# Computation of Burgers Vectors from Elastic Strain and Lattice Rotation Data

J. Cloete
Department of Engineering Science
University of Oxford
Parks Road, Oxford OX1 3PJ, UK
<EMAIL_ADDRESS>
& F. Hofmann
Department of Engineering Science
University of Oxford
Parks Road, Oxford OX1 3PJ, UK
<EMAIL_ADDRESS>
& E. Tarleton
Department of Materials
University of Oxford
Parks Road, Oxford OX1 3PH, UK
<EMAIL_ADDRESS>

###### Abstract

A theoretical framework for computation of Burgers vectors from strain and
lattice rotation data is presented, as well as implementation into a computer
program to automate the process. The efficacy of the method is verified using
simulated data of dislocations with known results. A 3D data set retrieved
from Bragg coherent diffraction imaging (BCDI) and a 2D data set from high
resolution transmission Kikuchi diffraction (HR-TKD) are used as inputs to
demonstrate the reliable identification of dislocation positions and accurate
determination of Burgers vectors from experimental data. For BCDI data, the
results found using our approach show very close agreement to those expected
from empirical methods. For the HR-TKD data the predicted dislocation position
and the computed Burgers vector showed fair agreement with the expected
result, which is promising considering the substantial experimental
uncertainties in this dataset. The method reported in this paper provides a
general and robust framework for determining dislocation positions and Burgers
vectors, and can be readily applied to data from other experimental
techniques.

_Keywords_ Dislocations $\cdot$ Burgers Vectors $\cdot$ BCDI $\cdot$ HR-TKD

##  1 Introduction

Crystal lattice defects such as dislocations play a key role in controlling
the properties of high performance materials central to modern life. Examples
include alloys for aerospace, nuclear and automotive applications [1, 2, 3].
Dislocations are also important in semiconductors, for example in photovoltaic
devices, for they act as impurity segregation sites that are detrimental to
solar cell performance [4, 5, 6, 7].

To optimise material performance, a detailed understanding of dislocations is
key. This is particularly important for tuning materials in order to modify
dislocation behaviour [8, 9, 10], e.g. by modifying the microstructural
landscape of the material. Indeed this is one of the main ways modern alloys
are optimised [11, 12, 13, 14].

Dislocations interact via the strain fields they cause [8, 9]. As such,
measurement techniques that allow us to image strain fields are of great
importance. It is possible to measure 2D strain fields surrounding
dislocations at the nano-scale using high resolution transmission electron
microscopy (HR-TEM) and scanning transmission electron microscopy [15, 16,
17], as well as high resolution transmission Kikuchi diffraction (HR-TKD) in
the scanning electron microscope [18, 19]. To measure 3D strain fields and
morphology of microcrystals, Bragg coherent diffraction imaging (BCDI) has
achieved successful results [20, 21, 22].

A key challenge when characterising dislocations is the accurate determination
of the Burgers vector [23] for specific dislocations. A classic approach is to
use $\bm{g}\cdot\bm{b}$ contrast, wherein dislocations cause little
diffraction contrast when the diffraction vector $\bm{g}$ is perpendicular to
the Burgers vector $\bm{b}$, i.e. when the invisibility criterion
$\bm{g}\cdot\bm{b}=0$ is met [9]. For materials with well-known
crystallography, multiple $\bm{g}$-vectors are required in TEM as there are
several possible directions that the Burgers vector can take, and only by
finding two $\bm{g}$-vectors for which the invisibility criterion is met can
the Burgers vector direction be determined. Even then, there is ambiguity to
the sign of the Burgers vector as it could be positive or negative, and this
ambiguity can only be resolved by detailed comparison with diffraction image
simulations [22, 19]. There is also no immediate indication of Burgers vector
magnitude from these techniques, and one must refer to the known
crystallographic structure of the material to determine this. A further
problem arises where the preferred Burgers vector directions are not known a
priori. In this case, $\bm{g}\cdot\bm{b}$ contrast cannot be easily applied as
there is no limited set of Burgers vectors to try.

Since measurements of the full lattice strain tensor are now readily possible
in 2D and 3D, an attractive idea is to use the measured strain fields at the
continuum scale to directly determine the Burgers vector of specific
dislocations. Two methods present themselves; the first being to integrate the
displacement gradient of the crystal lattice about a Burgers circuit enclosing
the dislocation, the second to integrate the computed Nye tensor (proposed by
Nye [24]) of the crystal across a surface through which the dislocation passes
[25].

Here, we first lay out the underlying theory and concepts used before
presenting a framework for determining dislocation Burgers vector direction
and magnitude from experimentally-measured strain and lattice rotation fields.
We also cover the key concepts for implementation of this framework into a
program. The Nye tensor approach is initially investigated, however this
method turns out not to be suitable for the task at hand. Thus, the framework
presented in this paper relies on integration of the displacement gradient
around a Burgers circuit. Initially we consider a mathematical model of the
displacement gradient surrounding an infinite straight mixed dislocation of
configurable sense $\bm{\xi}$ and Burgers vector $\bm{b}$. This is used to
verify the efficacy of the method and reliability of the approach as well as
its sensitivity to factors such as noise. Experimental data sets from BCDI and
HR-TKD techniques are then used to locate and characterise dislocations in
real material specimens. Comparisons to known results found from techniques
such as $\bm{g}\cdot\bm{b}$ contrast will then be made.

##  2 Theory and Methodology

###  2.1 Mathematical Definition of a Burgers Vector

(a)

(b)

Figure 1: a) A dislocation line of Burgers vector $\bm{b}$ and direction
defined by sense $\bm{\xi}$ enclosed by Burgers circuit $\bm{C}$. b)
Integration loop $\bm{C}$, now in a form suitable for numerical computation of
Burgers vector $\bm{b}$.

For any vector field, $\bm{V}$, that is continuous along the path of
integration, the closed-loop line integral of its gradient, $\bm{\nabla V}$,
must be $\bm{0}$. However if the vector field contains a discontinuity, for
example the displacement field $\bm{u}$ of a material containing dislocations,
it is possible for the result to be non-zero. In our case, the result is the
Burgers vector $\bm{b}$, which describes the discontinuity in the material
structure due to the presence of the dislocation [8, 9].

Therefore a mathematical definition for the Burgers vector $\bm{b}$ of a
dislocation is the line integral of the elastic displacement gradient
$\bm{\beta}=\bm{\nabla u}$ along the closed loop $\bm{C}$ that encloses the
dislocation (also known as the Burgers circuit, defined in the original
undistorted coordinate system) [8];

$\displaystyle\bm{b}=\oint_{C}\bm{\beta}\cdot d\bm{l}$ (1)

where

$\displaystyle d\bm{l}=\begin{bmatrix}dx\\\ dy\\\ dz\end{bmatrix}$ (2)

(Note that $\bm{\beta}=\bm{\varepsilon}+\bm{\omega}$, the elastic strain and
lattice rotation tensor fields respectively) [26, 25].

This means that, as long as we know the strain and lattice rotation fields
within a material, the Burgers vector $\bm{b}$ for dislocations enclosed
within some Burgers circuit $\bm{C}$ can be calculated.

