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![Interaction of electrons with crystal
lattices
¡ Strong interactions with both electron clouds and positively charged
nuclei
¡ Probability of scattering of electron beam is ~ four times higher than
for X-rays
¡ Electron scattering factor can be related to the X-ray scattering factor
by the Moth-Bethe formula:
¡ High probability of scattering leads to multiple scattering events
(dynamic scattering) è electron scattering must be treated with
dynamic scattering theory
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13
f el
(s) =
e
16π 2
ε0 s
2
[Z − f X
(s)] where s =
sinθ
λ
9](https://image.slidesharecdn.com/13-otherdiffractiontechniques-160518164001/85/UCSD-NANO106-13-Other-Diffraction-Techniques-and-Common-Crystal-Structures-9-320.jpg)
![Atomic sizes
¡ Some metals have more than one allotrope
¡ Fe exists in bcc and fcc structures
¡ Bcc lattice constant = 2.8664 Angstroms
¡ Fcc lattice constant = 3.6468 Angstroms
¡ If we treat Fe atoms as touching spheres, we can
derive associated atomic radii for the two
structures
¡ Bcc spheres are touching along [111]
¡ Fcc spheres are touching along [110]
¡ ~4% larger radius for fcc
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13
4rbcc = a2
+ a2
+ a2
→ rbcc =
3a
4
=1.242nm
4rfcc = a2
+ a2
→ rfcc =
2a
4
=1.289nm
28](https://image.slidesharecdn.com/13-otherdiffractiontechniques-160518164001/85/UCSD-NANO106-13-Other-Diffraction-Techniques-and-Common-Crystal-Structures-28-320.jpg)
Uploaded byUniversity of California, San Diego
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
The document discusses various diffraction techniques used to determine crystal structures, including X-ray diffraction (XRD), electron diffraction, and neutron diffraction. It provides details on each technique, such as the wavelength used, how the beams interact with materials, and common applications. Neutron diffraction is described as penetrating deeply into materials and providing information on magnetic structures. Examples of common crystal structures are also presented, such as face-centered cubic and hexagonal close-packed structures. Interstitial sites and packing fractions are discussed for relating structure to properties of materials.
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UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
- 1. Other diffraction techniquesand common crystal structures of real materials Shyue Ping Ong Department of NanoEngineering University of California, San Diego
- 2. Other diffraction techniques ¡XRD is the most common (and typically most economical) means of determining crystal structures. ¡ Electrons and neutrons are also widely used scattering techniques. ¡ Employs wave-particle duality of electrons and neutrons ¡ Transmission electron microscopes (TEM) are more expensive than XRD equipment, but still common ¡ Neutron diffraction is typically performed at national and international reactor facilities NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 2
- 3. Comparison of differentdiffraction techniques X-rays λ ~ 0.1nm Uncharged No magnetic dipole Scattered by atomic charge densities (~0.1nm) Electrons λ ~ 0.002nm Charged Possesses magnetic dipole Scattered by electronic and nuclear charge densities (~0.1nm) Neutrons λ ~ 0.1nm Uncharged Possess magnetic dipole Scattered by atomic nucleus (~0.0001nm) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 3
- 4. Neutron diffraction ¡Neutrons arescattered by crystalline solids ¡Uncharged: Penetrates deeply into most materials (allows for sample with large volumes ~ cm3) ¡Interacts with atomic nuclei and magnetic dipole moment of nuclei – can provide information on magnetic point and space group symmetry. ¡No interaction with electron cloud NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 4
