(l15) crystal systems f12

7/28/2019 (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

2


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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

(l15) crystal systems f12

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