UNIVERSITY OF SOUTH ALABAMA · rotation, then it is said to have a 2-fold rotation axis (360/180 =...
Transcript of UNIVERSITY OF SOUTH ALABAMA · rotation, then it is said to have a 2-fold rotation axis (360/180 =...
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GY 302: Crystallography & Mineralogy
UNIVERSITY OF SOUTH ALABAMA
Lecture 2: More Symmetry Operations, Bravais Lattices
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Last Time
A long winded introduction
Introduction to crystallography
Symmetry
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Crystal Systems The 6 (or 7) Crystal Systems.
Crystal System Axes Angles between axes Mineral examples
1 Cubic a = b = c = = =90° Halite, Galena, Pyrite
2 Tetragonal a = b ≠ c = = =90° Zircon
3a Hexagonal a1= a2=a3≠ c = =120° , =90° Apatite
3b Trigonal a= a1=b≠ c = =120° , =90° Quartz, Calcite
4 Orthorhombic a ≠ b ≠ c = = =90° Aragonite, Staurolite
5 Monoclinic a ≠ b ≠ c = =90° , ≠ 90° Gypsum, Orthoclase
6 Triclinic a ≠ b ≠ c ≠ ≠ ≠ 90° Plagioclase
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Crystal Systems
In order to fully understand how crystals are “put together”, We need to reconsider symmetry. In the last lab, we examined 3 symmetry operations: 1) Reflection (mirror planes)
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Crystal Systems
In order to fully understand how crystals are “put together”, We need to reconsider symmetry. In the last lab, we examined 3 symmetry operations: 1) Reflection (mirror planes) 2) Rotation (2, 3, 4, 6 fold axes)
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Crystal Systems
There are two other operations that we did not have a chance to discuss at the start of lab 1 (e.g., rotoinversion) 1) Reflection (mirror planes) 2) Rotation (2, 3, 4, 6 fold axes) 3) Inversion
Center of Inversion
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Crystal Systems
1) Reflection (mirror planes) 2) Rotation (2, 3, 4, 6 fold axes) 3) Inversion 4) Translation 5) Screw Rotation 6) Glide Rotation 7) Rotoinversion
Basic symmetry operations
Compound symmetry operations
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Today’s Agenda 1. Rotoinversion
2. Translational Symmetry 3. Bravais Lattices
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Symmetry in Crystals Rotation axes:
http://www.cartage.org.lb
2-fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2). Note that in these examples the axes we are referring to are imaginary lines that extend toward you perpendicular to the page or blackboard. A filled oval shape represents the point where the 2-fold rotation axis intersects the page.
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Symmetry in Crystals Rotation axes:
2-fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2). Note that in these examples the axes we are referring to are imaginary lines that extend toward you perpendicular to the page or blackboard. A filled oval shape represents the point where the 2-fold rotation axis intersects the page.
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Symmetry in Crystals Rotation axes:
http://www.cartage.org.lb
3-fold Rotation Axis - Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (360/120 =3), and they will repeat 3 times in a 360o rotation. A filled triangle is used to symbolize the location of 3-fold rotation axis.
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Symmetry in Crystals Rotation axes:
http://www.cartage.org.lb
6-fold Rotation Axis -If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of rotational symmetry (360/60=6). A filled hexagon is used as the symbol for a 6-fold rotation axis.
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Symmetry in Crystals Rotation axes:
http
://w
ww
.car
tage
.org
.lb
Although objects themselves may appear to have 5-fold, 7-fold, 8-fold, or 12-fold rotation axes, these are not possible in crystals. The reason is that the external shape of a crystal is based on a geometric arrangement of atoms and they must be able to completely fill in space in order to exist in nature (Think unit cells or floor tiles).
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Rotoinversion: A combination of rotation with a center of inversion.
http://www.cartage.org.lb
Symmetry in Crystals
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Symmetry in Crystals look down this axis
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Symmetry in Crystals
2 fold rotation
look down this axis
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Symmetry in Crystals
2 fold rotation
invert
look down this axis
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Rotoinversion: A combination of rotation with a center of inversion.
http://www.cartage.org.lb
2-fold Rotoinversion - The operation of 2-fold rotoinversion involves first rotating the object by 180 degrees then inverting it through an inversion center. This operation is equivalent to having a mirror plane perpendicular to the 2-fold rotoinversion axis.
