Crystallography- Advanced Topics Flashcards
What is crystallography good for?
Common language between scientists. Property prediction. Crystallography software. X-ray/electron diffraction patterns. Description of interfaces and grain boundaries. Description of 2D periodic sheet materials.
Two types of interfaces
Grain boundaries: between two identical but mis-oriented phases.
Phase boundaries: between two different mineral phases.
Zone axes for grain boundary
Two crystal orientation indices: [uvw]1 vs [uvw]2. Tells you the viewing direction of the image on the page. Can be the same for both grains either side of boundary
Extra crystallographic descriptions to define a grain boundary
Two crystal orientation indices. Two sets of Miller plane indices. Two d-spacings. Extra cross-interfacial volume expansion (or contraction) Angular relationship
Two sets of Miller plane indices for grain boundary
They specify the two surfaces touching at the interface. Each (hkl)1,2 being calculated with respect to its own bulk lattice. They define the grain boundary
Two d-spacings for grain boundary
Two d-spacings from each lattice projected onto the interface plane. Distance between two atoms along the line parallel to the grain boundary. Shows the periodicity at the boundary.
Cross-interfacial volume expansion for grain boundary
Translation of one lattice relative to the other. If a d-spacing perpendicular to grain boundary in one crystal is same as the corresponding d-spacing across the boundary, δV=0. If across the boundary it is bigger, then δV>0.
Angular relationship for grain boundary
Covers the tilt and twist component of the two grains relative to each other. α1 is angle between grain boundary and some plane of atoms in one grain. The two grains can have the same or different α1. α2 tells you about the twist in the boundary plane. α2=0 when no twist between grains
Lattice misfit at phase boundaries
μ=(a2-a1)/a2
Where a2 is the larger lattice constant of the two phases
Incoherent interface
There are no relaxations to accommodate lattice misfit.
Semi-coherent interface
Periodic array of misfit dislocations (MD) to accommodate lattice misfit. MDs are structural and not subject to T (annealing etc)
Real crystals in real space
Use metric calibrated coordinated axes often in Cartesian coordinate system (x,y,z all orthogonal) or another symmetry adapted (x,y,z) system such as monoclinic with oblique angles. Due to infinite periodicity of all crystals, all real space properties related to periodic order of crystals are delocalised over the infinite crystal extension.
Reciprocal crystals in reciprocal space
Coordinate axes measure reciprocal length (1/nm). Periodic repeat lengths, such as lattice constant a, are measured as frequencies. One point in reciprocal space describes the periodicity of the real space infinite lattice.
How do lattice constants compare in real and reciprocal space?
The longer one in real space is the shorter one in reciprocal space.
Real is in nm
Reciprocal is in 1/nm
Reciprocal space as analogue to frequency domain in electronics
A signal in real space for electronics gives a time periodicity in sec. The spectrum (reciprocal space) shows spectral frequency (v) in Hz. For a sinusoidal signal gives 3 peaks (-v,0,v). v0=1/T.
In crystallography the signal gives d-spacing in nm. Spectrum gives spatial frequency (κ) in 1/nm. Same 3 peaks at -κ,0,κ.
Reciprocal space as vector basis for diffraction patterns
Use electron microscope in diffraction mode or X-ray diffraction machine. Bragg equation sinθ=nλ/d. If θ roughly constant then 1/d=constant*κ. Measure distance r from centre to the point on the screen. r(screen)=λL/d=λLκ d is lattice distance λ is wavelength L is camera length κ is spatial frequency
Reciprocal space vector for Miller indexing
Reciprocal space vector g always perpendicular to its plane (hkl).
g=ha+kb+lc*
For real space was:
r=Ua+Vb+Wc
What 2D structures were known before graphene?
Monolayer materials connected to a substrate surface which exists for very many elements and compounds. Graphene was first freestanding 2D material known.
Freestanding 2D materials
Freestanding monolayers can be derived from naturally layered 3D candidate materials which can be exfoliated. Stability of material once extracted as a single layer against roll-up, zig-zagging, ribbling, structural collapse or chemical reaction is the core question. Apart from graphene there is BN, disulphides, diselenides, some TM oxides via cleavage. Graphene monolayers and others normally partially supported and slightly buckle if not connected to flattening substrate.
What can monolayers on a substrate be?
Gas phase absorband, chemical vapour deposit, terminating layers in stacking sequences, protein crystals.
Bond between 2D layer and substrate
Could be as strong as the intra-layer bond. The 2D crystals would adapt to the crystal structure of the substrate surface or build specific reconstructions.
The 5 different Bravais lattices in 2D
Also called Bravais-nets. Monoclinic primitive (mp) Orthorhombic/rectangular primitive (op) Orthorhombic centred (oc) Tetragonal/square primitive (tp) Hexagonal primitive (hp)
The 4 2D crystal systems
Same as Bravais lattices but without orthorhombic centred (oc)
Point groups in 2D materials
Only 10 of the 32 3D point groups needed to describe all possible compatible combinations of symmetry elements. Same 3D name:
1, 2, m, mm2, 4, mm4, 3, 3m, 6, mm6
Space groups in 2D
Materials are logically called plane groups. There are 17 of them combining the 5 Bravais nets with the 10 point groups
Carbon nanotube configurations
Roll up layer of graphene to form a tube. Can be rolled in any direction (2 orthogonal and any diagonal between them). One orthogonal is called zig-zag (rolled in direction along hexagon centres) and other is armchair (rolled in direction along centre, edge, centre, edge…). All directions apart from armchair and zig-zag are chiral. No special name for θ=45.
Roll-up vector
C=na1+ma2
a1 is vector from centre of one hexagon to centre of the one next to it on the same horizontal line to the right.
a2 is vector from centre of same starting hexagon to centre of one just up and right from it.
n and m are scalars. C=na1 for zig-zag. C=na1+na2 for armchair.
The non-chiral directions run along the two mirror planes parallel to a1 and (a1+a2)/2.
Nanotube diameter
D=|C|/π
D=(a(n^2+nm+m^2)^1/2)/π
