Crystallography- Advanced Topics

Question 1

What is crystallography good for?

Answer 1

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.

Question 2

Two types of interfaces

Answer 2

Grain boundaries: between two identical but mis-oriented phases.
Phase boundaries: between two different mineral phases.

Question 3

Zone axes for grain boundary

Answer 3

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

Question 4

Extra crystallographic descriptions to define a grain boundary

Answer 4

Two crystal orientation indices.
Two sets of Miller plane indices.
Two d-spacings.
Extra cross-interfacial volume expansion (or contraction)
Angular relationship

Question 5

Two sets of Miller plane indices for grain boundary

Answer 5

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

Question 6

Two d-spacings for grain boundary

Answer 6

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.

Question 7

Cross-interfacial volume expansion for grain boundary

Answer 7

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.

Question 8

Angular relationship for grain boundary

Answer 8

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

Question 9

Lattice misfit at phase boundaries

Answer 9

μ=(a2-a1)/a2

Where a2 is the larger lattice constant of the two phases

Question 10

Incoherent interface

Answer 10

There are no relaxations to accommodate lattice misfit.

Question 11

Semi-coherent interface

Answer 11

Periodic array of misfit dislocations (MD) to accommodate lattice misfit. MDs are structural and not subject to T (annealing etc)

Question 12

Real crystals in real space

Answer 12

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.

Question 13

Reciprocal crystals in reciprocal space

Answer 13

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.

Question 14

How do lattice constants compare in real and reciprocal space?

Answer 14

The longer one in real space is the shorter one in reciprocal space.
Real is in nm
Reciprocal is in 1/nm

Question 15

Reciprocal space as analogue to frequency domain in electronics

Answer 15

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.

Question 16

Reciprocal space as vector basis for diffraction patterns

Answer 16

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

Question 17

Reciprocal space vector for Miller indexing

Answer 17

Reciprocal space vector g always perpendicular to its plane (hkl).
g=ha+kb+lc*
For real space was:
r=Ua+Vb+Wc

Question 18

What 2D structures were known before graphene?

Answer 18

Monolayer materials connected to a substrate surface which exists for very many elements and compounds. Graphene was first freestanding 2D material known.

Question 19

Freestanding 2D materials

Answer 19

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.

Question 20

What can monolayers on a substrate be?

Answer 20

Gas phase absorband, chemical vapour deposit, terminating layers in stacking sequences, protein crystals.

Question 21

Bond between 2D layer and substrate

Answer 21

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.

Question 22

The 5 different Bravais lattices in 2D

Answer 22

Also called Bravais-nets.
Monoclinic primitive (mp)
Orthorhombic/rectangular primitive (op)
Orthorhombic centred (oc)
Tetragonal/square primitive (tp)
Hexagonal primitive (hp)

Question 23

The 4 2D crystal systems

Answer 23

Same as Bravais lattices but without orthorhombic centred (oc)

Question 24

Point groups in 2D materials

Answer 24

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

Question 25

Space groups in 2D

Answer 25

Materials are logically called plane groups. There are 17 of them combining the 5 Bravais nets with the 10 point groups

Question 26

Carbon nanotube configurations

Answer 26

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.

Question 27

Roll-up vector

Answer 27

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.

Question 28

Nanotube diameter

Answer 28

D=|C|/π

D=(a(n^2+nm+m^2)^1/2)/π