# Modeling Dislocation Contrasts Obtained by Accurate-Electron Channeling Contrast Imaging for Characterizing Deformation Mechanisms in Bulk Materials

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## Abstract

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## 1. Introduction

_{BSE}) modulation, thus generating several contrasts such as a bright line on a dark background [2] or a black line on a bright background [3].

_{BSE}profiles for both perfect and imperfect crystal, Spencer et al. [7] and Wilkinson et al. [6,8,9] used this Bloch wave-based model. They showed that, for the perfect crystal, the simulated profiles exhibit the main experimental features of the channeling pattern: bright band and dark edges. The same approach was also used by Reimer [10,11] for a perfect crystal where the multiple scatterings are treated as incoherent. These different approaches were extended to the case of an imperfect crystal containing a dislocation [6,7,8,9] or a stacking fault [12]. Despite their contribution to the theory of defects electron channeling contrasts [7,8,9,10,11,12], detailed calculations leading to an analytical expression of BSE signal as a function of experimental parameters are still missing. Furthermore, in most cases [7,8], theoretical results were not compared to the experiments. This can be illustrated from the dislocation profiles calculated for the Bragg condition, which exhibit an extra-pic of I

_{BSE}not observed experimentally [7,8].

_{BSE}as a function of physical parameters either relative to the material or governing the ECCI experiment. Our theoretical results show a good agreement with the experiments for several diffraction conditions.

## 2. Our Theoretical Approach for BSE Intensity Calculation for an Imperfect Crystal

_{D}(where z

_{D}is the mean depth of the dislocation), the distortion of the lattice planes near the dislocation does not depend on z but only on x, and it is given by $\frac{\mathsf{\partial}\mathbf{R}}{\mathsf{\partial}\mathrm{z}}{)}_{{\mathrm{z}=\mathrm{z}}_{\mathrm{D}}}$ (

**R**is the displacement field of the crystalline planes) [14].

_{BSE}in the case of an imperfect crystal containing a dislocation parallel to the sample surface, independently of the depth z, we take into account a new deviation parameter written by:

_{BSE}as a function of x (distance x away from the dislocation core), where the contrast associated to a dislocation is described by ${\mathrm{s}}_{\mathrm{D}}$ (containing all the effect of

**R**).

#### 2.1. Screw Dislocation

_{D}. This defect is characterized by a Burgers vector $\mathbf{b}$ and a line direction $\mathbf{u}$. At a distance x away from the dislocation core (in $\mathrm{x}\text{}=\text{}0)$, the crystal plane is deformed. The displacement field ${\mathbf{R}}_{\mathbf{screw}}$ is then defined in polar coordinate ($\mathsf{\xdf}$) as follows [15]:

_{D}), is given by:

#### 2.1.1. Deviation Parameter s = 0

**g**and

**s**vectors are, respectively, determined experimentally through the pseudo-band indexation of the High Resolution Selected Area Channeling Pattern (HR-SACP) assisted by Electron BackScatter Diffraction (EBSD) experiment [2,3].

_{BSE}given by Equation (8) is in good agreement with the experimental observations already reported in literature [3,7].

#### 2.1.2. Deviation Parameter s > 0

_{BSE}profiles calculated by Equation (8) with a deviation parameter slightly positive ($\mathrm{s}\text{}=\text{}0.01\text{}{\mathrm{nm}}^{-1}$) are represented in Figure 3a,b for the $+\mathbf{g}$ and $-\mathbf{g}$ diffractions, respectively. In this condition ($\mathrm{s}\text{}\text{}0$), both $\pm \mathbf{g}$ dislocation profiles present one intensity peak only. This is in agreement with the experimental ECC micrographs shown in Figure 3a’,b’: bright line on dark background. Note also that the maximum intensity does not coincide with the exact position of the dislocation core ($\mathrm{x}\text{}=\text{}0\text{}\mathrm{nm}$) but it is at $\mathrm{x}\text{}\approx \text{}\pm 4\text{}\mathrm{nm}$: it is displaced from one side of the dislocation position to the other side when changing from $+\mathbf{g}$ to $-\mathbf{g}$. This result is analogous to that obtained in TEM and can be used to characterize a dislocation configuration consisting of two parallel dislocations, such as dipole [3,16].

#### 2.1.3. Deviation Parameter s < 0

_{BSE}profiles calculated from our theoretical model for slightly negative deviation parameters ($\mathrm{s}\text{}=\text{}0.01{\mathrm{nm}}^{-1}$) and $\pm \mathbf{g}$ diffraction conditions are represented in Figure 3c,d. For the diffraction $+\mathbf{g}$, the curve contains a deep hollow and a peak corresponding to the black and white dislocation sides, respectively (Figure 3c’). This contrast is inverted with the inversion of the sign of $\mathbf{g}$ (Figure 3d,d’). For $\mathrm{s}\text{}\text{}0$, the BSE signal emitted from the zone of interest is high: bright background.

