Miller in Dice

Crystallography:Forms and Planes

Mineralogy

Carleton College

Miller Indices (hkl)

• The orientation of a surface or a crystal plane may be defined by considering how the plane (or indeed any parallel plane) intersects the main crystallographic axes of the solid. The application of a set of rules leads to the assignment of the Miller Indices, (hkl); a set of numbers which quantify the intercepts and thus may be used to

uniquely identify the plane or surface.

a

b

c

(100)

(001)

(010)

Miller Indices

• A set of parallel crystallographic planes is indicated by its Miller Index (hkl). The Miller Index of a plane is derived from the intercepts of the plane with the crystallographic axes.

a

b

c

(100)

(001)

(010)

Miller Indices: Example 1

• The intercepts of the plane are at 0.5a, 0.75b, and 1.0c

• Take the reciprocals to get (2, 4/3, 1)

• Reduce common factors to get Miller Index of (643)

ab

C1.0

0.50.75


• The intercepts of the plane are at 1a, infinity b, and 1.0c

• Take the reciprocals to get (1, 0, 1)

• Reduce common factors to get Miller Index of (101)

ab

C1.0


• The intercepts of the plane are at 1a, 1b, and 1.0c

• Take the reciprocals to get (1, 1, 1)

• Reduce common factors to get Miller Index of (111) QuickTime™ and a

GIF decompressorare needed to see this picture.



• The intercepts of the line are at 1a1, infinity 2a2, -2/3 a3 and infinity with a3

• Take the reciprocals to get (1, 1/2, -3/2, , 1/«)

• Reduce common factors to get Miller Index of (2 1 -3 0)

Hexagonal coordinates

• Except for (0001) plane, the geometry of this lattice requires both positive and negative terms in the index

• A quick check on the correctness of hexagonal indices is that the sum of the first two digits times (-1) should be equal to the third digit.

Stable Cleavage Planes and Forms

• The most stable surfaces are those with the lowest Miller Indices (e.g. 100, and 110). Surfaces with high Miller Indices have atoms with very incomplete coordination.

Stable Cleavage Planes and Forms

• For a hexagonal lattice, stable cleavage planes will be (-1010) and (0-110) to give cleavage angles of 120 degrees.

Crystal Forms

• A form is a set of crystal faces that result by applying the symmetry elements of the crystal to any face.

Crystal Forms

• Any group of crystal faces related by the same symmetry is called a form. There are 47 or 48 crystal forms depending on the classification used.

Crystal Forms, Open or Closed

• Closed forms are those groups of faces all related by symmetry that completely enclose a volume of space. It is possible for a crystal to have entirely faces of one closed form.


• Open forms are those groups of faces all related by symmetry that do not completely enclose a volume of space. A crystal with open form faces requires additional faces as well.


• There are 17 or 18 open forms and 30 closed forms.

Triclinic, Monoclinic and Orthorhombic Forms

• Pedion – A single face unrelated to any other by symmetry. Open

QuickTime™ and aGIF decompressor

are needed to see this picture.

Triclinic, Monoclinic and Orthorhombic Forms

• Pinacoid – A pair of parallel faces related by mirror plane or twofold symmetry axis. Open



Crystal Forms

• Dihedron – A pair of intersecting faces

related by mirror plane or twofold symmetry axis. Some crystallographers distinguish between domes (pairs of intersecting faces related by mirror plane) and sphenoids (pairs of intersecting faces related by twofold symmetry

axis). All are open forms



Crystal Forms, 3-, 4- and 6 Prisms • Prisms. A collection of faces all parallel to a symmetry axis.

All are open.



Crystal Forms, 3-, 4- and 6 Pyramids • Pyramid. A group of faces at symmetry axis. All are open.

The base of the pyramid would be a pedion.



Crystal Forms, 3-, 4- and 6 Dipyramids • Dipyramid. Two pyramids joined base to base along a mirror

plane. All are closed, as are all all following forms.



Scalenohedra and Trapezohedra • 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 isoceles triangles and a fourfold inversion axis joins the midpointsof the bases of the isoceles triangles.



Scalenohedra and Trapezohedra • Scalenohedron. A solid made up of scalene triangle faces (all

sides unequal)



Scalenohedra and Trapezohedra • Trapezohedron. A solid made of trapezia (irregular

quadrilaterals)



Scalenohedra and Trapezohedra • Rhombohedron. A solid with six congruent parallelogram faces. Can be

considered a cube distorted along one of its diagonal three-fold symmetry axes.



Tetartoidal, Gyroidal and Diploidal Forms

• Tetartoid – The general form for symmetry class 233. 12 congruent

irregular pentagonal faces. The name comes from a Greek root for one-fourth because only a quarter of the 48 faces for full isometric symmetry are present.


• Gyroid – The general form for symmetry class 432. 24 congruent

irregular pentagonal faces.

• Diploid – The general form for symmetry class 2/m3*. 24 congruent

irregular quadrilateral faces. The name comes from a Latin root for half, because half of the 48 faces for full isometric symmetry are present.


• Pyritohedron – Special form (hk0) of symmetry class 2/m3*. Faces are each

perpendicular to a mirror plane, reducing the number of faces to 12 pentagonal faces. Although this superficially looks like the Platonic solid with 12 regular pentagon faces, these faces are not regular.




Hextetrahedral Forms

• Tetrahedron – Four equilateral triangle faces (111)

• Trapezohedral Tristetrahedron – 12 kite-shaped faces (hll)

Hextetrahedral Forms

• Trigonal Tristetrahedron – 12 isoceles triangle faces (hhl). Like an

tetrahedron with a low triangular pyramid built on each face.

• Hextetrahedron – 24 triangular faces (hkl) The general form.

Crystal Forms:• Cube

– Six square faces (100).

• Octahedron

– Eight equilateral triangle faces (111)

• Rhombic Dodecahedron

– 12 rhombic faces (110)

• Trapezohedral Trisoctahedron – 24 kite-shaped faces (hhl). Note that the Miller indices for the two

trisoctahedra are the opposite of those for the tristetrahedra.

Crystal Forms:• Trigonal Trisoctahedron

– 24 isoceles triangle faces (hll). Like an octahedron with a low triangular pyramid built on each face.

• Tetrahexahedron

– 24 isoceles triangle faces (h0l). Like an cube with a low pyramid built on each face.

• Hexoctahedron – 48 triangular faces (hkl) The general form

Cubic Forms



Crystal Forms: Octahedral Example

• In Cubic symmetry, the face (111) will generate the faces (111), (-111), (11-1), (-1-1-1), (1-1-1), (-11, -1) and (-1-11). The resulting set of faces is designated (111) and is called an octahedron.



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