Transcript Crystallography: Forms and Planes

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
c
(001)
(100)
uniquely identify the plane or surface.
a
(010)
b
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.
c
(001)
(100)
a
(010)
b
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)
C
1.0
0.5
a
0.75
b
Miller Indices: Example 2
• 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)
C
1.0
a
b
Miller Indices: Example 2
• 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)
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Miller Indices: Example 3
Miller Indices: Example 3
• 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.
Crystal Forms, Open or Closed
• 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.
Crystal Forms, Open or Closed
• 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
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Triclinic, Monoclinic and Orthorhombic Forms
• Pinacoid
–
A pair of parallel faces related by mirror plane or twofold symmetry axis.
Open
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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
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Crystal Forms, 3-, 4- and 6 Prisms
• Prisms. A collection of faces all parallel to a symmetry axis.
All are open.
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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.
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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.
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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.
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Scalenohedra and Trapezohedra
• Scalenohedron. A solid made up of scalene triangle faces (all
sides unequal)
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Scalenohedra and Trapezohedra
• Trapezohedron. A solid made of trapezia (irregular
quadrilaterals)
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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.
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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.
Tetartoidal, Gyroidal and Diploidal Forms
• 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.
Tetartoidal, Gyroidal and Diploidal Forms
• 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.
Tetartoidal, Gyroidal and Diploidal Forms
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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
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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
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designated (111) and is
called an octahedron.