- 5. Applications of NeutronScattering ¡ Elastic nuclear scattering (Bragg scattering) ¡ Ability to locate light atomic species in presence of heavy atoms (no simple dependence on Z) ¡ Elastic magnetic scattering ¡ Probe magnetic structure ¡ Inelastic scattering ¡ Interaction of low-energy neutrons with vibrating crystal lattice and spin waves ¡ Probe magnetic phase transitions ¡ Isotopic substitution ¡ Isotopes can have different neutron scattering ¡ Tailor scattering factors to be more or less sensitive to a particular element NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 5
- 6. Neutron atomic scatterfactors ¡ No simple dependence on Z (unlike X-rays!) ¡ Variations in b due to resonance absorption in compound nucleus formation ¡ Negative scattering lengths for some elements (H, Li, Ti, V, Mn) ¡ V has smallest absolute value of b (i.e., mostly transparent to neutrons and is typically used as sample holder. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 6
- 7. Comparison of differentdiffraction techniques X-rays λ ~ 0.1nm Uncharged No magnetic dipole Scattered by atomic charge densities (~0.1nm) Electrons λ ~ 0.002nm Charged Possesses magnetic dipole Scattered by electronic and nuclear charge densities (~0.1nm) Neutrons λ ~ 0.1nm Uncharged Possess magnetic dipole Scattered by atomic nucleus (~0.0001nm) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 7
- 8. Electrons as adiffracting beam ¡ Electrons can be accelerated by electric fields ¡ Velocities can be significant proportion of speed of light – need relativistic corrections NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 λ = h mv = h 2m0eV(1+ e 2m0c2 V) = 1226.39 V + 0.97845×10−6 V2 Note that wavelengths are in pm! 8
- 9. Interaction of electronswith crystal lattices ¡ Strong interactions with both electron clouds and positively charged nuclei ¡ Probability of scattering of electron beam is ~ four times higher than for X-rays ¡ Electron scattering factor can be related to the X-ray scattering factor by the Moth-Bethe formula: ¡ High probability of scattering leads to multiple scattering events (dynamic scattering) è electron scattering must be treated with dynamic scattering theory NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 f el (s) = e 16π 2 ε0 s 2 [Z − f X (s)] where s = sinθ λ 9
- 10. Electron diffraction geometry ¡Wavelengths for electrons are much smaller ¡ Diffraction angles are much smaller (typically on the order of milliradians) ¡ Bragg condition can be approximated by ¡ Need to only look close to incident beam to find diffracted beam ¡ Ewald spheres are ~ 100-1000 times larger than X-rays (possible to orient crystal such that whole plane of reciprocal lattice points are tangent to sphere) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 2dhklθ = λ 10
- 11. Sample preparation ¡ Stronginteraction necessitates the use of very thin or small samples (~ few hundred nm) ¡ Unlike neutrons and X-rays, electrons are absorbed by matter ¡ Implications: ¡ Sample preparation is time-consuming and difficult ¡ Infinite crystal assumption is no longer valid and reciprocal lattice points can no longer be treated as zero-volume mathematical points (reciprocal point shape is the reciprocal of the extent of crystal – for thin films, this means a cylinder extended in direction of thin film) ¡ Reciprocal point volume can intersect Ewald sphere even when actual point is not on Ewald sphere – increases probability of diffraction NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 11
- 12. Transmission electron microscope (TEM) NANO106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 Heating W filament emits electrons Focuses and directs electron beam using magnetic fields Main image forming lens Fairly large - Can be > 3 stories high! - Vacuum is maintained as electrons cannot travel very far in air 12
- 13. Synchrontron X-ray sources ¡X-rays produced by circular motion of charged particles, e.g. electrons (recall that X-rays are produced by acceleration of charged particles) ¡ Properties ¡ High beam intensity: continuous source with five or more orders of magnitude higher intensities than X-ray tubes ¡ Broad radiation spectrum ¡ Strong polarization: Nearly 100% linearly polarized, circular or elliptical polarizations possible NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 13
