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 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 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 ( )

Plane Intercepting One Axis

Reduction of Indices

Planes Parallel to One Axis

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

Isometric (111) This plane represents a layer of close packing spheres in the conventional unit cell Source: http://www.chemsoc.org/exemplarchem/entries/2003/bristol_cook/millerindices.htm

Faces of a Hexahedron Miller Indices of cube faces

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

Faces of a Dodecahedron Six of the twelve dodecaheral faces

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

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 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 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 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 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 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 Crystal faces make simple rational intercepts on crystal axes

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

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 Calculate zone axis of (hkl), (pqr)

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

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

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 [111] {111} is equivalent to (111), , , , , , ,

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

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

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

Pedion Open form consisting of a single face

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

Benitoite The mineral benitoite has a set of two triangular faces which form a basal pinacoid Source: http://www.novagate.com/~ahines/rocks/vir_cris.htm

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

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

Apophyllite Pyramid Pyramid measures 4.45 centimeters tall by 5.1 centimeters wide at its base Source: http://www.equinoxbooksandgifts.com/assets/images/Apoph_pyramid_b_306._175.jpg

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

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

Citrine Quartz The six vertical planes are a prismatic form This is a rare doubly terminated crystal of citrine, a variety of quartz Source: http://www.faden.it/QUARZO_INT_FRAME/schede/QUARZO%20ALFA%20MODELLO.html

Vanadinite Forms hexagonal prismatic crystals Images: http://www.gamineral.org/members-photo-db-vanadinite.htm Forms hexagonal prismatic crystals

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 Photo: http://gaia.busanedu.net/~creative/2002/cyber/mineral/pages/m401802_052-galena.htm

Fluorite Image shows the isometric {111} form combined with isometric {100} Either of these would be closed forms if uncombined Image: http://www.northstarminerals.com/spar3858.jpg

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

Hanksite Tetragonal dipyramid

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 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 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 Photo: http://www.probertencyclopaedia.com/HJ.HTM#TETRAHEDRITE Tetrahedrite, a copper sulfide mineral

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 Octahedron Hexoctahedron Trigonal trisoctahedron

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 Photo: http://www.rasny.org/Pyrite/Slide6.JPG

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

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 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 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 Fig. 1. (left) An illustrated cutaway of the Earth’s interior reveals seismic anisotropy of the inner core. (middle) Representative elastic anisotropies of single-crystal hexagonal close packed (hcp; red dashed line) and body-centered cubic (bcc; red dash-dotted line) iron from theory [Belonoshko et al., 2008], prefer-oriented hcp iron from static experiments at 112 gigapascals (green diamond) [Antonangeli et al., 2004], and seismic observations in the inner core (blue lines) are shown for comparison. Elastic anisotropy could be explained if the bcc iron-based alloy in the inner core had preferable orientation along Earth’s rotation axis (similar to that of Atomium, right, a 102-meter-tall monument built for the 1958 World’s Fair in Brussels, Belgium, which forms the shape of an iron crystal magnified 165 billion times) or a dominant amount of prefer-oriented hcp iron. From EOS, v.90, #3, 1/20/09