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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

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 ( )

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

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),

,

,

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

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

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

Vanadinite

- 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

Fluorite

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

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

- 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

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

- From EOS, v.90, #3, 1/20/09

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