Mapping the full lattice strain tensor of a single dislocation by High Angular Resolution Transmission Kikuchi Diffraction (HR-TKD)

The full lattice strain tensor and lattice rotations induced by a dislocation in pure tungsten were mapped using high-resolution transmission Kikuchi diffraction (HR-TKD) in a SEM. The HR-TKD measurement agrees very well with a forward calculation using an elastically isotropic model of the dislocation and its Burgers vector. Our results demonstrate that the spatial and angular resolution of HR-TKD in SEM is sufficiently high to resolve the details of lattice distortions near individual dislocations. This capability opens a number of new interesting opportunities, for example determining the Burgers vector of an unknown dislocation in a fast and straightforward way.

Importantly, HR-EBSD can access the spatial distribution of GND density and lattice strain at the nano-scale near interesting features, such as grain boundaries [18][19][20], indents [21,22] , second phases [20,23] or slip bands [24]. The spatial resolution of HR-EBSD is governed by the electron interaction volume with estimates of the probed volume ranging from several tens to hundred nanometers in bulk material [25]. This, and experimental issues with drift, have prevented the study of the strain fields associated with individual dislocations using HR-EBSD, though statistical analysis by Wilkinson et al [26] indicates that sufficient spatial resolution should be available to probe the lattice strains near dislocations.
Dislocations are one of the most important lattice defects in crystalline materials. Thus far detailed characterisation of dislocations has mostly relied on TEM for determination of dislocation type, Burgers vector (b) and associated strain fields. The two most common methods for measuring lattice strain at the atomic scale are geometric phase algorithms (GPA) [27,28] and nano-beam diffraction in TEM [29][30][31]. Both offer a strain sensitivity of ~ and a spatial resolution of 2 to 3 nm [32]. However, only the 2D in-plane strain tensor can be measured from these techniques, and the measurement must be performed on a certain zone axis, placing stringent requirements on sample preparation. By using transmission Kikuchi diffraction [25] (TKD) in a SEM, i.e. detecting the Kikuchi pattern from the bottom surface of a thin foil, the absolute spatial resolution can be improved to ~10 nm [25], while the effective spatial resolution falls to 2-4 nm [33] . Here we show that by combining TKD with HR-EBSD approaches in the SEM, it becomes possible to measure the full deviatoric strain tensor associated with an individual dislocation. These measurements are compared to predicted strain fields, calculated using an isotropic elasticity model. Our results show that HR-TKD provides a convenient and reliable way of probing nano-scale strain fields, with sufficient sensitivity to study strains associated with specific dislocations.
Ultra-high purity tungsten foil (99.99% in purity and 120 μm thick) was punched into 3 mm diameter discs. The samples were thinned to electron transparency by twin-jet electropolishing (0.5 wt% NaOH aqueous solution, 0 °C, 14 V). g b analysis was performed on a JEOL 2100 TEM. TEM bright field images under 8 independent g vectors from 4 zone axes were acquired. A Zeiss Merlin SEM with a Bruker eFlash detector was then used to carry out HR-TKD measurements (20 kV, 3 nA). The TKD setup is shown in Fig. 1 (a). The sample was tilted -45° to the electron beam, a TKD pattern size of 800 600 selected and a scanning step size of 4 nm used. The cross-correlation analysis of the Kikuchi patterns was done by the XEBSD matlab code described by Britton & Wilkinson [34,35].
The anticipated spatial variation of the deviatoric lattice strain tensor and lattice rotations in the vicinity of dislocation were calculated using isotropic elasticity. This is reasonable since tungsten is almost perfect elastically isotropic [36]. Since a thin foil was used for the measurement, the surface relaxation was taken into account. For simplicity, we assume that the dislocation is straight with line direction normal to the foil surface, i.e. along the -z direction (see supplementary Fig. S1). For a dislocation with arbitrary Burgers vector b that meets the free surface at (0, 0, 0), the displacement field at (x, y, z) can be found by decomposing the Burgers vector into components along x, y, z directions: where are unit vectors along x, y, and z directions, are the corresponding coefficients. can then be expressed as, where is total displacement along the direction ( refers to x, y or z). is the displacement along induced by dislocation with Burgers vector .
