P. 306 – olivine information

1. P. 306 – olivine information Chemistry Crystallography Physical Properties Optical Properties

2. Crystal Faces • Common crystal faces relate simply to surfaces of unit cell • Often parallel to the faces of the unit cell • Isometric minerals often are cubes • Hexagonal minerals often are hexagons • Other faces are often simple diagonals – at uniform angles – to the unit cell faces

3. These relationships were discovered in 18th century and codified into laws: • Steno’s law • Law of Bravais • Law of Huay

4. Steno’s Law • Angle between equivalent faces on a crystal of some minerals are always the same • Can understand why • Faces relate to unit cell, crystallographic axes, and angular relationships between faces and axes • Strictly controlled by symmetry of the crystal system and class of that mineral

5. Law of Bravais • Common crystal faces are parallel to lattice planes that have high lattice node densities

6. All faces parallel unit cell – high density of lattice nodes Monoclinic crystal Fig. 2-21 T has intermediate density of lattice nodes – fairly common and pronounced face on mineral t c Faces A, B, and C intersect only one axis – principal faces Face T intersects two axes a and c, but at same unit lengths Face Q intersects A and C at ratio 2:1 b a Q has low density, rare face

7. Law of Haüy • Crystal faces intersect axes at simple integers of unit cell distances on the crystallographic axes

8. Lengths can be absolute or relative: • Absolute distance -lengths have units (typically Å) and are not integers • Unit cell distances - typically small integers, e.g., 1 to 3, occasionally higher • 1 unit length is the absolute length of crystallographic axis • Allows a naming system to describe planes in the mineral (faces, cleavage, atomic planes etc.) Miller Indices

9. Miller Indices • Shorthand notation for where the faces intercept the crystallographic axes • Miller Index • Set of three integers (hkl) • Inversely proportional to where face or crystallographic plane (e.g. cleavage) intercepts axes

10. General form is (hkl) where • h represents the a intercept • k represents the b intercept • l represents the c intercept • h, k, and l are ALWAYS integers

11. Unit cell: each side is one “unit” length Fig. 2-22 Consider face t: Imagine you extend face t until it intercepts crystallographic axes How many unit lengths out along the crystallographic axes?

12. For face t: Axial intercepts in terms of unit cell lengths: a = 12 b = 12 c = 6 Fig. 2-22 Face t, without the rest of the form

13. Imagine the face is fit within the unit cell so that the maximum intercept is 1 unit length; The intercepts for a:b:c would be 1:1:1/2 Miller indices are the inverse of the intercepts Inverting give (112) – note that the higher the number the closer to the origin Fig. 2-22 Face t is the (112) face

14. Algorithm for calculation

15. What about faces that parallel axes? • For example, intercepts a:b:c could be 1:1:∞ • With algorithm, miller index would be: • (hkl) = (1/1 1/1 1/∞) = (110) • If necessary you need to clear fractions • E.g. intercepts for a:b:c = 1:2:∞ • Invert: 1/1 1/2 1/∞ • Clear fractions: 2(1 ½ 0) = (210)

16. Some intercepts can be negative – they intercept negative axes • E.g. a:b:c = 1:-1:½ • Here (hkl) = 1/1:1/-1:1/½ = (112)

17. Fig. 2-23 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

18. 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)

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

20. 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