###  2.2 The Nye Tensor Approach

On the continuum scale, Nye defined the dislocation tensor (or ‘Nye tensor’)
$\bm{\alpha}^{Nye}$ as a geometrical relation between lattice curvature and
the distribution of geometrically-necessary dislocations (GNDs); dislocations
required to accommodate the plastic strain gradients in a material [8, 24,
25]. The net Burgers vector $\bm{B}$ for a set of dislocations passing through
a surface $\bm{S}$ can be defined by the surface integral

$\displaystyle\bm{B}=\iint_{S}\bm{\alpha}^{Nye}\cdot d\bm{S}$ (3)

One can apply Stokes’ Theorem to the line integral from Eq.(1), which results
in the following surface integral [25];

$\displaystyle\bm{b}=\iint_{S}(\bm{\nabla}\times\bm{\beta})^{T}\cdot d\bm{S}$
(4)

where $\bm{S}$ is the surface enclosed by Burgers circuit $\bm{C}$.

The resulting integrand is in fact the negative of the Nye tensor,
$\bm{\alpha}^{Nye}=-(\bm{\nabla}\times\bm{\beta})^{T}$, which can be more
easily interpreted as

$\displaystyle\bm{\alpha}^{Nye}=\begin{bmatrix}\partialderivative{\beta_{12}}{z}-\partialderivative{\beta_{13}}{y}&\partialderivative{\beta_{13}}{x}-\partialderivative{\beta_{11}}{z}&\partialderivative{\beta_{11}}{y}-\partialderivative{\beta_{12}}{x}\\\
\partialderivative{\beta_{22}}{z}-\partialderivative{\beta_{23}}{y}&\partialderivative{\beta_{23}}{x}-\partialderivative{\beta_{21}}{z}&\partialderivative{\beta_{21}}{y}-\partialderivative{\beta_{22}}{x}\\\
\partialderivative{\beta_{32}}{z}-\partialderivative{\beta_{33}}{y}&\partialderivative{\beta_{33}}{x}-\partialderivative{\beta_{31}}{z}&\partialderivative{\beta_{31}}{y}-\partialderivative{\beta_{32}}{x}\end{bmatrix}$
(5)

Note that the appearance of the minus sign is due to ambiguity caused by
alternate definitions of the Burgers vector. The Nye tensor is actually
related to the plastic displacement gradient (or slip tensor) $\bm{\beta}^{p}$
via $\bm{\alpha}^{Nye}=(\bm{\nabla}\times\bm{\beta}^{p})^{T}$ [25], such that
upon substitution into Eq.(3) and applying Stokes’ Theorem,

$\displaystyle\bm{B}=\oint_{C}\bm{\beta}^{p}\cdot d\bm{l}$ (6)

However, it must be true that

$\displaystyle\oint_{C}(\bm{\beta}+\bm{\beta}^{p})\cdot d\bm{l}=\bm{0}$ (7)

as total displacement around a closed loop $\bm{C}$ must be zero [25].
Therefore it must be the case that $\bm{B}=-\bm{b}$; in other words the two
expressions for the Burgers vector are opposite in direction. Since the sign
of the Burgers vector is determined by the sign of the sense $\bm{\xi}$, which
is itself arbitrary [8, 9], the impact of the conflicting definitions is minor
but should be borne in mind. In this paper Eq.(1) as defined by Anderson et
al. [8] is the authoritative definition.

Nye related $\bm{\alpha}^{Nye}$ to the dislocation distribution inside a
material [8, 24, 25]; for each possible slip system $\lambda$, given $q$
dislocations per unit area each with Burgers vector $\bm{b}$ and sense
$\bm{\xi}$ threading the plane,

$\displaystyle\alpha^{Nye}_{km}=qb_{k}\xi_{m}$ (8)

Let GND density $\bm{\rho}$ be defined such that $\rho_{m}=q\xi_{m}$;
therefore

$\displaystyle\bm{\alpha}^{Nye}=\sum_{i=1}^{\lambda}(\bm{b}^{i}\otimes\bm{\rho}^{i})$
(9)

One might argue that if $\bm{\alpha}^{Nye}$ can be found, the Burgers vector
$\bm{b}$ can be evaluated using the surface integral as opposed to the line
integral. An immediate benefit would be improved numerical integration, as one
could integrate across an entire surface of voxels rather than just a line.
The significantly increased number of data points used would hopefully reduce
error due to noise and discretisation of the data.

However, upon further investigation it has been deduced that the Nye tensor
approach to calculating the Burgers vector is not feasible for the
applications covered in this paper. This is most easily explained by first
examining the mathematical models of infinite straight dislocations.

###  2.3 Mathematical Model of an Infinite Straight Mixed Dislocation

It should be noted that the mathematical dislocation model used in this paper
is singular; meaning that the discontinuity associated with the dislocation is
confined to an infinitesimal line (the dislocation line, in fact).

Let $x$, $y$ and $z$ be the three dimensions in a Cartesian coordinate system,
with respective unit basis vectors $\bm{e_{x}}$, $\bm{e_{y}}$ and
$\bm{e_{z}}$. The constant $v$ is the hypothetical Poisson’s ratio of the
material, and $b$ is the magnitude of the Burgers vector. Angle $\alpha$ is
the angle between the sense and Burgers vector, such that $\alpha=0\degree$
corresponds to a screw dislocation and $\alpha=90\degree$ to an edge
dislocation.

For an infinite straight mixed dislocation with sense $\bm{\xi}$ =
$\mathbf{e_{z}}$ and Burgers vector $\mathbf{b}$ confined to the $x-z$ plane
(see Fig. 11 in appendix AA.3), the corresponding displacement gradient
$\bm{\beta}$ is given by

$\displaystyle\bm{\beta}=\bm{\nabla
u}=\begin{bmatrix}\partialderivative{u_{x}}{x}&\partialderivative{u_{x}}{y}&\partialderivative{u_{x}}{z}\\\
\partialderivative{u_{y}}{x}&\partialderivative{u_{y}}{y}&\partialderivative{u_{y}}{z}\\\
\partialderivative{u_{z}}{x}&\partialderivative{u_{z}}{y}&\partialderivative{u_{z}}{z}\end{bmatrix}=\begin{bmatrix}\beta_{11}&\beta_{12}&0\\\
\beta_{21}&\beta_{22}&0\\\ \beta_{31}&\beta_{32}&0\end{bmatrix}$ (10)

where

$\displaystyle\beta_{11}=-\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{y}{(x^{2}+y^{2})^{2}}\left[(3-2v)x^{2}+(1-2v)y^{2}\right],$
(11)
$\displaystyle\beta_{12}=\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{x}{(x^{2}+y^{2})^{2}}\left[(3-2v)x^{2}+(1-2v)y^{2}\right],$
(12)
$\displaystyle\beta_{21}=-\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{x}{(x^{2}+y^{2})^{2}}\left[(1-2v)x^{2}+(3-2v)y^{2}\right],$
(13)
$\displaystyle\beta_{22}=\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{y}{(x^{2}+y^{2})^{2}}\left[(1+2v)x^{2}-(1-2v)y^{2}\right],$
(14)
$\displaystyle\beta_{31}=-\frac{b\cos(\alpha)}{2\pi}\frac{y}{(x^{2}+y^{2})},$
(15)
$\displaystyle\beta_{32}=\frac{b\cos(\alpha)}{2\pi}\frac{x}{(x^{2}+y^{2})}.$
(16)

Derivations for the above expressions as well as information on applying
rotation matrices to the dislocation can be found in appendix A and are based
on the elastic dislocation solution reported by Anderson et al [8].

Note that we can convert the result into strain and lattice rotation tensors
using the definitions
$\bm{\varepsilon}=\frac{1}{2}\left(\bm{\beta}+\bm{\beta}^{T}\right)$ and
$\bm{\omega}=\frac{1}{2}\left(\bm{\beta}-\bm{\beta}^{T}\right)$ [26, 25].