The displacement field of an edge or screw dislocation meeting a free surface can be obtained by superposing the displacement of the dislocation in an infinite body and the displacement induced by image forces due to the traction free surface condition, Here we use the solution for and given by Anderson et al [37] and Yoffe [38] (also provided in the supplementary material). The components of the 3D strain tensor of the dislocation at any position (x, y, z) are obtained by differentiation: The lattice rotation can be obtained as As the TKD patterns are dominated by a tens of nanometers thick surface layer, the reported elastic strain and lattice rotations are an average over a depth of 50 nm. In SEM, dislocations can be imaged with high-energy primary electrons (PE) without needing to set up specific diffraction conditions. Fig. 1 Fig. 1 (b). A dislocation close to a grain boundary (2~3° misorientation) was selected for TEM g analysis and HR-TKD (red rectangle in Fig. 1 (b)) as the grain boundary provides a convenient reference for judging drift during the HR-TKD measurement.    (Table 1). Table 1 The g b table for visibility (v) and invisibility (i) of dislocations in bcc crystal.
Strain maps for the 6 components of the 3D strain tensor (upper triangle and diagonal in the matrix map) and 3 lattice rotations (lower triangle in the matrix map) near the dislocation, measured by HR-TKD, are shown in Fig. 3(a). The strains and rotations are plotted in the microscope coordinate frame shown in the upper left corner, which is the same as used in Fig.   1 and 2. The deviatoric, rather than full, lattice strain tensor is measured, as HR-TKD is not sensitive to lattice dilation [8]. However, the volumetric strain can be calculated by assuming stress along the out of plane direction to be zero (see supplementary  It is interesting to note that the separation between the tensile and compressive strain peak in the strain map is ~25nm. The isotropic elasticity model, on the other hand, predicts no separation between these maxima (see Supplementary Fig. S3). The elasticity model has a singularity at the dislocation core, however, at >10 b (~2.7 nm) away from the core [39], elasticity is valid. This suggests that the large separation between extreme values we observe is due to the finite size of the electron interaction volume. To enable a better comparison between the measured and predicted strain fields, the forward calculated lattice strain and lattice rotation maps were convolved with a 2D Gaussian function (σ = 5 nm). This estimate of the probe resolution is consistent with previous reports of TKD spatial resolution [25]. Fig. 3  In summary, we have demonstrated that, using tungsten as a case study, the full deviatoric lattice strain tensor and rotation field due to an individual dislocation can be quantitatively mapped using HR-TKD. The experimentally measured lattice distortions are in remarkably good agreement with those expected from a forward calculation using an isotropic elasticity model of the dislocation. Our results suggest that the combination of strain field simulation and HR-TKD may offer a straightforward approach to determining Burgers vector magnitude, direction and sign. In principle, this is similar to black-white contrast simulations used in TEM, but rather than interpreting the intensity contrast caused by the strain fields, the strain itself is directly used for the analysis.

Strain Field Calculation for dislocations at a free surface.
The coordinate convention used for simulations of the dislocation displacement fields is shown in supplementary Fig. S1. The displacement field of a screw dislocation (b = , is a unit vector along z direction) in an infinite medium is The displacement field caused by the relaxation of a screw dislocation (b = ) normal to the free surface, due to the traction free boundary condition, has been provided by Yoffe [1], where R √ , and is the magnitude of the Burgers vector b.
The total displace field of the screw dislocation at (x, y, z) is here represent x, y, z coordinates system.
The displacement field of an edge dislocation (b = , is unit vector along x direction) in an infinite medium is where is poison ratio.
The displacement field caused by surface relaxation of an edge dislocation (b = ) normal to the free surface is The total displacement field of the edge dislocation at the surface is ( ) .
In a similar way, we can obtain the total displacement field, , of an edge dislocation with b = ( is unit vector along y direction) and normal to the surface.

Supplementary figures
Fig. S1 The coordinate system setup for electron diffraction measurements and simulations of the elastic strain fields associated with the dislocation.

Fig. S2
The volumetric strain indirectly calculated from HR-TKD measurement (left) and the elasticity calculation (right).

Fig. S3
The comparison between the HR-TKD and elasticity model before convolution.