GY 302: Crystallography & Mineralogy
GY 302: Crystallography & Mineralogy
UNIVERSITY OF SOUTH ALABAMA
Lecture 3: Miller Indices & Point Groups
Lecture 3: Miller Indices & Point Groups
A combination of rotation with a center of inversion.
http://www.cartage.org.lb
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.
Rotoinversion
Translation: Repetition of points by lateral displacement.Consider 2 dimensional translations:
a
b
Unit Mesh orPlane Lattice
Symmetry in Crystals
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.•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 origine.g., 1a, 2a, 3a…
(a, ∞, ∞)1a
2a3a
Miller Indices
(1a, ∞, ∞)
•In the x direction, the first plane terminates at point 1a. It continues indefinitely in the y and z directions
Miller Indices•Likewise, this plane in yellow can be designated
( ∞, 1, ∞) And the plane in green can be designated
( ∞,∞, 1)
( ∞, 1, ∞)
( ∞, ∞, 1)
( 1, ∞, ∞)
Miller IndicesBy convention, Miller Indices are reciprocals of the parameters of each crystal face
( ∞, 1, ∞)
( ∞, ∞, 1)
( 1, ∞, ∞)
Miller IndicesBy convention, Miller Indices are reciprocals of the parameters of each crystal face
( ∞, 1, ∞)
( ∞, ∞, 1)
( 1, ∞, ∞)
Pink Face = 1/1, 1/∞, 1/∞1, 0, 0
Yellow Face = 1/∞, 1/1, 1/∞0, 1, 0
Green Face = 1/∞, 1/∞, 1/1 0, 0, 1
Miller IndicesMiller 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 IndicesThe opposite sides of each face are designated with negative signs
Pink Face = (-1 0 0)
Yellow Face = (0 -1 0)
Green Face = (0 0 -1)
(0 -1 0)
(0 0 -1)
(-1 0 0)
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•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
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•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 fold-rotoinversion axis
The Point Groups•There are 3 possible orthorhombic Point Groups; all only have either 2-fold rotational axes or 2 fold-rotational 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-rotoinversionaxes
Each symmetry operation has a symbol…
m - mirror planes1, 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
•3 2-fold rotational axes (A2)•3 mirror planes (m)•center of inversion
Example 1: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Step 1: Write down a number representing each uniquerotational axis
2 2 2
Example 1: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Step 2: Write an “m” for every unique mirror plane*
2m 2m 2m
* those not produced by other symmetry operations
Example 1: Orthorhombic crystal
Hermann-Mauguin Class Symbols
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
Example 2: Orthorhombic crystal
Hermann-Mauguin Class Symbols
•1 2-fold rotational axes (A2)•2 mirror planes (m)
Example 2: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Step 1: Write down a number representing each uniquerotational axis
2
Example 2: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Step 2: Write an “m” for every unique mirror plane
2 m m
Example 2: Orthorhombic crystal
Hermann-Mauguin Class Symbols
Step 3: Mirror planes perpendicular to rotational axes?
No
2 m m
Example 3: Tetragonal crystal
Hermann-Mauguin Class Symbols
•1 4-fold rotational axes (A4)•4 2-fold rotational axes (A2)•5 mirror planes (m)•center of inversion
Example 3: Tetragonal crystal
Hermann-Mauguin Class Symbols
Step 1: Write down a number representing each unique rotational axis*
4 2 2
* here 2 of the 2-fold rotational axis are generated by 4 fold rotation; they are not unique
Example 3: Tetragonal crystal
Hermann-Mauguin Class Symbols
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
Example 3: Tetragonal crystal
Hermann-Mauguin Class Symbols
Step 3: Mirror planes perpendicular to rotational axes?
Yes
4/m 2/m 2/m
Example 4: Isometric crystal
Hermann-Mauguin Class Symbols
•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
Example 4: Isometric crystal
Hermann-Mauguin Class Symbols
Step 1: Write down a number representing each unique rotational axis*
4 3 2
* in high symmetry crystals, most axes are not unique. Here only 1 of each axes is unique.
Example 4: Isometric crystal
Hermann-Mauguin Class Symbols
* none of the mirror planes is perpendicular to the 3-fold rotoinversion axes
Steps 2/3: Write an “m”for every unique mirror plane. Determine if they are perpendicular to the axes*
4/m 3 2/m