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P. 306 – olivine information. Chemistry. Crystallography. Physical Properties. Optical Properties. 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
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P. 306 – olivine information Chemistry Crystallography Physical Properties Optical Properties
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
These relationships were discovered in 18th century and codified into laws: • Steno’s law • Law of Bravais • Law of Huay
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
Law of Bravais • Common crystal faces are parallel to lattice planes that have high lattice node densities
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
Law of Haüy • Crystal faces intersect axes at simple integers of unit cell distances on the crystallographic axes
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
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
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
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?
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
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
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)
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)
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
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)
Klein and Hurlbut Fig. 2-33 (1120) (1121) (1010) (100) (110) (111)
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