To find $\bm{\alpha}^{Nye}$ we populate Eq.(5) with the elements of
$\bm{\beta}$. The only non-zero terms are

$\displaystyle\partialderivative{\beta_{11}}{y}=\partialderivative{\beta_{12}}{x}=-\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{1}{(x^{2}+y^{2})^{3}}\left[(3-2v)x^{4}-6x^{2}y^{2}-(1-2v)y^{4}\right],$
(17)
$\displaystyle\partialderivative{\beta_{21}}{y}=\partialderivative{\beta_{22}}{x}=-\frac{b\sin(\alpha)}{4\pi(1-v)}\frac{2xy}{(x^{2}+y^{2})^{3}}\left[(1+2v)x^{2}-(3-2v)y^{2}\right],$
(18)
$\displaystyle\partialderivative{\beta_{31}}{y}=\partialderivative{\beta_{32}}{x}=\frac{b\cos(\alpha)}{2\pi}\frac{(y^{2}-x^{2})}{(x^{2}+y^{2})^{2}}$
(19)

and these terms all cancel, resulting in $\bm{\alpha}^{Nye}=\bm{0}$.

In fact, the Nye tensor is always $\bm{0}$ for individual (singular)
dislocations. This may at first seem counter-intuitive, however the reason is
that the Nye tensor is evaluated at an infinitesimal point (as we are taking
the curl of a tensor field, $\bm{\beta}$), and all of the discontinuities
associated with the (singular) dislocation are confined to an infinitesimally
thin line (i.e. the dislocation core). Essentially, the Nye tensor corresponds
to having an infinitesimally small Burgers circuit. While this works
mathematically across regions with dislocation presence defined by a density
per unit area, it cannot work when we consider individual dislocations such
that dislocation density is effectively zero everywhere and infinite at the
dislocation cores.

Even for experimentally measured dislocations, deviations from linear
elasticity theory will be limited to a small region around the dislocation
core [9], thus using Nye tensor to determine the Burgers vector of an
individual dislocation from experimental data is not feasible.

###  2.4 Numerical Computation

Elastic strains and lattice rotations obtained experimentally are measured as
discrete voxels of data rather than being defined by algebraic functions (as
with the mathematical model). Therefore, rather than integrating analytically,
we must make use of numerical techniques.

To compute Burgers vector $\bm{b}$ we perform the line integral in Eq.(1).
Note that, in Cartesian coordinates, $\bm{\beta}\cdot d\bm{l}$ can be written
as

$\displaystyle\bm{\beta}\cdot
d\bm{l}=\begin{bmatrix}\beta_{11}&\beta_{12}&\beta_{13}\\\
\beta_{21}&\beta_{22}&\beta_{23}\\\
\beta_{31}&\beta_{32}&\beta_{33}\end{bmatrix}\cdot\begin{bmatrix}dx\\\ dy\\\
dz\end{bmatrix}=\begin{bmatrix}\beta_{11}dx+\beta_{12}dy+\beta_{13}dz\\\
\beta_{21}dx+\beta_{22}dy+\beta_{23}dz\\\
\beta_{31}dx+\beta_{32}dy+\beta_{33}dz\end{bmatrix}$ (20)

Therefore, integrating $\bm{\beta}$ with respect to $\bm{l}$ requires
integrating each of the nine components of $\bm{\beta}$ with respect to either
$x$, $y$ or $z$. The limits for, and equations relating, these three variables
are determined by the choice of Burgers circuit $\bm{C}$.

The simplest Burgers circuit design that will be effective for dislocations in
any orientation in 3D space is the loop $\bm{C}$ shown in Fig. 1 b). The
Burgers circuit encloses a cube (or cuboid), and is designed such that a
right-hand screw dislocation pointing in the direction of a positive
coordinate axis gives a positive result for $\bm{b}$ (this should be borne in
mind when considering the signs of computed Burgers vector directions).

For integration along loop $\bm{C}$ shown, along each of the six straight
lines two of $dx$, $dy$ or $dz$ will be zero. Thus the line integral of
displacement gradient $\bm{\beta}$ is reduced to a total of 18 integrals of
one variable (six for each of the $x$-, $y$\- and $z$-components of $\bm{b}$)
that can each be computed individually. This is ideal for numerical
integration.

For data that is discrete and exists across a set of voxels, we can define a
Burgers circuit that passes through the centres of a chain of voxels to form
integration loop $\bm{C}$. We can then integrate the discrete displacement
gradient data numerically, for example by using MATLAB’s trapz function or a
variation thereof, and travelling along the loop of voxels. This numerical
approach was used to produce a MATLAB program that computes Burgers vectors
from arrays of 3D or 2D elastic strain and lattice rotation data. Further
information on implementation can be found in appendix B.

##  3 Tests - Mathematical Models

###  3.1 Test Methodology

Figure 2: Integration loop $\bm{C}$, now with vertex coordinates defined by
the coordinate limits in the $x$, $y$ and $z$ directions. The dislocation line
for Case 8 is also present, passing through the origin and with $\bm{b}$ and
$\bm{\xi}$ as shown. Note that in this particular case the dislocation line
passes through the edges of the integration loop, which causes an error in
computation.

To test the reliability of Burgers vector determination, we initially consider
a mathematical model of an infinite, straight, mixed dislocation. Nine cases
were chosen and used to confirm that the program can accurately compute the
Burgers vector for dislocations of various orientations. The vertices of
Burgers circuit $\bm{C}$ are defined by the coordinate limits shown in Fig. 2.

A set of parameters must be chosen not only to generate the mathematical model
of the dislocation, but also to prepare the program for the specific set of
input data, as the size of the voxel spacing must be taken into account for
numerical integration.

The parameters for the test cases were as follows:
Burgers Vector Magnitude, $b=1$ $\mathrm{\SIUnitSymbolAngstrom}$
Voxel Size $=5$ $\mathrm{n}\mathrm{m}$ (cube)
Poisson’s Ratio, $\upsilon=0.3$
Noise Coefficient, $\eta=0$
Burgers Circuit limits = [-102.5 $\mathrm{n}\mathrm{m}$, 102.5
$\mathrm{n}\mathrm{m}$] (expect for Case 9)

Note that the voxel spacing and coordinate limits used were such that the
Burgers circuit $\bm{C}$ was defined around a cube with a side-length of 42
pixels and centred on the origin. Each of the dislocations also passed through
the origin.

The following cases were considered, where the angles $\alpha$, $\psi$,
$\theta$ and $\phi$ define the directions of Burgers vector $\bm{b}$ and sense
$\bm{\xi}$, as depicted in Fig. 11 in appendix AA.3:

  1. 1.

Z-Screw; $\bm{\xi}=\bm{e_{z}}$, $\bm{b}=b\bm{e_{z}}$ ($\alpha=0\degree$ ,
$\psi=0\degree$, $\theta=0\degree$, $\phi=0\degree$)

  2. 2.

X-Screw; $\bm{\xi}=\bm{e_{x}}$, $\bm{b}=b\bm{e_{x}}$ ($\alpha=0\degree$ ,
$\psi=0\degree$, $\theta=90\degree$, $\phi=0\degree$)

  3. 3.

Y-Screw; $\bm{\xi}=\bm{e_{y}}$, $\bm{b}=b\bm{e_{y}}$ ($\alpha=0\degree$ ,
$\psi=0\degree$, $\theta=90\degree$, $\phi=90\degree$)

  4. 4.

