chp2 handouts

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E. Goeke, Fall 2006

Crystallography

Chapter 2

E. Goeke, Fall 2006

What is crystallography?

• Deals with the symmetry of crystals and crystal structures• Provides a descriptive method of describing the symmetry

of crystals• Warning: Perkins has condensed this material into one

chapter, so it comes quickly and without much background

E. Goeke, Fall 2006

Symmetry• The ordered arrangement of atoms in mineral structures is

defined by a lattice– Lattice = 3D dimensional network of atoms/molecules– Lattice node = intersection of lattice lines– Unit cell = smallest volume that contains all of the

elements• Three types of symmetry:

– Reflection– Rotation– Inversion

• Point of symmetry is the center of the crystal or the originof the unit cell

E. Goeke, Fall 2006

Reflection

E. Goeke, Fall 2006

Rotation• Rotation occurs around an axis (A)• There are five possible rotations in nature:

– 1-fold (A1 or 1)– 2-fold (A2 or or 2)– 3-fold (A3 or or 3)– 4-fold (A4 or or 4)– 6-fold (A6 or or 6)

• 5-fold, 7-fold, 8-fold don’t appear in nature

http://www.tulane.edu/~sanelson/eens211/introsymmetry.htmE. Goeke, Fall 2006

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Inversion

• A line drawn through the origin will find identical featureson the other side

• Indicated by the letter i

http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm

E. Goeke, Fall 2006

E. Goeke, Fall 2006

Example

• Take each number and determine the 2D symmetry of it

l 2 3 4 5 6 7 8 9 0

E. Goeke, Fall 2006

Further symmetry work

• ID the inversion points, rotations, and mirror planes of thefollowing alphabet in 2D (similar to the number exercise)

A B C D E F G �H I J K L M

N O P Q R S T U V W X Y Z

E. Goeke, Fall 2006

Rotoinversion

• Combination of inversion and rotation• Notated by a bar over the rotation

– 1-fold rotoinversion = A1 = 1 = i– 2-fold rotoinversion = A2 = 2 = m– 3-fold rotoinversion = A3 = 3 = A3 + i– 4-fold rotoinversion = A4 = 4– 6-fold rotoinversion = A6 = 6 = A3 + m

E. Goeke, Fall 2006

Hermann-Mauguin & Point Groups

• The presence of some symmetry elements requires thepresence of others--therefore we only notate the explicitelements and not the implied ones

• Hermann-Maugin notation includes:– 2 = 2-fold rotation axis– 4/m = 4-fold rotation axis with a mirror plane

perpendicular to the axis– 3 = 3-fold rotoinversion

• The explicit elements are used to define the 32 pointgroups (Table 2.2 in your textbook)

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How many symmetry elements?

E. Goeke, Fall 2006

4 three-fold axescubic

1 four-fold rotation or rotoinversionaxis

tetragonal

1 three-fold or six-fold axishexagonal

3 two-fold rotation axes and/or 3 morthorhombic

two-fold rotation and/or m

one-fold rotation with or without i

common symmetry elements

triclinic

monoclinic

crystal system

E. Goeke, Fall 2006

Crystallographic axes

• Crystal axes = frame of reference to describe the crystalstructure; arbitrarily determined

• Origin = intersection of the crystal axes• Length of axes is proportional to lattice spacing

E. Goeke, Fall 2006

4 3-fold axesα = β = γ = 90°a = b = ccubic

1 4-fold rotation orrotoinversion axis

α = β = γ = 90°a = b ≠ ctetragonal

1 3-fold or 6-fold axis60° btw a’s, β =90°

a1 = a2 = a3, a≠ c

hexagonal

3 2-fold rotation axesand/or 3 m

α = β = γ = 90°a ≠ b ≠ corthorhombic

2-fold rotation and/or 1m

1-fold rotation w/ orw/out i

common symmetryelements

α ≠ β ≠ γ ≠ 90°a ≠ b ≠ ctriclinic

α = γ = 90°, β >90°

a ≠ b ≠ cmonoclinic

angles betweenaxes

axis lengthscrystal system

E. Goeke, Fall 2006

cubic

tetragonal

hexagonal

orthorhombic

monoclinic

triclinic

crystal system

3 2-fold or 4-fold = a, b, cH-M symbols: 1st = a, b & c, 2nd = long diagonals through theunit cell, 3rd = edge-to-edge diagonals

