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Crystal Systems

GLY 4200

Fall, 2015

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

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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]

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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 ( )

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Plane Intercepting One Axis

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Reduction of Indices

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Planes Parallel to One Axis

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Isometric System

• All intercepts are at distance a

• Therefore (1/1, 1/1, 1/1,) = (1 1 1)

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Isometric (111)

• This plane represents a layer of close packing spheres in the conventional unit cell

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Faces of a Hexahedron

• Miller Indices of cube faces

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Faces of an Octahedron

• Four of the eight faces of the octahedron

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Faces of a Dodecahedron

• Six of the twelve dodecaheral faces

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Octahedron to Cube to

Dodecahedron

• Animation shows the conversion of one form to another

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Negative Intercept

• Intercepts may be along a negative axis

• Symbol is a bar over the number, and is read “bar 1 0 2”

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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

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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)

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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

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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

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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  Haüy

• Crystal faces make simple rational intercepts on crystal axes

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• Rene Just  Haüy, 1743-1822• French Mineralogist• Write Traité de minéralogie

in 1801 (5 volumes)

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Law of Bravais

• Common crystal faces are parallel to lattice planes that have high lattice node density

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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

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Generalized Zone Axis Calculation

• Calculate zone axis of (hkl), (pqr)

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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, [ ]2 1 4

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Common Mistake

• Zero x Anything is zero, not “Anything’

• Every year at least one student makes this mistake!

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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 [ ]2 1 4

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Zone Axis Diagram

• [001] is the zone axis (100), (110), (010) and related faces

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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),

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( )1 0 0 ( )0 1 0, ,( )0 0 1

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Isometric {111}

• {111} is equivalent to (111), ( )111 ( )111 ( )111

( )111 ( )111 ( )111( )111

, , ,

, , ,

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Closed Form – Isometric {100}

• Isometric form {100} encloses space, so it is a closed form

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Closed Form – Isometric {111}

• Isometric form {111} encloses space, so it is a closed form

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Open Forms –Tetragonal {100} and

{001}

• Showing the open forms {100} and {001}

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Pedion

• Open form consisting of a single face

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Pinacoid

• Open form consisting of two parallel planes

• Platy specimen of wulfenite – the faces of the plates are a pinacoid

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Benitoite

• The mineral benitoite has a set of two triangular faces which form a basal pinacoid

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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

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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

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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

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Pyramids

• A group of faces intersecting at a symmetry axis• All pyramidal forms are open

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Apophyllite Pyramid

• Pyramid measures 4.45 centimeters tall by 5.1 centimeters wide at its base

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Uvite

• Three-sided pyramid of the mineral uvite, a type of tourmaline

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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

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Diprismatic Forms

• Upper – Trigonal prism

• Lower – Ditrigonal prism – note that the vertical axis is an A3, not an A6

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Citrine Quartz

• The six vertical planes are a prismatic form

• This is a rare doubly terminated crystal of citrine, a variety of quartz

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Vanadinite

• Forms hexagonal prismatic crystals

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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

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Fluorite

• Image shows the isometric {111} form combined with isometric {100}

• Either of these would be closed forms if uncombined

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Dipyramids

• Two pyramids joined base to base along a mirror plane

• All are closed forms

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Hanksite

• Tetragonal dipyramid

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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.

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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

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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

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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)

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Forms Related to the Octahedron

• Hexoctahedron

• Trigonal trisoctahedron

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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

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Tetrahexahedron

• A 24-faced closed form with a general form symbol of {0hl}

• It is clearly related to the cube

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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

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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)

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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

• From EOS, v.90, #3, 1/20/09

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