Z-Edge; $\bm{\xi}=\bm{e_{z}}$, $\bm{b}=b\bm{e_{x}}$ ($\alpha=90\degree$ ,
$\psi=0\degree$, $\theta=0\degree$, $\phi=0\degree$)

  5. 5.

Mixed 1 ($\alpha=45\degree$ , $\psi=90\degree$, $\theta=90\degree$,
$\phi=45\degree$)

  6. 6.

Mixed 2 ($\alpha=45\degree$ , $\psi=0\degree$, $\theta=0\degree$,
$\phi=45\degree$)

  7. 7.

Sum of Three Screws (Z-Screw + X-Screw + Y-Screw)

  8. 8.

Error Case ($\alpha=0\degree$ , $\psi=0\degree$, $\theta=135\degree$,
$\phi=0\degree$)

  9. 9.

Error Case ($\alpha=0\degree$ , $\psi=0\degree$, $\theta=135\degree$,
$\phi=0\degree$), new z-axis limits = [-122.5 $\mathrm{n}\mathrm{m}$, 122.5
$\mathrm{n}\mathrm{m}$]

Cases 1 through 4 as well as 7 should be straightforward to visualise, and
visualisations of Cases 5 and 6 are provided in Fig. 3 and Case 8 in Fig. 2.

(a)

(b)

Figure 3: A visualisation of Cases 5 (a) and 6 (b), each with the dislocation
line, sense $\bm{\xi}$, and Burgers vector $\bm{b}$ shown.

The robustness of the program was also tested. Using Case 7 as a basis, three
tests were performed. In Test 1, the Burgers circuit side length was
decreased, two voxels at a time, from 42 to 2. In Test 2, the fractional
distance of the dislocation line from the Burgers circuit centre was increased
from 0 (centre) to 1 (Burgers circuit edge). In Test 3, the noise coefficient
$\eta$ was exponentially increased from $1\times 10^{-6}$ to $7.86\times
10^{-4}$ and the number of concentric Burgers circuits used for removing
outliers (with $x$-, $y$\- and $z$-components beyond a certain number of
standard deviations from the mean values) and averaging was increased from 1
to 10. For each of the tests, the change in computed Burgers vector is
reported as the angular error and percentage magnitude error compared to the
ground truth.

Experimental noise/uncertainty was modelled using a standard Gaussian
distribution multiplied by the noise coefficient $\eta$ and superposed onto
each element of the strain and lattice rotation tensors. The distribution was
truncated to within the limits of 1 standard deviation, as otherwise
unrealistically large magnitudes of noise may appear in the data. The noise
applied to one element of data was independent to the noise applied to
adjacent elements of data.

###  3.2 Results

Case No. |  True Magnitude ($\mathrm{\SIUnitSymbolAngstrom}$) |  True Direction |  Computed Magnitude ($\mathrm{\SIUnitSymbolAngstrom}$) |  Computed Direction |  Magnitude Percentage Error (%) |  Angular Error (°)
---|---|---|---|---|---|---
1. |  1 |  0 0 1 |  1.0000 |  0 0 1 |  2.4620e-6 |  0.0000
2. |  1 |  1 0 0 |  1.0000 |  1 0 0 |  2.4620e-6 |  0.0000
3. |  1 |  0 1 0 |  1.0000 |  0 1 0 |  2.4620e-6 |  0.0000
4. |  1 |  1 0 0 |  1.0000 |  1 0 0 |  2.4620e-6 |  0.0000
5. |  1 |  0 1 0 |  1.0000 |  0.0000 1.0000 0 |  7.4046e-6 |  3.0783e-6
6. |  1 |  0.5 0.5 0.7071 |  1.0000 |  0.5000 0.5000 0.7071 |  2.4620e-6 |  0.0000
7. |  1.7321 |  0.5774 0.5774 0.5774 |  1.7321 |  0.5774 0.5774 0.5774 |  2.4620e-6 |  0.0000
8. |  1 |  0.7071 0 -0.7071 |  0 |  NaN NaN NaN |  N/A |  N/A
9. |  1 |  0.7071 0 -0.7071 |  1.0000 |  0.7071 0 -0.7071 |  3.1500e-5 |  0.0000
Table 1: Results for the input cases using the simulated strain and lattice
rotation data. Note that zero noise is present for any of these cases.

From Table 1, we see that in all cases (bar number 8), the computed Burgers
vector is virtually identical to the expected result. Both magnitude
percentage error and angular error between the computed and expected results
are negligible for the input parameters given. This is as expected for the
"noise-free" case, where the strain and lattice rotation fields had been
mathematically determined such that the expected Burgers vectors would be
computed exactly. Obviously this is not the case for experimental data, where
the program can only provide an estimate of the Burgers vector given the
measured strains and lattice rotations.

In interesting error case (8) arises when the dislocation line intersects the
closed loop that defines the Burgers circuit. Upon adjusting the Burgers
circuit such that the $z$-axis coordinate limits are [-122.5
$\mathrm{n}\mathrm{m}$, 122.5 $\mathrm{n}\mathrm{m}$] (such that the cuboid
was 50 voxels wide in the $z$-direction) this intersection no longer occurs
and the Burgers vector is computed correctly. Therefore, care must be taken to
ensure that the Burgers circuit limits are defined such that the intersection
with the dislocation line is avoided, as well as ensuring that the dislocation
line does indeed pass through the chosen Burgers circuit.

Figure 4: Test 1 (applied to Case 7) - Variation of computed Burgers vector
with Burgers circuit size. Angular error was found to be 0 for all sizes
tested.

Fig. 4 shows the results of Test 1 when applied to Case 7. It was seen that
decreasing Burgers circuit size did not inherently increase angular error but
did increase percentage error in magnitude. This effect was small for larger
circuits but reached 1% error at around 5 voxels wide and increased to 10% at
around 3 voxels wide. It should be noted that linear elasticity theory is
valid at distances beyond a few Burgers vectors from the dislocation centre
[9], and because voxel spacing is typically in the range of several nanometers
we can assume that linear elasticity is valid for all voxels (unless the
dislocation passes through the centre of one). Given this understanding, the
reason for the deterioration in Burgers vector magnitude must be due to the
numerical integration.

Reducing Burgers circuit size means that there are fewer voxels of data across
which to integrate, and thus the integral is of a lower resolution. This will
result in more substantial changes in strain and lattice rotation between
adjacent voxels, leading to more significant error from numerical integration.
This has a more severe impact for Burgers circuits that are small in size, as
a single voxel has a proportionally greater contribution to the integral.

Additionally, and perhaps more importantly, smaller Burgers circuits will have
their sides closer to the dislocation centre, and so the changes in value from
voxel to voxel will be larger. Therefore the spatial gradients of the strains
and lattice rotations at voxels on the Burgers circuit will tend to be steeper
and are not properly captured when performing numerical integration. As a
result, the error in the numerical integration will be greater in magnitude.

This effect is also evident when considering the case where the distance
between the dislocation lines and the edge of a Burgers circuit with constant
size is varied, such as when Test 2 was applied to Case 7 (see Fig. 5). Here,
despite the number of voxels contributing to the Burgers circuit (and
therefore input data) staying the same, there was an increase in percentage
magnitude error similar to that observed in Test 1 as the dislocation is moved
closer to one side of the Burgers circuit. There was also an increase in
angular error (reaching 0.1°at a ratio of around 0.9), which is likely due to
an asymmetry in strain and lattice rotation magnitudes on opposite sides of
the Burgers circuit. The side closer to the dislocation suffers from large
changes in magnitude between adjacent voxels and therefore steeper spatial
strain and lattice rotation gradients, resulting in greater error in the
numerical integral. Meanwhile, the farther side benefits from smaller voxel-
to-voxel changes in the strain and lattice rotation components.

