1.3 crystal faces and miller indices
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MetalurgiTRANSCRIPT
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P. 306 – olivine information
Chemistry
Crystallography
Physical Properties
Optical Properties
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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
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These relationships were discovered in 18th century and codified into laws: Steno’s law Law of Bravais Law of Huay
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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
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Law of Bravais
Common crystal faces are parallel to lattice planes that have high lattice node densities
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Fig. 2-21
All faces parallel unit cell – high density of lattice nodes T has
intermediate density of lattice nodes – fairly common and pronounced face on mineral
Q has low density, rare face
Monoclinic crystal
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
a b
c t
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Law of Haüy Crystal faces intersect axes at simple
integers of unit cell distances on the crystallographic axes
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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
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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
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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
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Fig. 2-22
Imagine you extend face t until it intercepts crystallographic axes
Unit cell: each side is one “unit” length
How many unit lengths out along the crystallographic axes?
Consider face t:
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Fig. 2-22For face t:
Axial intercepts in terms of unit cell lengths:a = 12b = 12c = 6
Face t, without the rest of the form
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Fig. 2-22
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
Face t is the (112) face
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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)
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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)
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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
a
b
c
-b
-c
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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)
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Klein and Hurlbut Fig. 2-
33
(100)(1010)
(110) (111)
(1120) (1121)
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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