Transcript Introduction to Environmental Geochemistry

Crystal Systems
GLY 4200
Fall, 2012
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William Hallowes Miller
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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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Plane Intercepting One Axis
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Reduction of Indices
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Planes Parallel to One Axis
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Isometric System
• All intercepts are
at distance a
• Therefore
(1/1, 1/1, 1/1,) =
(1 1 1)
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Isometric (111)
• This plane
represents a layer
of close packing
spheres in the
conventional unit
cell
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Faces of a Hexahedron
• Miller Indices
of cube faces
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Faces of an Octahedron
• Four of the eight
faces of the
octahedron
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Faces of a Dodecahedron
• Six of the twelve
dodecaheral faces
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Octahedron to
Cube to
Dodecahedron
• Animation shows the conversion of one form to
another
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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
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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
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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 Huay
• Crystal faces make simple rational
intercepts on crystal axes
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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
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direction of the line, this does not matter
Generalized Zone Axis Calculation
• Calculate zone axis of (hkl), (pqr)
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Zone Axis Calculation
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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, [214 ]
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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
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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 [ 214 ]
This is simply the same direction, in the
opposite sense
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Zone Axis
Diagram
• [001] is the zone
axis (100),
(110), (010) and
related faces
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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),
1
( 10 0 ) , ( 010 ) , ( 0 01 )
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Isometric [111]
• {111} is equivalent to (111),( 111) , ( 111) , ( 111) ,
( 111) , ( 111 ) , ( 111) , ( 111)
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Closed Form – Isometric {100}
• Isometric form
{100} encloses
space, so it is a
closed form
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Closed Form – Isometric {111}
• Isometric form
{111} encloses
space, so it is a
closed form
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Open Forms –
Tetragonal
{100} and
{001}
• Showing the
open forms
{100} and {001}
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Pedion
• Open form
consisting of a
single face
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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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Benitoite
• The mineral
benitoite has a set
of two triangular
faces which form a
basal 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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Pyramids
• A group of faces intersecting at a symmetry
axis
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• All pyramidal forms are open
Apophyllite Pyramid
• Pyramid measures
4.45 centimeters
tall by 5.1
centimeters wide at
its base
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Uvite
• Three-sided
pyramid of the
mineral uvite, a
type of tourmaline
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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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Vanadinite
• Forms
hexagonal
prismatic crystals
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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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Dipyramids
• Two pyramids joined base to base
along a mirror plane
• All are closed forms
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Hanksite
• Tetragonal
dipyramid
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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
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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
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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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Forms Related to the Octahedron
• Hexoctahedron
• Trigonal
trisoctahedron
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Pyritohedron
• The pyritohedron is a 12faced 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
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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 .
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Application to the Core
• From EOS, v.90, #3, 1/20/09
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