1. It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces c -b b a -c
Fig. 2-23

2. Hexagonal Miller index
There need to be 4 intercepts (hkil)
h = a1
k = a2
i = a3
l = c
Two a axes have to have opposite sign of other axis so that
h + k + i = 0
Possible to report the index two ways:
(hkil)
(hkl)

3. (1120)
(1121)
(1010)
(100)
(110)
(111)
Klein and Hurlbut Fig. 2-33

4. Assigning Miller indices
Prominent (and common) faces have small integers for Miller Indices
Faces that cut only one axis
(100), (010), (001) etc
Faces that cut two axes
(110), (101), (011) etc
Faces that cut three axes
(111)
Called unit face

5. Zones, Forms, Habits
Quantitative description of orientation in minerals – use Miller indices:
Zone - Lines, or linear directions within minerals
Form - Shapes of three dimensional objects
Qualitative description of mineral shapes:
Habit

6. Crystal Habit
Qualitative terminology to describe individual minerals and aggregates of minerals
Shape of individual minerals
Intergrowths of several mineral grains
Shape of masses of grains

7. Colloform
finely crystalline, concentric mineral layer
Globular – (spherulitic)
radiating, concentrically arranged acicular minerals
Reniform
kidney shaped
Botryoidal
like a bunch of grapes
Mammillary
similar, but larger than botryoidal, breast-like or portions of spheres
Drusy
Surface covered with layer of small crystals

8. Drusy quartz
Globular hematite

9. Terminology useful for describing general shapes of minerals
(asbestos: amphiboles and pyroxenes)
(table-like)
(knife like – kyanite)
(Mica)
Fig. 2-47

10. Fibrous tremolite: amphibole Ca(Mg,Fe)5Si8O22 (OH)2
Bladed kyanite
Al2SiO5

11. Zones Collection of common faces Parallel to some common line
Line called the zone axis
Identified by index [hkl]
Zone axis parallels intersection of edges of faces

12. Zone axis intesects (001) lattice nodes = [001]
Intersection of faces
= [001] Zone c
Faces = (110), (110), (110), (110)
Zone axis intesects (001) lattice nodes
= [001] -a -b
Note typo in first edition a b -c
Fig 2-30

13. Other linear crystallographic directions
Lattice node
Other linear crystallographic directions
Includes crystallographic axes
Referenced to intersection of lattice nodes
For example: location of rotation axes or other linear features
Fig 2-27

14. Form Formal crystallographic nomenclature of the shape of minerals
Description
Collection of crystal faces
Related to each other by symmetry
Identified by index: {hkl}
Values for h, k and l are determined by one of the faces

15. Example There are six faces in a cube (a kind of form):
(100), (010), (001), (100), (010), (001)
All faces parallel two axes and are perpendicular to one axis
Form is written with brackets
Uses miller index of one face
Generally positive face
E.g., {001} c
(001)
(010)
(100) b a
Isometric form
{001}

16. Possible to determine the shape of a form with:
Miller index of one face in form
Point symmetry of the crystal class
The form is created by operating point symmetry on the initial face
Number of faces in a form depends on crystal class

17. – called a rhombic prism Rhombus – an equilateral parallelogram
Face parallel to a axis
Mirror parallel to (010)
Mirror parallel to (001)
{011} form in crystal class with point symmetry 2/m 2/m 2/m (Orthorhombic)
– called a rhombic prism
Rhombus – an equilateral parallelogram
Prism – a crystal form whose faces are parallel to one axis
Fig. 2-29

18. Triclinic system: Point group (i.e. crystal class) = 1
Symmetry content = (1A1)
{111} has only 2 faces

19. Isometric system: Point group (crystal class) = 4/m 3 2/m
Symmetry content = 3A4, 4A3, 6A2, 9m
{111} has 8 faces
Form is an octahedron

20. Isometric system Point group (crystal class) = 4
Symmetry content = 1A4
{111} has 4 faces
Form is a tetrahedron
(111) C b a

21. Minerals must have more than one form if they have an open form
Two types of forms:
Open form – does not enclose a volume
Closed form – encloses a volume
Minerals must have more than one form if they have an open form
Minerals may have only one closed form
Mineral could have more than 1 form, closed or open

22. Open Form Closed Form Prism Requires additional forms Cube
Does not require additional forms, but may contain them

23. Example of multiple forms
Cube {001}, octahedron {111}, and 3 prisms{110}, {101}, {011}
All forms have 4/m 3 2/m symmetry c
Two combined closed forms, plus 3 additional open forms
Prisms
{110}
{101}
{011}
{111} = octahedron b
{001} = cube a

24. Isometric forms 15 possible forms 4 common ones
Cube {001} – 4/m 3 2/m symmetry
Octahedron {111} – 4/m 3 2/m symmetry
Tetrahedron {111} – 4 symmetry
Dodecahedron {110}

25. Both isometric forms: {111} {111} Tetrahedron Octahedron
(111) b a b a
Tetrahedron
Octahedron
{111}
{111}
Crystal class = 4
Crystal class = 4/m 3 2/m

26. Non-isometric form 10 types of forms Pedion (open) Pinacoid (open)
Single face
No symmetrically identical face
Pinacoid (open)
Two parallel faces
Related by mirror plane or inversion
Dihedron (open - 2 types)
Two non-parallel face
Related by mirror (dome) or 2-fold rotation (sphenoid)

27. Note: dome switches handedness Sphenoid retains handedness
Fig. 2-31

28. Prism (open) Pyramid (open) Dipyramid (closed)
3, 4, 6, 8 or 12 faces
Intersect with mutually parallel edges forming a tube
Pyramid (open)
3, 4, 6, 8, or 12 faces
Intersect at a point
Dipyramid (closed)
6, 8, 12, 16, or 24 faces
Two pyramids at each end of crystal
All of these forms are named on the basis of the shape of the cross section
Total of 21 different forms

