Structure factors and crystal stacking

Size: px

Start display at page:

Download "Structure factors and crystal stacking"

Rosalind Phillips
5 years ago
Views:

1 Structure factors and crystal stacking Duncan Alexander EPFL-CIME 1 Contents Atomic scattering theory Crystal structure factors Close packed structures Systematic absences Twinning and stacking faults in diffraction Discriminating crystal phases by diffraction and imaging 2

2 Crystals: translational periodicity & symmetry Repetition of translated structure to infinity 3 Elastic scattering theory 4

3 Scattering theory - Atomic scattering factor Consider coherent elastic scattering of electrons from isolated atom Differential elastic scattering! cross section: Atomic scattering factor 5 Scattering theory - Huygen s principle Periodic array of scattering centres (atoms) Plane electron wave generates secondary wavelets Secondary wavelets interfere =>! strong direct beam and multiple orders of diffracted beams from constructive interference 6

4 Scattering theory - Huygen s principle Periodic array of scattering centres (atoms) Plane electron wave generates secondary wavelets k D1 Secondary wavelets interfere => strong direct beam and multiple orders of diffracted beams from constructive interference 6 Scattering theory - Huygen s principle Periodic array of scattering centres (atoms) Plane electron wave generates secondary wavelets k D1 k D2 Secondary wavelets interfere => strong direct beam and multiple orders of diffracted beams from constructive interference 6

5 Scattering theory - Huygen s principle Periodic array of scattering centres (atoms) Plane electron wave generates secondary wavelets k D1 k D2 Atoms closer together => scattering angles greater k D2 Secondary wavelets interfere =>! strong direct beam and multiple orders of diffracted beams from constructive interference => Reciprocity! k D1 6 Structure factor 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: Note fi is a function of s and (h k l) 7

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

A - B - A -B Cubic close packed/ face centred cubic: A - B - C - A - B - C - A - B - C Animations from: http://departments.kings.

html 8 Stacking of close packed structures 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;

6 Stacking of close packed structures 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: 8 Stacking of close packed structures 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 9

$reflections with h + k + l = odd absent: " Reciprocal lattice of FCC is BCC and vice-versa " What do such systematic absences mean for diffraction?$

7 Structure factor and forbidden reflections Consider FCC lattice with lattice point coordinates: 0,0,0;!,!,0;!,0,!; 0,!,! Calculate structure factor for plane (h k l) (assume single atom motif): z where: For atomic structure factor f find: Since: y For h k l all even or all odd: For h k l mixed even and odd: x 10 Systematic absences " 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 for! the non-absent planes (h k l). 11

$Twinning in diffraction Example: FCC twins Stacking of close-packed {1 1 1} planes reversed at twin boundary: A B C A B C A B C A B C A B C A B C B A C B A C View on [1 1 0] zone axis: {1 1 1}$

8 Twinning in diffraction Example: FCC twins Stacking 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 View on [1 1 0] zone axis: {1 1 1} planes: Twinning in diffraction Example: FCC twins Stacking 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 View on [1 1 0] zone axis: {1 1 1} planes: A B B A

$Twinning in diffraction Example: Co-Ni-Al shape memory FCC twins observed on [1 1 0] zone axis (1 1 1) close-packed twin planes$ $crystal we can note that the diffracted beam for plane (1 0 0) is parallel to the lattice vector [1 0 0]; makes life easy However, not$

9 Twinning in diffraction Example: Co-Ni-Al shape memory FCC twins observed on [1 1 0] zone axis (1 1 1) close-packed twin planes overlap in SADP Images provided by Barbora Bartová, CIME 13 Scattering from non-orthogonal crystals 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: z a y (1 0 0) planes a x [1 0 0] g => care must be taken in reciprocal space! 14

$Stacking faults in diffraction Stacking fault: error in stacking sequence Example: intrinsic$ $transform with diffraction spots creates streaking in wurtzite diffraction pattern SADP on$ $crystal phases Use diffraction and diffraction contrast imaging to discriminate different$ crystal phases Example of Ni3Al-based superalloy γ -phase precipitate: primitive cubic (Ni

crystal phases Example of Ni3Al-based superalloy γ -phase precipitate: primitive cubic (Ni

10 Stacking faults in diffraction 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 SADP on [1 1 00] zone axis Bright-field g = Dark-field g = g g 15 Discriminating crystal phases Use diffraction and diffraction contrast imaging to discriminate different crystal phases Example of Ni3Al-based superalloy γ -phase precipitate: primitive cubic (Ni on face centres, Al on corners) γ-phase matrix: FCC (Ni, Al disordered on sites)

$Discriminating crystal phases " Use diffraction and diffraction contrast imaging to discriminate$ different crystal phases " Example of Ni3Al-based superalloy Dark-field image g = 1 0 0 0 0 0 1 0 0

different crystal phases " Example of Ni3Al-based superalloy Dark-field image g = 1 0 0 0 0 0 1 0 0

11 Discriminating crystal phases " Use diffraction and diffraction contrast imaging to discriminate different crystal phases " Example of Ni3Al-based superalloy Bright-field image Discriminating crystal phases " Use diffraction and diffraction contrast imaging to discriminate different crystal phases " Example of Ni3Al-based superalloy Dark-field image g =

$Discriminating crystal phases " Use diffraction and diffraction contrast imaging to discriminate different crystal phases " Example of Ni3Al-based superalloy Dark-field image g = 2 0 0 0 0 0 1 0 0 2$

12 Discriminating crystal phases " Use diffraction and diffraction contrast imaging to discriminate different crystal phases " Example of Ni3Al-based superalloy Dark-field image g = Summary 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 Electron diffraction and diffraction contrast imaging are also used to discriminate and localise different crystal phases Note: dynamical scattering can give rise to appearance of certain systematic absences by a process called double diffraction 20

Imaging with Diffraction Contrast

$Imaging with Diffraction Contrast$ Imaging with Diffraction Contrast Duncan Alexander EPFL-CIME 1 Introduction When you study crystalline samples TEM image contrast is dominated by diffraction contrast. An objective aperture to select either