Miller indices/crystal forms/space groups

Crystal Morphology

• How do we keep track of the faces of a crystal?

• Sylvite a= 6.293 Å

• Fluorite a = 5.463 Å

• Pyrite a = 5.418 Å

• Galena a = 5.936 Å

Crystal MorphologyHow do we keep track of the faces of a crystal?

Remember, face sizes may vary, but angles can't

Note: Note: “interfacial “interfacial angle”angle” = the angle = the angle between the faces between the faces measured like thismeasured like this

120o

120o

120o 120o 120o

120o

120o

Crystal MorphologyHow do we keep track of the faces of a crystal?

Remember, face sizes may vary, but angles can't

Thus it's the orientation & angles that are the best source of our indexing

Miller Index is the accepted indexing method

It uses the relative intercepts of the face in question with the crystal axes

Crystal MorphologyGiven the following crystal:

aa

bb

cc

2-D view2-D viewlooking down clooking down c

aabb

Crystal MorphologyGiven the following crystal:

aabb How reference faces?How reference faces?

aa face? face?bb face? face?-a-a and and -b-b faces? faces?

Crystal MorphologySuppose we get another crystal of the same mineral with 2 other sets of faces:

How do we reference them?

aabb

b

a

wx

y

z

Miller Index uses the relative intercepts of the faces with the axes

b

a

w

x

y

z

b

a

x

y

Pick a reference face that intersects both axes

Which one?

Which one?

b

a

w

x

y

z

b

a

x

y

Either x or y. The choice is arbitrary. Just pick one.

Suppose we pick x

MI process is very structured (“cook book”)

b

a

x

y

a b ca b c

unknown face (unknown face (yy))

reference face (reference face (xx))1122

1111

11

invertinvert 2211

1111

11

clear of fractionsclear of fractions 22 11 00

Miller index ofMiller index offace face yy using using xx as as the a-b reference facethe a-b reference face(2 1 0)(2 1 0)

What is the Miller Index of the reference face?

b

a

x

y

a b ca b c

unknown face (unknown face (xx))

reference face (reference face (xx))1111

1111

11

invertinvert 1111

1111

11

clear of fractionsclear of fractions 11 11 00

Miller index ofMiller index ofthe reference face the reference face is always 1 - 1is always 1 - 1

(1 1 0)(1 1 0)

(2 1 0)(2 1 0)

b

a

x

y

a b ca b c

unknown face (unknown face (xx))

reference face (reference face (yy))2211

1111

11

invertinvert 1122

1111

11

clear of fractionsclear of fractions 11 22 00

What if we pick y as the reference. What is the MI of x?

(1 1 0)(1 1 0)

Miller index ofMiller index ofthe reference face the reference face is always 1 - 1is always 1 - 1

(1 2 0)(1 2 0)

c

ba

O

YX

Z

A

B

C

3-D Miller Indices (an unusually complex example)3-D Miller Indices (an unusually complex example)

aa bb cc

unknown face (unknown face (XYZXYZ))

reference face (reference face (ABCABC))21

4

Miller index of Miller index of face face XYZXYZ using using

ABCABC as the as the reference facereference face

3

invertinvert 12

4

3

clear of fractionsclear of fractions (1(1 3)3)44

Miller indices

• Always given with 3 numbers – A, b, c axes

• Larger the Miller index #, closer to the origin

• Plane parallel to an axis, intercept is 0

What are the Miller Indices of face Z?

b

a

w(1 1 0)

(2 1 0)

z

The Miller Indices of face z using x as the reference

b

a

w(1 1 0)

(2 1 0)

z

a b ca b c

unknown face (z)unknown face (z)

reference face (reference face (xx))1111

11

Miller index ofMiller index offace face zz using using xx ( (or or any faceany face) as the ) as the reference facereference face

11

invertinvert 1111

11

11

clear of fractionsclear of fractions 11 0000

(1 0 0)

b

a

(1 1 0)

(2 1 0)

(1 0 0)

What do you do with similar facesWhat do you do with similar faceson opposite sides of crystal?on opposite sides of crystal?

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)

• If you don’t know exact intercept:– h, k, l are generic notation for undefined

intercepts

You can index any crystal face

Crystal habit

• The external shape of a crystal– Not unique to the mineral– See Fig. 5.2, 5.3, and 5.4

Crystal Form = a set of symmetrically equivalent facesbraces indicate a form {210}

b

a

(1 1)

(2 1)

(1 0)

(0 1)

(2 1)(2 1)

(2 1)

(1 1)(1 1)

(1 1)

(0 1)

(1 0)

Form = a set of symmetrically equivalent faces

braces indicate a form {210}

Multiplicity of a form depends on symmetry

Form = a set of symmetrically equivalent faces

braces indicate a form {210}

What is multiplicity?

pinacoid prism pyramid dipryamid

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

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

Form = a set of symmetrically equivalent faces

braces indicate a form {210}Quartz = 2 forms:Quartz = 2 forms:

Hexagonal prism (m = 6)Hexagonal prism (m = 6)Hexagonal dipyramid (m = 12)Hexagonal dipyramid (m = 12)

Isometric forms include

CubeOctahedron

Dodecahedron

111

111 _

111 __

111 _

110

101 011

011 _

110

_

101 _

Crystal forms

• Forms can be open or closed– Cube block demo

• Forms on stereonets– Cube faces on stereonet

• General form– {hkl} not on, parallel, or perpendicular to any

symmetry element

• Special form– On, parallel, or perpendicular to any symmetry

element

• Rectangle block– Find symmetry, plot symmetry, plot special face,

general face, determine multiplicity

Space groups

• Point symmetry: symmetry about a point– 32 point groups, 6 crystal systems

• Combine point symmetry with translation, you have space groups– 230 possible combinations

SymmetryTranslations (Lattices)

A property at the atomic level, not of crystal shapes

Symmetric translations involve repeat distances

The origin is arbitrary

1-D translations = a rowa

aa is the is the repeat vectorrepeat vector

SymmetryTranslations (Lattices)

2-D translations = a net

a

b

Pick Pick anyany point point

Every point that is exactly n repeats from that point is an Every point that is exactly n repeats from that point is an equipointequipoint to the original to the original

Translations

There is a new 2-D symmetry operation when we consider translations

The Glide Plane:

A combined reflection

and translation

Step 1: reflectStep 1: reflect(a temporary position)(a temporary position)

Step 2: translateStep 2: translate

repeatrepeat

• 32 point groups with point symmetry

• 230 space groups adding translation to the point groups

3-D translation

• Screw axes: rotation and translation combined