Symmetry in Crystals
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Rotoinversion: A combination of rotation with a center of inversion.
http://www.cartage.org.lb
3-fold Rotoinversion - This involves rotating the object by 120o (360/3 = 120), and inverting through a center. A cube is good example of an object that possesses 3-fold rotoinversion axes.
Symmetry in Crystals
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Rotoinversion: A combination of rotation with a center of inversion.
http://www.cartage.org.lb
4-fold Rotoinversion - This involves rotation of the object by 90o then inverting through a center. Note that an object possessing a 4- fold rotoinversion axis will have two faces on top and two identical faces upside down on the bottom, if the axis is held in the vertical position.
Symmetry in Crystals
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Rotoinversion: A combination of rotation with a center of inversion.
http://www.cartage.org.lb
6-fold Rotoinversion - involves rotating the object by 60o
and inverting through a center. Note that this operation is identical to having the combination of a 3-fold rotation axis perpendicular to a mirror plane.
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 1 dimensional translations:
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 1 dimensional translations:
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 1 dimensional translations:
a
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 1 dimensional translations:
a
a
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 1 dimensional translations:
a
a
a
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
a
a
b
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
a
a
a b
b
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
a
a
a
a
a
b
b
b
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
a
a
a
a
a
b
b
b
b
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
a
a
a
a
a
b
b
b
b
a
a
a
b
b
b
b
Symmetry in Crystals
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Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations:
a
b
Unit Mesh or Plane Lattice
Symmetry in Crystals
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Symmetry in Crystals
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Symmetry in Crystals
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Symmetry in Crystals
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Consider 3 dimensional translations:
Orthorhombic a ≠ b ≠ c = = =90°
One of 14 possible 3D or Bravais Lattices
Symmetry in Crystals
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Orthorhombic a ≠ b ≠ c = = =90°
Another possible 3D or Bravais Lattices
Symmetry in Crystals
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Symmetry in Crystals
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Bravais Lattices
Bravais Lattices in the Isometric System
Primative (P)
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Bravais Lattices
Bravais Lattices in the Isometric System
Body Centered (I)
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Bravais Lattices
Bravais Lattices in the Isometric System
Face Centered (F)
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The 14 Bravais Lattices
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Bravais Lattices
Bravais Lattices in crystals are more or less synonymous with unit cells as used by chemists… … but unit cells are defined as the smallest repeating unit that forms an infinite crystalline material
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Unit Cells
NaCl (Halite)
-Na+
-Cl-
Sour
ce: w
ww
.chm
.bris
.ac.
uk
Face-centered isometric crystal
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Unit Cells
NaCl (Halite)
-Na+
-Cl-
Sour
ce: w
ww
.chm
.bris
.ac.
uk
Face-centered isometric crystal
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CaF2 (Fluorite)
-Ca2+
-F-
Sour
ce: h
ttp:\\
staf
f.ais
t.go.
jp
Unit Cells
Body-centered isometric crystal
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CaF2 (Fluorite)
-Ca2+
-F-
Sour
ce: h
ttp:\\
staf
f.ais
t.go.
jp
Unit Cells
Body-centered isometric crystal
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CaCO3 (Aragonite)
-Ca2+
-O2-
- C4+
Orthorhombic Sour
ce: h
ttp:\\
staf
f.ais
t.go.
jp
Unit Cells
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CaCO3 (Calcite)
-Ca2+
-O2-
- C4+
Trigonal
Sour
ce: h
ttp:\\
staf
f.ais
t.go.
jp
Unit Cells
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K(Mg,Fe)3AlSi3O10(OH,F)2 (Biotite)
Monoclinic
Sour
ce: h
ttp:\\
staf
f.ais
t.go.
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Unit Cells
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Today’s Homework 1. Get your notes in order
Next Week 1. Miller Indices and Point Groups (3)
2. Labs: Isometric & Hexagonal Wooden Models
Lab Today
1. In class activity: symmetry in models