#### 2.2. Edge Dislocation

_{D}produces a local deformation of the crystalline planes nearby its core (see Figure 4). Such distortion is described by its displacement field, written in polar coordinate, as follows [15]:

_{BSE}. The edge dislocation generates a white/black contrast. However, for $\mathrm{s}\text{}\text{}0$, the profile presents only a single peak consistent with experimental observations. The maximum intensity is situated at a position $\mathrm{x}\text{}\approx \text{}-6\text{}\mathrm{nm}$ away from the dislocation core. Concerning the case of $\mathrm{s}\text{}\text{}0$, the I

_{BSE}profile show a hollow with a slight peak. All profiles are also reversed, following the inversion of the $\mathbf{g}$ sign regardless of the deviation parameter $\mathrm{s}$.

#### 2.3. Extinction Criteria

_{D}. Nevertheless, the position of the dislocation is located in the [z

_{1}, z

_{2}] range (see Figure 1), therefore the $\mathbf{b}\times \mathbf{u}$ term is not null. For $\mathbf{g}\xb7\mathbf{b}\text{}=\text{}0$ and $\mathbf{g}\xb7\mathbf{b}\times \mathbf{u}\text{}\ne \text{}0$, in the [z

_{1}, z

_{2}] range, except z

_{D}, the calculated profile for an edge dislocation displays a low intensity peak ${\mathsf{\eta}}_{\mathrm{O}.\mathrm{C}.}\approx \text{}2,7\text{}\mathrm{a}.\mathrm{u}$ (with respect to the background level) surrounded by two hollows. Such residual contrast (Figure 5b) is characteristic of an edge dislocation observed by TEM under these diffraction conditions [17].

#### 2.4. Quantitative Comparisons with Experimental Profiles

## 3. Conclusions

_{BSE}around dislocations without resorting to numerical methods. An analytical formula of the BSE signal as a function of the different physical parameters governing the ECCI experiment has been proposed for the first time to our knowledge. The BSE contrast profiles, produced by screw and edge dislocations parallel to the sample surface, display the same appearance for the diffraction conditions. For a deviation parameter s = 0 (Bragg condition) and s < 0, the theoretical BSE curves show hollows and peaks of intensity corresponding to the black and white dislocation sides, respectively. The inversion of

**g**leads to the profile inversion (hollow becomes peak and vice versa). For s > 0, the bright dislocation contrast is envisaged in the modeled profile by the intensity peak. The latter (dislocation image) does not coincide with the exact dislocation position (x = 0) and it is displaced to the opposite side when

**g**is reversed. Moreover, our theoretical model confirms the use of the invisibility criteria in ECCI.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of a dislocation parallel to the surface and located at a depth z

_{D}. Deformed planes, perpendicular to the surface, are at a distance x away from the dislocation core.

**Figure 2.**I

_{BSE}profiles modeled, for a screw dislocation parallel to the surface, with a deviation parameter $\mathrm{s}\text{}=\text{}0$ for the diffractions (

**a**) $+\mathbf{g}$ and (

**b**)$\text{}-\mathbf{g}$ with their corresponding experimental ECC micrographs (

**a’**) and (

**b’**).

**Figure 3.**I

_{BSE}profiles modeled, for a screw dislocation parallel to the surface for $+\mathbf{g}$ and $-\mathbf{g}$, with $\mathrm{s}\text{}\text{}0$ (

**a**,

**b**), and $\mathrm{s}\text{}\text{}0$ (

**c**,

**d**), and their corresponding experimental ECC micrographs (

**a’**–

**d’**).

**Figure 4.**Schematic of an edge dislocation parallel to the surface and located at a depth z

_{D}. Deformed planes, perpendicular to the surface, are at a distance x away from the dislocation core.

**Figure 5.**I

_{BSE}profiles modeled for the extinction conditions: (a) $\mathbf{g}\xb7\mathbf{b}\text{}=\text{}0$, $\mathbf{g}\xb7\mathbf{b}\times \mathbf{u}\text{}=\text{}0$, and (b)$\text{}\mathbf{g}\xb7\mathbf{b}\text{}=\text{}0$, $\mathbf{g}\xb7\mathbf{b}\times \mathbf{u}\text{}\ne \text{}0$.

**Figure 6.**Fitted (blue line) and experimental (black squares) I

_{BSE}profiles and their corresponding ECC micrographs obtained for (

**a**,

**a’**):

**g**= (01-1) and $\mathrm{s}\text{}\text{}0$, (

**b**,

**b’**):

**g**= (020) and $\mathrm{s}\text{}\text{}0$ and (

**c**,

**c’**):

**g**= (2-1-1) and s = 0, respectively.

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KRIAA, H.; GUITTON, A.; MALOUFI, N.
Modeling Dislocation Contrasts Obtained by Accurate-Electron Channeling Contrast Imaging for Characterizing Deformation Mechanisms in Bulk Materials. *Materials* **2019**, *12*, 1587.
https://doi.org/10.3390/ma12101587

**AMA Style**

KRIAA H, GUITTON A, MALOUFI N.
Modeling Dislocation Contrasts Obtained by Accurate-Electron Channeling Contrast Imaging for Characterizing Deformation Mechanisms in Bulk Materials. *Materials*. 2019; 12(10):1587.
https://doi.org/10.3390/ma12101587

**Chicago/Turabian Style**

KRIAA, Hana, Antoine GUITTON, and Nabila MALOUFI.
2019. "Modeling Dislocation Contrasts Obtained by Accurate-Electron Channeling Contrast Imaging for Characterizing Deformation Mechanisms in Bulk Materials" *Materials* 12, no. 10: 1587.
https://doi.org/10.3390/ma12101587