3. Anisotropic blurring by dislocations

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1 Dynamical Simulation of EBSD Patterns of Imperfect Crystals 1 G. Nolze 1, A. Winkelmann 2 1 Federal Institute for Materials Research and Testing (BAM), Berlin, Germany 2 Max-Planck- Institute of Microstructure Physics, Halle, Germany 1. Introduction 2. General pattern blurring 3. Anisotropic blurring by dislocations

$approach of dynamical electron diffraction theory Based on ideal crystals!$ Computation time depends on the number of atoms per unit cell, their periodic number, the

Computation time depends on the number of atoms per unit cell, their periodic number, the

$chapter 2 in Electron Backscatter Diffraction in Materials Science, Springer, 2009 W 2 C, P$

2 EBSD pattern simulations realistic intensity simulation of EBSD patterns using a Bloch wave approach of dynamical electron diffraction theory Based on ideal crystals! Computation time depends on the number of atoms per unit cell, their periodic number, the considered number of reflectors, the kinetic energy of the electrons, the angular resolution of the intensity simulation etc. (cf. chapter 2 in Electron Backscatter Diffraction in Materials Science, Springer, 2009 W 2 C, P 3m Geometrical simulation experiment Intensity simulation 11/14/2013

3 simulation experiment Periodic number effect W Al Rule: The higher the periodic number the stronger the dynamical diffraction effects!

$Intensity simulation experiment The pattern simulation of Gold shows numerous dynamical diffraction$ effects. Bands are not visible by sharp edges anymore and display many dark and bright spots.

effects. Bands are not visible by sharp edges anymore and display many dark and bright spots.

4 Exception: Gold? Intensity simulation experiment The pattern simulation of Gold shows numerous dynamical diffraction effects. Bands are not visible by sharp edges anymore and display many dark and bright spots. In contrast, experimental patterns display unusual flat band profiles, typical for Cu, Fe etc. Intensity fluctuations along the bands are much lower than theoretically predicted. Why? Crystal defects?

5 Crystal defects Point defects: vacancy (Schottky defect), interstitials, Frenkel defect (interstitial and vacancy nearby), substitutions Linear defects: dislocations screw (b line) or edge (b line) Planar defects: grain boundaries, twins, antiphase boundaries, stacking faults Bulk defects: voids, precipitates Dislocation classification Statistically stored dislocations (SSD), geometrically necessary dislocations (GND), and grain boundary dislocations (GBD). The dislocation density is a total length of dislocations in a unit crystal volume. The dislocation density of annealed metals is about m 2. After work hardening the dislocation density increases up to m². Further increase of dislocation density causes crack formation and fracture. Hull & Bacon (2001), Introduction to Dislocations 11/14/2013 5

6 Geometrically necessary dislocations (GNDs) (simple P-lattice) During deformation dislocations will be used for material transport. Some of the mobile dislocations remain in the crystal and cause lattice rotations which can be measured by EBSD. Analyzing the misorientation angle the number of dislocations can be derived considering their character (Burgers vector). Patterns also often appear blurred because of the smaller far-ordering and the existing SSDs. GNDs cover only a small fraction of dislocations! 11/14/2013 6

Statistically stored dislocations (SSDs) Starting from a perfect lattice the

reduction of far ordering. No change in mean lattice parameter or in lattice orientations.

7 Statistically stored dislocations (SSDs) Starting from a perfect lattice the implementation of lattice defects (dislocations, interstitials, vacancies) causes a reduction of far ordering. No change in mean lattice parameter or in lattice orientations. A distinct distance between adjacent atoms changes to a smooth distribution of distances what causes the pattern blurring. Often this is equated with the effect of the Debye-Waller factors which reduce the diffracted intensity with decreasing interplanar distance d hkl. 11/14/2013 7

Influence of a mix of defects (cubic P-lattice,

atoms results in a very sharp intensity distribution.

structure will affect the intensities by a blurred

Idea: Is it possible to describe the defect

8 Influence of a mix of defects (cubic P-lattice, hypothetic) A crystal structure of well-aligned atoms results in a very sharp intensity distribution. A transition of a regular structure into a defect structure will affect the intensities by a blurred pattern only. The band positions are not affected. Idea: Is it possible to describe the defect structure by a statistical displacement of the atomic positions? 11/14/2013 8

generated and the structure factor results to: (0.03,0.

9 Simplification Shift of the atom out of (0,0,0) D=(0,0,0) (0.01,0.0075,0.005) (0.02,0.015,0.01) B=0.5 Using the space-group symmetry 192 atoms per unit cell with an occupation factor of 1/48 for each position will be generated and the structure factor results to: (0.03,0.0225,0.015) (0.04,0.03,0.02) F hkl 192 i1 k i f i exp(2 i( hx T iso i i ky B i exp 4d 2 i lz i )) T iso i k i occupation factor f i atomic scattering factor B i Debye-Waller factor (temperature factor) 9

Result for Gold experiment simulation A surprisingly good

using an average displacement of ~4% out of the origin.

pattern are mainly caused by excess deficiency effects and

10 Result for Gold experiment simulation A surprisingly good description of the experimental Au pattern has been found using an average displacement of ~4% out of the origin. Residual deviations between experimental and simulated pattern are mainly caused by excess deficiency effects and the inhomogeneous distribution of the BSE signal at 70 degrees.

