Abstract
Interpreting diffuse intensities in electron diffraction patterns can be challenging in samples with high atomic-level complexity, as often is the case with multi-principal element alloys. For example, diffuse intensities in electron diffraction patterns from simple face-centred cubic (fcc) and related alloys have been attributed to short-range order1, medium-range order2 or a variety of different {111} planar defects, including thin twins3, thin hexagonal close-packed layers4, relrod spiking5 and incomplete ABC stacking6. Here we demonstrate that many of these diffuse intensities, including \({}^{1}{ / }_{3}\){422} and \({}^{1}{ / }_{2}\){311} in ⟨111⟩ and ⟨112⟩ selected area diffraction patterns, respectively, are due to reflections from higher-order Laue zones. We show similar features along many different zone axes in a wide range of simple fcc materials, including CdTe, pure Ni and pure Al. Using electron diffraction theory, we explain these intensities and show that our calculated intensities of projected higher-order Laue zone reflections as a function of deviation from their Bragg conditions match well with the observed intensities, proving that these intensities are universal in these fcc materials. Finally, we provide a framework for determining the nature and location of diffuse intensities that could indicate the presence of short-range order or medium-range order.
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Data availability
All experimental diffraction patterns and images used in this work are presented here. Additional information is available from the corresponding authors.
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Acknowledgements
This research was in part funded by Fundação de Amparo à Pesquisa do Estado de São Paulo (grant nos. 2021/04302-8 and 2022/02770-7). This work was conducted through the International Nuclear Energy Research Initiative of the US Department of Energy (contract 2011-01-K). Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the US Department of Energy under contract DE-AC52-06NA25396. Much of the characterization was conducted using the microscopes in the Shared Instrumentation Center at the Colorado School of Mines. We thank the Laboratory of Structural Characterization, Department of Materials Engineering, Federal University of São Carlos, for use of its general facilities. We also thank M. Twigg for the discussion regarding the dynamical electron diffraction theory.
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F.G.C. wrote part of the manuscript, conceptualized some of the diffraction experiments and did part of the experiments and calculations. C.M. wrote part of the manuscript, conceptualized some of the diffraction experiments and did part of the experiments. R.F. helped to conceptualize the research and wrote part of the manuscript. M.K. conceptualized the research, did some of the experiments and wrote part of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Analysis of the pure Al sample.
Transmission electron microscopy images (a, c, d) and SADP (b) taken from the pure aluminum sample. The grain highlighted in the Bright Field image taken down the [111] zone axis in (a) was analyzed in the region indicated by the yellow circle, the scale bar is 1 micron. The sample was then tilted to a two-beam condition for the 422 planes, and oriented to the Bragg condition of the \({}^{1}{ / }_{3}\){422} extra reflection as shown in (b). The arrows denote the location of the intensities at the \({}^{1}{ / }_{3}\) and \({}^{2}{ / }_{3}\) positions and the line denotes where the “Kikuchi line” of the extra intensity would be if observable. A BF(b)/DF(c) pair of images was then acquired from the highlighted region in (a) using the \({}^{1}{ / }_{3}\){422} extra reflection. A long exposure time (of about 90 s) was used for the DF image. Note that no defects (e.g., stacking faults or dislocations) can be seen in the image inside of the highlighted grain. Furthermore, in the DF image no specific contrast suggesting SRO regions is observed. The scale bar in (c,d) is 300 nm.
Extended Data Fig. 2 Diffraction from incomplete stackings.
Supercells were constructed by elongating an FCC lattice along [001] and [111] directions. The results can be used to interpret the effects of surface steps on the extra intensities observed in <100> and <111> SADPs. The stacking of planes in the first supercell is {…AB…}, in (a), a complete stacking is shown, with the last plane being of the “B” type plane, while the incomplete stacking in (b) ends on an “A” type plane. The same is true for the complete {…ABC…} stacking of [111] planes given in (c) and the incomplete stacking in (d). As shown in (a,b), when a step is present on the (001) plane, the region with incomplete cells (b) should have a non-zero structure factor for the {110} reflections (or \({}^{1}{ / }_{2}\){220}) on the [001] SADP. The same is true for \({}^{1}{ / }_{3}\){422} reflections on [111] SADPs when incomplete stacking is present as given in (d).
Extended Data Fig. 3 HOLZ projections on other SADPs.
SADPs from the Alloy 690 specimen aged at 475 °C for 3,000 h down the zone axes (a) [113], (b) [123], (c) [012] and (d) [013]. Note, yellow, blue and red indices are ZOLZ, positive HOLZ and negative HOLZ reflections, respectively.
Extended Data Fig. 4 Crystallographic relation between <112> and <111> Zas.
(a) Stereographic triangle showing the zone axes containing {422} and {311} reflections that were examined and (b) schematic representation of the tilting experiment showing the <112> SADPs relative to the <111> (taken from Co-28.5Cr-6Mo) and overlaid on an EBSD Kikuchi map. The {111} reflections were tracked from the B = [112] pattern into the B = [111] pattern.
