# High Angular Resolution EBSD (HR-EBSD)

## Introduction:

High angular resolution electron backscatter diffraction (HR-EBSD) provides measurements of residual elastic strain (i.e. stress with Hooke's law) and lattice rotation with very high precision. Conventional, Hough-based, EBSD has a precision of ~0.5° for lattice rotation. High resolution EBSD measures angular changes in the diffraction pattern with a precision of ~0.006°. This improved precision means deviatoric elastic strains can be measured as well as lattice rotations, with a precision of 1x10^{-4} in strain.

**NEW! --> Slides from the M&M 2016 FIG Meeting are now available.**

**Notes from RMS EBSD 2016 workshop are now available. Notes from the MAS EBSD 2016 workshop are now available (these are slightly more detailed than the RMS ones).**

## Method

An EBSD pattern is a direct projection of the crystal lattice. Each Kikuchi band reflects diffraction of a particular lattice plane. If the lattice is distorted, then the band will typically move on the screen. This means that a change in the interplanar angles will be observed as a change in angles within the diffraction pattern. These angles can be measured through measurement of pattern shifts. These pattern shifts must be measured with high precision to access elastic strain information.

In practice, the scanning electron microscope is set up in the usual manner for an EBSD experiment. Patterns are captured at high resolution (i.e. minimal binning) and with minimal noise (i.e. longer exposure times) and stored to disc for offline image analysis. [Step 1 of the figure].

The Hough based EBSD data is used as our first starting point to explore grain structure within our samples. A single reference point within each grain is selected and all patterns within that grain are compared with the reference. Ideally the reference point is chosen as a point of known strain, so that later calculations can describe the strain state of each grain in absolute terms. Often this is not possible and simply variations in strain within each grain are reported.

Regions of interest (ROI) from the reference pattern are extracted. For a one mega pattern we typically extract 256x256 pixel ROIs. The intensity distributions within these ROIs will be cross correlated with ROIs from each test pattern. The intensity distributions are converted into the Fourier domain and filtered. This process is repeated within the test pattern. [Step 2 of the figure].

Cross correlation is a numerical technique which compares the best 'fit' between two data sets. For numerical efficiency we perform cross correlation in the Fourier domain. The cross correlation function describes a series of measurements of the best fit of the reference and test ROI allowing for small translations of the test pattern with respect to the reference. A peak in the cross correlation function represents the 'best translation' to map the test and reference ROIs onto each other. This peak is upsampled to extract the peak position as a best 'guess' of the subpixel shift. This guess is possible as we know the local gradients of the cross correlation function and can therefore estimate where the peak is actually located. [Step 3 of the figure.]

Cross correlation between all the reference ROIs and test ROIs is performed to obtain pattern shifts for each ROI. These shifts form the basis of the strain measurement.

We construct a description of the distortion of the unit cell that is a best fit from the geometry of shifts observed within the diffraction pattern. Key to this measurement is the precise location of the pattern centre, which is the shortest distance between the diffraction source point on the sample and the diffraction pattern on the screen. [Step 4 of the figure.]

Once the description of the unit cell distortion is created this can be separated into the elastic strain and lattice rotation components through standard tensor approaches.

Unfortunately, a change in volume of the unit cell does not result in a change in the angle between bands within the diffraction pattern. This renders HR-EBSD insensitive to hydrostatic strain changes. We apply a traction free boundary condition, assuming that the out of plane normal stress is equal to zero, and knowledge of Hooke's law and the crystal orientation to solve for the full 3D strain tensor.

The modern variant of the technique, required for strain analysis in metallic crystals, requires an additional remapping step. This in effect is used to accomodate for significant distortions in the pattern in the presence of large lattice rotations. A first pass of the cross correlation method is used to estimate the lattice rotation which is required and a second pass, where the intensity distributions of the test pattern are remapped to be closer to the reference and then cross correlated to measure precise strain variations. [Step 6 of the figure.]

For a more complete discussion of the method, and the source of the image here, see the review by Britton et al. [1]

## History:

HR-EBSD is a technique that was initially developed by Angus Wilkinson, Graham Meaden and David Dingley who wrote the "WMD" papers in 2006 outlining the premise of the technique [2.3].

There have been numerous contributions and improvements to the technique since these papers, though the method has largely remained unchanged.

Some key highlights in the literature are:

(1) Error analysis using simulated patterns by Villert et al. [4]

(2) Robust fitting to solve for the strain tensor by Britton et al. [5]

(3) Remapping algorithms introduced by Maurice et al. [6] and Britton et al. [7]

(4) Introduction of GND analysis with HR-EBSD by Wilkinson and Randman for indents in Fe (BCC). [8]

(5) Introduction of GND analysis with HR-EBSD by Karamched et al. for carbides in Ni superalloys (FCC). [9]

(6) Introduction of GND analysis with HR-EBSD by Britton et al. for indents in Ti (HCP). [10]

(7) Reference pattern problems explored by Britton et al. hinting that use of a simulated diffraction pattern is very difficult (and has likely not been successfully used to date). [11]

(8) Tracing to NIST standards through AFM and Raman analysis by Vaudin et al. [12, 13]

(9) Discussion of precision and accuracy and understanding detectors by Britton et al. [14, 15]

(10) Analysis of pattern overlap and accuracy for EBSD by Tong et al. [16]

## References:

[1] Britton, Jiang, Karamched, and Wilkinson (2013) JOM

[2] Wilkinson, Meaden and Dingley (2006) Ultramicroscopy

[3] Wilkinson, Meaden and Dingley (2006) Materials Science and Technology

[4] Villert, Maurice, Wyon and Fortunier (2009) Journal of Microscopy-Oxford

[5] Britton and Wilkinson (2011) Ultramicroscopy

[6] Maurice, Driver and Fortunier (2012) Ultramicroscopy

[7] Britton and Wilkinson (2012) Ultramicroscopy

[8] Wilkinson and Randman (2010) Phil. Mag.

[9] Karamched and Wilkinson (2011) Acta Mat.

[10] Britton, Liang, Dunne and Wilkinson (2010) Proc. Roy. Soc. A

[11] Britton, Maurice, Fortunier Driver, Day, Meaden, Dingley, Mingard, and Wilkinson (2010) Ultramicroscopy

[12] Vaudin, Gerbig, Stranick and Cook (2008) Applied Physics Letters

[13] Vaudin, Stan, Gerbig, and Cook (2011) Ultramicroscopy

[14] Britton, Jiang, Clough, Tarleton, Kirkland and Wilkinson (2013) Ultramicroscopy v135, p126-135

[15] Britton, Jiang, Clough, Tarleton, Kirkland and Wilkinson (2013) Ultramicroscopy v135, p136-141

[16] Tong, Jiang, Wilkinson and Britton (2015) Ultramicroscopy