4-fold = c; a & b = 2-fold or lines ⊥ to mH-M symbols: 1st = c, 2nd = a & b, 3rd = symmetry axis thatbisects the angle btw a & b

3-fold or 6-fold = cH-M symbols: 1st = c, 2nd = a’s, 3rd = symmetry axis that bisectsangle between a1 and a2

2-fold axes or lines ⊥ m, match w/ a, b, cH-M symbols: 1st = a, 2nd = b, 3rd = c

2-fold or line ⊥ m = b; prominent = c

no sym restrictions; prominent = c

crystal axes

E. Goeke, Fall 2006

Triclinic• Only two classes:1. Pedial = 1

– No symmetry, all xtal faces uniqueand unrelated to other faces

– Pedion = xtal face unrelated to anyother face by symmetry

2. Pinacoidal = 1– Pinacoid = pairs of faces related to

each other due to an i or m– Microcline, plagioclase, turquoise,

wollastonite

http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm

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Monoclinic• Sphenoidal = 2

– Sphenoids = non-parallel faces related toeach other by a 2-fold rotational axis

• Domatic = m– Domes = non-parallel faces related to

each other by a m-plane• Prismatic = 2/m

– Pinacoid & prism faces– Prisms = 3, 4, 6, 8 or 12 identical faces

that are all parallel to the same line; 4identical faces in this class

– Micas, azurite, chlorite, cpx, epidote,gypsum, malachite, kaolinite, orthoclase,talc http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm E. Goeke, Fall 2006

Orthorhombic• Rhombic-dispenoid = 222

– Dispenoid = 4 triangular faces– Epsomite

• Rhombic-pyramidal = mm2– No center of symmetry, so faces on the top are

not repeated on the bottom of the xtal– Pyramid = collection of 3, 4, 6, 8 or 12 faces that

intersect at one point; 4 identical faces in thisclass

– Hemimorphite• Rhombic-dipyramidal = 2/m2/m2/m

– Dypyramid = two pyramids related by a m or a 2-fold rotation; consist of 6, 8, 12, 16 or 24 faces; 4faces on the top and 4 on the bottom in this class

– Aragonite, barite, cordierite, olivine, sillimanite,topaz, andalusite, anthophyllite, stibnite, sulfur


E. Goeke, Fall 2006

Tetragonal• Tetragonal-pyramidal = 4

– No pyramid faces on bottom due tolack of m planes

– Wulfinite• Tetragonal-disphenoid = 4

– 2 identical faces on the top andbottom offset by 90°

– No m planes• Tetragonal-dipyramidal = 4/m

– 4 identical faces top & bottom dueto m plane

– Scheelite and scapolitehttp://www.tulane.edu/~sanelson/eens211/32crystalclass.htm E. Goeke, Fall 2006

• Tetragonal-trapezohedral = 422– No m planes

• Ditetragonal-pyramidal = 4mm– 8 face pyramid on top

• Tetragonal-scalenohedral = 42m– Chalcopyrite and stannite

• Ditetragonal-dipyramidal =4/m2/m2/m– Most symmetry in tetragonal

system– 8 faces on top & bottom– Zircon, vesuvianite, anatase,

cassierite, apophyllite


E. Goeke, Fall 2006

a

a

a

a

a

a

a

a

bbb

γ

γ

γ

Lattice nodePlane Lattice

• Plane lattice = repeated translations parallel to a & bproduce a repeating pattern of dots extending to infinity inthe ab plane

• Translational symmetry = repetition of a point/unit cell in aspecific distance and angle in space

E. Goeke, Fall 2006

a

-a-b

b

c

-c

β α

γ

a

b

cβ

α

γ

Space Lattice

• Space lattice = repeated translations of a, b, & c in 3D

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E. Goeke, Fall 2006

Crystal Faces

• Law of Haüy = crystal faces will intercept thecrystallographic axes in a simple, rational fashion

• Law of Bravais = crystal faces are more likely to developwhen they intercept large number of lattice nodes

• 4 general considerations about xtal face growth:

E. Goeke, Fall 2006

1. Xtal faces grow along planes defined by points in thelattice--points are either atoms or molecules

http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htm

E. Goeke, Fall 2006

2. Angle between xtal faces is determined by the latticepoint spacing

http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htmE. Goeke, Fall 2006

3. All xtals of the samecomposition will havethe same lattice spacing-> Steno’s Law = anglebetween equivalentfaces on the samemineral will always bethe same

