Crystal MorphologyRemember:Space groups for atom symmetry Point groups for crystal face symmetry

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

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

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

Crystal MorphologyBecause 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 AnglesQuartz

Crystal MorphologyDiff planes have diff atomic environments

Crystal MorphologyCrystal symmetry conforms to 32 point groups 32 crystal classes in 6 crystal systemsCrystal 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 MorphologyCrystal Axes: generally taken as parallel to the edges (intersections) of prominent crystal faces

Crystal MorphologyCrystal 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 hexagonalWith x-ray crystallography we can determine the internal structure and the unit cell directly and accuratelyThe crystallographic axes determined by XRD and by the face method nearly always coincideThis is not coincidence!!

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

Crystal MorphologyHow do we keep track of the faces of a crystal?Remember, face sizes may vary, but angles can'tNote: interfacial angle = the angle between the faces measured like this

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 indexingMiller Index is the accepted indexing methodIt uses the relative intercepts of the face in question with the crystal axes

Crystal MorphologyGiven the following crystal:abc2-D viewlooking down cab

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

Crystal MorphologySuppose we get another crystal of the same mineral with 2 other sets of faces:How do we reference them?abbawxyz

Miller Index uses the relative intercepts of the faces with the axesbawxyzbaxyPick a reference face that intersects both axesWhich one?

Which one?baxyEither x or y. The choice is arbitrary. Just pick one.Suppose we pick x

MI process is very structured (cook book) a b cunknown face (y)reference face (x)12111

What is the Miller Index of the reference face? a b cunknown face (x)reference face (x)11111(2 1 0)

a b cunknown face (x)reference face (y)21111What if we pick y as the reference. What is the MI of x?(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)The choice is arbitraryWhat is the difference?

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

The technique above requires that we graph each faceA simpler (?) way is to use trigonometry141o148o??Measure the interfacial anglesinterfacial angles

The technique above requires that we graph each faceA simpler (?) way is to use trigonometry141o148o39o58otan 39 = a/b = 0.801tan 58 = a/b = 1.600

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

The Miller Indices of face z using x as the referencebaw(1 1 0)(2 1 0)za b cunknown face (z)reference face (x)1111(1 0 0)

ba(1 1 0)(2 1 0)(1 0 0)Can you index the rest?

ba(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)

cbaOYXZABC3-D Miller Indices (an unusually complex example)abcunknown face (XYZ)reference face (ABC)212423

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 itIf the unit face is correctly selected, the ratios should be the sameIf not, will be off by some multiple - i.e. picked (211) and called it (111)Best to change itMineralogy texts listed axial ratios long before XRDWe had to change some after XRD developed

Form = a set of symmetrically equivalent facesbraces indicate a form {210}ba(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 facesbraces indicate a form {210}Multiplicity of a form depends on symmetry {100} in monoclinic, orthorhombic, tetragonal, isometric

Form = a set of symmetrically equivalent facesbraces indicate a form {210}F. 2.36 in your text (p. 49-52)pinacoidprismpyramiddipryamidrelated by a mirror or a 2-fold axisrelated by n-fold axis or mirrors

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

Isometric forms include

CubeOctahedron

Dodecahedron

Octahedron to Cube to DodecahedronClick on image to run animation

All three combined:

ZoneAny group of faces || a common axisUse of h k l as variables for a, b, c intercepts(h k 0) = [001]If the MIs 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

BUT doesn't say which faceEither is OK

*A note on directional properties: H, n, thermal exp...