ERTH 2001: Symmetry

1
ERTH 2001 Symmetry Putnis Ch. 1 (handout), Nesse
Ch. 2 GIS Lab Module on Crystallography

• Nature of Crystalline Matter
• Symmetry Operations
• Unit Cell
• Bravais Lattice
• other details (crystal systems, Miller indices)
• in GIS Crystallography Module and Lab Notes

2
ERTH 2001 Symmetry
Nature of Crystalline Matter most minerals
are crystals most Earth processes involve
transformations (reactions, strain, etc.)
involving crystalline matter (/-
fluid) crystal - a solid characterised by a
regular (ordered) internal
arrangement of atoms that persists over
distances that are great compared with
distances between individual atoms
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ERTH 2001 Symmetry
Nature of Crystalline Matter crystal - a solid
characterised by a regular (ordered)
internal arrangement of atoms that persists
over distances that are great
compared with distances between
individual atoms crystal structure - assemblage
of atoms arranged in a regular
manner in 3-D, together with their bonding
arrangement (geometry
chemistry) crystal lattice - imaginary
geometrical framework describing the
3-D arrangment of atoms in the crystal
structure lattice points - locations (e.g., atoms
or polyatomic groups) within ordered
2-D or 3-D structure (e.g., crystal) that have
an identical environment when viewed
in the same orientation
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ERTH 2001 Symmetry
Symmetry - a fundamental property of crystalline
matter   - correspondence in size, form, and
arrangement of the parts of a recurring
array - in minerals, the arrangement of atoms,
ions, or radicals within the unit cell defines
the symmetry      elements of symmetry - simple
geometric operations that change the position
and/or orientation of a distinctive and
recurring form, shape, or figure of the array
(motif) but not its size or shape    e.g.,
reflection across a mirror plane
rotation around an axis also referred
to as symmetry operations
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds translation
- movement of a point along a straight line
     reflection - across mirror
planes      rotation - about an axis
number of repetitions in 360 1-fold 2-fold 3-fol
d 4-fold 6-fold axes    inversion - through
the geometric centre of the lattice -
equivalent to reflection rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds translation
- movement of a point along a straight
line generates linear, planar, or 3-D array
?????????    translation of a point or set of
points in 2-D generates a plane lattice (tiling
or "tesselation") ???? ????? ? ? ? ?
? ? ? ? ? ?
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds translation
- movement of a point along a straight
line generates linear, planar, or 3-D
array    translation of a point or set of points
in 2-D generates a plane lattice   translation
of a point or set of points in 3-D generates
a space lattice which can be described in terms
of an appropriate co-ordinate system (3 or 4
axes, not necessarily orthogonal) the 14
possible types of space lattice are referred to
as the Bravais lattices (Lab Manual p.14)
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds
the 14 Bravais lattices (Lab Manual
p.14)
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds reflection
- across mirror planes (m) - generates
mirror-images of points at equivalent
distances from mirror plane ? ?
? glide plane - combines translation
reflection (g)
m
?
m
??
?
?
g
?
?
?
?
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
90 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
180 rotation
90 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
180 rotation
270 rotation
90 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
180 rotation
360 rotation
270 rotation
90 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
180 rotation
360 rotation
270 rotation
90 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
start
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
90 rotation
start
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
90 rotation
start
180 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
90 rotation
270 rotation
start
180 rotation
360 rotation
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ERTH 2001 Symmetry
Symmetry Operations - 4 basic kinds

rotation - about an axis number of
repetitions in 360 1-fold 2-fold
3-fold 4-fold 6-fold rotation axes
corresponding symbols
X
90 rotation
270 rotation
start
180 rotation
360 rotation
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ERTH 2001 Symmetry
Unit Symmetry Operations - 4 basic
kinds inversion - through the geometric
centre of the lattice - equivalent to
reflection rotation - requires 3-D
lattice - equivalent to centre of
symmetry notation bar over number corresponding
to rotation axis e.g., 1 2 3 4 6
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ERTH 2001 Symmetry
Unit Cell "repeat unit" of a given crystal
lattice small volumes of identical size,
orientation, and atomic constitution that, taken
together, fill all the space within the
crystal - a parallel-sided box with lattice
points at each corner - can be translated and
repeated many times to produce entire lattice
without leaving any gaps - has the symmetry of
the entire lattice (highest possible) - normally
chosen to be the simplest possible 3-D
arrangement from which the entire crystal
structure can be generated
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ERTH 2001 Symmetry
Unit Cell "repeat unit" of a given crystal
lattice small volumes of identical size,
orientation, and atomic constitution that, taken
together, fill all the space within the
crystal In minerals, the unit cell consists
of - an ordered arrangement of anions and
cations that.... ....forms the basic repeat unit
of the crystal lattice ....has the chemical
composition of the mineral as a whole ....is
chemically balanced (total ve charges total
-ve charges)
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ERTH 2001 Symmetry
Unit Cell "repeat unit" of a given crystal
lattice consists of..... - a parallel-sided
box with lattice points at each corner - can be
translated and repeated many times to produce
entire lattice without leaving any gaps - has
the symmetry of the entire lattice (highest
possible) - normally chosen to be the simplest
possible 3-D arrangement from which the entire
crystal structure can be generated In minerals,
the unit cell consists of - an ordered
arrangement of anions and cations
that.... ....forms the basic repeat unit of the
crystal lattice ....has the chemical composition
of the mineral as a whole ....is chemically
balanced (total ve charges total -ve charges)
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ERTH 2001 Symmetry
Unit Cell the 14 Bravais lattices
(Lab Manual p.14)
the basic "body plans" of mineral unit cells
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ERTH 2001 Symmetry
Unit Cell examples
which of the following are acceptable unit cells?
2
4
1
3
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ERTH 2001 Symmetry
Unit Cell examples
which of the following are acceptable unit cells?
5
3
1
6
4
2
chemical formula?
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ERTH 2001 Symmetry
Unit Cell examples
which of the following are acceptable unit cells?
1
4
3
2
5
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ERTH 2001 Symmetry
Unit Cell examples
which of the following are acceptable unit cells?
1
4
3
2
5
note 1 hexagonal cell ? 3 rhombic cells
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ERTH 2001 Symmetry
Unit Cell the 14 Bravais lattices
(Lab Manual p.14)
the basic "body plans" of mineral unit cells