GY 302: Crystallography & Mineralogy GY 302: Crystallography & Mineralogy

GY 302: Crystallography & Mineralogy

GY 302: Crystallography & Mineralogy

UNIVERSITY OF SOUTH ALABAMA

Lecture 3: Miller Indices & Point Groups

Lecture 3: Miller Indices & Point Groups

Last Time

1. Rotoinversion2. Translational Symmetry

3. Bravais Lattices

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




The 14 BravaisLattices

Unit Cells

NaCl(Halite)

-Na+

-Cl-

Sour

ce: w

ww

.chm

.bris

.ac.

uk

Face-centered isometric crystal

Today’s Agenda

1. Miller Indices2. Point Groups (32 of them)

3. Hermann-Mauguin Class Symbols

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 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•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 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•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


Miller Indices

Hexagonal crystal forms related to Miller Indices


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


Symmetry Parameters: A2 = 2 fold rotational axes; A3 = 3 fold rotational axes; A4 = 4 fold rotational axes; m = mirror planes


Name of the crystal form


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:

Hermann-Mauguin Class Symbols

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.


Example 1: Orthorhombic crystal




•3 2-fold rotational axes (A2)•3 mirror planes (m)•center of inversion



Step 1: Write down a number representing each uniquerotational axis

2 2 2



Step 2: Write an “m” for every unique mirror plane*

2m 2m 2m

* those not produced by other symmetry operations



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





•1 2-fold rotational axes (A2)•2 mirror planes (m)



Step 1: Write down a number representing each uniquerotational axis

2



Step 2: Write an “m” for every unique mirror plane

2 m m



Step 3: Mirror planes perpendicular to rotational axes?

No

2 m m

Example 3: Tetragonal crystal




•1 4-fold rotational axes (A4)•4 2-fold rotational axes (A2)•5 mirror planes (m)•center of inversion



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



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



Step 3: Mirror planes perpendicular to rotational axes?

Yes

4/m 2/m 2/m

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



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.



* 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

Thursday’s Lecture1.1. StereoprojectionsStereoprojections (another assignment)(another assignment)

2.2. Point Group ProjectionsPoint Group Projections

Tuesday’s Lab

1.1. Isometric/Hexagonal modelsIsometric/Hexagonal models

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