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UNIVERSITY OF SOUTH ALABAMA
GY 302: Crystallography &
Mineralogy
Lecture 3:
Miller Indices & Point Groups
Last Time
Rotoinversion
Translational Symmetry
3. Bravais Lattices
1.
2.
Rotoinversion
A combination of rotation with a center of inversion.
e.g., 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.
http://www.cartage.org.lb
Symmetry in Crystals
Translation: Repetition of points by lateral displacement.
Consider 2 dimensional translations:
b
a
Unit Mesh or
Plane Lattice
Symmetry in Crystals
Symmetry in Crystals
Symmetry in Crystals
The 14
Bravais
Lattices
Unit Cells
NaCl
Source: www.chm.bris.ac.uk
(Halite)
+
-Na
-Cl
Face-centered isometric crystal
Today’s Agenda
Miller Indices
2.
Point Groups (32 of them)
Hermann-Mauguin Class Symbols
1.
3.
Miller Indices
•
Crystal facies can be identified using a set of coordinates.
Miller Indices
•
•
Crystal facies can be identified using a set of coordinates.
The most widely used scheme is that by Miller (Miller
Indices)
Miller Indices
•
•
Crystal facies can be identified using a set of coordinates.
The most widely used scheme is that by Miller (Miller
Indices)
Miller Indices
•Consider
the
plane in pink
(a, ∞, ∞)
Miller Indices
•Consider
the
plane in pink.
•It’s actually one
of an infinite
number of parallel
planes each a
consistent distance
from the origin
(a, ∞, ∞)
Miller Indices
•Consider
the
plane in pink.
•It’s actually one
of an infinite
number of parallel
planes each a
consistent distance
from the origin
e.g., 1a, 2a, 3a…
1a
(a, ∞, ∞)
2a
3a
Miller Indices
•In
the x direction,
the first plane
terminates at point
1a. It continues
indefinitely in the
y and z directions
(1a, ∞, ∞)
Miller Indices
•This
plane can be
designated
(1a, ∞, ∞)
or better yet
(1, ∞, ∞)
(1a, ∞, ∞)
Miller Indices
•Likewise,
this
plane in yellow
can be designated
( ∞, 1, ∞)
And the plane in
green can be
designated
( ∞,∞, 1)
( ∞, ∞, 1)
( ∞, 1, ∞)
( 1, ∞, ∞)
Miller Indices
By convention, Miller Indices are reciprocals of the
parameters of each crystal face
( ∞, ∞, 1)
( ∞, 1, ∞)
( 1, ∞, ∞)
Miller Indices
By convention, Miller Indices are reciprocals of the
parameters of each crystal face
Pink Face = 1/1, 1/∞, 1/∞
1, 0, 0
Yellow Face = 1/∞, 1/1, 1/∞
0, 1, 0
( ∞, ∞, 1)
( ∞, 1, ∞)
( 1, ∞, ∞)
Green Face = 1/∞, 1/∞, 1/1
0, 0, 1
Miller Indices
Miller Indices are placed in parentheses with no
commas and no fractions*
Pink Face = (1 0 0)
Yellow Face = (0 1 0)
Green Face = (0 0 1)
* e.g., if you got (1 ¾ ½), you would convert this to (4 3 2)
Miller Indices
The opposite sides of each face are designated with
negative signs
(-1 0 0)
Pink Face = (-1 0 0)
Yellow Face = (0 -1 0)
Green Face = (0 0 -1)
(0 -1 0)
(0 0 -1)
Miller Indices
•This
time, the
plane of interest
cuts two of the
crystallographic
axes.
•The
Miller Index?
Miller Indices
•This
time, the
plane of interest
cuts two of the
crystallographic
axes.
•The
Miller Index?
(1 1 0)
Miller Indices
•This
plane cuts all
three crystallographic
axes.
•The
Miller Index?
Miller Indices
•This
plane cuts all
three crystallographic
axes.
•The
Miller Index?
(1 1 1)
Miller Indices
•Tricky;
this plane
cuts two of the
crystallographic
axes, but not
equidimensionally
•
Miller Indices
•Tricky;
this plane
cuts two of the
crystallographic
axes, but not
equidimensionally
•The
coordinates
of the plane are:
(1/2, 1, 0)
Miller Indices
•Tricky;
this plane
cuts two of the
crystallographic
axes, but not
equidimensionally
•The
coordinates
of the plane are:
(1/2, 1, 0)
Multiple by 2 to
get Miller Indices = (1 2 0)
Miller Indices
Miller Indices
Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p
Miller Indices
Isometric crystal forms
related to Miller Indices
Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p
Miller Indices
Hexagonal crystal forms
related to Miller Indices
Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p
The Point Groups
The Point Groups
•There
are 32 possible combinations of symmetry
operations (the point groups or crystal classes)
The Point Groups
•There
are 32 possible combinations of symmetry
operations (the point groups or crystal classes)
•Each point group will have crystal faces that define the
symmetry of the class (the crystal forms)
The Point Groups
•There
are 32 possible combinations of symmetry
operations (the point groups or crystal classes)
•Each point group will have crystal faces that define the
symmetry of the class (the crystal forms)
•The point groups are best appreciated through the use
of stereo net projections (Thursdays Lecture… Oh Boy!)
