Scattering and Diffraction

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Scattering and Diffraction Andreas Kreyssig, Alan Goldman, Rob McQueeney Ames Laboratory Iowa State University All rights reserved, 2018.

Atomic scale structure - crystals Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers crystalline SiO2 Adapted from Fig. 3.18(a), Callister 6e. Noncrystalline materials... atoms have no regular packing occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline Distance between atoms ~ Å (10-10 m) noncrystalline SiO2 Adapted from Fig. 3.18(b), Callister 6e. 26

How we do study crystal structures? X-rays Visible light Resolution ~ wavelength So, 10-10 m resolution requires λ ~ 10-10 m

Diffraction Interference of two waves Double slit diffraction Constructive Destructive You can also do this with light (as well as neutrons and electrons). 2 slits 2 slits and 5 slits

Diffraction from periodic structures d(h k l ) Interference of waves crystalline SiO2 d(hkl) Constructive I Bragg equation: 2θ

Crystal structures and diffraction - PDF database PDF_DataBase 1; 2; 3; 4

Crystal structures and diffraction - PDF database PDF_DataBase 5; 6

Crystal structures and diffraction - PDF database PDF_Hanawalt 1; 2; PDF_Fink 1; 2

PDF database - Example: growth of PrAuSi out of Sn flux Which elements can/must be present? E. D. Mun: PM721-Ex1b

PDF database - Example: growth of PrAuSi out of Sn flux Oops is Au really present? E. D. Mun: PM721-Ex1c

PDF database - Example: growth of PrAuSi out of Sn flux Expected traces of Sn, but is the main phase right? E. D. Mun: PM721-Ex1e

PDF database - Example: growth of PrAuSi out of Sn flux Expected traces of Sn, but is the main phase right? E. D. Mun: PM721-Ex1f

PDF database - Example: growth of PrAuSi out of Sn flux If your phase is not in the database search for isostructural compounds... E. D. Mun: PM721-Ex1g

Diffraction from periodic structures 7 Crystal systems: with symmetry (cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, triclinic) 14 Bravais lattices [above + centering (body, base, face)] 230 Space groups (14 Bravais lattices + 32 crystallographic point groups)

International Tables for Crystallography InternationalTables E1

International Tables for Crystallography InternationalTables E1

International Tables for Crystallography InternationalTables E2

International Tables for Crystallography InternationalTables 7_1

International Tables for Crystallography InternationalTables 7_2

International Tables for Crystallography InternationalTables 8

International Tables for Crystallography InternationalTables 9

International Tables for Crystallography Cullity 35, 36

International Tables for Crystallography InternationalTables E4

International Tables for Crystallography InternationalTables E3

Problems describing a structure Rhombohedral unit cell InternationalTables 84

Problems describing a structure Rhombohedral unit cell InternationalTables 146_11

Problems describing a structure Rhombohedral unit cell InternationalTables 146_11; 21

Problems describing a structure Rhombohedral unit cell InternationalTables 146_11; 12a; 21; 22

Problems describing a structure Origin of cell InternationalTables 129_11

Problems describing a structure Origin of cell InternationalTables 129_11; 21

Problems describing a structure Origin of cell InternationalTables 129_11; 12; 21; 22

Reciprocal space For an infinite 3D lattice defined by primitive vectors (a 1, a 2, a 3 ) we can define a reciprocal lattice generated by: For real space vector R = m 1 a 1 + m 2 a 2 + m 3 a 3 and reciprocal vector G = n 1 b 1 + n 2 b 2 + n 3 b 3 with all m s and n s integer is e ig R = 1 (G R = 2π x integer) G is normal to sets of planes of atoms. Each point (n 1, n 2, n 3 ) or (hkl) in the reciprocal lattice corresponds to a set of planes in the real space lattice.

Reciprocal space and Miller indices (0 K 0) (H 0 0) (100) Reflection = diffraction from planes of atoms spaced 2π/a apart (200) Reflection = diffraction from planes of atoms spaced 2π/2a apart

Diffraction from periodic structures d(h k l ) Interference of waves crystalline SiO2 d(hkl) Constructive k i G k f I Bragg equation: 2θ

Diffraction from periodic structures Ewald construction d(h k l ) crystalline SiO2 d(hkl) k i k f Q hkl 2q k i Scattering triangle: Q hkl = k f - k i Lifshin_31_1

Diffraction from periodic structures Ewald construction k f Q hkl k i k f Q hkl 2q k i Laue equation: G Q hkl = k f - k i Lifshin_31_1

Diffraction Basic equations Bragg equation: Laue equation: G Q hkl = k f - k i Structure Amplitude:

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