Lecture 12 Crystallography

Internal Order and 2-D Symmetry
Plane Lattices
Planar Point Groups
Plane Groups
Internal Order and Symmetry
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

Repeated and symmetrical arrangement
(ordering) of atoms and ionic complexes in
minerals creates a 3-dimensional lattice array
Arrays are generated by translation of a unit cell
– smallest unit of lattice points that define the
basic ordering
Spacing of lattice points (atoms) are typically
measured in Angstroms (= 10-10 m) About the
scale of atomic and ionic radii
Two-Dimensional Plane Lattice
Generating an 2D Lattice Array (Plane Lattice)
involves translation of a motif in two
directions; possible directions not unique
Translation in two directions: x and y axes
Angle between x and y axes is called g gamma
Translation distance: a along x and b along y
Replacing motifs with points (or nodes) creates
a plane lattice
Unit Cell defined by a choice
of lengths and directions.
Symmetry
Translations (Lattices)
A property at the atomic level, not of crystal
shapes
Symmetric translations involve repeat
distances
The origin is arbitrary
1-D translations = a row
Symmetry
Translations (Lattices)
A property at the atomic level, not of crystal shapes
Symmetric translations involve repeat distances
The origin is arbitrary
1-D translations = a row
a
a is the repeat vector
Symmetry
Translations (Lattices)
2-D translations = a net
a
b
Symmetry
Translations (Lattices)
2-D translations = a net
a
b
A 2-D Unit Cell
Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern
Symmetry
Translations (Lattices)
2-D translations = a net
b
a
Pick any point
Every point that is exactly n repeats from that point is an equipoint to the original
There are 5 Types of Plane Lattices
Memorize these names and rules
Preferred
Symmetry Elements of Planar Motifs: Planar Point Groups
A point group is a group of geometric
symmetries that keep at least one point fixed.
Point groups have labels that
are similar to Hermann
Mauguin symbols.
For example: 2mm shown has
the a axis with a two fold
rotational axis, and b and c
have mirrors
10 Possible symmetry
combinations; called Planar
Point Groups
Limitations of rotational
symmetries: (1,2,3,4, & 6)
dark lines added “found mirrors”
Translations
The lattice and point group symmetries interrelate,
because both are properties of the overall
symmetry pattern
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Translations
The lattice and point group symmetry interrelate,
because both are properties of the overall
symmetry pattern
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Good unit cell choice.
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Choose:
9 Smallest9
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Most orthogonal
Most in line with symmetry
2 Nodes per Lattice Vector
Most Primitive (non-centered)
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6
9
Defining a 2-D Unit Cell
Rules that help us
Choose the:
Smallest
Most orthogonal
Most in line with symmetry
Use 2 Nodes per Lattice Vector
Pick the Most Primitive (noncentered)
total 17 point groups
Translations
There is a new 2-D
symmetry operation
when we consider
translations
The Glide Line, g:
A combined reflection
and translation
repeat
Step 2: translate
Step 1: reflect
(a temporary position)
There are 5 unique 2-D plane lattices.
Oblique Net
Diamond Net
Rectangular P Net
a  b
g 90o
a = b
g  90o, 120o, 60o
a b
g = 90o
Rectangular
C Net
a
a
Square Net
a1 = a2
g = 60o
a1 = a2
g = 90o
a b
g = 90o
b
b
Hexagonal Net
a2
b
g
g
a
g
a
g
g
a2
p2
p2mm
a1
a1
p2mm
p6mm
There are also 17 2-D Plane Groups that combine translations
with compatible symmetry operations. The bottom row are
examples of Plane Groups that correspond to each lattice type
Note: p refers to a primitive cell, as opposed to c, a 2-end
(opposite ends) centered cell. More on this in 3-D
p4mm
17 Plane Groups
10 H-M Point Groups and 5 Lattices
combine to form 17 Plane Groups.
Lecture 13 3-D Crystallography

3-D Internal Order & Symmetry
Space (Bravais) Lattices
Space Groups

So far we examined the five 2-D plane lattices and
combined them with the 10 planar point groups to
generate the 17 2-D plane (space) groups. Next we
study the 14 Bravais 3-D lattices and combine them
with the 32 3-D point groups to generate 230 3-D
space groups.
3-D Translations and Lattices

Different ways to combine 3 axes
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Translations compatible with 32 3-D point groups
(~ crystal classes)
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32 Point Groups fall into 6 systems
3-D Translations and
Lattices
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Different ways to combine 3 axes
Translations compatible with 32 3-D point
groups (~ crystal classes)
32 Point Groups fall into 6 Crystal Systems
+c
3-D Lattice Types
Name
axes
angles
Triclinic
abc
g  90o
Monoclinic
abc
g = 90 90
Orthorhombic
abc
g = 90o
a1 = a 2  c
g = 90
a1 = a2 = a3  c
 = 90 g120
a1 = a 2 = a 3
g 90o
a1 = a 2 = a 3
g = 90o
Tetragonal
Hexagonal
Hexagonal (4 axes)
Rhombohedral
Isometric
o
o
+a
g
o
o

o

+b
Axial convention:
“right-hand rule”
Unit Cell Types
in 14 Bravais
Lattices
P – Primitive; nodes at corners
only
C – Side-centered; nodes at
corners and in center of one set of
faces
F – Face-centered; nodes at
corners and in center of all faces
I – Body-centered; nodes at
corners and in center of cell
On combining 7 Crystal Classes with 4 possible
unit cell types we get 14 Bravais Lattices
c
c
c
b
a
b
b
P
P
a I
Monoclinic
g 90o 
a b c
a
Triclinic
g
a b c
c
a
b
P
C
F
Orthorhombic
 g 90o a b c
I
=C
c
c
a2
a1
a2
P
a1
I
Tetragonal
 g 90o a1 = a2 c
P or C
R
Hexagonal
Rhombohedral
90og 0o g90o
a1 = a2 = a3
a1a2c
a3
a2
a1
P
F
I (body-centered)
Isometric
 g 90o a1 = a2 = a3
Crystal Axes Conventions
Triclinic:
No symmetry constraints.
No reason to choose C (white) when can choose simpler P (blue)
Do so by convention, so that all mineralogists do the same
Crystal Axes Conventions
+c
+a

g

+b
Axial convention:
“right-hand rule”
System Conventions
System Conventions
System Conventions
System Conventions
3-D Space Groups
As in the 17 2-D Plane Groups, the 3-D point group
symmetries can be combined with translations to create the
230 3-D Space Groups
Also as in 2-D there are some new symmetry elements that
combine translation with other operations
Glides: Reflection + translation
Screw Axes: Rotation + translation
A point group is a group of geometric symmetries that keep at least one point fixed.
A space group is some combination of the translational symmetry of a unit cell including
lattice centering, the point group symmetry operations of reflection, rotation and
rotoinversion, and the screw axis and glide plane symmetry operations. The combination
of all these symmetry operations results in a total of 230 unique space groups describing
all possible crystal symmetries.
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Hexagonal
Isometric
230 Space
Groups
Notation indicates lattice type
(P,I,F,C) and HermannMauguin notation for basic
symmetry operations (rotation
and mirrors)
Screw Axis notation as
previously noted
Glide Plane notation indicates
the direction of glide – a, b, c,
n (diagonal) or d (diamond)