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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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Isometric (111)
• This plane represents a layer of close packing spheres in the conventional unit cell
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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
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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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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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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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Pinacoid
• Open form consisting of two parallel planes
• Platy specimen of wulfenite – the faces of the plates are a 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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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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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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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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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 .