From these observations, it should be noted that, for an accurate computation
of the Burgers vector, one must ensure that the dislocation is close to the
centre of the Burgers circuit and that the Burgers circuit is of a
sufficiently large size. The requirements will ultimately depend on the
quality of the input data, as discussed in sections 4 and 5.

(a)

(b)

Figure 5: Test 2 (applied to Case 7) - Variation of computed Burgers vector
with fractional distance of dislocation from Burgers circuit centre. Note the
increase of both percentage error in magnitude (a) and angular error (b) as
the ratio approaches 1.

(a)

(b)

Figure 6: Test 3(A) (applied to Case 7) - Variation of computed Burgers vector
with noise (single loop). Note the roughly linear general trend of increase of
both percentage error in magnitude (a) and angular error (b) as the noise
magnitude increases, with the line of best fit gradients being 0.927 and 0.971
respectively. The deviation of these gradients from 1 is a result of the
inherent randomness of the noise.

(a)

(b)

Figure 7: Test 3(B) (applied to Case 7) - Variation of computed Burgers vector
with noise and no. concentric Burgers circuits. The percentage error in
magnitude (a) and angular error (b) tend to increase as the noise magnitude
increases and no. concentric Burgers circuits decreases.

Fig. 6 shows the variation of computed Burgers vector with noise, with results
produced by applying Test 3 to Case 7. It was found that increasing noise
magnitude caused a roughly linear increase in both percentage error in
magnitude and angular error, with the line of best fit gradients of the log-
log graphs being 0.927 and 0.971 respectively. Note that the slight deviation
of these gradients from 1 has resulted from the inherent randomness of the
noise. At $\eta=1\times 10^{-4}$ there was roughly 3% error in magnitude and
4°angular error increasing to 6% error in magnitude and 8°angular error at
$\eta=2\times 10^{-4}$. These are realistic levels of uncertainty in
experimental data (see sections 4 and 5). It should be noted that the
simulated noise is defined entirely by the truncated Gaussian distribution,
and the noise magnitude at one voxel is entirely independent of that in
adjacent voxels.

As the noise magnitude approached $\eta=1\times 10^{-3}$ the computed Burgers
vector deteriorated to the point that it had little relation to the expected
result, likely as a result of the magnitudes of the strain field components
becoming negligible compared to the noise itself. For very noisy data it can
be beneficial to use smaller Burgers circuits in order to reduce the
accumulation of noise in the numerical integration.

Fig. 7 shows a general trend that increasing the number of concentric Burgers
circuits decreased the errors caused by noise. The concentric Burgers circuits
are all arranged such that they lie within surfaces $S_{x}$, $S_{y}$ and
$S_{z}$ in Fig. 1 b), and are as large as possible without crossing or
overlapping at any points. The results obtained are as expected, as more
computations of the Burgers vector allows for greater confidence that an
individual result may be anomalous due to excessive noise (with $x$-, $y$\-
and $z$-components beyond a certain number of standard deviations from the
mean values) and so can be discarded before averaging. Additionally, as the
theoretical mean of the noise is zero, increasing the number of computed
Burgers vectors to use for averaging should also generally result in a
reduction in the effects of noise via mutual cancellation. However, due to the
inherently random nature of the noise, these trends are never ideal and
introducing another Burgers circuit that happens to be more significantly
influenced by noise can actually worsen the final result. An example of this
would be the fluctuation between high and low percentage error in magnitude as
the number of concentric Burgers circuits is increased at the highest noise
coefficient used in Test 3, i.e. the top row of elements in Fig. 7 a).

##  4 Tests - BCDI Data

Bragg coherent diffraction imaging (BCDI) allows the 3D-resolved experimental
characterisation of morphology and lattice strain in microcrystals [20, 21].
Lattice reflections of a microcrystal illuminated with a coherent x-ray beam
are used to measure coherent x-ray diffraction patterns (CXDPs), which are in
turn used to determine the effective (complex) electron density in the sample.
The phase of the complex electron density of a reflection in the $hkl$
direction, $\phi_{hkl}(\bm{r})$, is related to the displacement field of the
crystal lattice structure, $\bm{u}(\bm{r})$, by the expression
$\phi_{hkl}(\bm{r})=\bm{q}_{hkl}\cdot\bm{u}(\bm{r})$, where $\bm{q}_{hkl}$ is
the reflection Bragg vector [27]. By measuring at least three linearly-
independent crystal reflections $\bm{u}(\bm{r})$ can be recovered, and the
displacement gradient $\bm{\beta}$ and thus the strain and lattice rotation
fields calculated.

The data used for this test was taken from a micron-sized sample of high-
purity tungsten containing a set of five dislocations and measured using BCDI.
The data was then processed to determine the strain and lattice rotations
$\bm{\varepsilon}$ and $\bm{\omega}$. For further information on the
manufacture of the material specimen, the BCDI measurements, the processing of
the data and the strain and lattice rotation fields, refer to Hofmann et al.
[22].

The locations of the five dislocations were first determined and plotted using
a standard Burgers circuit size of 5x5x5 voxels rastered over the 3D sample
volume. Fig. 8 shows the specimen shape and known dislocations superposed on
the field of quivers, with the quivers depicting the direction and magnitude
of the computed Burgers vector at each point.

(a)

(b)

(c)

Figure 8: The 3D plot produced by rastering a 5x5x5 voxel Burgers circuit over
the domain to build up a Burgers vector map, using the BCDI data as an input.
The regions of black quivers display computed Burgers vectors above a cutoff
magnitude. The material specimen shape and the experimentally-observed
dislocation lines are also plotted. Numbers labelling each dislocation have
been added and three views of the sample from different directions are shown.
The coordinate axes are all 500 $\mathrm{n}\mathrm{m}$ for reference scale.
Note that the origin of the coordinate system is at the centre of the material
specimen, not at the centre of the triad.

As can be seen in Fig. 8, the approach was able to reliably identify the 3D
path of dislocations in an automated fashion. Increasing the size of the
standard Burgers circuit that iterates through the data resulted in more
accurate Burgers vectors near the centres of the dislocation lines at the cost
of thicker regions of computed Burgers vectors around each dislocation line.
The reasoning for the latter effect is that because the Burgers circuit is
larger, it catches dislocations from a greater distance away. Note that the
size of 5x5x5 voxels was used to achieve thin regions of computed Burgers
vectors at the expense of accuracy, and increasing the size to 9x9x9 would
significantly reduce the magnitude and angular error.

The field of Burgers vectors is stored in one array, and it is straightforward
to use the position of a computed Burgers vector in the array to determine the
Cartesian coordinates of the centre of the Burgers circuit used to calculate
it. Another (preferably larger) Burgers circuit that surrounds this point can
then be manually chosen to allow for closer inspection of the Burgers vector.

Upon initial inspection on the resulting 3D plot, the dislocations were then
examined more carefully. Large limits for the Burgers circuit were used at
first and they were then tightened over a process of iteration to achieve a
size of 9x9x9 voxels, which was still sufficient to produce accurate results.
The final size used for dislocation 3 was actually 8x9x9 voxels to better fit
the shape of the dislocation line.

