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
Crystallography
Chapter 2
E. Goeke, Fall 2006
E. Goeke, Fall 2006
Reflection
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
E. Goeke, Fall 2006
E. Goeke, Fall 2006
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
1
Inversion
• A line drawn through the origin will find identical features
on the other side
• Indicated by the letter i
http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
E. Goeke, Fall 2006
Example
Further symmetry work
• Take each number and determine the 2D symmetry of it
l
2
3
4
5
6
7
E. Goeke, Fall 2006
8
9
• ID the inversion points, rotations, and mirror planes of the
following alphabet in 2D (similar to the number exercise)
0
ABCDEFGH
IJKLM
NOPQRSTUVWXYZ
E. Goeke, Fall 2006
Rotoinversion
E. Goeke, Fall 2006
Hermann-Mauguin & Point Groups
• 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 = A 3 + i
– 4-fold rotoinversion = A4 = 4
– 6-fold rotoinversion = A6 = 6 = A 3 + m
• The presence of some symmetry elements requires the
presence of others--therefore we only notate the explicit
elements 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 point
groups (Table 2.2 in your textbook)
E. Goeke, Fall 2006
E. Goeke, Fall 2006
2
How many symmetry elements?
crystal system
common symmetry elements
triclinic
one-fold rotation with or without i
monoclinic
two-fold rotation and/or m
orthorhombic
3 two-fold rotation axes and/or 3 m
hexagonal
1 three-fold or six-fold axis
tetragonal
1 four-fold rotation or rotoinversion
axis
cubic
4 three-fold axes
E. Goeke, Fall 2006
Crystallographic axes
• Crystal axes = frame of reference to describe the crystal
structure; arbitrarily determined
• Origin = intersection of the crystal axes
• Length of axes is proportional to lattice spacing
E. Goeke, Fall 2006
crystal system
axis lengths
angles between
axes
common symmetry
elements
triclinic
a≠b≠c
α ≠ β ≠ γ ≠ 90°
1-fold rotation w/ or
w/out i
monoclinic
a≠b≠c
α = γ = 90°, β >
90°
2-fold rotation and/or 1
m
orthorhombic
a≠b≠c
α = β = γ = 90°
3 2-fold rotation axes
and/or 3 m
hexagonal
a1 = a 2 = a3, a 60° btw a’s, β =
≠c
90°
1 3-fold or 6-fold axis
tetragonal
a=b≠c
α = β = γ = 90°
1 4-fold rotation or
rotoinversion axis
cubic
a=b=c
α = β = γ = 90°
4 3-fold axes
E. Goeke, Fall 2006
E. Goeke, Fall 2006
Triclinic
crystal system
crystal axes
triclinic
no sym restrictions; prominent = c
monoclinic
2-fold or line ⊥ m = b; prominent = c
orthorhombic
2-fold axes or lines ⊥ m, match w/ a, b, c
H-M symbols: 1st = a, 2nd = b, 3rd = c
hexagonal
3-fold or 6-fold = c
H-M symbols: 1st = c, 2nd = a’s, 3rd = symmetry axis that bisects
angle between a 1 and a 2
tetragonal
4-fold = c; a & b = 2-fold or lines ⊥ to m
H-M symbols: 1st = c, 2nd = a & b, 3rd = symmetry axis that
bisects the angle btw a & b
cubic
3 2-fold or 4-fold = a, b, c
H-M symbols: 1st = a, b & c, 2nd = long diagonals through the
unit cell, 3rd = edge-to-edge diagonals
E. Goeke, Fall 2006
• Only two classes:
1. Pedial = 1
– No symmetry, all xtal faces unique
and unrelated to other faces
– Pedion = xtal face unrelated to any
other 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
E. Goeke, Fall 2006
3
Orthorhombic
Monoclinic
• Sphenoidal = 2
– Sphenoids = non-parallel faces related to
each 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; 4
identical 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
• 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 this
class
– Hemimorphite
• Rhombic-dipyramidal = 2/m2/m2/m
– Dypyramid = two pyramids related by a m or a 2fold rotation; consist of 6, 8, 12, 16 or 24 faces; 4
faces 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
http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
• Tetragonal-trapezohedral = 422
– No m planes
• Ditetragonal-pyramidal = 4mm
– 8 face pyramid on top
• Tetragonal-scalenohedral = 42m
Tetragonal
• Tetragonal-pyramidal = 4
– No pyramid faces on bottom due to
lack of m planes
– Wulfinite
• Tetragonal-disphenoid = 4
– Chalcopyrite and stannite
