Unformatted text preview: Symmetry
Symmetry Symmetry of minerals derive from their “highly ordered atomic arrangement” Patterns of atoms repeat “infinitely” in all directions Ordering of atoms may form external crystal faces – reflects internal symmetry Minerals usually lack crystal faces Still contain ordered internal arrangement Material without orderly arranged framework of atoms are not minerals Terms
Terms Used to describe crystal faces: Euhedral – well developed crystal faces
Subhedral – crystal faces present, but are not perfect
Anhedral – without faces Crystals
Crystals A homogeneous solid with longrange three dimensional order
Minerals must be crystals – e.g., they are crystalline
Some crystals are not minerals Why not? How would these solids not fit our definition of a mineral? What would be an example of this? Terms
Terms Non crystalline material – no internal atomic symmetry present
Amorphous – these are solids that lack an ordered arrangement of atoms, i.e. not crystalline
Examples: glass – can be natural (e.g. frozen lavas)
mineraloids natural material that are not minerals because they have lost their atomic ordering. Processes that can create Processes that can create mineraloids Metamict – destruction of structure through radiation
Fulgarite – melting of sand through lightning strike
Clinker – melted material from fires in coal bed seams Observing internal ordering
Observing internal ordering Internal ordering of atoms can be seen through symmetry present in minerals
How is symmetry observed? Macroscopically: observable ONLY if euhedral
Microscopically: usually observable with petrographic microscopes
Atomic scale: Always observable with Xray diffraction Granite Photomicrograph
Granite Photomicrograph Quartz
Feldspars Microscopic image Plagioclase
Kspar Biotite
Horneblende
Accessory Mineral XRay diffractometer
XRay diffractometer Observations at atomic scale Measures distance between atoms
Determine three dimensional arrangement of atoms in solids All specimens of every mineral has identical symmetry If symmetry different between two specimens, then they HAVE to be different minerals E.g. calcite and aragonite Remember – easily observed so… An extremely powerful diagnostic tool BUT: Different minerals may have different symmetry What is symmetry?
What is symmetry? Comparison: Correspondence in size, shape and position of parts on opposite sides of a dividing line or median plane or about a center or axis
Action: A rigid motion of a geometric figure that determines a onetoone mapping onto itself, i.e. a symmetry operation Two types of symmetry operations
Two types of symmetry operations
Translational – i.e. through a volume 1. Continues to infinity Repetition (also called “point symmetry”) 2. Around a point, line or plane
Returns to original starting place Preview of Coming Attractions:
Preview of Coming Attractions: 1. Describe translation operations and resulting shapes 1st in 2D – easier
Then in 3D – needed to understand mineral classification scheme 2. We’ll find for 3D symmetry there are only 14 unique translational shapes The 14 shapes define 6 CRYSTAL SYSTEMS
Basis of organization (and classification) of minerals 3. Describe different repetition operations Translation in a plane (2D)
Translation in a plane (2D) Repetition of a single point or pattern The resulting pattern is called a plane lattice Repeated in one and two directions
Each point referred to as a lattice node The lattice extends to infinity in the plane Each plane lattice has a unique shape/pattern called a unit mesh A single lattice node:
e.g. an atom or collection of atoms
Translation in 1D Figure 2.1 Plane lattice: translation in two directions The parallelogram is is the “unit mesh” Nomenclature
Nomenclature Plane lattice axes parallel edges of unit mesh
For unit meshes there are only 2 axes
Axes used to define space Labeled “a” and “b”, or for negative directions –a and –b.
Angle between axes is γ Plane symmetry operations
Plane symmetry operations Only 5 unique plane lattices produced by infinite translation in 2 dimensions
Only 4 unique shapes to the unit mesh One shape has two distinct lattices Primitive (p) – nothing at center of mesh Centered (c) – node at center of mesh Figure 2.2
ules for Plane lattice trans
extends to lation
infinity Unit mesh defines the symmetry (and chemistry if mineral) Square Rules for Plane lattice trans
extends to lation
infinity Unit mesh defines the symmetry (and chemistry if mineral) Hexagon (contains a rhombus) Parallelogram
Primitive Rectangle Center Rectangle Note – these are the same shaped unit mesh Translation in 3D
Translation in 3D Needed to build three dimensional objects: e.g., minerals
Analogous to 2D lattices, but slightly different nomenclature: Arrangement of nodes – Space Lattice (2D = plane lattice)
Volume outlined – Unit Cell (2D = unit mesh)
Edges of unit cell – Crystallographic Axes (2
D same) For unit cells there are three axes Unit Cell
Unit Cell Smallest volume of crystalline structure that contains all:
1.