Chosen Burgers circuit coordinate limits (as depicted in Fig. 2) for each
Dislocation ($\mathrm{n}\mathrm{m}$):

  1. 1.

x = [42.5,82.5], y = [82.5,42.5], z = [222.5,262.5]

  2. 2.

x = [-177.5,-137.5], y = [137.5,97.5], z = [-72.5,-32.5]

  3. 3.

x = [67.5,102.5], y = [-317.5,-357.5], z = [52.5,92.5]

  4. 4.

x = [-92.5,-52.5], y = [-242.5,-282.5], z = [82.5,122.5]

  5. 5.

x = [-67.5,-27.5], y = [-257.5,-297.5], z = [152.5,192.5]

It should be noted that the chosen Burgers circuit limits are merely examples,
and in fact there are a great many that can be chosen in order to achieve
accurate computation of the Burgers vectors.

Strain and lattice rotation data was also extracted by regions unaffected by
the presence of defects and the standard deviation (and therefore effectively
the noise magnitude) was computed for each component of strain and lattice
rotation.

Dislocation |  True Direction (Lattice Coordinates) |  True Direction (Lab Coordinates) |  Computed Magnitude ($\mathrm{\SIUnitSymbolAngstrom}$) |  Computed Direction |  Angular Error (°)
---|---|---|---|---|---
1 |  $\frac{1}{2}[\overline{1}\overline{1}\overline{1}]$ |  -0.2799 -0.5126 0.8117 |  2.7065 |  -0.2752 -0.5195 0.8090 |  0.5027
2 |  $\frac{1}{2}[\overline{1}11]$ |  0.7854 0.5804 0.2151 |  2.7143 |  0.7841 0.5808 0.2187 |  0.2205
3 |  $[\overline{1}00]$ |  0.4405 0.0587 0.8958 |  3.0959 |  0.4406 0.0507 0.8962 |  0.4590
4 |  $\frac{1}{2}[\overline{1}11]$ |  0.7854 0.5804 0.2151 |  2.7166 |  0.7856 0.5787 0.2191 |  0.2493
5 |  $\frac{1}{2}[\overline{1}\overline{1}\overline{1}]$ |  -0.2799 -0.5126 0.8117 |  2.6977 |  -0.2760 -0.5150 0.8115 |  0.2626
Table 2: Results for the five dislocations present in the microcrystal
observed by BCDI.

From Table 2 it can be seen that the computed directions for Burgers vectors
are close to the directions expected from $\bm{q}_{hkl}\cdot\bm{b}$ analysis
[22], with angular error being within half a degree for all five dislocations.
The computed magnitudes are also very consistent, and by taking a mean
(adjusted for relative Burgers vector magnitudes in crystal lattice
coordinates), the results predict a lattice parameter of
$a=3.121$$\mathrm{\SIUnitSymbolAngstrom}$. This is $1.38$% below the standard
value of $3.1652$$\mathrm{\SIUnitSymbolAngstrom}$ [28, 29]. Using the simple
assumption of a Gaussian distribution for the noise, the average standard
deviation and therefore noise magnitude was found to be $\eta=1.0396\times
10^{-4}$, which would have suggested an angular error of 6°and percentage
error in magnitude of 3%, given the results of Fig. 6. Thus the Burgers
vectors computed using BCDI data are well within the expected uncertainty.

##  5 Tests - HR-TKD Data

High (angular) resolution electron backscatter diffraction (HR-EBSD) allows
the measurement of strains and lattice rotations at the nano-scale. A scanning
electron microscope (SEM) is used to record electron backscatter diffraction
(EBSD) patterns over an array of points to measure the crystallographic
orientation at each point [30, 31]. A cross-correlation based approach using a
reference diffraction pattern allows for a significant improvement in angular
resolution, with small shifts of features in the patterns relative to the
reference used to determine lattice rotations and distortions [30, 32, 33, 34,
35]. This approach is referred to as HR-EBSD.
The transmission Kikuchi diffraction (TKD) variation of the technique operates
by detecting the Kikuchi pattern from the bottom surface of a thin foil, which
allows for a significantly improved spatial resolution over the conventional
EBSD method [18].

However, this method allows only for the measurement of the 2D in-plane
deviatoric strain tensor, and was taken as an average over a depth of half the
foil thickness in the case of the test data used [19]. Through the assumption
of plane stress the full strain tensor required for computation of the Burgers
vector can be calculated [26], which is reasonable given the specimen
thickness of 40 $\mathrm{n}\mathrm{m}$[19].

Here we examine HR-TKD data that has been recently reported from a single
dislocation in tungsten [19]. The material used was a high-purity tungsten
sample electro-polished to electron transparency. The dislocation observed was
assumed to be straight with sense normal to the foil surface. The data was
processed to generate maps for the components of the strain and lattice
rotation tensors. For further information on the manufacture of the material
specimen, the HR-TKD measurements taken, the processing of the data and the
strain and lattice rotation fields, refer to the associated paper by Yu et al.
[19]. It should be noted that a grain boundary is present in the the top left
of the data set, manifesting as a diagonal line of significant strain and
lattice rotation.

As with the BCDI data, the position of the dislocation centre was first found
by rastering a 7x7 pixel Burgers circuit over the domain to build up a Burgers
vector map.

(a)

(b)

Figure 9: a) Full strain and lattice rotation tensor for the HR-TKD data (from
Yu et al. [19]). b) 2D plot produced using the HR-TKD data as an input and
then rastering a 7x7 pixel Burgers circuit over the domain to build up a
Burgers vector map. The reference Burgers vector has also been added for
comparison (magenta). Positions of the grain boundary and the dislocation are
also annotated onto the plot.

As depicted by Fig. 9, our approach was able to discern the location of the
dislocation as well as the grain boundary nearby. The region of black quivers
identifies the position of the dislocation, as well as estimates for the
direction of the computed Burgers vector. Also visible is a magenta quiver
representing the expected Burgers vector direction from experimental
measurement (and true magnitude from reference data). There is good agreement
in direction and magnitude amongst the estimates, and fair agreement in
direction between the estimates and the result expected by $\bm{g}\cdot\bm{b}$
contrast [19].

Upon determination of the dislocation coordinates, a closer inspection was
performed. The outermost Burgers circuit had the limits x = [20,40], y =
[33,13] in nanometers. Several concentric Burgers circuits were used and the
variation of percentage error in magnitude and angular error with Burgers
circuit side length was recorded.

Strain and lattice rotation data was also extracted from regions unaffected by
the presence of the dislocation or grain boundary and the standard deviation
(and therefore effectively the noise magnitude) was computed for each
component of strain and lattice rotation.

(a)

(b)

Figure 10: Variation of Computed Burgers Vector with Burgers Circuit Size (HR-
TKD Data). The values of percentage error in magnitude (a) and angular error
(b) at 17 pixels wide roughly match the values anticipated from Fig. 6.

Fig. 10 shows that angular error tended to reduce with Burgers circuit side
length to a minimum of 4.12°(with percentage error in Burgers vector magnitude
of 42% below the expected result of 2.7411$\mathrm{\SIUnitSymbolAngstrom}$for
tungsten [28, 29]) at a Burgers circuit width of 9 pixels before increasing
rapidly. There is also a local minimum of 16.9°angular error at 17 pixels.
Meanwhile, the percentage error in magnitude was a minimum of 5.88% at 17
pixels wide before increasing steadily as side length decreased (note that the
magnitude of the Burgers vector decreased with decreasing side length). The
significant percentage magnitude error for small Burgers circuit side length
(for this data set, roughly 13 pixels and below) agrees with previously-
discussed observations (see Fig. 4). Assuming a Gaussian distribution for the
noise, the average standard deviation and therefore noise magnitude was found
to be $\eta=2.7071\times 10^{-4}$, which is considerable. Observing Fig. 6,
this would predict a percentage error of magnitude of at least 8% and angular
error of at least 12°. This roughly agrees with the results for the Burgers
circuit of 17 pixels wide, an apparent "sweet-spot" in Burgers circuit size
for this specific case.

