We begin by showing results from a classical DFXM scan of the misorientation of the crystal for a single 2D slice through the crystal. Fig. 2a presents the rotation COM (center-of-mass) map (i.e. rocking curve COM map²⁸). In this COM map, we see three primary subgrains (orange, yellow, and blue) that are separated by boundaries, each of which appear as nearly vertical lines that discontinuously change the local orientation. By comparing the COM image in Fig. 2a to a single image from the same layer that satisfies the weak-beam condition from the () scan in Fig. 2b, we see that the orientational shifts across the boundaries correspond to apparently dotted lines in the raw image. The dotted lines shown with the yellow arrow indicating an array of discrete dislocations whose line vectors are steeply inclined with respect to the observation plane²⁹. The COM map shows that all three subgrains have rather homogeneous angular spreads, except in locations that have internal dislocations within the cell, as indicated in Fig. 2a,b by the yellow circle. These are localized areas of strong intensity, characteristic of the strain field of isolated dislocations. We focus on the row of dislocations with an overlaid yellow dashed line that separates the yellow and orange subgrains. For this boundary, we demonstrate the spacing between the dislocations by plotting the intensity-trace depicted by the yellow line in Fig. 2b as a plot in Fig. 2c. The average distance between the dislocations along the boundary is 4.1 m. This misorientation across the boundary as determined from the COM map is = , while the Burgers vector has a magnitude of b =2.86 Å.

We note that classical dislocation theory predicts a misorientation of for a dislocation boundary of Burgers vector, b, and spacing, D³⁰. Our measurement of the crystal misorientation across the boundary and the corresponding dislocation spacing we measure from our weak-beam image fit this model precisely. At present, dislocations with spacing as low as 150 nm can be resolved, corresponding (in Al) to small-angle grain boundaries with 0.109 misorientation. See the Supp. Mat. for more on the resolution of the instrument.

From the 1D illumination in Fig. 2b, the small dots for each dislocation indicate that each dislocation line slices through the 2D observation plane defined by the 1D X-ray line-beam illumination. As demonstrated in Supplementary Material, the 3D position of dislocation lines cannot be traced simply by making the incident beam larger. Instead, we compile a spatial 3D map of dislocations by stacking the weak-beam image from each layer that are analogous to Fig. 2b.

The resulting 3D dislocation structures resolved with our section-DFXM approach are shown in Fig. 3 for the full volume probed at the highest magnification. Fig. 3 shows that dislocations in the probed 3D volume self organize into preferential structures. The map comprises clearly defined lines that are analogous to those seen via dark-field TEM at smaller scales: they represent the dislocation lines, as measured by the locally high strain and orientation components that become asymptotic immediately surrounding the core structure²⁹. The dislocations identified in this volume clearly pack in hierarchical structures: a large collection of dislocations is present, and furthermore, in some cases, the dislocations pack into long-range boundary structures (e.g. those on the right) that separate different subgrains of the crystal. To understand the mechanics of a crystal at the mesoscale, we explore different types of dislocation packing arrangements in the crystal, interpreting key details of the boundaries at the scale of the boundary planes and the component dislocations.

We show an annotated version of the 3D dislocations from Fig. 3 in Fig. 4 to present in detail the structure within five crystalline regions (identified by five colored boxes) that are characteristic sections of each well-defined dislocation boundary (DB) we describe in this work. A clear picture now emerges on the self-organization of the dislocation structures in the probed volume. We observe that dislocations pack along well-defined planes within 3D, even after long annealing times at temperatures close to melting. From a first glance, the dihedral angles of the triple junctions are far from 120 in contrast to what conventional growth models predict. Below, we zoom in on individual boundaries shown in Fig. 4 and analyze in detail to extract more information on the self-organization process. Inlays in Fig. 4 obtained from zooming in on each separate low-angle boundary plane demonstrate that we can resolve full structures. In particular, we can resolve the defect plane that separates sub-domains of the crystal, and by projecting each boundary along different vectors we can identify the relevant in-plane and out-of-plane directions crystallographically. As described above, this allowed us to solve for the zone axes for the 5 primary boundaries in this structure, as labeled in Fig. 4. To further refine our assignments, we then isolated each DB and viewed each one along the possible zone-axis vectors, verifying that the appropriate vector corresponds to the one with the largest spatial extent within our view, and two normal vectors that constrain the plane to linear features. Table  1 show the results for all of the boundaries in Fig. 4.

Boundary B1 (red) is the primary boundary that slices down the middle of the characterized volume; the normal vector defines the B1 plane. The inlay in Fig. 4 shows that B1 is comprised of clearly defined, straight linear features that indicate the direction of the dislocation lines in the boundary.

A closer look at the bottom of Fig. 5a shows that B1 forms a triple-junction with boundaries B1 and B1. The boundary dislocations in B1 and B1 bend around the junction point, making both become curved planes over a region. B1 discretely changes at a “kink point” that makes the boundary flatten into a planar boundary that is normal to the [012] vector with straight dislocations that point along the vector. The dislocations in B1 gradually bend onto the new (120) plane (with dislocation line vectors that primarily point along , and have some possible kinks). Based on the angle between the vectors normal to the planes, the angle at the triple-junction is immediately surrounding the B1 and B1 junction, but the bend of the B1 plane ultimately shifts the long-range boundary angle to as measured further away from the junction.

