crystallography lv

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crystallography lv
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Lattice planes
Useful concept for crystallography diffraction
Think of sets of planes in lattice - each plane
in set parallel to all others in set. All
planes in set equidistant from one another
Infinite number of set of planes in lattice
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Keep track of sets of planes by giving
them names - Miller indices
(hkl)
Lattice planes
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Miller indices (hkl) Choose cell, cell
origin, cell axes
origin
b
a
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Miller indices (hkl) Choose cell, cell
origin, cell axes Draw set of planes of interest
origin
b
a
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Miller indices (hkl) Choose cell, cell
origin, cell axes Draw set of planes of
interest Choose plane nearest origin
origin
b
a
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Miller indices (hkl) Choose cell, cell
origin, cell axes Draw set of planes of
interest Choose plane nearest origin Find
intercepts on cell axes 1,1,8
origin
b
1
a
1
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Miller indices (hkl) Choose cell, cell
origin, cell axes Draw set of planes of
interest Choose plane nearest origin Find
intercepts on cell axes 1,1,8 Invert
these to get (hkl) (110)
origin
b
1
a
1
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Exercises
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Lattice planes
Two things characterize a set of lattice
planes interplanar spacing (d) orientation
(defined by normal)
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Strange indices
For hexagonal lattices - sometimes see 4-index
notation for planes (hkil) where i - h - k
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Zones
2 intersecting lattice planes form a zone
plane (hkl) belongs to zone uvw if hu kv lw
0
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1h2 k1k2 l1l2 ) also in same
zone.
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Zones
zone axis uvw is ui vj wk
Example zone axis for (111) (100) - 011
(011) in same zone? hu kv lw 0 00 11
- 11 0
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1h2 k1k2 l1l2 ) also in same
zone.
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Reciprocal lattice
Real space lattice
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Reciprocal lattice
Real space lattice - basis vectors
a
a
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Reciprocal lattice
Real space lattice - choose set of planes
(100) planes
n100
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Reciprocal lattice
Real space lattice - interplanar spacing d
(100) planes
d100
1/d100
n100
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Reciprocal lattice
Real space lattice gt the (100) reciprocal
lattice pt
(100) planes
d100
n100
(100)
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Reciprocal lattice
The (010) recip lattice pt
n010
(100) planes
d010
(010)
(100)
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Reciprocal lattice
The (020) reciprocal lattice point
n020
(020) planes
d020
(010)
(020)
(100)
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Reciprocal lattice
More reciprocal lattice points
(010)
(020)
(100)
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Reciprocal lattice
The (110) reciprocal lattice point
(100) planes
n110
d110
(010)
(020)
(110)
(100)
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Reciprocal lattice
Still more reciprocal lattice points
(100) planes
(010)
(020)
(100)
the reciprocal lattice
(230)
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Reciprocal lattice
Reciprocal lattice notation
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Reciprocal lattice
Reciprocal lattice for hexagonal real space
lattice
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Reciprocal lattice
Reciprocal lattice for hexagonal real space
lattice
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Reciprocal lattice
Reciprocal lattice for hexagonal real space
lattice
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Reciprocal lattice
Reciprocal lattice for hexagonal real space
lattice
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Reciprocal lattice