Upon further observation of the original strain and lattice rotation data
[19], it was found that the grain boundary nearby had significant influence,
especially on the $\varepsilon_{13}$ field. Increasing Burgers circuit size
caused the integration loop to pass over the regions of intense strain and
lattice rotation close to the grain boundary, almost crossing the grain
boundary itself. Although theoretically this should not have affected Burgers
vector calculation as the displacement field of the grain boundary was
continuous across the path of integration, the discretisation of the data and
numerical integration cause errors that worsen when the changes in magnitudes
between adjacent pixels (i.e. the strain and lattice rotation gradients)
increase. Therefore in this case it is not necessarily true that a larger
Burgers circuit improves the result for computed Burgers vector, as
illustrated by Fig. 10. The low angular error for smaller Burgers circuits of
7-9 pixels wide may support this also.

It was also noted that there was poor agreement of the measured
$\varepsilon_{23}$ and $\omega_{23}$ components with simulated distortion
fields for the known dislocation [19].

Thus, even for this rather noisy experimental dataset, the approach presented
in this paper can provide reasonable values for Burgers vector magnitude and
direction.

##  6 Conclusion

We have presented an approach for determining dislocation Burgers vector
magnitude and direction from experimentally measured lattice strain and
rotation data. Applying this approach using simulated data worked flawlessly
except for extreme cases where the Burgers circuit is very small or the
dislocation line is very close to the integration loop. The addition of noise
causes a degradation of the result that is proportional to the noise
magnitude, though the computed Burgers vector still shares good agreement with
the expected result with simulated noise at magnitudes typically found in
experimental data. The use of concentric Burgers circuits can help to reduce
errors in cases of very high noise.

For the BCDI data, the approach was able to reliably identify the 3D paths of
all five dislocations in an automated fashion as well as accurately compute
Burgers vectors with directions within half a degree of angular error to those
expected (given appropriate Burgers circuit inputs). The computed magnitudes
also showed good agreement with the standard value for the lattice parameter
of tungsten.

The results using the HR-TKD data were also promising, as although deviation
from the expected Burgers vector was much greater than for the BCDI data,
there were a number of significant detrimental factors affecting computation
and a result that still shows fair agreement was recovered. For this data also
the rastering of a Burgers circuit across the sample space could be used to
successfully identify the dislocation position.

Thus it has been demonstrated that the determination of Burgers vectors using
strain and lattice rotation fields is a promising technique and is not only
feasible for simulated data but also for a wide range of experimental data
such as BCDI and even HR-TKD. It should be noted, however, that the resolution
and noise of the data can have a significant impact on the results due to the
numerical integration required for this technique.

## Data Access

The program created, data used and tests performed are freely available from
https://github.com/JacquesCloete/Burgers-Vector-Calculator.

## Author Contributions

J. Cloete formulated, applied, and tested the approach reported in the paper,
and wrote the manuscript. F. Hofmann and E. Tarleton designed the research
project, provided supervision, and edited the manuscript. All authors have
approved the final form of the paper.

## Declaration of No Competing Interests

All authors declare there are no competing interests.

## Funding

The research was supported by the Engineering Undergraduate Research
Opportunities Programme (EUROP), hosted by the Department of Engineering
Science, University of Oxford. F. Hofmann acknowledges funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant Agreement No. 714697).

## Acknowledgements

We thank Hongbing Yu for provision of the HR-EBSD data, and the Department of
Engineering Science, University of Oxford for hosting this project.

##  Appendix A Mathematical Models of Singular Dislocations

###  A.1 Infinite Straight Edge Dislocation

Let $x$, $y$ and $z$ be the three dimensions in a Cartesian coordinate system,
with respective unit basis vectors $\bm{e_{x}}$, $\bm{e_{y}}$ and
$\bm{e_{z}}$. The constant $v$ is the hypothetical Poisson’s ratio of the
material, and $b$ is the magnitude of the Burgers vector.

Consider an infinite straight edge dislocation, with sense $\bm{\xi}$ =
$\mathbf{e_{z}}$ and Burgers vector $\mathbf{b}$ = $b\mathbf{e_{x}}$. The
discontinuity of the dislocation is present at $x=0$.

Anderson et al. [8] provide an apt derivation of the components of
displacement field $\bm{u}$, which are found to be

$\displaystyle
u_{x}=\frac{b}{2\pi}\left[\tan^{-1}\left(\frac{y}{x}\right)+\frac{xy}{2(1-v)(x^{2}+y^{2})}\right],$
(21) $\displaystyle
u_{y}=-\frac{b}{2\pi}\left[\frac{1-2v}{4(1-v)}\ln\left(x^{2}+y^{2}\right)+\frac{x^{2}-y^{2}}{4(1-v)(x^{2}+y^{2})}\right].$
(22)

Note that for this edge dislocation $u_{z}=0$.

To find the elastic displacement gradient $\bm{\beta}$, the gradient of the
displacement field, the spatial partial derivatives are taken:

$\displaystyle\beta_{11}=\partialderivative{u_{x}}{x}=-A\frac{y}{(x^{2}+y^{2})^{2}}\left[(3-2v)x^{2}+(1-2v)y^{2}\right],$
(23)
$\displaystyle\beta_{12}=\partialderivative{u_{x}}{y}=A\frac{x}{(x^{2}+y^{2})^{2}}\left[(3-2v)x^{2}+(1-2v)y^{2}\right],$
(24)
$\displaystyle\beta_{21}=\partialderivative{u_{y}}{x}=-A\frac{x}{(x^{2}+y^{2})^{2}}\left[(1-2v)x^{2}+(3-2v)y^{2}\right],$
(25)
$\displaystyle\beta_{22}=\partialderivative{u_{y}}{y}=A\frac{y}{(x^{2}+y^{2})^{2}}\left[(1+2v)x^{2}-(1-2v)y^{2}\right],$
(26) $\displaystyle\beta_{13}=\beta_{23}=\beta_{31}=\beta_{32}=\beta_{33}=0$
(27)

where

$\displaystyle A=\frac{b}{4\pi(1-v)}$ (28)

With the elastic displacement gradient thus represented by

$\displaystyle\bm{\beta}_{edge}=\begin{bmatrix}\beta_{11}&\beta_{12}&0\\\
\beta_{21}&\beta_{22}&0\\\ 0&0&0\end{bmatrix}$ (29)

###  A.2 Infinite Straight Screw Dislocation

Consider an infinite straight screw dislocation, with sense $\bm{\xi}$ =
$\mathbf{e_{z}}$ and Burgers vector $\mathbf{b}$ = $b\mathbf{e_{z}}$ (note the
dislocation is therefore right-handed). Let the plane of slip be confined to
the positive x-axis (this is an important clarification for when we combine
our results to produce a mixed dislocation).