The curvature of the boundaries near the triple junction suggests that the stabilizing force usually predicted from triple junctions may not be valid in this system. Instead, we interpret that the curvature surrounding this junction may arise from a localized impurity that pinned a single dislocation during the annealing process, then was stabilized by the three surrounding boundaries, B1, B1 and B1. The curvature in these boundaries near the junction suggests the innermost dislocation has the highest energy, thereby distorting the topology of the boundary planes at that site.

Beyond B1 and B1, Fig. 5 shows two other boundaries that are clearly defined in the high-z regions of the dislocation structures. Another boundary, B2 (green), intersects B1 in the upper 50 m z region of the volume, intersecting at an angle of (Fig. 5a). B2’s plane is defined by its normal vector, with dislocations that pack significantly closer together, as observed in the inlay, packed along the vector. Near the region where the B1 and B2 planes intersect, a third boundary—B3 (cyan)—nears the edge of the dislocation structures, with a normal vector along . The spacing between the dislocations in B3 is the smallest of all five boundary structures, imposing the highest misorientation angle. As corroborated by the COM map shown for the top layer in Fig. 2a, the misorientation across the boundary, indicating a  nm spacing that makes the dislocations difficult to differentiate based on the thresholds used in our segmentation methods. We note that with higher precision afforded by Bayesian inference, future implementations of this method could improve the resolution significantly³¹. From Fig. 5, the projected image along the axis shows that B3 never intersects boundaries B1 and B2. Note that some of the dislocations shown in Fig. 2b do not appear in these volumes as they only satisfy the Bragg condition, thus become visible, at certain tilts. These 3D maps generated from the 4D scans () are measured at a fixed value.

Going beyond the well-formed boundaries, we also note that this weak-beam 3D DFXM scan allows us to map the isolated (lone) dislocations quite effectively as well. Figures 3 and  5 show an interesting and complex dislocation structure between B1 and B3. This structure includes curved dislocations that appear to form a complex boundary shape with significant curvature. It is possible that this dislocation structure connects to B1 and B2, forming another triple junction. The irregular character of this dislocation structure indicates DFXM’s ability to characterize structures with complexity beyond a classical boundary. For example, one of the dislocations in this structure stretches down 100 m along direction before it truncates in a partial loop centered around . While a precise analysis of this unusual dislocation tangle is beyond the scope of this work, we note that Section-DFXM provides a new approach to characterize these complex structures using the 3D image segmentation techniques to resolve a deeper view of complex topologies.

If a boundary is not associated with long-range stresses and is thus a low-energy-dislocation structure, the dislocation arrangement in the boundary should fulfill the Frank equation³²:

where and are are the density and line direction of the dislocation with Burgers vector . The boundary plane normal and misorientation axis are and , respectively. is an arbitrary vector that lies in the boundary plane. The Frank equation relates the net Burgers vector content of the boundary free of long-range stresses to the crystallographic misorientation between the domains separated by the boundary.

All of the boundaries in the observation volume (Fig. 3) have straight parallel dislocation lines as the dominant feature. Some indications of crossing dislocation lines may be seen but their densities are low. A boundary consisting of dislocations of only one Burgers vector must lie on the plane with the Burgers vector as the normal³². In fcc this implies boundary planes of the {110} family. As seen in Table 1 this is the case for boundary B1. The classical boundary of this type is a tilt boundary consisting of edge dislocations with dislocation line along <11>, which enter the boundary by glide. By contrast, the dislocation line for B1 is [21], which does not lie in any slip plane. It can be inferred that the dislocations did not enter the boundary by slip and that climb enabled by the high temperature was involved in its formation^30,33.

For the rest of the boundaries, the parallel dislocations must have different Burgers vectors to fulfill the Frank equation. The Burgers vectors of each dislocation cannot be identified at present, though Table 1 shows the average Burgers vectors for all boundaries. For B3, the boundary plane and dislocation line directions are consistent with the Frank equation as a tilt boundary, with equal densities of dislocations with Burgers vectors of [10] and [01].

For the remaining boundaries, B1, B1 and B2, the boundary planes and dislocation line directions are symmetrically equivalent with planes of {012} and line directions of <221> to <321>. With the (01) plane of boundary B2 as an example, the Frank equation was employed to establish that a boundary on the (02) plane consisting of two sets of parallel dislocation lines along [15] with Burgers vectors [01] and [01] fulfills the equation if the density of the first Burgers vector is larger than the density of the other by a factor of about 2.3. The angle between the experimentally observed boundary plane and the one obtained using the Frank equation is and the theoretical dislocation line lies in between those determined experimentally for the three symmetric boundaries. These deviations may be due to the presence of a small density of additional dislocations. Analogous analyses for B1 and B1 can be made.