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1 Crystal Systems GLY 4200 Fall, 2012

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William Hallowes Miller 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 2

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3 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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4 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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5 Plane Intercepting One Axis

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6 Reduction of Indices

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7 Planes Parallel to One Axis

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8 Isometric System All intercepts are at distance a Therefore (1/1, 1/1, 1/1,) = (1 1 1)

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9 Isometric (111) This plane represents a layer of close packing spheres in the conventional unit cell

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10 Faces of a Hexahedron Miller Indices of cube faces

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11 Faces of an Octahedron Four of the eight faces of the octahedron

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12 Faces of a Dodecahedron Six of the twelve dodecaheral faces

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13 Octahedron to Cube to Dodecahedron Animation shows the conversion of one form to another

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14 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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15 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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16 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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17 Miller Index from X-ray Data Given Halite, a = nm Given axis intercepts from X-ray data x’ = nm, y’ = nm, z’ = nm Calculate the intercepts in terms of the unit cell magnitude

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18 Unit Cell Magnitudes a’ = /0.5640, b’ = 1.128/0.5640, c’ = / a’ = , b’ = 2.000, c’ = Invert: 1/0.4998, 1/2.000, 1/1.501 = 2,1/2, 2/3

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19 Clear Fractions Multiply by 6 to clear fractions 2 x 6 =12, 0.5 x 6 = 3, x 6 = 4 (12, 3, 4) Note that commas are used to separate double digit indices; otherwise, commas are not used

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20 Law of Huay Crystal faces make simple rational intercepts on crystal axes

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21 Law of Bravais Common crystal faces are parallel to lattice planes that have high lattice node density

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22 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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23 Generalized Zone Axis Calculation Calculate zone axis of (hkl), (pqr)

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24 Zone Axis Calculation Given planes (120), (201) 1│ │0 2│ │1 (2x1 - 0x0, 0x2-1x1, 1x0-2x2) = The symbol for a zone axis is given as [uvw] So,

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25 Common Mistake Zero x Anything is zero, not “Anything’ Every year at least one student makes this mistake!

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26 Zone Axis Calculation 2 Given planes (201), (120) 2│ │1 1│ │0 (0x0-2x1, 1x1-0x2,2x2-1x0) = Zone axis is This is simply the same direction, in the opposite sense

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27 Zone Axis Diagram [001] is the zone axis (100), (110), (010) and related faces

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28 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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29 Isometric [111] {111} is equivalent to (111),,,,,,,

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30 Closed Form – Isometric {100} Isometric form {100} encloses space, so it is a closed form

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31 Closed Form – Isometric {111} Isometric form {111} encloses space, so it is a closed form

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32 Open Forms – Tetragonal {100} and {001} Showing the open forms {100} and {001}

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33 Pedion Open form consisting of a single face

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34 Pinacoid Open form consisting of two parallel planes Platy specimen of wulfenite – the faces of the plates are a pinacoid

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

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36 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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37 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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38 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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39 Pyramids A group of faces intersecting at a symmetry axis All pyramidal forms are open

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40 Apophyllite Pyramid Pyramid measures 4.45 centimeters tall by 5.1 centimeters wide at its base

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

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42 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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43 Diprismatic Forms Upper – Trigonal prism Lower – Ditrigonal prism – note that the vertical axis is an A 3, not an A 6

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44 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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45 Vanadinite Forms hexagonal prismatic crystals

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46 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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47 Fluorite Image shows the isometric {111} form combined with isometric {100} Either of these would be closed forms if uncombined

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48 Dipyramids Two pyramids joined base to base along a mirror plane All are closed forms

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49 Hanksite Tetragonal dipyramid

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50 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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51 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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52 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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53 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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54 Forms Related to the Octahedron Hexoctahedron Trigonal trisoctahedron

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55 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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56 Tetrahexahedron A 24-faced closed form with a general form symbol of {0hl} It is clearly related to the cube

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57 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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58 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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59 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.

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Application to the Core From EOS, v.90, #3, 1/20/09 60

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