- 14. Example applications ofSynchroton XRD ¡In-situ study of crystallization with heating ¡ High-flux allows for diffraction patterns to be taken in short amount of time ¡Study of superlattice orderings, e.g., ordered α’-FeCo ¡ Superlattice reflections can be several orders of magnitude less intense than fundamental reflections NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 14
- 15. Common crystal structures NANO 106- Crystallography of Materials by Shyue Ping Ong - Lecture 13 15
- 16. Concept of tessellation ¡Tessellation– the filling of space using geometric shapes with no overlap and no gaps ¡ 2D - Tiling using polygons ¡ 3D – Space filling using polyhedra ¡Frequently used in the way that materials scientists think about crystal structures ¡Topic is vastly and mathematically beautiful, but in this course, we will focus only a few simple concepts and build some common crystal structures NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 16
- 17. Regular polygons andpolyhedra ¡Regular polygons and polyhedra have identical sides and angles. ¡2D ¡ Equilateral triangle ¡ Square ¡ Regular hexagon ¡3D (also known as platonic solids) ¡ Tetrahedron ¡ Cube ¡ Octahedron ¡ Icosahedron ¡ Pentagonal dodecahedron NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 17
- 18. 2D tilings ¡ Regulartilings (monohedral tilings) ¡ Labeled using the Schläfli symbols (nm, where n is number of the sides of the polygon and m is the number of polygons meeting at a vertex) ¡ Archimedean tilings ¡ Still uses regular polygons, but more than one type allowed. ¡ All vertices remain identical. ¡ Only 8 additional possible tilings ¡ K-uniform and other tilings ¡ Further relaxation of number of unique vertices and other conditions. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 36 44 63 18
- 19. Stackings of 36tilings ¡ 3D crystals can be constructed by stackings of 2D tilings. ¡ An important class of structures – the closed-packed structures – comprise of stacking of 36 tilings NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 • A 36 tiling has plane group p6mm • If we consider a single layer of atoms occupying “A” sites, two kinds of interstitial sites are created – “B” sites and “C” sites. • Subsequent layers can occupy either B or C sites to form close-packed structures, and the process can be repeated to generate infinite variations of stacking 19
- 20. Two common formsof closed packing ¡ Hexagonal close-packed ¡ ABAB stacking ¡ Face-centered cubic ¡ ABCABC stacking NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 Fcc perspective views 20
- 21. Interstitials in close-packedstructures ¡ Tetrahedral interstitial ¡ Four-fold coordinated ¡ Formed by having an atom atop three atoms ¡ Octahedral interstitial ¡ Formed when there are no atoms sitting directly on top of the interstitial ¡ Six-fold coordinated ¡ Larger than tetrahedral site ¡ Interstitial can be labeled as α, β, or γ depending on whether the site is formed by B-C, A-C or A-B planes respectively. ¡ (Recall that NaCl can be thought of as having an fcc lattice of Na with Cl sitting in the oct interstitial) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 21
- 22. Notations for close-packedstructures ¡ ABC notation ¡ Can be modified to describe compounds (e.g., CdI2 with Cd occupying tetrahedal interstices between close-packed I planes can be denoted as α BC… ¡ E.g., SiC has many polytypes that differ simply in the stacking sequence of Si (C simply occupies the tetrahedral interstitials). ¡ Ramsdell notation ¡ Total number of close-packed layers followed by a letter that indicates whether the lattice type is cubic (C), hexagonal (H), or rhombohedral (R). If two or more structures have the same lattice type and the same repeat period, a subscript a, b, c, or 1, 2, 3 is used to distinguish between structures. ¡ E.g., SiC has two hexagonal polytypes with stacking sequences ABCACB . . .and ABCBAB . . . for the Si atoms. They are distinguished by their subscripts as 6H1 and 6H2, respectively. ¡ Does not specify actual stacking sequence NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 22
