04 - crystallogaphy iii miller indices-faces-forms-edited

Determine Point Symmetry

1. Assign crystallographic axes

2. Determine crystal system

3. Look for symmetry elements

4. Assign Hermann-Mauguin symbol by finding symmetry elements in the standard H-M directions.

3D H-M Notation:

System First Second Third

Triclinic (only 1 and 1 possible)

Monoclinic b-axis [010]

Orthorhombic a-axis [100] b-axis [010] c-axis [001]

Tetragonal c-axis [001] a-axis [100] a1a2 [110]

Trigonal c-axis [001] a-axis [100]

Hexagonal " " a1a2 [110]

Cubic c-axis [001] abc [111] ab [110]

e.g.: mm2; 2/m 2/m 2/m; 432

Worked example block:

m

Crystal system: Monoclinic

Crystal Class: 2/m

Triclinic System :

Lecture 3

Crystal Systems

Monoclinic System :

Lecture 3

Crystal Systems

Orthorhombic System :

Lecture 3

Crystal Systems

Tetragonal System :

Crystal Systems

Hexagonal System :

Lecture 3

Crystal Systems

Isometric System :

Lecture 3

Crystal Systems

Determine Point Symmetry

1. Assign crystallographic axes

2. Determine crystal system

3. Look for symmetry elements

4. Assign Hermann-Mauguin symbol

Point Group? 2/m

Crystal system? MonoclinicMonoclinic b-axis [010]

Point Group? 2/m 2/m 2/m

Crystal system? OrthorhombicOrthorhombic a-b-c

c

b

a

See, I fixed it!

Classes Distinguished by Centres of Symmetry :

Lecture 3

Crystal Systems

Crystal morphology

• Crystal Faces = limiting surfaces of growth–Depends in part on shape of building units & physical cond. (T, P, matrix, nature & flow direction of solutions, etc.)

Crystal Morphology

Observation: The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane

Observation: 1) The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane.

2) Because faces have direct relationship to the internal structure, they must have a direct and consistent angular relationship to each other

Crystal MorphologyNicholas Steno (1669): Law of Constancy of Interfacial Angles

QuartzQuartz

120o

120o

120o 120o 120o

120o

120o

Imperfect crystals

Law of constant interfacial angles: “The angles between symmetrically equivalent faces of a crystal are all the same”

The Contact Goniometer

The Reflection Goniometer

Problem 2:

Crystals don’t always fit into nice orthogonal XYZ coordinate systems…

The stereonet: A handy tool for 3-D geometry

Stereonets are used for:

1. Recording absolute angles between crystal faces

2. Choosing and showing crystallographic axes.

3. Keeping track of point symmetry elements.

4. Recording the orientations of lines and planes in space generally

Structural Geology

Next problem: Relative directions in crystals

• “Down the a-axis”• “Equally between the a and c axes”• You know, kinda sideways to that weird

lookin’ face there.

We need an exact method for describing lines and planes relative to crystal axes and lattices.

(p. 133 in Klein)

Miller Index:

Describes a plane or a line perpendicular to the plane in a lattice in terms of axis intercepts in a lattice.

a

cb

Example: A plane parallel to the c-axis, that intersects the a and b axes at equallattice spacings. Calculate the miller index by taking the inverse of theIntercepts in each direction: 1/1, 1/1, 1/ (110)

(110) 1/1, ½, 1/ (210)

(010)

Some aspects of Miller Indices

• Parallel planes or lines in a crystal have the same miller index.

• Miller indices are always reduced to integers (same thing!)

a

cb

(010) (010) (010)

(110)

(111) (221)

Indexing crystals

If you can pick the Miller index of a prominent face you can determine indices for all other faces and the relative lengths of the lattice spacings.

Miller Indices for directions

a = [100]

b = [010]

c = [001]

In between a and b?

?

ab = [110]

[110]

Miller indices:• Negative numbers indicated by a bar above the numeral (e.g.: (001))

• When using a variable or general MI, we name them hkl.

• In Hexagonal crystals, 4 MI (hkil) are often given: i = -h-k.

• Square braces (e.g.: [001]) for particular directions within a crystal.

• Round braces (e.g.: (001) describe a direction that is normal (perpendicular) to particular planes within a crystal.

• Curly braces (e.g.: {001}) describe a set of faces related by symmetry operations.

• This is called a crystal form (more later).

b

a

(1 1 0)

(2 1 0)

(1 0 0)

Can you index the rest?Can you index the rest?

b

a

(1 1 0)

(2 1 0)

(1 0 0)

(0 1 0)

(2 1 0)(2 1 0)

(2 1 0)

(1 1 0)(1 1 0)

(1 1 0)

(0 1 0)

(1 0 0)

Crystal Forms

Form = a set of symmetrically equivalent faces

Form = a characteristic mineral shape


pinacoid prism pyramiddipyramid

related by a mirror related by a mirror or a 2-fold axisor a 2-fold axis

|| faces related || faces related by n-fold axis or by n-fold axis or mirrorsmirrors

inclined faces inclined faces related by n-fold related by n-fold axis or mirrorsaxis or mirrors


Quartz = 2 forms:Quartz = 2 forms:Hexagonal prismHexagonal prismHexagonal dipyramidHexagonal dipyramid

Herkimer “diamond”, Herkimer, NY.:Displays only or mostly the dipyramid.

Crystal Forms: Two types

• Open

• Closed

Triclinic, Monoclinic, Orthorhombic

• PedionSingle face with no symmetrical equivalents

• PinacoidTwo faces related by an inversion center

• DomeTwo faces related by a mirror plane

• SphenoidTwo faces related by 2-fold rotation

All open!

How do you make a crystal out of open forms?Combine them…

m

Triclinic

All pedions All pinacoids

Point group? 1 1

Monoclinic

Pinacoid faces (010)(“prismatic”)

Point group? 2/m

Dihedral faces (011), (110)(Sphenoids)

Combinations

Orthorhombic forms

Important rotational forms:

• Prism (open): A collection of 3, 4, 6, 8, or 12 faces that intersects a set of mutually parallel edges (a zone), forming a tube (open).

• Pyramid (open): A collection of 3, 4, 6, 8, or 12 nonparallel faces that can intersect at a point. The base is not part of the form.

• Dipyramid (closed): Two pyramids, one on each end of the crystal, related by reflection across the base of the pyramid.

Rotational forms:

Crystal Zone:

The plane normal to the intersection lines of a group of prismatic faces.

Here:Zone: (001)Zone Axis: [001]

Trapezohedron• (Closed) Consists of 6, 8, or 12 faces, each of which is a

trapezoid. the faces on top of the crystal are offset in relation to the ones on the bottom

Rhombohedron• (Closed) 6 faces, each of which is rhomb shaped: A

rhomboherdon looks like a stretched or shortened cube.

3 2/m

Tetrahedron• (Closed) 4 Triangular faces. In the tetragonal case they are

identical isosceles triangles, in the orthorhombic case they are pairs of different isosceles triangles.

43m22242m

Tetragonal forms

Scapolite4/m

Zircon4/m 2/m 2/m

Chalcopyrite43m

Trigonal forms

Dolomite Quartz Tourmaline Calcite

Hexagonal forms

Nepheline Apatite Beryl Zincite 6 6/m 6/m 2/m 2/m 6mm

Isometric

Today’s outline:

• Steno’s law• Miller indices• Continue 3D symmetry basics• 32 Crystal classes (cont’d)

Next time:• The isometric system