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1.3 Crystal Faces and Miller Indices

1.3 Crystal Faces and Miller Indices - Crystal Faces Common...

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Unformatted text preview: Crystal Faces Common crystal faces relate simply to surface of unit cell Common shapes are similar to the outline of the unit cell Other faces are often simple diagonals through lattice Isometric minerals often are cubes Hexagonal minerals often are hexagons These relationship were discovered in 18th century and made into laws: Steno's law Law of Bravais Law of Huay Steno's Law Angle between equivalent faces on a crystal of some mineral are always the same Can understand why Faces relate to unit cell, crystallographic axes, and angular relationships between faces and axes Strictly controlled by crystal system and class 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 T has intermediate density of lattice nodes fairly common and pronounced face on mineral 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 Q has low density, rare face Fig. 221 Law of Hay Crystal faces intersect axes at simple integers of unit cell distances on the crystallographic axes Allows a naming system to describe planes in the mineral (faces, cleavage, atomic planes etc.) Miller Indices Absolute distance (units of length, commonly ) are not integers Unit cell distances typically small integers, e.g., 1 to 3, occasionally higher 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 Fig. 222 Face t extends until it intercepts crystallographic axes Face t miller index is (112) Unit cell Axial intercepts in terms of unit cell lengths: a = 12 b = 12 c = 6 If face "shrunk" to fit within the unit cell, the intercepts for a:b:c would be 1:1:1/2 Miller indices are the inverse of the intercepts Inverting give (112) Algorithm for calculation What about faces that parallel axes? With algorithm, miller index would be: For example, intercepts a:b:c could be 1:1: (hkl) = (1/1 1/1 1/) = (110) If necessary you need to clear fractions E.g. for a:b:c = 1:2: Invert: 1/1 1/2 1/ Clear fractions: 2(1 1/) = (210) Some intercepts can be negative they intercept negative axes E.g. a:b:c = 1:1: Here (hkl) = 1/1:1/1:1/ = (112) It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces Fig. 223 There need to be 4 intercepts (hkil) Hexagonal Miller index Two a axes have to have opposite sign of other axis so that h = a1 k = a2 i = a3 l = c Possible to report the index two ways: h + i + k = 0 (hkil) (hkl) (1010) (100) (1120) (110) (1121) (111) Klein and Hurlbut Fig. 233 Miller indices of real crystals Need to know the absolute lengths of the crystallographic axes Need to know the crystal systems to assign "unit" distances Sulfur unit distances True distances that represent 1 "unit" distance a = 10.47 b = 12.87 c = 24.49 Orthorhombic A common form of S Atomic arrangement within a unit cell. Note that the unit cell contains 128 atoms of sulfur; z = 128 Augite a common pyroxene (Ca, Mg silicate) Mononclinic a b c = = 90 90 Imagine a face with following axial intercepts: a = 3.82 cm b = 3.5 cm c = 2.07 cm Calculate axial ratio This is done by dividing unit cell dimension by the b unit cell dimension For augite unit cell dimensions are: a = 9.73 b = 8.91 c = 5.25 Fig. 226 Axial ratio = 9.73/8.91:8.91/8.91:5.25/8.91 = 1.09:1.00:0.59 Determine miller index by dividing axial ratios by values for axial intercept (hkl) = (1.09/3.82 1/3.5 0.59/2.07) = (0.29 0.29 0.29) Divide by 0.29 gives (111) Alternative method Find axial intercepts for a, b, and c axes Divide by unit cell lengths Prominent (and common) faces have small integers for Miller Indices Faces that cut only one axis Assigning Miller indices Faces that cut two axes (100), (010), (001) etc (110), (101), (011) etc (111) Called unit face Faces that cut three axes ...
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