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# Crystal Systems - PowerPoint PPT Presentation

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

GLY 4200

Fall, 2012

• 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

• 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]

• 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 ( )

• All intercepts are at distance a

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

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

• Miller Indices of cube faces

• Four of the eight faces of the octahedron

• Six of the twelve dodecaheral faces

• Animation shows the conversion of one form to another

• Intercepts may be along a negative axis

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

• 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

• 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)

• 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

• 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

• 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

• Crystal faces make simple rational intercepts on crystal axes

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

• 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

• Calculate zone axis of (hkl), (pqr)

• 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,

• Zero x Anything is zero, not “Anything’

• Every year at least one student makes this mistake!

• 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

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

• 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),

,

,

• {111} is equivalent to (111),

,

,

,

,

,

,

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

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

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

• Open form consisting of a single face

• Open form consisting of two parallel planes

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

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

• 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

• 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

• 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

• A group of faces intersecting at a symmetry axis

• All pyramidal forms are open

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

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

• 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

• Upper – Trigonal prism

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

• The six vertical planes are a prismatic form

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

• Forms hexagonal prismatic crystals

• Galena is isometric, and often forms cubic to rectangular crystals

• Since all faces of the form {100} are equivalent, this is a closed form

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

• Either of these would be closed forms if uncombined

• Two pyramids joined base to base along a mirror plane

• All are closed forms

• Tetragonal dipyramid

• 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.

• 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

• 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

• 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)

• Hexoctahedron

• Trigonal trisoctahedron

• 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

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

• It is clearly related to the cube

• 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 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)

• 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 .

• From EOS, v.90, #3, 1/20/09