(L15) Crystal Systems F12
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Transcript of (L15) Crystal Systems F12
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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 introducedthe concept now known as the Miller
Indices
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Notation
Lattice points are not enclosed100
Lines, such as axes directions, are shown insquare 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 withthe lattice axes are located
The reciprocals of these values are taken toobtain 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 areat distance a
Therefore(1/1, 1/1, 1/1,) =
(1 1 1)
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Isometric (111)
This planerepresents a layer
of close packingspheres in the
conventional unit
cell
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Faces of a Hexahedron
Miller Indicesof cube faces
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Faces of an Octahedron
Four of the eightfaces of the
octahedron
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Faces of a Dodecahedron
Six of the twelvedodecaheral faces
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Octahedron to
Cube to
Dodecahedron
Animation shows the conversion of one form toanother
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Negative
Intercept Intercepts may
be along a
negative axis Symbol is a
bar over the
number, and isread bar 1 02
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Miller Index from Intercepts
Let a, b, and c be the intercepts of a planein 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
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 unitcell 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
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 rationalintercepts on crystal axes
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Law of Bravais
Common crystal faces are parallel to latticeplanes 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 respectiveMiller Indices
Crystallographic direction through the center ofa crystal which is parallel to the intersectionedges of the crystal faces defining the crystalzone
This is equivalent to a vector cross-product
Like vector cross-products, the order of theplanes in the computation will change the result
However, since we are only interested in the
direction of the line, this does not matter
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Generalized Zone Axis Calculation
Calculate zone axis of (hkl), (pqr)
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Zone Axis Calculation
Given planes (120) , (201) 12 0 1 20
20 1 2 01 (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
Given planes (201) , (120) 20 1 2 01
12 0 1 20 (0x0-2x1, 1x1-0x2,2x2-1x0) = -2 1 4
Zone axis is
This is simply the same direction, in theopposite sense
[ ]214
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Zone Axis
Diagram [001] is the zone
axis (100),
(110), (010) andrelated faces
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Form Classes of planes in a
crystal which are
symmetrically equivalent
Example the form {100}for a hexahedron isequivalent to the faces
(100), (010), (001),
1
( )100 ( )010, ,( )001
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Isometric [111]
{111} is equivalent to (111),( )111 ( )111 ( )111
( )111 ( )111 ( )111( )111
, , ,
, , ,
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Closed FormIsometric {100}
Isometric form{100} encloses
space, so it is aclosed form
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Closed FormIsometric {111}
Isometric form{111} encloses
space, so it is aclosed form
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Open Forms
Tetragonal{100} and
{001}
Showing theopen forms
{100} and {001}
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Pedion
Open formconsisting of a
single face
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Pinacoid
Open formconsisting of two
parallel planes
Platy specimen ofwulfenitethefaces of the plates
are a pinacoid
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Benitoite
The mineralbenitoite has a set
of two triangularfaces which form a
basal pinacoid
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Dihedron
Pair of intersecting faces related by mirrorplane or twofold symmetry axis
Sphenoids - Pair of intersecting faces relatedby two-fold symmetry axis
Dome - Pair of intersecting faces related by
mirror plane
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Dome
Open form consisting of twointersecting planes, related
by mirror symmetry
Very large gem golden topazcrystal 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 twointersecting 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 thedrawing are pinacoids
Top and bottom faces are twodifferent pedions
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Pyramids
A group of faces intersecting at a symmetryaxis
All pyramidal forms are open
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Apophyllite Pyramid
Pyramid measures4.45 centimeters
tall by 5.1centimeters wide at
its base
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Uvite
Three-sidedpyramid of the
mineral uvite, atype of tourmaline
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Prisms
A prism is a set of faces that run parallel to anaxes in the crystal
There can be three, four, six, eight or eventwelve faces
All prismatic forms are open
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Diprismatic Forms
UpperTrigonalprism
LowerDitrigonalprismnote thatthe vertical axis is
an A3, not an A6
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Citrine Quartz
The six vertical planes area prismatic form
This is a rare doublyterminated 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 rectangularcrystals
Since all faces of the
form {100} areequivalent, this is a
closed form
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Fluorite
Image shows the isometric {111} formcombined with isometric {100}
Either of these would be closed forms ifuncombined
Di id
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Dipyramids
Two pyramids joined base to basealong a mirror plane
All are closed forms
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Hanksite
Tetragonaldipyramid
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Disphenoid
A solid with four congruenttriangle faces, like a distorted
tetrahedron
Midpoints of edges are twofoldsymmetry axes
In the tetragonal disphenoid, the
faces are isosceles triangles anda fourfold inversion axis joins
the midpoints of the bases of
the isosceles triangles.
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Dodecahedrons
A closed 12-faced form Dodecahedrons can be
formed by cutting off
the edges of a cube
Form symbol for adodecahedron is
isometric{110}
Garnets often displaythis form
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Tetrahedron
The tetrahedron occurs inthe class bar4 3m and has
the form symbol {111}(theform shown in the drawing)or {1 bar11}
It is a four faced form thatresults form three bar4axes 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 withthe general form symbol
{hhl}
The diploid is the generalform {hkl} for the diploidalclass (2/m bar3)
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Forms Related to the Octahedron
Hexoctahedron
Trigonaltrisoctahedron
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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 theform have 5 sides
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Tetrahexahedron
A 24-faced closedform with a general
form symbol of{0hl}
It is clearly related
to the cube
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Scalenohedron
A scalenohedron is a closedform with 8 or 12 faces In ideally developed faces
each of the faces is a scalene
triangle
In the model, note thepresence of the 3-fold
rotoinversion axisperpendicular to the 3 2-foldaxes
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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
mm)
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Rhombohedron
A rhombohedron is 6-faced closedform wherein 3 faces on top areoffset by 3 identical upside down
faces on the bottom, as a result of a
3-fold rotoinversion axis
Rhombohedrons can also resultfrom 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
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Application to the Core
From EOS, v.90, #3, 1/20/09