Assuming that $u_{z}$ increases uniformly with angle $\theta$ about the
dislocation core measured from the positive x-axis, the components of the
elastic displacement $\bm{u}$ can be found by inspection [9];

$\displaystyle u_{x}=0;\quad u_{y}=0$ (30) $\displaystyle
u_{z}(r,\theta)=b\frac{\theta}{2\pi}\quad[0<\theta\leq 2\pi]$ (31)

Using the geometric relation between $x$, $y$ and $\theta$, the third
expression can be re-written as

$\displaystyle
u_{z}(x,y)=\begin{cases}\frac{b}{2\pi}\tan^{-1}\left(\frac{y}{x}\right)&[0<\theta\leq\frac{\pi}{2}]\\\
\frac{b}{2\pi}\left[\pi+\tan^{-1}\left(\frac{y}{x}\right)\right]&[\frac{\pi}{2}<\theta\leq\frac{3\pi}{2}]\\\
\frac{b}{2\pi}\left[2\pi+\tan^{-1}\left(\frac{y}{x}\right)\right]&[\frac{3\pi}{2}<\theta\leq
2\pi]\end{cases}$ (32)

These all have the same partial derivatives, which are found to be

$\displaystyle\partialderivative{u_{z}}{x}=\beta_{31}=-\frac{b}{2\pi}\frac{y}{(x^{2}+y^{2})},$
(33)
$\displaystyle\partialderivative{u_{z}}{y}=\beta_{32}=\frac{b}{2\pi}\frac{x}{(x^{2}+y^{2})}$
(34)

To find the elastic displacement gradient $\bm{\beta}$, note that it will be
the case that

$\displaystyle\beta_{11}=\beta_{12}=\beta_{13}=\beta_{21}=\beta_{22}=\beta_{23}=\beta_{33}=0$
(35)

and thus

$\displaystyle\bm{\beta}_{screw}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\
\beta_{31}&\beta_{32}&0\end{bmatrix}$ (36)

###  A.3 Infinite Straight Mixed Dislocation

A mixed dislocation with sense $\bm{\xi}$ = $\mathbf{e_{z}}$ and Burgers
vector $\bm{b}$ constrained to the $x-z$ plane can be described by linear
combination of the previously-described edge and screw cases. Projection of
the Burgers vector onto the z-axis and x-axis gives
$\bm{b}_{screw}=\bm{b}\cos(\alpha)$ and $\bm{b}_{edge}=\bm{b}\sin(\alpha)$
respectively, where $\alpha$ is the angle subtended from the z-axis to the
Burgers vector. The displacement gradients of the two cases can be superposed
because they are independent of each other in linear isotropic elasticity [9].
It follows, therefore, that the displacement gradient of the mixed dislocation
is given by

$\displaystyle\bm{\beta}=\bm{\beta}_{edge}\sin(\alpha)+\bm{\beta}_{screw}\cos(\alpha)$
(37)

where $\bm{\beta}_{edge}$ and $\bm{\beta}_{screw}$ are as described in the
previous sections.

To rotate the mixed dislocation such that both $\bm{\xi}$ and $\bm{b}$ can
take any orientation, it was found that applying 3 consecutive rotations about
coordinate axes performs this task effectively. Each of the rotations causes a
rotation of the entire displacement gradient field, however they are applied
in an order such that the following procedure is effectively carried out:

  1. 1.

Rotate $\bm{b}$ anticlockwise by angle $\psi$ about the z-axis. This, paired
with the aforementioned use of angle $\alpha$, allows $\bm{b}$ to take any
orientation relative to $\bm{\xi}$. Note $\bm{\xi}$ is unaffected.

  2. 2.

Rotate $\bm{\xi}$ anticlockwise by angle $\theta$ about the y-axis. This
allows $\bm{\xi}$ to take any orientation within the $x-z$ plane. Note both
$\bm{\xi}$ and $\bm{b}$ are affected equally.

  3. 3.

Rotate $\bm{\xi}$ anticlockwise by angle $\phi$ about the z-axis. Note both
$\bm{\xi}$ and $\bm{b}$ are affected equally. This, paired with the previous
rotation, allows $\bm{\xi}$ to take any orientation.

The method is summarised visually in Figure 11.

(a)

(b)

Figure 11: Summary of the rotation of the mixed dislocation. a) Setting the
orientation of $\bm{b}$ relative to $\bm{\xi}$. b) Setting the orientation of
$\bm{\xi}$.

The three rotations can each be described separately by rotation matrices,
which can be combined to form an overall rotation matrix:

$\displaystyle\bm{R}_{1}=\begin{bmatrix}\cos(\psi)&-\sin(\psi)&0\\\
\sin(\psi)&\cos(\psi)&0\\\ 0&0&1\end{bmatrix},$ (38)
$\displaystyle\bm{R}_{2}=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\\
0&1&0\\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix},$ (39)
$\displaystyle\bm{R}_{3}=\begin{bmatrix}\cos(\phi)&-\sin(\phi)&0\\\
\sin(\phi)&\cos(\phi)&0\\\ 0&0&1\end{bmatrix},$ (40)
$\displaystyle\bm{R}=\bm{R}_{3}\bm{R}_{2}\bm{R}_{1}$ (41)

$\bm{R}$ rotates a tensor expressed in the dislocation frame
$(x^{\prime},y^{\prime},z^{\prime})$ to the global frame $(x,y,z)$. The tensor
field $\bm{\beta}(x,y,z)$ of a dislocation at arbitrary orientation, is then
found by rotating the field expressed in the dislocation frame
$\bm{\beta}^{\prime}(x^{\prime},y^{\prime},z^{\prime})$

$\displaystyle\bm{\beta}(x,y,z)=\bm{R}\bm{\beta}^{\prime}\left(x^{\prime},y^{\prime},z^{\prime}\right)\bm{R}^{T}$
(42)

The variables $x^{\prime}$, $y^{\prime}$ and $z^{\prime}$ are the field point
coordinates in the rotated dislocation frame and are found by rotating the
field points in the global frame $x$, $y$, $z$ in the opposite direction

$\displaystyle\begin{bmatrix}x^{\prime}\\\ y^{\prime}\\\
z^{\prime}\end{bmatrix}=\bm{R}^{T}\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}$
(43)

Thus the required expressions for an infinite straight mixed dislocation of
any orientation have been derived.

##  Appendix B Implementation

The program is split into a Plotter and a Calculator:

  1. 1.

The Plotter iterates through a chosen set of data and computes an approximate
Burgers vector at every voxel/pixel position using a Burgers circuit of
standard shape and size. The set of Burgers vectors can then be plotted as
quivers in a 3D or 2D space, and the geometry of the material specimen itself,
known dislocation lines, etc. can be superposed onto the plot if implemented
by the user.

  2. 2.

The Calculator allows for manual input of Burgers circuit limits to target a
dislocation or group of dislocations and accurately compute the Burgers vector
therein.

The two halves of the program can be used effectively in conjunction with each
other, with the Plotter able to find dislocations and give estimates of their
Burgers vectors and the Calculator then being used to investigate the
dislocations in detail. The Calculator may also be configured to iterate
through several concentric Burgers circuits to remove outliers and find an
average computed Burgers vector.

The program can be adjusted to suit a wide variety of data; all it requires
are the elastic strain and lattice rotation fields (in the form of evenly-
spaced data arrays). The implementation of the data will naturally have to be
adjusted depending on how the data is originally stored.

As well as working with experimental data, the program features a mathematical
model for an infinite straight mixed dislocation with sense and Burgers vector
being able to take any direction as described in appendix AA.3.

Further information can be found within the supplementary folder containing
the program.

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