X-ray, Neutron and e-beam scattering Introduction Why scattering? Diffraction basics Neutrons and x-rays Techniques Direct and reciprocal space Single crystals Powders CaFe 2 As 2 an example
What is the atomic scale structure? Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers Noncrystalline materials... atoms have no regular packing occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline Distance between atoms ~ Å (10-9 m) Si crystalline SiO2 Adapted from Fig. 3.18(a), Callister 6e. Oxygen noncrystalline SiO2 Adapted from Fig. 3.18(b), Callister 6e. d(h k l ) 26 d(hkl)
How do we study crystal structures? X-rays Visible light Resolution ~ wavelength So, 10-9 m resolution requires λ ~ 10-9 m
Diffraction Interference of two waves Double slit diffraction You can also do this with light (as well as neutrons and electrons)! 2 slits 2 slits and 5 slits
Diffraction from periodic structures Diffraction nλ = 2dsinθ I θ θ d 2θ Bragg 2θ
Neutrons and x-rays Neutrons λ ~ interatomic spacings E ~ elementary excitations Penetrates bulk matter (neutral particle) Strong contrasts possible (H/D) Scattered strongly by magnetic moments Low brilliance of sources Large samples Strong absorption for some elements (e.g. Cd, Gd, B) X-rays λ ~ interatomic spacings E >> elementary excitations Strong absorption for low energy photons Not for high energies (e.g. 100keV) Weak scattering for light elements, magnetic moments and little contrast for hydrocarbons High brilliance of x-ray sources High resolution; small samples; high degree of coherence Lets us look at weak scattering processes
Techniques Single crystal diffraction Powder diffraction Anomalous (resonant) x-ray scattering Small angle scattering Reflectivity and surface scattering T x-ray source C detector S specimen
The linguistics of scattering from periodic crystals 7 crystal systems (cubic, tetragonal, orthorhombic, monoclinic, trigonal, hexagonal) 14 Bravais lattices (above + centering (body, base, face)) 230 periodic space groups (14 Bravais lattices + 32 crystallographic point groups)
Lost in 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 R = m 1 a 1 + m 2 a 2 + m 3 a 3 and G = m 1 b 1 + m 2 b 2 + m 3 b 3 e ig R = 1; G R = 2π x integer G is normal to sets of planes of atoms Each point (hkl) in the reciprocal lattice corresponds to a set of planes (hkl) in the real space lattice.
Miller indices and reciprocal space (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
Reciprocal space, angle space and diffraction nλ = 2d hkl sinθ k k 0 = G; k = k 0 = 2π/λ Ewald sphere; Radius = 2π/λ 2θ For single crystal diffraction both the detector angle and the sample orientation matter
Things to keep in mind about single crystal measurements Because low-energy x-rays are strongly absorbed by most materials: we are looking at only the first few microns from the surface. Surface quality matters quite a bit. Surface oxidation/contamination Is this characteristic of the bulk? High energy x-rays level the playing field. What are the consequences of using high enegy x-rays? Extinction effects Multiple scattering effects We are probing only a small volume of reciprocal space. What else is out there?
Powder diffraction The intensity at discrete points in reciprocal space are now distributed over a sphere Of radius G or 2π/d hkl (sample angle is irrelevant)
Point detection Rietveld Refinement Intensity 10 12 14 Q 2D area detectors 5 10 15 Q ( Å)
PDF measurements Structure function G( r) 2 = π 0 Q[ S( Q) 1] sin QrdQ Raw data PDF
What happens if your sample is made of disordered dodecahedra? Sit on an atom and look at your neighborhood G(r) gives the probability of finding a neighbor at a distance r PDF is experimentally accessible PDF gives instantaneous structure.
Things to keep in mind about powder diffraction measurements Easy to do, but make sure you have a good powder!!! Powder diffraction is excellent for getting the big picture but since intensities are spread over a sphere, small (but perhaps important) details are missed. Lose information about anisotropy
CaFe 2 As 2 an example
Peak positions: Temperature dependent studies
Peak Positions: Pressure dependent studies
Peak Widths strain, crystallite size and mosaic Intensity (counts) 10000 1000 101 004 103 112 006 114 202 211 116 007 008 CaFe 2 As 2 Si standard Sn traces 206 215 301 311 310 224 Counts / s 400 200 (1 1 10) Rocking curve FWHM = 0.017 deg 100 222 0 20 30 40 50 60 70 80 2θ (deg) Powder after grinding Fig. X1 45.60 45.65 45.70 θ (deg) Single crystal mosaic
Integrated Intensities
What do you learn? From peak positions Lattice parameters and how they change with environmental conditions (e.g. temperature and pressure) From peak widths Crystal quality (e.g. mosaic) Presence of strain (e.g. longitudinal widths) From integrated intensities Contents of the unit cell Positions of atoms within the unit cell Thermal parameters (thermal disorder)