2. 3. Chemical information
Structural (symmetry and atomic arrangement) information
Edges define the lengths of and angles between crystallographic axes Crystallographic Axes
Crystallographic Axes Axes are named a, b, and c (positive) or 1 2 3 a , a , a if the same length 2 They may be a, b, and –c, or –a , a ,
3 1 a (negative)
Intersect at a point called origin Note that the axes are the same lengths as the edge of the unit cell Example of a Unit cell Translation in third dimension
Unit mesh (parallelogram) a angle b = γ
b angle c = α
a angle c = β
Figure 2.4 The crystallographic axes are DEFINED by the lengths of the edges of the unit cell P. 306 – olivine information
P. 306 – olivine information
Crystallographic information Bravais Lattices and Crystal Bravais Lattices and Crystal Systems If you take the 5 plane lattices and repeat systematically in 3D, they make up 14 unique arrangement of nodes: Bravais lattices 14 Bravais lattices can be combined to 6 unique shapes that outline unit cells Crystal systems 6 Crystal systems
6 Crystal systems Triclinic
Monoclinic
Orthorhombic
Hexagonal
Tetragonal
Isometric Each crystal system has a uniquely shaped unit cell Edge of the unit cell corresponds to “unit” lengths along the 3 edges of the unit cell Defines the lengths and arrangements of the crystallographic axes in the system Each mineral has specific and mostly unvarying “unit” length
Unit lengths have absolute values, typically a few Å (1 Å = 0.1 nanometer = .0001 micron) Angles between the axes are also specific and mostly unvarying for each mineral Why 14 Bravais lattices, but only 6 Why 14 Bravais lattices, but only 6 systems? Systems have “extra” lattice nodes, but retain the unit shape of the system, e.g., Primative (p) – lattice points only at corners of unit cell
Body centered (i) – additional lattice point at center of cell (innenzentrierte)
Face centered (c) – additional lattice points on two opposite sides
Face centered (f) – additional lattice points on every face Triclinic Monoclinic
P P C Orthorhombic
C
P F I Tetragonal
P Hexagonal
P I 14 Bravais lattices
Broken into 6 crystal systems
Unit cells of each crystal system Isometric
P I F
Fig. 210 Arrangement of crystallographic Arrangement of crystallographic axes Triclinic Triclinic – three inclined a ≠ b ≠ c and α ≠ β ≠ γ ≠ 90º
Formed by translating an oblique plane lattice a distance “c” at an angle not equal to α or β There is no formal convention to how minerals correspond to the unit cell axes, generally: c is parallel to prominent elongation
b is down and to right
a is down and to front Triclinic bravais lattice
a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90º Fig. 25
Fig. 25 Monoclinic
Monoclinic One inclined
Translate rectangular plane lattice not perpendicular to other axes, and a distance not equal to a or b a ≠ b ≠ c, α = γ = 90º ≠ β
90º Monoclinic bravais lattice
a ≠ b ≠ c, α = γ = 9 0 º ≠ β Fig. 26a and 27a
Fig. 26a and 27a Orthorhombic
Orthorhombic Translate rectangular plane lattice at right angles to other axes, but a distance not equal to a or b a ≠ b ≠ c, α = β = γ = 90º
90º
No convention about actual lengths of unit cell
No
sides and axes.
sides
Commonly c < a < b, but not always Primitive Face Centered
P C I F
Body Centered
Orthorhombic bravais lattice
a ≠ b ≠ c, α = β = γ = 90º = 90º Face Centered Fig. 26 & 27
Fig. 26 & 27 Hexagonal
Hexagonal Translate hexagonal plane lattice perpendicular to a1 and a2, but a distance that is not equal to a1 and a2. Angles between a axes are 120º
Angle between a and c = 90º Hexagonal bravais lattice
a1 = a2 = a3 ≠ c
angles 120º & 90º Translation is 1/3C up and 2/3a*cos30 back
Forms a rhombohedron, but also can use the hexagon as the unit cell (most common)
Fig. 28
Fig. 28 Tetragonal
Tetragonal Translate a square plane lattice a distance “c” not equal to a1 and a2 and perpendicular to the lattice
1 2 a = a ≠ c
α = β = γ = 90º
90º Tetragonal bravais lattice
a = b ≠ c, α = β = γ = 90º = 90º
Fig. 29
Fig. 29 Isometric
Isometric Translate a square plane lattice a distance equal to “a” and perpendicular to the lattice
1 2 3 a = a = a
α= β= γ Isometric bravais lattice
a = b = c, α = β = γ = 90º = 90º Fig. 29
Fig. 29 Summary – 6 Crystal Summary – 6 Crystal Systems
Triclinic a ≠ b ≠ c α ≠ β ≠ γ Monoclinic a ≠ b ≠ c α = γ = 9 0 º ≠ β
α= β= γ=
90º
Angles are 120º and 90º
α= β= γ=
90º
α= β= γ=
90º Orthorhombic a ≠ b ≠ c
Hexagonal a1 = a2 = a3 ≠ c Tetragonal a = b ≠ c Isometric a = b = c Repetition Symmetry
Repetition Symmetry Crystallographic axes (i.e. unit cell dimensions) give minerals their symmetry
Two types of symmetry: Translational – e.g. generates lattices (we’ve seen this)
Point symmetry (repetition symmetry) – Defines a finite shape called a “form”
Ultimately will get to defining 32 point groups, aka 32 crystal classes Point symmetry defined by specific operations Operations are procedures that generate the symmetry
Limited number (4) of possible operations If you know the point symmetry you can identify which of the 6 crystal systems the mineral crystallizes in
Eliminates roughly 5/6 of all minerals Point Symmetry
Point Symmetry How point symmetry works: Repetition of motif around point, line or plane
Motifs are geometric shape – represents crystal face, group of atoms, single atom etc.
