Chapter 7 Structure Factors and Crystal Stacking 2019 · 2019-07-09 · Duncan Alexander: Structure factors and crystal stacking LSME, EPFL Amplitude of a diffracted beam from a unit

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Duncan Alexander: Structure factors and crystal stacking LSME, EPFL

Structure factors and crystal stacking

1

Duncan AlexanderEPFL-CIME


Contents

• Atomic scattering theory

• Crystal structure factors

• Close packed structures

• Systematic absences

• Twinning and stacking faults in diffraction

• Ring diffraction patterns (nanocrystalline and amorphous)

• Measuring epitaxial relationships

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Repetition of translated structure to infinity

Crystals: translational periodicity & symmetry

3


Repetition of translated structure to infinity

Crystals: translational periodicity & symmetry

3


Elastic scattering theory

4


Consider coherent elastic scattering of electrons from isolated atom

Differential elastic scatteringcross section:

Atomic scattering factor

Scattering theory - Atomic scattering factor

5


Amplitude of a diffracted beam from a unit cell:

ri: position of each atom => ri: = xi a + yi b + zi c

K = g: K = h a* + k b* + l c*

Define structure factor:

Intensity of reflection:

Structure factor

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Note fi is a function of s and (h k l)


Stacking of close packed structures

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● For monoatomic compounds face centred cubic (FCC) and hexagonal close packed (HCP) are the most dense arrangements of atoms possible

● Both consist of hexagonal rafts of atoms called close packed planes

● These rafts stack together in sequences:

Hexagonal close packed:A - B - A - B - A -B

Cubic close packed/face centred cubic:

A - B - C - A - B - C - A - B - C

Animations from: http://departments.kings.edu/chemlab/animation/clospack.html


Stacking of close packed structures

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● FCC has a crystal structure of:

Cubic lattice(a = b = c, α = β = γ = 90º)

Lattice points:0,0,0; ½,½,0; ½,0,½; 0,½,½

● HCP has a crystal structure of:

Hexagonal lattice(a = b ≠ c, α = β = 90º, γ = 120º)

Lattice points:0,0,0; 2⁄3,1⁄3,½

● Both can have > 1 atom in the motif that combines with the lattice point, e.g.:– zinc blende structure (AaBbCc) packing based on FCC– wurtzite structure (AaBbAaBb) packing based on HCP


Consider FCC lattice with lattice point coordinates:0,0,0; ½,½,0; ½,0,½; 0,½,½

x

z

y

Calculate structure factor for plane (h k l) (assume single atom motif):

x

z

y

x

z

y

x

z

y

Structure factor and forbidden reflections

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where:

For atomic structure factor f find:

Since:

For h k l all even or all odd:

For h k l mixed even and odd:


Systematic absences

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● Face-centred cubic: reflections with mixed odd, even h, k, l absent:

● Body-centred cubic: reflections with h + k + l = odd absent:

● Reciprocal lattice of FCC is BCC and vice-versa

● What do such systematic absences mean for diffraction?

When we have them we only see diffraction spots forthe non-absent planes (h k l).


Twinning in diffraction

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Example: FCC twinsStacking of close-packed {1 1 1} planes reversed at twin boundary:

A B C A B C A B C A B C➔ A B C A B C B A C B A C


A B C A B C A B C A B C

View on [1 1 0] zone axis:

{1 1 1} planes:

1 -1 1

0 0 2

1 -1 -1



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A B C A B C A B C A B C➔ A B C A B C B A C B A C


A B C A B C A B C A B C

View on [1 1 0] zone axis:

{1 1 1} planes:

1 -1 1

0 0 2

1 -1 -1

{1 1 1} planes:{1 1 1} planes:

1 -1 1A

1 -1 -1B

A B

Duncan Alexander: Structure factors and crystal stacking LSME, EPFL 12

Example: Co-Ni-Al shape memory FCC twins observed on [1 1 0] zone axis

Images provided by Barbora Bartová, CIME

(1 1 1) close-packed twin planes overlap in SADP



With scattering from the cubic crystal we can note that the diffracted beam for plane (1 0 0)is parallel to the lattice vector [1 0 0]; makes life easy

However, not true in non-orthogonal systems - e.g. hexagonal:

x

yz

120

a

a

(1 0 0) planes

yz

120

a

a

[1 0 0]

(1 0 0) planes

yz

120

a

a

[1 0 0] g1 0 0

(1 0 0) planes

=> care must be taken in reciprocal space!

Scattering from non-orthogonal crystals

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Stacking faults in diffraction

SADP on [1 1 00] zone axis Bright-field g = 1 -1 0 0 Dark-field g = 1 -1 0 0

g g

● Stacking fault: error in stacking sequence

● Example: intrinsic stacking fault in wurtzite ZnO:– one unit cell of zinc blende structure in sequence: …AaBbAaBbAaBbCcAaBbAaBb…

● Creates thin slice of material; the convolution of its Fourier transform with diffraction spots creates streaking in wurtzite diffraction pattern

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Duncan Alexander: TEM Imaging and diffraction examples LSME, EPFL

Ring diffraction patterns

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If selected area aperture selects numerous, randomly-oriented nanocrystals,SADP consists of rings sampling all possible diffracting planes

- like powder X-ray diffraction

Example: “needles” of contaminant cubic MnZnO3 - which XRD failed to observe!Note: if scattering sufficiently kinematical, can compare intensities with those of X-ray PDF files


Nanocrystalline sample image/diffraction

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Image modeDiffraction mode

Bright field image setup - select direct beam with objective aperture



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Image modeDiffraction mode

Bright field image setup - select direct beam with objective aperture

Contrast from different crystals according to diffraction condition



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Image mode

Dark field image setup - select some transmitted beams with objective aperture

Diffraction mode

Only crystals diffracting strongly into objective aperture give bright contrast in image



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Image mode

Dark field image setup - select some transmitted beams with objective aperture

Diffraction mode

Only crystals diffracting strongly into objective aperture give bright contrast in image


Amorphous diffraction pattern

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Crystals: short-range order and long-range order

Vitrified germanium (M. H. Bhat et al. Nature 448 787 (2007)

Example:

Amorphous materials: no long-range order, but do have short-range order(roughly uniform interatomic distances as atoms pack around each other)

Short-range order produces diffuse rings in diffraction pattern

Figure from Williams & Carter“Transmission Electron Microscopy”


Measuring epitaxial relationships

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SADP excellent tool for studying orientation relationships across interfaces

Example: Mn-doped ZnO on sapphire

Sapphire substrate Sapphire + film

Zone axes:[1 -1 0]ZnO // [0 -1 0]sapphire

Planes:c-planeZnO // c-planesapphire

Duncan Alexander: Structure factors and crystal stacking LSME, EPFL 21

• The sequence of stacking of atoms in a crystal structure determines which crystal planes diffract or are systematic absences

• Specific changes in stacking sequence such as twinning and stacking faults can be identified and localised by a combination of electron diffraction and diffraction contrast imaging

• Sampling of many randomly oriented nanocrystals by selected area aperture gives ring pattern, with one ring for each family of diffracting planes

• Zone axis diffraction patterns can be used to characterise orientation relationships between neighbouring crystals

Summary on Structure Factors and Crystal Stacking

Documents

Chapter 7 Structure Factors and Crystal Stacking 2019 · 2019-07-09 · Duncan Alexander: Structure factors and crystal stacking LSME, EPFL Amplitude of a diffracted beam from a unit