Crystal Morphology Remember: Space groups for atom symmetry Point groups for crystal face symmetry 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

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

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

Nicholas Steno (1669): Law of Constancy of Interfacial Angles Crystal Morphology Nicholas Steno (1669): Law of Constancy of Interfacial Angles Quartz

Diff planes have diff atomic environments Crystal Morphology Diff planes have diff atomic environments A note on directional properties: H, n, thermal exp...

Crystal Morphology Crystal symmetry conforms to 32 point groups  32 crystal classes in 6 crystal systems Crystal faces act just as our homework: symmetry about the center of the crystal so the point groups and the crystal classes are the same

Crystal Morphology Crystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces a b c

Crystal Morphology Crystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces The more faces the better  prism faces & quartz c-axis, halite cube, etc. We must also keep symmetry in mind: c = 6-fold in hexagonal With x-ray crystallography we can determine the internal structure and the unit cell directly and accurately The crystallographic axes determined by XRD and by the face method nearly always coincide This is not coincidence!!

How do we keep track of the faces of a crystal? Crystal Morphology How do we keep track of the faces of a crystal?

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 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)

Which choice is correct? 1) x = (1 1 0) y = (2 1 0) 2) x = (1 2 0) y = (1 1 0) b a x y The choice is arbitrary What is the difference?

What is the difference? axial ratio = a/b = 0.80 y unit cell shape if y = (1 1 0) unit cell shape if x = (1 1 0) b b a a x b y a axial ratio = a/b = 0.80 axial ratio = a/b = 1.60

The technique above requires that we graph each face A simpler (?) way is to use trigonometry Measure the interfacial angles b a x y b a w x y z 148o ? ? interfacial angles 141o

The technique above requires that we graph each face A simpler (?) way is to use trigonometry b a x y b a w x y z tan 39 = a/b = 0.801 tan 58 = a/b = 1.600 58o 148o 39o 141o

What are the Miller Indices of all the faces if we choose x as the reference? 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

b Can you index the rest? (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)

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

Demonstrate MI on cardboard cube model

We can get the a:b:c axial ratios from the chosen (111) face We can also determine the true unit cell by XRD and of course determine the a:b:c axial ratios from it If the unit face is correctly selected, the ratios should be the same If not, will be off by some multiple - i.e. picked (211) and called it (111) Best to change it Mineralogy texts listed axial ratios long before XRD We had to change some after XRD developed

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 {100} in monoclinic, orthorhombic, tetragonal, isometric

Form = a set of symmetrically equivalent faces braces indicate a form {210} F. 2.36 in your text (p. 49-52) 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 _

Octahedron to Cube to Dodecahedron Click on image to run animation

All three combined:

Zone Any group of faces || a common axis Use of h k l as variables for a, b, c intercepts (h k 0) = [001] If the MI’s of 2 non-parallel faces are added, the result = MI of a face between them & in the same zone

BUT doesn't say which face (100) Which?? (010) (110)?

BUT doesn't say which face (100) (010) (210) (110) (100) Which?? (010) (110)? (100) (010) (110) (120) Either is OK