The motif is repeated until returned to original location
An “operation” or “mapping” of motif onto itself Point Symmetry operations
Point Symmetry operations Four types of symmetry operations Reflection
Rotation
Inversion
Rotoinversion Reflection
Reflection Created by a “mirror plane”
Generates mirror image on opposite sides of plane
Characteristics: Creates only one new motif E.g. reflection of new motif across plane ends up at the original location Handedness of motif changes (e.g. left to right hand) Notation = m Mirror image – reflection across a mirror plane {}
Note – handedness changes with this face Fig. 211
Fig. 211 Rotation
Rotation Motif is repeated by rotation around an axis
Five possible rotation axes: One fold = 360° rotation
Two fold = 180° rotation
Three fold = 120° rotation
Four fold = 90° rotation
Six fold = 60° rotation Notation = An where n =1, 2, 3, 4, 6 depending on number of rotations. Fig. 212
Fig. 212 4fold rotation, 90º 2fold rotation, 180º 3fold rotation, 120º Various rotation symmetry Note symbols for rotations 6fold rotation, 60º Inversion
Inversion Symmetry through a point, also “center of symmetry”
A line drawn from a point through the center will hit an identical point equal distance on the opposite side of the center
If a mineral has a center of symmetry, every point on mineral has a point exactly opposite point
Notation = i Center of symmetry No center of symmetry,
e.g. quartz, tourmaline, topaz
All are piezoelectric
Fig. 213
Fig. 213 Rotoinversion
Rotoinversion Combination of rotation and inversion
Rotation may be 1, 2, 3, 4, or 6
Notation Ān, where n = number of rotations
Only Ā4 is unique, others can be duplicated with other operations Ā1= i Ā2 = m Ā3 = A3 + i Ā4 is unique, this symmetry cannot be produced any other way
Ā6 = A3 + m = i = m = A3 + i Unique
=A3 + m Fig. 214
Fig. 214 Multiple Point Symmetry Multiple Point Symmetry Shapes may have more than one type of symmetry
If more than one, the different symmetries must relate to each other Example of multiple symmetry Example of multiple symmetry operations Consider a form with the following symmetry Center of symmetry
Three 2fold rotation axes
Three mirror planes
In shorthand = i, 3A2, 3m What does the form look like?
Where are the symmetry elements?
How do you write the symmetry elements? Orthorhombic system: a ≠ b ≠ c, α = β = γ = 90º 3 two fold rotations
3 mirrors
1 center of symmetry Why not some other system, e.g. isometric or triclinic? Fig. 215
Fig. 215 The total amount of symmetry (i, 3A2, 3m) also is described as: 2/m 2/m 2/m
Describes 3 2fold rotation axes perpendicular to mirrors
This symmetry has to have a center in it.
This type of notation called “Hermann
Mauguin” 32 point groups
32 point groups In general, all the symmetry operations can be combined in only a 32 unique ways Called “Point Groups” First consider number of possible combinations in plane symmetry In two dimension, there are no inversions, only rotations and mirrors
Combinations of rotations and mirrors produce only 10 unique combinations Point groups in 2D:
only 10 possible combinations
symmetry operations only mirror and rotation Fig. 216
Fig. 216 In three dimensions, the rotations and mirrors are expanded because they also have inversions
In this case there are 32 unique symmetry combinations
Called 32 point groups
Each mineral crystallizes with the symmetry of one of the point groups
Called Crystal Classes Each crystal class falls into one of the 6 crystal systems The classes occur in the system depending on their common symmetry elements
E.g. Table 2.2 Triclinic
Monoclinic
Orthorhombic Hexagonal Tetragonal Isometric Table 22
Table 22
• Symmetry of the 32 point groups
• Common symmetry elements within crystal systems
• Orientation of crystallographic axes Two additional symmetry Two additional symmetry operations Glide plane Screw Axis Translation plus a mirror
Translation plus a rotation (2, 3, 4, or 6 fold) Note: 2fold screw axis differs from glide plane Glide symmetry Silica tetrahedron, e.g. pyroxenes
Fig. 2.19 & 2.20
Fig. 2.19 & 2.20 3fold screw axis • May be 2, 3, 4, or 6 – fold
• May be right of left spiral
• Enantiomorphic forms How to use this to your advantage
How to use this to your advantage Identify crystal class Determine if crystal has center of symmetry
Identify all mirror planes
Identify rotation axes and number of rotations
Compile all symmetry P. 306 – olivine information
P. 306 – olivine information
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