1 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 origin of 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 (A 1 or 1) – 2-fold (A 2 or or 2) – 3-fold (A 3 or or 3) – 4-fold (A 4 or or 4) – 6-fold (A 6 or or 6) • 5-fold, 7-fold, 8-fold don’t appear in nature http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm E. Goeke, Fall 2006
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What is crystallography? Crystallography · Tetragonal • Tetragonal-pyramidal = 4 –No pyramid faces on bottom due to lack of m planes –Wulfinite • Tetragonal-disphenoid =
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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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E. Goeke, Fall 2006
Inversion
• A line drawn through the origin will find identical featureson the other side
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
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
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
{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