29. Three types – seven modifiers – total of 21 forms
Prisms
Open
Pyramids
Open
Closed
Dipyramids
Cross section
Rhombic
Tetragonal
Trigonal
Hexagonal
Ditetragonal
Ditrigonal
Dihexagonal
Fig. 2-32

30. Trapezohedrons (closed)
6, 8, 12 faces
each a trapezoid (plane shape with 4 unequal sides)
Named according to number of faces
Scalenohedron (closed)
8 or 12 faces
Each a scalene triangle (no two angles are equal)

31. Rhombohedrons (closed)
6 faces, each rhomb shaped (4 equal sides, no 90 angles)
Looks like a stretched or shortened cube
Tetrahedron (closed)
4 triangular faces

32. Fig. 2-33

33. Combining forms Restrictions on types of forms within a crystal
All forms must be in the same crystal system
All forms must have symmetry of one crystal class, for example:
Tetragonal prism has a single 4-fold rotation, only found in tetragonal crystal class with single 4-fold rotation axis
Pedions never occur in mineral with center of symmetry

34. Multi-faced forms are not composed of several simpler forms
A cube is not 6 pedions or 3 pinacoids

35. Special relationships between forms
Enantiomorphous forms
Positive and negative forms

36. Enantiomrophous Form Enantiomorphic forms contain a screw axis
Axis may rotate to the right or left
The two forms generated are mirror images of each other

37. Almost always left handed Through time convert to right handed
3-fold screw axis
Atomic scale rotation
Enantiomorphous forms result from either right or left spiral of screw axis
Amino acids:
Almost always left handed
Through time convert to right handed
Age-dating tool 0 < D/L < 1
May be 2-fold, 4-fold, or 6-fold
Fig. 2.20

38. Enantiomorphous Forms
Must lack center of symmetry and mirrors
Forms are related to each other by a mirror
right and left handed forms
Individual crystal either right or left handed, but not both
Quartz is common example

39. Enantiomorphous Forms
Crystal are mirror images of each other, but there are no mirror images in the crystals
Fig. 2-34

40. Positive and Negative forms
Created by rotation of a form
Rotation not present in the form itself
Two forms related to each other by mirror planes
Mirror planes missing within the form itself
Two possible rotations:
60º on 3-fold rotation axis
90º on 4- or 2-fold rotation axis

41. Positive and negative faces in quartz crystal
Quartz lacks center of symmetry
Quartz Crystal
Fig. 2-35

42. Forms in the Six Crystal System
Forms control orientation of crystallographic axes of the 6 crystal system
Systematic relationship between form, symmetry present, and Hermann-Mauguin symbols
Following slides show these relationships

43. Triclinic Common symmetry: 1-fold rotation
Table 2.2
c-axis parallels prominent zone axis
b and a axes parallel crystal edges
a and b typically > 90º
Single Hermann-Mauguin symbol
Common minerals: plagioclase and microcline

44. Triclinic
c = zone axis
Pedions
Pinacoid b a 1 1
Fig. 2-36

45. Monoclinic Common symmetry: 2-fold rotation and/or single mirror plane
b axis commonly parallel the 2-fold rotation and/or perpendicular to mirror plane
c axis parallel to prominent zone
a axis down and to front so b > 90
Single H-M symbol (2, m, or 2/m)
Common minerals: amphiboles, pyroxenes, micas

46. Monoclinic
2-fold rotation axis
Fig. 2-37

47. Orthorhombic
Common symmetry: 3 2-fold rotations and/or 3 mirror planes
Crystal axes are parallel to 2-fold rotations or perpendicular to mirror planes, or both
Any axis could have any symmetry
Reported in H-M notation:
1st = a axis, 2nd = b axis, 3rd = c axis
E.g. mm2 – a ^ mirror, b ^ mirror, c parallel 2-fold rotation

48. Orthorhombic c c c b b
222 a b a a
2/m2/m2/m
mm2
Fig. 2-38

49. Tetragonal
Common symmetry: single 4-fold rotation, or 4-fold rotoinversion
c axis always the single 4-fold rotation axis
a and b coincide with 2-fold rotation or ^ mirror (if present)
H-M symbol:
1st = c axis
2nd = b and a axes
3rd = symmetry on [110] and [110] axis at 45º to a and b axes

50. Example 42m C = 4-fold rotoinversion
a and b axes [100] and [010] are 2-fold rotation
There are mirrors ^ to [110] and [110]

51. c
42m
Positive and negative tetragonal tetrahedron
Note – tetragonal so a = b ≠ c, this is not an isometric form b a
Fig. 2-39

52. Hexagonal
Common symmetry: 1 3-fold axis (trigonal division) or 1 6-fold axis (hexagonal division)
c axis parallel to 6-fold or 3-fold rotation
a axes parallel to 2-fold rotation or perpendicular to mirror
H-M symbols written with 1st = c axis, 2nd parallel a axes, 3rd perpendicular to a

53. c
A prism and multiple dipyramids a2
-a3 a1
6/m2/m2/m
Figure 2-41

54. Isometric Common symmetry 4 3-fold axes
3 equivalent symmetry axes coincide with crystallographic axes
(e.g. for cube, it’s the 4 fold rotations)
Symmetry either 2-fold or 2-fold
H-M symbols;
1st crystallographic axes
2nd diagonal axes [111]
3rd center of one edge to center of another edge [110]

55. 4/m32/m - Isometric 3 c b a
2/m
4/m
Fig. 2-44