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1.3 Crystal Faces and Miller Indices

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

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Unformatted text preview: Crystal Faces Crystal Faces Common crystal faces relate simply to surface of unit cell Common shapes are similar to the outline of the unit cell Isometric minerals often are cubes Hexagonal minerals often are hexagons Other faces are often simple diagonals through lattice These relationship were discovered in 18th century and made into laws: Steno’s law Law of Bravais Law of Huay Steno’s Law 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 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 c a Faces A, B, and C intersect only one axis – principal faces b 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 Fig. 2­21 Law of Haüy Law of Ha Crystal faces intersect axes at simple integers of unit cell distances on the crystallographic axes Absolute distance (units of length, commonly Å) are not integers Unit cell distances typically small integers, e.g., 1 to 3, occasionally higher Allows a naming system to describe planes in the mineral (faces, cleavage, atomic planes etc.) Miller Indices 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 Fig. 2­22 Fig. 2­22 Face t extends until it intercepts crystallographic axes Unit cell How many unit lengths out along the crystallographic axes? Fig. 2­22 Fig. 2­22 Axial intercepts in terms of unit cell lengths: a = 12 b = 12 c = 6 Fig. 2­22 Fig. 2­22 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) Face t is the (112) face Face u is the (112) face Algorithm for calculation 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 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) 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 Fig. 2­23 Hexagonal Miller index Hexagonal Miller index There need to be 4 intercepts (hkil) Two a axes have to have opposite sign of other axis so that h = a1 k = a2 i = a3 l = c h + k + i = 0 Possible to report the index two ways: (hkil) (hkl) (1010) (100) (1120) (110) (1121) (111) Klein and Hurlbut Klein and Hurlbut Fig. 2­33 Assigning Miller indices Assigning Miller indices Prominent (and common) faces have small integers for Miller Indices Faces that cut only one axis Faces that cut two axes (100), (010), (001) etc (110), (101), (011) etc Faces that cut three axes (111) Called unit face ...
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