The Point Groups
•There
are 5 possible isometric Point Groups; all either
have 4 3-fold rotational axes or 4 3-fold-rotoinversion
axes
Hermann-Mauguin class symbol; more on this shortly
The Point Groups
•There
are 5 possible isometric Point Groups; all either
have 4 3-fold rotational axes or 4 3-fold-rotoinversion
axes
Symmetry Parameters: A2 = 2 fold rotational axes; A3 = 3 fold rotational axes;
A4 = 4 fold rotational axes; m = mirror planes
The Point Groups
•There
are 5 possible isometric Point Groups; all either
have 4 3-fold rotational axes or 4 3-fold-rotoinversion
axes
Name of the crystal form
The Point Groups
•There
are 5 possible isometric Point Groups; all either
have 4 3-fold rotational axes or 4 3-fold-rotoinversion
axes
The Point Groups
•There
are 12 possible hexagonal and trigonal Point
Groups; the former has at least one 6-fold rotational
axis, the later at least one 3-fold rotational axis
The Point Groups
•There
are 7 possible tetragonal Point Groups; all either
have a single 4-fold rotational axis or a 4 foldrotoinversion axis
The Point Groups
•There
are 3 possible orthorhombic Point Groups; all
only have either 2-fold rotational axes or 2 foldrotational axes and mirror planes
The Point Groups
•There
are also 3 possible monoclinic Point Groups; all
only have a single 2-fold rotational axis or a single
mirror plane
The Point Groups
•Lastly
we have the 2 triclinic Point Groups. They only
contain 1-fold rotational axes or 1 fold-rotoinversion
axes
Hermann-Mauguin Class Symbols
Each symmetry operation has a symbol:
Hermann-Mauguin Class Symbols
Each symmetry operation has a symbol…
m
- mirror planes
1, 2, 3, 4, 6 - rotational axes (1-fold, 2-fold, 3-fold….etc.)
1, 2, 3, 4, 6 - rotoinversion axes (1-fold, 2-fold, ...etc.)
i
- inversion
… which are used to classify and name the Point Groups.
Hermann-Mauguin Class Symbols
Example 1: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Example 1: Orthorhombic crystal
•3
2-fold rotational axes (A2)
•3 mirror planes (m)
•center of inversion
Hermann-Mauguin Class Symbols
Example 1: Orthorhombic crystal
Step 1: Write down a number
representing each unique
rotational axis
222
Hermann-Mauguin Class Symbols
Example 1: Orthorhombic crystal
Step 2: Write an “m” for every
unique mirror plane*
2m 2m 2m
* those not produced by other symmetry operations
Hermann-Mauguin Class Symbols
Example 1: Orthorhombic crystal
Step 3: Mirror planes
perpendicular to rotational
axes are put in a denominator
position relative to the
rotational axes
2/m 2/m 2/m
Hermann-Mauguin Class Symbols
Example 2: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Example 2: Orthorhombic crystal
•1
2-fold rotational axes (A2)
•2 mirror planes (m)
Hermann-Mauguin Class Symbols
Example 2: Orthorhombic crystal
Step 1: Write down a number
representing each unique
rotational axis
2
Hermann-Mauguin Class Symbols
Example 2: Orthorhombic crystal
Step 2: Write an “m” for every
unique mirror plane
2mm
Hermann-Mauguin Class Symbols
Example 2: Orthorhombic crystal
Step 3: Mirror planes
perpendicular to rotational
axes?
No
2mm
Hermann-Mauguin Class Symbols
Example 3: Tetragonal crystal
Hermann-Mauguin Class Symbols
Example 3: Tetragonal crystal
•1
4-fold rotational axes (A4)
•4 2-fold rotational axes (A2)
•5 mirror planes (m)
•center of inversion
Hermann-Mauguin Class Symbols
Example 3: Tetragonal crystal
Step 1: Write down a
number representing each
unique rotational axis*
422
* here 2 of the 2-fold rotational axis are generated by 4 fold rotation; they are not unique
Hermann-Mauguin Class Symbols
Example 3: Tetragonal crystal
Step 2: Write an “m” for
every unique mirror
plane*
4 m 2 m 2m
* here 2 of the 5 mirror planes are not unique. They are generated by 4 fold rotation
Hermann-Mauguin Class Symbols
Example 3: Tetragonal crystal
Step 3: Mirror planes
perpendicular to
rotational axes?
Yes
4/m 2/m 2/m
Hermann-Mauguin Class Symbols
Example 4: Isometric crystal
Hermann-Mauguin Class Symbols
Example 4: Isometric crystal
•3
4-fold rotational axes (A4)
•4 3-fold rotoinversion axes (A3)
•6 2-fold rotational axes (A2)
•9 mirror planes (m)
•center of inversion
Hermann-Mauguin Class Symbols
Example 4: Isometric crystal
Step 1: Write down a
number representing each
unique rotational axis*
432
* in high symmetry crystals, most axes are not unique. Here only 1 of each axes is unique.
Hermann-Mauguin Class Symbols
Example 4: Isometric crystal
Steps 2/3: Write an “m”
for every unique mirror
plane. Determine if they
are perpendicular to the
axes*
4/m 3 2/m
* none of the mirror planes is perpendicular to the 3-fold rotoinversion axes
Thursday’s Lecture
1.
Stereoprojections (another assignment)
2. Point Group Projections
Tuesday’s Lab
1.
Isometric/Hexagonal models