or 6 lower faces
  • The trapezohedron results from 3-, 4-, or 6-fold axes combined with a perpendicular 2-fold axis
  • Bottom - Grossular garnet from the Kola Peninsula (size is 17 mm)
rhombohedron
Rhombohedron
  • A rhombohedron is 6-faced closed form wherein 3 faces on top are offset by 3 identical upside down faces on the bottom, as a result of a 3-fold rotoinversion axis
  • Rhombohedrons can also result from a 3-fold axis with perpendicular 2-fold axes
  • Rhombohedrons only occur in the crystal classes bar3 2/m , 32, and bar3 .
application to the core
Application to the Core
  • From EOS, v.90, #3, 1/20/09
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