Figure 6-1 shows schematically the block of atoms used in all atomistic studies of the½[111] screw dislocation. The atoms in the block form a bcc lattice oriented
according to the coordinate system shown in Figure 6-1. Periodic boundary conditions have been applied along the [111] direction so that the block consists of three consecutive, periodically repeated, (111) planes. As far as the relaxation is concerned, the block
comprises two regions: An “active” region in the center surrounded by an “inert” region. The dislocation is centered in the active region in which all the atoms are relaxed using the steepest decent molecular statics method. The relaxation was always regarded as complete when the maximum force on any atom in this region was less than 10-4 eV/Å.
The validity of this condition was thoroughly tested by trying a variety of maximum forces. In the inert region the atoms are at fixed positions displaced away from the ideal bcc lattice in accordance with the anisotropic elastic displacement field of the dislocation [120]. In all our calculations the active region contained 711 atoms and the inert region 858 atoms. This size of the block has been used in earlier calculations [72] and tested to be large enough even when the dislocation starts to move under the effect of an applied stress. This arrangement corresponds to the simulation of an infinitely long ½[111] screw dislocation located in the bulk of the material.
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Figure 6-1. Schematic of the block of atoms used for studying dislocation core structure and its changes induced by external stresses.
In order to interpret the dislocation core structure we use in the following the differential displacement maps, originally introduced in [121], to depict displacements of atoms induced by the presence of the dislocation. In these maps the atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line, [111]; circles represent atoms in the ideal bcc lattice within one period, with coloring
distinguishing the three successive (111) planes. Two types of displacements are shown in the maps. First are the displacements parallel to the Burgers vector and the dislocation line, [111], and these are called screw displacements. Second are the displacements perpendicular to the Burgers vector and the dislocation line, and for this reason called edge displacements.
The screw components of the relative displacements of neighboring atoms, produced by the dislocation, are represented by arrows drawn along the line connecting these atoms in the [111] projection. The lengths of these arrows are proportional to the magnitudes of these displacements and normalized such that they are equal to the
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separation of these atoms in the [111] projection when the magnitude of their relative displacement is equal to 1
6[111] =b/ 3. The arrows really indicate out-of-plane
displacements but are drawn in the (111) plane since the picture is two-dimensional. The edge components of the relative displacements of neighboring atoms, produced by the dislocation, are again represented by arrows. These are centered in between the corresponding neighboring atoms but now they point in the directions of the
displacements in the plane of the plot, i.e. in the (111) plane. The lengths of these arrows are again proportional to the magnitudes of the edge displacements. However, it should be noted that the magnitudes of edge displacements are about ten times smaller than those of screw displacements since the dislocation is screw.
The maps of the screw displacements were found to be practically the same for all the metals studied and when using both BOPunscr and BOPscr. This implies that the screw
displacements in the dislocation cores are not dependent on the material and relate merely to the bcc crystal structure. Hence, we show here in Figure 6-2(a) only one example, the map of the screw components of the ½[111] screw dislocation in Nb studied using BOPscr.
The associated coordinates system and {110} and {112} planes of the [111] zone are seen in the rosette shown in Figure 6-2(b). The largest screw components are confined to three intersecting {110} planes of the [111] zone; this is highlighted by light grey line segments on the differential displacement map. As explained in Section 4.5, there are two possible core configurations of the ½[111]screw dislocation, both invariant with respect to the [111] three-fold screw axis. However, one is also invariant with respect to the [101] diad (reflection in the (111) followed by reflection in the (121) plane) and the other is not. The latter one occurs in two energetically equivalent configurations related
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to each other by the [101] diad, and this core is called degenerate. The core invariant with respect to the [101] diad is called non-degenerate. As seen in Figure 6-2(a), for all the metals studied the core found using the developed BOPs is non-degenerate, which agrees with the prediction based on the analysis of γ-surfaces made earlier in this Thesis (Sections 4.5 and 5.3) and found in DFT based calculations for Ta [98, 99], Fe and W [122, 123] and for the transition metals studied in this Thesis [124].
Figure 6-2. (a) Map of screw displacements of the core of the ½[111] screw dislocation in Nb modeled using BOPscr. (b) Coordinate system used and the rosette of planes of the
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In contrast with the screw components of the core displacements, the edge components are different for different materials and also differ when using BOPunscr and
BOPscr, respectively. The edge components of the core structures calculated using both
BOPunscr and BOPscr are shown in Figures 6-3 to 6-9 for all metals studied. Since the edge
components are much smaller than the screw components, the lengths of the arrows have been multiplied by the factor of 15 in all these figures. Importantly, for group 5 and group 6 metals, visible discrepancy in edge components is found between calculations using BOPunscr and BOPscr, respectively. This clear difference is not only in the magnitude of
the edge displacements (the lengths of the arrows) but also in the directions of these displacements (directions of the arrows). This implies that the relative in-plane displacements of the neighboring atoms varies in both magnitude and direction from material to material and also depends on whether BOPunscr and BOPscr have been
employed. This indicates that screening of bond integrals has a significant influence on the edge components of the core displacements that may in turn affect the dislocation glide under applied stresses, as will be discussed later. As for Fe, the edge components of the core displacements calculated using BOPunscr and BOPscr are almost identical. This is
so, presumably, because in Fe the magnetic part of the BOPs plays the dominant role rather than the attractive bond part, and the magnetic parts are very similar in both BOPunscr and BOPscr.
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Figure 6-3. Edge components of the core displacements of the ½[111] screw dislocation in V calculated using BOPunscr and BOPscr.
Figure 6-4. Edge components of the core displacements of the ½[111] screw dislocation in Nb calculated using BOPunscr and BOPscr.
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Figure 6-5. Edge components of the core displacements of the ½[111] screw dislocation in Ta calculated using BOPunscr and BOPscr.
Figure 6-6. Edge components of the core displacements of the ½[111] screw dislocation in Cr calculated using BOPunscr and BOPscr.
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Figure 6-7. Edge components of the core displacements of the ½[111] screw dislocation in Mo calculated using BOPunscr and BOPscr.
Figure 6-8. Edge components of the core displacements of the ½[111] screw dislocation in W calculated using BOPunscr and BOPscr.
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Figure 6-9. Edge components of the core displacements of the ½[111] screw dislocation in Fe calculated using BOPunscr and BOPscr.
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