- 23. Crystal structures ofelements NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 23
- 24. Parent, derivative andsuperlattice structures ¡Derivative structures are those that are derived from simpler structures by substitution of one atom for another ¡Interstitial structures are obtained by ordered occupation of subsets of interstitial sites in simple structures NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 24
- 25. Definition of Superlattice ¡Consider two lattice’s whose basis vectors are related by: ¡ If Mij are integers and det(M) = 1, then the two lattices coincide. ¡ If Mij are integers and det(M) > 1, then the lattice defined by A’ is a superlattice of that defined by A. ¡ If Mij are integers and det(M) < 1, then the lattice defined by A’ is a sublattice of that defined by A. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 !A = a1 ! a2 ! a3 ! " # $ $ $ $ $ % & ' ' ' ' ' = M11 M12 M13 M21 M22 M23 M31 M32 M33 " # $ $ $$ % & ' ' '' a1 a2 a3 " # $ $ $$ % & ' ' '' = MA 25
- 26. Classification of structures ¡Strukturbericht symbols ¡ Letter followed by number ¡ Sequentialnumbering by order of discovery,e.g., A2 refers to bcc structure. ¡ Frequently used in materials science literature ¡Pearson symbol ¡ Bravais lattice symbol (a, m, o, t, h, c) + centering (P, C, I, F) + # of atoms in unit cell. ¡ E.g., NaCl has Pearson symbol cF8. ¡ Not unique. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 26
- 27. Structure of metals ¡Derivative structures are based on fcc, bcc, and hcp parent structures ¡ fcc (A1) and hcp (A3) – Stackings of 36 tiles. ¡ bcc (A2) – Stacking of 44 tiles. ¡ fcc and bcc are derivatives of the simple cubic structure, which is rarely observed for elements (though common in compounds) ¡ fcc and hcp are close-packed – packing fraction of 74.05%. bcc is not close-packed – packing fraction of 68.02%. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 27
- 28. Atomic sizes ¡ Somemetals have more than one allotrope ¡ Fe exists in bcc and fcc structures ¡ Bcc lattice constant = 2.8664 Angstroms ¡ Fcc lattice constant = 3.6468 Angstroms ¡ If we treat Fe atoms as touching spheres, we can derive associated atomic radii for the two structures ¡ Bcc spheres are touching along [111] ¡ Fcc spheres are touching along [110] ¡ ~4% larger radius for fcc NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 4rbcc = a2 + a2 + a2 → rbcc = 3a 4 =1.242nm 4rfcc = a2 + a2 → rfcc = 2a 4 =1.289nm 28
- 29. Interstitial sites infcc and bcc ¡ How many of each type of interstitials are there in the bcc and fcc unit cells? ¡ Bcc ¡ # oct sites = 6 x 0.5 + 12 * 0.25 = 6 ¡ # tet sites = 6 x 4 x 0.5 = 12 ¡ Fcc ¡ # oct sites = 1 + 3 x 4 x 0.24 = 4 ¡ # tet sites = 8 NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 bcc fcc 29
- 30. Packing fractions ¡ Ratioof the volume occupied by atoms to the total volume of unit cell ¡ Kepler’s conjecture: No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements – recently proved with help of computers by Thomas Hale NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 pfcc = 4× 4 3 πrfcc 3 a3 = 74.05% pbcc = 2× 4 3 πrbcc 3 a3 = 68.02% 30
- 31. Alloys ¡ Substitutional solidsolutions ¡ Same crystal structure as components ¡ Lattice constant governed by Vegard’s law (with some deviations) ¡ Interstitial alloys ¡ One component atom sits in interstitial sites of another large component atom ¡ Lattice constant determined by strain effects NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 31