Extended Data Fig. 5 Intensity profiles from ZOLZ/FOLZ exclusive rows.
Intensity profile for reflections present in a pure Al SADP acquired using 40 kV and 120 kV accelerating voltages. As the electron wavelength increases, the radius of the Ewald Sphere decreases so the nFOLZ intensities drop faster compared to SADPs acquired using higher accelerating voltages. (a) The SADP acquired using 40 kV. The intensity profile within the highlighted region was extracted and is shown in (b). The nFOLZ reflections are weaker than the pFOLZ and ZOLZ at a much smaller distance than what is observed for the 200 kV SADP. In (c), the experimental 120 kV SADP of pure aluminum is plotted on a logarithmic intensity scale. The intensity profiles of the three rows indicated by the colored dotted lines are plotted in (d). A schematic showing indexing of each row is provided in (e); each line was selected to only contain either FOLZ, pFOLZ or nFOLZ reflections. As shown by the profiles, at position 5, the intensities of the pFOLZ and nFOLZ are comparable, however when moving further away from the (000) reflection the pFOLZ becomes more intense than the nFOLZ (as seen in reflection 3), and even further away the pFOLZ becomes stronger than the ZOLZ reflections (as seen in reflections 1 and 2).
Extended Data Fig. 6 Additional intensities from ordered phases.
Simulated SADPs of an FCC lattice containing ordered precipitates. (a) [001] and (b) [112] SADPs of the FCC fundamental reflections as well as the positions of the reflections of 4 commonly observed ordered versions of this phase. As indicated in the figure, the \({}^{1}{ / }_{2}\){311} reflection does not correspond to the position where any of those phases would display reflections.
Extended Data Fig. 7 [411] SADP with projected intensity from FOLZ.
A [411] SADP from pure Al (a) without Precession Electron Diffraction (PED) and (b) with 1° PED. (c) SADP simulation including the ZOLZ reflections in green and two rows of the HOLZ reflections in blue and red (immediately below and above the ZOLZ, respectively). In (d) the simulation is overlapped with the PED pattern. The PED was used to remove the effect of the Kikuchi lines as discussed in the Methods section. As shown in the figure, the extra diffuse intensities are no longer observed concentrated in the \({}^{1}{ / }_{2}\){311} positions. Instead, they split into the two sets of HOLZ reflections indicated in (b). Note that even the asymmetry of the HOLZ reflections below and above the {311}, expected from the simulation, can be seen in the image. The fact that the extra diffuse intensity shifts position with sample tilt confirms they are not due to SRO, the [411] zone axis can be reached by a 47° tilt from the [121] zone axis. Therefore, for this effect to be noticeable, the tilt needs to be sufficiently large to substantially displace the reflection.
Extended Data Fig. 8 Ruling out other possible sources for the additional intensities.
(a,b) Cryostage TEM down the <111> zone of solution annealed Alloy 690 aged at 475 °C for 3,000 h. Examined at (a) −192 °C and (b) 23 °C, showing no obvious change in the \({}^{1}{ / }_{3}\){422} diffuse intensities. (c,d) An examination of Kikuchi band intersections and their contribution to \({}^{1}{ / }_{3}\){422} diffuse scattering in B = [111] SADPs. (c) A Kikuchi band simulation overlaid on an experimental SADP, showing the (d) decoupling of the Kikuchi band intensities and diffuse scattering. (e-i) Precession experiments on the pure aluminum sample to remove dynamical effects on the [111] SADP. The angles of (e) 0, 0.5 (f), 1.5 (g) and 3 (h) degrees were used. The intensity profile shown in (h) is shown in (i), with the arrows indicating the additional intensities that, although less evident, are still present. (j-l) TEM examination of the effect of foil thickness on \({}^{1}{ / }_{3}\){422} diffuse scattering in Alloy 690. (j) BFTEM from region of interest, showing the two locations where the thickness was measured, along with (k,l) the selected area diffraction patterns corresponding to these two locations. In (l) the SADP was specifically taken slightly off-axis, to ensure Kikuchi band intersections were not contributing to the \({}^{1}{ / }_{3}\){422} intensities. The chemical composition of the Alloy 690 used in this work is given on table (m).
Extended Data Fig. 9 Multi-beam simulations.
Results of the multi-beam simulation for the {220}, {440}, {660}, {880} and {11-1} reflections on the [111] SADP. As shown, the intensities change considerably with sample thickness. The average intensity value over the 0–100 nm thickness range was considered for Extended Data Table 1 and is shown in the figure as a straight line.
Supplementary information
Supplementary Video 1
Tilting experiment explanation. This supplementary video provides further details and an explanation on how the tilting experiment was performed.
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Coury, F.G., Miller, C., Field, R. et al. On the origin of diffuse intensities in fcc electron diffraction patterns. Nature 622, 742–747 (2023). https://doi.org/10.1038/s41586-023-06530-6
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DOI: https://doi.org/10.1038/s41586-023-06530-6
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