UCSD NANO106 - 03 - Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations
Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets...
-
Upload
sheryl-powell -
Category
Documents
-
view
230 -
download
1
Transcript of Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets...
![Page 1: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/1.jpg)
crystallograp
hy lv
crystallograp
hy lv
![Page 2: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/2.jpg)
Useful concept for crystallography & diffractionUseful concept for crystallography & diffraction
Lattice planesLattice planes
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
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 latticeInfinite number of set of planes in lattice
dd
d - interplanarspacing
d - interplanarspacing
![Page 3: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/3.jpg)
Keep track of sets of planes by giving them names - Miller indices
(hkl)
Keep track of sets of planes by giving them names - Miller indices
(hkl)
Lattice planesLattice planes
![Page 4: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/4.jpg)
Miller indices (hkl)
Choose cell, cell origin, cell axes:
Miller indices (hkl)
Choose cell, cell origin, cell axes:
a
b
originorigin
![Page 5: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/5.jpg)
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interest:
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interest:
a
b
originorigin
![Page 6: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/6.jpg)
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest origin:
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest origin:
a
b
originorigin
![Page 7: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/7.jpg)
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes:
1,1,∞
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes:
1,1,∞a
b
originorigin
1
1
![Page 8: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/8.jpg)
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes
1,1,∞
Invert theseto get (hkl)
(110)
Miller indices (hkl)
Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes
1,1,∞
Invert theseto get (hkl)
(110)
a
b
originorigin
1
1
![Page 9: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/9.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 10: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/10.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 11: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/11.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 12: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/12.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 13: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/13.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 14: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/14.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 15: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/15.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 16: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/16.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 17: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/17.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 18: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/18.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 19: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/19.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 20: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/20.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 21: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/21.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 22: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/22.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 23: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/23.jpg)
Lattice planesLattice planes
Exercises Exercises
![Page 24: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/24.jpg)
Lattice planesLattice planes
Two things characterize a set of lattice planes:
interplanar spacing (d)
orientation (defined by normal)
Two things characterize a set of lattice planes:
interplanar spacing (d)
orientation (defined by normal)
![Page 25: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/25.jpg)
Strange indicesStrange indices
For hexagonal lattices - sometimes see 4-index
notation for planes (hkil)where i = - h - k
For hexagonal lattices - sometimes see 4-index
notation for planes (hkil)where i = - h - k
a1
a2
a3
(110)(1120)
![Page 26: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/26.jpg)
ZonesZones
2 intersecting lattice planes form a zone2 intersecting lattice planes form a zone
zoneaxiszoneaxis
zoneaxiszoneaxis zone axis
[uvw] is ui + vj + wk
zone axis [uvw] is ui + vj + wk
i j k h1 k1 l1 h2 k2 l2
i j k h1 k1 l1 h2 k2 l2
plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0 plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1+h2 k1+k2 l1+l2 ) also in same zone.
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1+h2 k1+k2 l1+l2 ) also in same zone.
![Page 27: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/27.jpg)
ZonesZones
zone axis [uvw] is ui + vj + wk
zone axis [uvw] is ui + vj + wk
(011) in same zone? hu + kv + lw = 00·0 + 1·1 - 1·1 = 0
(011) in same zone? hu + kv + lw = 00·0 + 1·1 - 1·1 = 0
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1+h2 k1+k2 l1+l2 ) also in same zone.
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then
(h1+h2 k1+k2 l1+l2 ) also in same zone.
Example: zone axis for
(111) & (100) - [011]
Example: zone axis for
(111) & (100) - [011] i j k
h1 k1 l1 h2 k2 l2
i j k h1 k1 l1 h2 k2 l2
i j k 1 1 1 1 0 0
i j k 1 1 1 1 0 0
(100)(100)
(111)(111) [011][011]
![Page 28: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/28.jpg)
Reciprocal latticeReciprocal lattice
Real space latticeReal space lattice
![Page 29: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/29.jpg)
a
a
Reciprocal latticeReciprocal lattice
Real space lattice - basis vectorsReal space lattice - basis vectors
![Page 30: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/30.jpg)
(100)planes
n100
Reciprocal latticeReciprocal lattice
Real space lattice - choose set of planesReal space lattice - choose set of planes
![Page 31: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/31.jpg)
(100)planes
n100
d100
1/d100
Reciprocal latticeReciprocal lattice
Real space lattice - interplanar spacing dReal space lattice - interplanar spacing d
![Page 32: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/32.jpg)
(100)planes
n100
d100
(100)
Reciprocal latticeReciprocal lattice
Real space lattice ––> the (100) reciprocal lattice ptReal space lattice ––> the (100) reciprocal lattice pt
![Page 33: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/33.jpg)
(100)planes
n010
d010
(100)
(010)
Reciprocal latticeReciprocal lattice
The (010) recip lattice ptThe (010) recip lattice pt
![Page 34: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/34.jpg)
The (020) reciprocal lattice pointThe (020) reciprocal lattice point
(020)planes
n020
d020
(100)
(010) (020)
Reciprocal latticeReciprocal lattice
![Page 35: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/35.jpg)
More reciprocal lattice pointsMore reciprocal lattice points
(100)
(010) (020)
Reciprocal latticeReciprocal lattice
![Page 36: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/36.jpg)
The (110) reciprocal lattice pointThe (110) reciprocal lattice point
(100)planes
(100)
(010) (020)
n110d110
(110)
Reciprocal latticeReciprocal lattice
![Page 37: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/37.jpg)
Still more reciprocal lattice pointsStill more reciprocal lattice points
(100)planes
(100)
(010) (020)
the reciprocal lattice(230)
Reciprocal latticeReciprocal lattice
![Page 38: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/38.jpg)
Reciprocal lattice notationReciprocal lattice notation
Reciprocal latticeReciprocal lattice
![Page 39: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/39.jpg)
Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice
Reciprocal latticeReciprocal lattice
![Page 40: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/40.jpg)
Reciprocal latticeReciprocal lattice
Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice
![Page 41: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/41.jpg)
Reciprocal latticeReciprocal lattice
Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice
![Page 42: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/42.jpg)
Reciprocal latticeReciprocal lattice
Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice
![Page 43: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to.](https://reader036.fdocuments.in/reader036/viewer/2022081506/56649dcf5503460f94ac36f0/html5/thumbnails/43.jpg)
Reciprocal latticeReciprocal lattice