Anisotropic blurring (W) Three patterns acquired at adjacent position in an EBSD

an anisotropic blurring, and is rotated down.

the intensities of the {110} planes which should be identical.

11 Anisotropic blurring (W) Three patterns acquired at adjacent position in an EBSD map. Left and right they are nearly perfect whereas the centered W-pattern shows an anisotropic blurring, and is rotated down. The loss of symmetry is clearly visible around the 111 zone axis if one compares the intensities of the {110} planes which should be identical. Also the intensity distribution in 111 is blurred along one of the {110}. Is this possible to explain by dislocations? 11/14/

12 Lattice deformations by edge dislocation b y 1 xy arctan 2 2 x 2(1 ) x y Dx y x y b 1 2 ln (1 ) x y Dy 2 Dz 0 Poisson ratio The positions of the black circles result from the given equations. The red circles displaying the inserted half plane must be added manually. The distortions of a crystal structure (here a simple cubic primitive arrangement can be calculated using Hooks law. This description works well, except of the dislocation core. Disadvantage: It breaks the translation symmetry required for a crystal. 11/14/

Edge dislocation in fcc example: edge dislocation loop Alternative way: In place of a real calculation a translationally symmetric, monoclinic super cell has been generated by subgroup relationships,

13 Edge dislocation in fcc example: edge dislocation loop Alternative way: In place of a real calculation a translationally symmetric, monoclinic super cell has been generated by subgroup relationships, and then modified using PowderCell. 110 c The used super cell has a space group symmetry of P 2/m and describes a Frank loop removing a part of an atomic layer in {111} c ={120} m. 001 c Translationally symmetric super cell with a001 c, and b110 c. The cubic cell is shown as light-blue, transparent box, i.e. the monoclinic super cell has a dimension of 10 a c x 7 2 a c x ½ 2 a c A slightly tilted bigger volume element nicely shows the effective atomic displacement. 11/14/

14 Intensity variation caused by edge dislocation loops (example: Gold) cubic reference pattern 110 c 010 m b The small displacement (Dx,Dy,0) for a limited number of atoms causes as expected an anisotropic blurring of Kikuchi bands. 11/14/

$c 010 m b A decrease of the fraction of displaced$

15 Intensity variation caused by edge dislocation loops (example: Gold) cubic reference pattern 110 c 010 m b A decrease of the fraction of displaced atoms results in a less blurred signal. 11/14/

The two colored layers display the stacking sequence along 110 c.

16 Screw dislocations in fcc materials Displacement: (0,0,Dz) Dz b 2 angle anticlockwise b=½ 110 c around the Burgers vector b; defined by x and y 001 c Volume element representing the screw dislocation (blue arrow). The two colored layers display the stacking sequence along 110 c. The volume can be described by a only two layers as content of the blackframed cell. However, this cell is again not translational-symmetric along x and y.

Screw dislocations in fcc materials b b b b b=½ 110 c 001 c By increasing the cell dimension using a mm2 point symmetry the resulting orthorhombic super cell is now

17 Screw dislocations in fcc materials b b b b b=½ 110 c 001 c By increasing the cell dimension using a mm2 point symmetry the resulting orthorhombic super cell is now translation-symmetric and can be applied as unit cell as usual. The dislocation density remains because the volume but also the number of dislocations has been increased fourfold.

Screw dislocation: 89x86x3 Å 3 dislocation

distribution around symmetryequivalent zone axes

18 Screw dislocation: 89x86x3 Å 3 dislocation density m c 001 o b 110 c 010 o b The implementation of 4 screw dislocations in a super cell causes certain deviations of the intensity distribution around symmetryequivalent zone axes (related to the cubic symmetry of the phase). 11/14/

Screw dislocation: 122x121x3 Å 3 dislocation density 6 10-15 m 2 110 c 001 o b 110 c 010 o b Smaller atomic displacement

19 Screw dislocation: 122x121x3 Å 3 dislocation density m c 001 o b 110 c 010 o b Smaller atomic displacement caused by a smaller dislocation density reduces the blurring of Kukuchi bands perpendicular to the Burgers vector. 11/14/

Effect of atomic displacements Ideal cubic structure Screw

20 Effect of atomic displacements Ideal cubic structure Screw dislocation The displacement of atoms has a unique component along 110. This singular direction is clearly visible in the pattern simulation. Edge dislocation Since the majority of atom displacements are parallel to a single 111 they are invisible at the perpendicularly aligned /14/

21 Conclusions EBSD patterns are not only affected by elastic lattice distortions (elastic strain) but also by atomic displacements caused by remaining dislocations, e.g. after plastic deformation. An atomic displacement approach delivers a sufficient description of the EBSD pattern blurring by statistically stored dislocations. The consideration of dislocations show that the mean Burgers vector in combination with the dislocation character cause anisotropic blurring of Kikuchi bands. In general: The interpretation is complex because of the superimposition of intrinsic and extrinsic defects, e.g. caused by sample prep, inhomogeneous coatings etc. 11/14/

Planar Defects in Materials. Planar Defects in Materials

Planar Defects in Materials. Planar Defects in Materials Classification of Defects in Solids: Planar defects: Stacking faults o {311} defects in Si o Inversion domain boundaries o Antiphase boundaries (e.g., super dislocations): analogous to partials but in