4. Lattice symmetry willdetermine the anglesbetween xtal faces --even in distorted orimperfect xtals


E. Goeke, Fall 2006

Unit Cells• Lengths for crystallographic axes are based on the size of

the unit cell• Unit cell includes all of the required points on the lattice

needed to repeat the lattice in an infinite array• Arbitrary definition of unit cell, but following rules are

good to follow:– Edges of unit cell should match with symmetry of

lattice– Edges of unit cell should also be related by the

symmetry of lattice– The smallest possible cell size that contains all of the

elements should be chosen

E. Goeke, Fall 2006

Which is the best unit cell?


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E. Goeke, Fall 2006

Crystal Face Intercepts• Also called “Weiss Parameters”• Intercepts are always relative and do not

indicate any actual length• Faces can be moved parallel to themselves

without changing the intercept• Three cases:1. Intercepts only one crystallographic axis

(e.g. ∞a, ∞b, 1c)2. Intersects two crystallographic axes (e.g. 1a,

1b, ∞c)3. Intersects all three axes (e.g. 1a, 1b, 1c)• Convention states that you take the largest

face that intersects all 3 axes and assign it1a, 1b, 1c = unit face

http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htmE. Goeke, Fall 2006

Miller Indices• Convenient method to describe the orientation of planes

(e.g. xtal faces, crystallographic planes, cleavage planes)• Three step process:1. Determine the xtal face intercepts2. Invert the intercepts3. Clear fractions• (hkl) is the normal form for Miller indices

– h = a-axis, k = b-axis, l = c-axis– (hkl) indicate the index is for a specific face or

crystallographic plane– [hkl] is used for crystallographic directions– {hkl} is used for xtal forms

E. Goeke, Fall 2006

What are the intercepts?

http://britneyspears.ac/physics/crystals/wcrystals.htm E. Goeke, Fall 2006

Invert the intercepts

1a, 1b, 1c

1/2a, ∞b, ∞c

1a, 1b, ∞c

-1a, ∞b, ∞c

∞a, 1b, ∞c

(001)1/∞, 1/∞, 1/1∞a, ∞b, 1c

Miller IndicesInversionIntercept

E. Goeke, Fall 2006

Miller-Bravais Indices

• Miller indices work well for all thextal system except for the hexagonalsystem

• Use a four number index, instead ofthree (hkil)

• h + k + i = 0

http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htm E. Goeke, Fall 2006

Crystal Forms• Crystal form = set of crystal faces related to one another

via symmetry• Symmetry of xtal will determine the number of related

faces

http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htm

{111} = 8 related faces: (111), (11 1), (1 11), etc.{113} = 8 related faced: (113), (1 13), (11 3), etc.

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E. Goeke, Fall 2006

Forms you should know…• Pedion = one-faced form• Pinacoid = two-faced form related by i• Prism = 3+ faces related by rotation• Pyramids = 3+ faces related by rotation that meet (or could meet)

at a point• Dipyramids = 6+ faces; two pyramids related by a m• Tetrahedron = in 43m class, either {111} or {1 11}; 4 faces• Octahedron = 8-faced form due to 3 four-fold rotation axes + ⊥ m

planes; {111}• Dodecahedron = 12-faced form by cutting corner off cube; {110}• Pyritohedron = 12-faced form with no four-fold axes; 2/m 3 class;

{h0l} or {0kl}; each face has 5 sides• Cube = hexahedron = 6 equal faces; {100}• Rhombohedron = 6 faces related by 3-fold rotoinversion or 3-fold

rotation + ⊥ 2-fold rotationE. Goeke, Fall 2006

Pedion

PinacoidDihedron

Tetrahedron Cube Octahedron

Dodecahedron

Pyritohedron

http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm

E. Goeke, Fall 2006 E. Goeke, Fall 2006

Crystal Habit

• Euhedral = idiomorphic = automorphic = idioblastic =well-defined xtal faces

• Subhedral = hypidiomorphic = hypautomorphic =subidioblastic = irregular xtal form, but with some well-defined faces

• Anhedral = allotriomorphic = xenomorphic = xenoblastic =without well-defined xtal faces

E. Goeke, Fall 2006

euhedral

subhedral

anhedral

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