• Ditetragonal-dipyramidal =
4/m2/m2/m
– Most symmetry in tetragonal
system
– 8 faces on top & bottom
– Zircon, vesuvianite, anatase,
cassierite, apophyllite
– 2 identical faces on the top and
bottom offset by 90°
– No m planes
• Tetragonal-dipyramidal = 4/m
– 4 identical faces top & bottom due
to m plane
– Scheelite and scapolite
http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
E. Goeke, Fall 2006
Lattice node
Plane Lattice
Space Lattice
a
a
a
a
E. Goeke, Fall 2006
c
a
a
γ
a
γ
a
γ
-b
-a
α
β
b
b
b
• Plane lattice = repeated translations parallel to a & b
produce a repeating pattern of dots extending to infinity in
the ab plane
• Translational symmetry = repetition of a point/unit cell in a
specific distance and angle in space
E. Goeke, Fall 2006
a
c
γ
a
γ
β
α
b
b
-c
• Space lattice = repeated translations of a, b, & c in 3D
E. Goeke, Fall 2006
4
1. Xtal faces grow along planes defined by points in the
lattice--points are either atoms or molecules
Crystal Faces
•
•
•
Law of Haüy = crystal faces will intercept the
crystallographic axes in a simple, rational fashion
Law of Bravais = crystal faces are more likely to develop
when they intercept large number of lattice nodes
4 general considerations about xtal face growth:
E. Goeke, Fall 2006
2. Angle between xtal faces is determined by the lattice
point spacing
http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htm
E. Goeke, Fall 2006
http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htm
E. Goeke, Fall 2006
3. All xtals of the same
composition will have
the same lattice spacing
-> Steno’s Law = angle
between equivalent
faces on the same
mineral will always be
the same
4. Lattice symmetry will
determine the angles
between xtal faces -even in distorted or
imperfect xtals
http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htm
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
Which is the best unit cell?
http://www.tulane.edu/~sanelson/eens211/crystalmorphology&symmetry.htm
E. Goeke, Fall 2006
E. Goeke, Fall 2006
5
Crystal Face Intercepts
Miller Indices
•
•
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 it
1a, 1b, 1c = unit face
•
Convenient method to describe the orientation of planes
(e.g. xtal faces, crystallographic planes, cleavage planes)
• Three step process:
1. Determine the xtal face intercepts
2. Invert the intercepts
3. 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
E. Goeke, Fall 2006
http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htm
What are the intercepts?
Invert the intercepts
Intercept
Inversion
Miller Indices
∞a, ∞b, 1c
1/∞, 1/∞, 1/1
(001)
∞a, 1b, ∞c
-1a, ∞b, ∞c
1a, 1b, ∞c
1/2a, ∞b, ∞c
1a, 1b, 1c
http://britneyspears.ac/physics/crystals/wcrystals.htm
E. Goeke, Fall 2006
Miller-Bravais Indices
Crystal Forms
• Crystal form = set of crystal faces related to one another
via symmetry
• Symmetry of xtal will determine the number of related
faces
• Miller indices work well for all the
xtal system except for the hexagonal
system
• Use a four number index, instead of
three (hkil)
• h+k+i=0
http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htm
E. Goeke, Fall 2006
{111} = 8 related faces: (111), (111), (111), etc.
{113} = 8 related faced: (113), (113), (113), etc.
E. Goeke, Fall 2006
http://www.tulane.edu/~sanelson/eens211/axial_ratios_paramaters_miller_indices.htm
E. Goeke, Fall 2006
6
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 rotation
E. Goeke, Fall 2006
Pedion
Dihedron
Pinacoid
Tetrahedron
Cube
Octahedron
Pyritohedron
Dodecahedron
http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm
E. Goeke, Fall 2006
Crystal Habit
• Euhedral = idiomorphic = automorphic = idioblastic =
well-defined xtal faces
• Subhedral = hypidiomorphic = hypautomorphic =
subidioblastic = irregular xtal form, but with some welldefined faces
• Anhedral = allotriomorphic = xenomorphic = xenoblastic =
without well-defined xtal faces
E. Goeke, Fall 2006
E. Goeke, Fall 2006
subhedral
euhedral
anhedral
E. Goeke, Fall 2006
7