**3. Results and Discussion**

Strain determination is based on the shifts of the HOLZ lines in the strained specimen relative to the unstrained pattern. Specific orientations are more sensitive to the changes in the strain state. The <411> zone axis has been shown to be highly sensitive to changes in strain of face centered cubic (FCC) crystals [31]. The <411> CBED patterns were acquired in small volumes of the cyclically deformed copper very close to and remote from a dislocation heterogeneity (dislocation walls) and in the direction of applied stress ([001]). A channel with five locations where a CBED pattern was acquired is illustrated in Figure 4. The closest a CBED pattern could be acquired from a dislocation heterogeneity was approximately 30 nm. Below 30 nm, the dislocation tangles within the walls are too close to the electron probe, causing perturbations within the CBED pattern.

Lattice parameter determination was performed using the normalized distance between different HOLZ line intersections. Normalization by using the ratios of the distance between different intersections was used to adjust for differences in magnification. The comparison of simulated and experimental patterns is achieved by using the chi-squared equation, which is the typical refinement method for producing the best match between the simulated and experimental patterns [27]. The chi-squared equation is defined as:

$$\chi^2 = \sum\_{i}^{N} \frac{1}{d\_{is}} (d\_{is} - d\_{ix})^2 \tag{1}$$

where *N* is the number of data points, *ds* is the normalized distance between two intersections in the simulated pattern, and *dx* is the normalized distance between the same intersections of the experimental pattern. As stated earlier, the CBED pattern of an unstrained copper single crystal has also been recorded and compared with the simulated patterns to determine the observed lattice parameter of the undeformed copper. Consequently, the strain can be evaluated by comparing the lattice parameter of the cyclically deformed copper with the unstrained value. The strain was converted into stress using the elastic modulus along the [001] direction E [001] = 66.6 ± 0.5 GPa [32], accounting for the cubic elasticity anisotropy of copper.

**Figure 4.** Convergent beam electron diffraction (CBED) patterns were recorded across channels to assess the internal stresses in the direction of applied stress [001]. (**a**) A channel illustrating the locations where the CBED patterns were acquired. (**b**) The <411> CBED pattern that was recorded closer to the wall and (**c**) in the middle of the channel.

Figure 5 shows the lattice parameter measurements and stress calculations corresponding to different positions within the channel. The data are obtained from four channels in two TEM foils. The horizontal axis shows the distance from the walls normalized by the channel width. The average channel width is 0.36 μm and the wall width is about 0.12 μm. Minor scatterings exist in the lattice parameters of different channels that is <sup>±</sup>2 <sup>×</sup> 10−<sup>4</sup> nm. Identical values of lattice parameters in each channel show that the internal stresses are homogenous in the channel and close to the walls of the labyrinth dislocation microstructure. Comparing the lattice parameter of cyclically deformed copper single crystal with that of unstrained copper indicates that the internal stresses are minimal. Considering Brown's note on permanent softening, this weak backstress is in line with the fact that the measured cyclic softening in our test was weak [9,10]. Of course, it is possible that internal stresses exist and are less than the measurement error. The accuracy of lattice parameter measurements is about <sup>±</sup><sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> nm. The error in stress measurements is then approximately <sup>±</sup>18 MPa that is 6.5% of the applied axial stress of 275 MPa.

**Figure 5.** The obtained lattice parameters and corresponding axial stress values in different locations within a dislocation channel. The lattice parameter of an unloaded copper was found to be 0.3592 nm.