- 32. Hume-Rothery rules 1. Atomicsize factor: The range of solid solubility will be restricted if the atomic radii differ by more than about 15%. 2. Electronegativity valency effect: Large electronegativity differences between components of a binary alloy can promote charge transfer and differences in the covalency, ionicity, or metallicity of the bonds. This leads to bond energy differences between A−A, A−B, and B−B bond energies in the alloy. A strong proclivity for A−B bond formation can lead to the formation of stable compounds. 3. Relative valency effect: A metal of lower valency is more likely to dissolve in a metal of higher valency than vice versa. This rule is not universally obeyed. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 32
- 33. Derivative and superlatticestructures in alloys ¡ At high temperatures, alloys are typically disordered. As temperature is decreased, some alloys undergo a disorder-order phase transition resulting in an ordered solid solution or superlattice or superstructure. ¡ fcc (Fm-3m) derivatives: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 L10: CuAu, FePt, FePd L12: Cu3Au,Au3Cd, AlCo3 Tetragonal P4/mmm Cubic Pm-3m 33
- 34. XRD patterns NANO 106- Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 34
- 35. Fcc- interstitial substitution ¡Diamondcubic (A4) ¡ Tet interstial in fcc occupied ¡ Common for many Group IV semiconductors (Si, Ge) ¡Rocksalt (B1) ¡ Oct interstitial occupied by a different atom ¡ Common for many ionic compounds (NaCl, LiF, etc.) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 35
- 36. Bcc derived structures ¡CsCl structure ¡ 2x2x2 bcc superlattices (characteristics of both fcc and bcc) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 D03: BiF3, Fe3Si, AlF3 L21: A2BC – Cu2MnAl 36
- 37. Diamond cubic derivedstructures ¡Zincblende (B3) ¡ III-V and II-VI compounds ¡ ZnS, AsGa, InSb ¡Fluorite – AB2 (C) ¡ CaF2 ¡Anti-fluorite ¡ Similar to fluorite, but with cations and anions reversed ¡ K2O NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 37
- 38. Hcp derived structures ¡Interstitialoccupation in tet or oct sites ¡ Wurtzite ZnO (B4) structure – occupation of one of tet sites in hcp ¡ B81 NiAs structure – occupation of oct sites NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 38
- 39. Types of ceramics ¡Restrict our discussion to binary ceramics (two components) ¡ Ionic ceramics ¡ Compounds of an electropositive cation M with an electronegative anion X, e.g., Al2O3, MgO, ZrO2, NaCl ¡ Packing and coordination governed by Pauling’s rules as discussed earlier in the course ¡ Covalent ceramics ¡ Covalent bonding ¡ Examples: GaAs, SiC, ZnO, SiO2 ¡ Note that the ionic/covalent classification is not a strict one – all structures have some ionic and covalent character NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 39
- 40. Halide salts ¡ Monovalentcations and anions ¡ Two main types of structure ¡ Rocksalt or NaCl structure (fcc derivative) ¡ CsCl structure (bcc derivative) ¡ Significantly larger radius of Cs compared to Na responsible for the difference in structures ¡ CsCl in fact exists in both the α-CsCl form, as well as a rocksalt structure at higher temperatures NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 40
- 41. Oxides of Fe ¡FeO(wustite) ¡ Rocksalt (NaCl) structure ¡Fe3O4 (Magnetite) ¡ Spinel structure ¡Fe2O3 ¡ α (hematite) ¡ γ (maghemite) ¡ β and δ are two other allotropes NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 41
- 42. Close-packed sulfides andoxides ¡Zincblende ZnS structure discussed previously ¡Corundum Al2O3 ¡ Mineral name for ruby / emerald / sapphire ¡ Almost hcp O2- with Al3+ occupying 2/3 of oct sites NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 42
- 43. Ternary oxide structures ¡Perovskite ABO3 (E21) ¡ Prototype CaTiO3 ¡ Large family of crystalline ceramics ¡ Idealized perovskite is cubic, but more typically is tetragonal or orthorhombic due to tilting or distortion ¡ Many technological applications rely on perovskite structures ¡ Solar ¡ Ferroelectrics ¡ Catalysts ¡ Superconductors ¡ …. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 43
- 44. Ternary oxide structures ¡SpinelAB2O4 ¡ Prototype MgAl2O4 ¡ Fcc sublattice of O2-, with A occupying a fraction of tet sites and B occupying a fraction of oct sites. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 13 44