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 Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't Note: “interfacial angle” = the angle between the faces measured like this

Crystal Morphology Miller Index is the accepted indexing method How 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

Given the following crystal: Crystal Morphology Given the following crystal: 2-D view looking down c b a b a c

Given the following crystal: Crystal Morphology Given the following crystal: a b How reference faces? a face? b face? -a and -b faces?

Crystal Morphology Suppose we get another crystal of the same mineral with 2 other sets of faces: How do we reference them? b w x y b a a z

Miller Index uses the relative intercepts of the faces with the axes Pick a reference face that intersects both axes Which one? b b w x x y y a a z

Either x or y. The choice is arbitrary. Just pick one. Which one? Either x or y. The choice is arbitrary. Just pick one. Suppose we pick x b a w x y z b x y a

MI process is very structured (“cook book”) a b c unknown face (y) 1 1  reference face (x) 2 1 1 b a x y invert 2 1  clear of fractions 2 1 Miller index of face y using x as the a-b reference face (2 1 0)

What is the Miller Index of the reference face? a b c unknown face (x) 1 1  reference face (x) 1 1 1 b a x y invert 1  clear of fractions 1 Miller index of the reference face is always 1 - 1 (1 1 0) (2 1 0)

What if we pick y as the reference. What is the MI of x? a b c unknown face (x) 2 1  reference face (y) 1 1 1 b a x y invert 1 2  clear of fractions 1 2 Miller index of the reference face is always 1 - 1 (1 2 0) (1 1 0)

Miller index of face XYZ using ABC as the reference face 3-D Miller Indices (an unusually complex example) a b c c unknown face (XYZ) 2 2 2 reference face (ABC) 1 4 3 C invert 1 2 4 3 Z clear of fractions (1 3) 4 Miller index of face XYZ using ABC as the reference face O A X Y B a b

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

1 The Miller Indices of face z using x as the reference a b c 1 ¥ ¥ 1 unknown face (z) 1 ¥ ¥ reference face (x) 1 1 1 invert 1 ¥ b w (1 1 0) clear of fractions 1 (2 1 0) Miller index of face z using x (or any face) as the reference face (1 0 0) a z

What do you do with similar faces on opposite sides of crystal? b (1 1 0) (2 1 0) (1 0 0) a

b (0 1 0) (1 1 0) (1 1 0) (2 1 0) (2 1 0) (1 0 0) a (1 0 0) (2 1 0) (2 1 0) (1 1 0) (1 1 0) (0 1 0)

Demonstrate MI on cardboard cube model

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 faces braces indicate a form {210} b (0 1) (1 1) (1 1) (2 1) (2 1) (1 0) a (1 0) (2 1) (2 1) (1 1) (1 1) (0 1)

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 or a 2-fold axis related by n-fold axis or mirrors

Form = a set of symmetrically equivalent faces braces indicate a form {210} Quartz = 2 forms: Hexagonal prism (m = 6) Hexagonal dipyramid (m = 12)

Isometric forms include Cube Octahedron Dodecahedron 111 _ __ 110 101 011 _

Crystal forms Forms can be open or closed Forms on stereonets Cube block demo Forms on stereonets Cube faces on stereonet

General form Special form Rectangle block {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 4/m, 2/m, 2/m

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

Symmetry Translations (Lattices) 1-D translations = a row A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row a  a is the repeat vector

Symmetry Translations (Lattices) 2-D translations = a net b a Pick any point Every point that is exactly n repeats from that point is an equipoint to the original

Translations There is a new 2-D symmetry operation when we consider translations The Glide Plane: A combined reflection and translation repeat Step 2: translate Step 1: reflect (a temporary position)

32 point groups with point symmetry 230 space groups adding translation to the point groups

3-D translation Screw axes: rotation and translation combined