Following the Legros et al. [26] method of chi-squared analysis, an effort was made to assess the local changes of the lattice parameter within a single channel. Since the lattice parameter of the unstrained copper was observed to be 0.3592 nm, channel four with the lattice parameter of 0.3592 nm in the cyclically deformed copper was chosen for this analysis. HOLZ lines of simulated CBED patterns corresponding to the lattice parameter of 0.35920 nm were considered as the reference point for chi-squared analysis. The data illustrated in Figure 6 is a chi-squared fit between the aforementioned HOLZ lines intersection ratios for cyclically deformed (yellow) and undeformed (red) copper. Chi-squared analysis can "refine" the strain measurement below 10−<sup>4</sup> and provide increased resolution of the elastic strain, although precise values of the strain within in the 10−<sup>5</sup> range is not possible. The chi-squared results plotted in Figure 6 are somewhat qualitative. Although the changes in the lattice parameter in a channel are minimal and less than 1 <sup>×</sup> <sup>10</sup>−<sup>4</sup> nm, the data shown in Figure <sup>6</sup> can show that the difference between the lattice parameter values of the cyclically deformed copper and the unstrained copper are slightly higher in the proximity of the walls in comparison with the values in the channel interior. This result is consistent with the composite model but with much lower values of internal stresses (less than 6.5% of the applied stress). It should also be noted that the aforementioned differences in the chi-squared values might be due to the higher quality of HOLZ lines in the middle of the channels as opposed to the vicinity of the walls. Considering the two renowned theories that rationalize the Bauschinger effect (Composite model and Orowan-Sleeswyk mechanism), it appears that the dominant characteristics of the Bauschinger effect may need to include the Orowan–Sleeswyk [6] mechanism type of explanation since both the maximum dipole height measurements and the lattice parameter assessment through CBED analysis suggest a relatively homogenous stress state. As stated earlier, no internal stresses are involved in the Orowan-Sleeswyk mechanism where the Bauschinger effect is rationalized by the lower lineal density of obstacles in reverse direction of straining.

It should be noted that dislocations may eject out of the surface in the thin areas of the TEM foil. This will result in stress relaxation and can subsequently alter the values of the internal stress. This is rather challenging since thin areas of the specimen are to be used for acquiring high quality HOLZ lines in a CBED pattern. It must be emphasized that these relaxations may be negligible as the labyrinth pattern with similar characteristics such as dislocation density, channel, and wall width were observed both in the thin regions as well as thicker areas. The dislocation density in the thinner regions (approximately 130 <sup>±</sup> 10 nm) where CBED patterns were recorded was 8.2 <sup>×</sup> 1014 m/m3 in the walls and 1.8 <sup>×</sup> 1013 m/m3 in the channels. This is very close to the wall dislocation density of 8.6 <sup>×</sup> <sup>10</sup><sup>14</sup> <sup>m</sup>/m3 and channel dislocation density of 1.5 <sup>×</sup> 1013 m/m3 in relatively thicker regions (approximately 0.23 μm) where dislocation densities were measured [1]. Although copper has a

fairly low stacking fault energy of 60 mJ/m2 and the labyrinth microstructure characteristics including dislocation densities are consistent in the thick and thin regions, relaxations caused by dislocations ejecting the thin regions of the foil cannot be completely neglected.

**Figure 6.** Change in the [411] CBED pattern higher order Laue zone (HOLZ) lines of cyclically deformed copper in a channel (and near the walls) and undeformed copper relative to simulated pattern with lattice parameter of 0.35920 nm. The data is shown as a χ<sup>2</sup> analysis fit between the HOLZ lines of the aforementioned CBED patterns.

As stated earlier, there have been few studies on internal stress assessment using the CBED technique in creep deformed and fatigued polycrystals and single crystals oriented in single slip. Such studies on cyclically deformed single crystals oriented in multiple slip are missing in the literature. The current study shows a uniform stress state within the crystal since the lattice parameters are almost identical near the dislocation walls and within the channel. This is similar to the findings of earlier studies on fatigued copper and silicon oriented for single slip and creep deformed aluminum and copper [21–26]. All of the deformation conditions of the current study along with other studies that used CBED to assess LRIS are summarized in Table 1.


**Table 1.** Summary of studies on creep and cyclically deformed materials that utilized CBED to assess long range internal stresses.
