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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

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Crystal systems

Crystal Systems

GLY 4200

Fall, 2012

William hallowes miller

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



  • 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

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

Plane Intercepting One Axis

Reduction of indices

Reduction of Indices

Planes parallel to one axis

Planes Parallel to One Axis

Isometric system

Isometric System

  • All intercepts are at distance a

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

Isometric 111

Isometric (111)

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

Faces of a hexahedron

Faces of a Hexahedron

  • Miller Indices of cube faces

Faces of an octahedron

Faces of an Octahedron

  • Four of the eight faces of the octahedron

Faces of a dodecahedron

Faces of a Dodecahedron

  • Six of the twelve dodecaheral faces

Octahedron to cube to dodecahedron

Octahedron to Cube to Dodecahedron

  • Animation shows the conversion of one form to another

Negative intercept

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

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



  • 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

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

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

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

Law of Huay

  • Crystal faces make simple rational intercepts on crystal axes

Law of bravais

Law of Bravais

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

Zone axis

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

Generalized Zone Axis Calculation

  • Calculate zone axis of (hkl), (pqr)

Zone axis calculation

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

Common Mistake

  • Zero x Anything is zero, not “Anything’

  • Every year at least one student makes this mistake!

Zone axis calculation 2

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

Zone Axis Diagram

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

Crystal systems


  • 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 1111

Isometric [111]

  • {111} is equivalent to (111),







Closed form isometric 100

Closed Form – Isometric {100}

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

Closed form isometric 111

Closed Form – Isometric {111}

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

Open forms tetragonal 100 and 001

Open Forms –Tetragonal {100} and {001}

  • 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

Crystal systems


  • 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

Apophyllite pyramid

Apophyllite Pyramid

  • 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

Diprismatic forms

Diprismatic Forms

  • Upper – Trigonal prism

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

Citrine quartz

Citrine Quartz

  • 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

Forms related to the octahedron

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 octahedron1

Forms Related to the Octahedron

  • 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 .

Application to the core

Application to the Core

  • From EOS, v.90, #3, 1/20/09

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